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PHOTONICS)LABORATORY)
)Manual)
)ECE)5137))
Betty)Lise)Anderson)Bradley)D.)Clymer)Stuart)A.)Collins,)Jr.)Lawrence)J.)Pelz)Steven)Ringel)
)Department)of)Electrical)and)Computer)
Engineering))
Copyright)2012,)The)Ohio)State)University)
!
!"#$%&'(&)'*!%*!+
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1
PART I: INTRODUCTION TO SEVERAL PHOTONICTECHNOLOGIES
SafetyOptical Sensing
Fiber Optic CommunicationAcousto-Optic Modulation
Laser Diode PhysicsQuantum Well Detection
Liquid CrystalsSolar Cells
EE 737 Photonics Laboratory Manual Safety
2
SAFETY
"When the beam struck my eye I heard a distinct popping
sound, caused by a laser-induced explosion at the back of my
eyeball.* My vision was obscured almost immediately by streams of
blood floating in the vitreous humor, and by what appeared to be
particulate matter suspended in the vitreous humor. It was like
viewing the world through a round fishbowl full of glycerol into
which a quart of blood and a handful of black pepper have been
partially mixed. There was local pain within a few minutes of the
accident, but it did not become excruciating. The most immediate
response after such an incident is horror. As a Vietnam War
Veteran, I have seen several terrible scenes of human carnage, but
none affected me more than viewing the world through my
bloodfilled eyeball. In the aftermath of the accident I went into
shock, as is typical in personal injury accidents.
"As it turns out, my injury was severe but not nearly as bad
as it might have been. I was not looking directly at the prism from
which the beam had reflected, so the retinal damage is not in the
fovea. The beam struck my retina between the fovea and the optic
nerve, missing the optic nerve by about three millimeters. Had the
focused beam struck the fovea, I would have sustained a blind spot
* The author had been using a relatively low power neodymium-yag laser.
He had not wearing protective goggles, even though they were available in the
laboratory because the goggles tend to result in tunnel vision, they fog up, and
they can become uncomfortable during long hours in the lab.
EE 737 Photonics Laboratory Manual Safety
3
in the center of my field of vision. Had it struck the optic nerve, I
probably would have lost the sight of that eye.
"The beam did strike so close to the optic nerve, however, that it
severed nerve-fiber bundles radiating from the optic nerve. This has
resulted in a crescent-shaped blind spot many times the size of the
lesion.... The effect of the large blind area is much like having a finger
placed over one's field of view of my damaged eye, although the blood
streamers have disappeared. These "floaters" are more a daily hindrance
than the blind areas, because the brain tries to integrate out the blind area
when the undamaged eye is open. There is also recurrent pain in the eye,
especially when I have been reading too long or when I get tired." [1]
During the course of this laboratory, you will be working with some
dangerous equipment. In order to protect yourself, your classmates and your
beloved instructor, you will want to know what the hazards are, how dangerous
they are, and what precautions to take. While protecting yourself and the people
around you must be your first priority, there are also a variety of interesting
ways to damage the equipment, and we will concern ourselves with that kind of
safety as well.
The dangers to people fall into three primary categories as far as this
laboratory is concerned: radiation hazards (laser beams), electrical hazards (high
voltage), and chemical hazards.
The equipment is also sensitive to radiation, electricity, and chemicals. For
example, a highly focused high-power laser beam can damage the surface of a
mirror. Some of the devices you'll be handling are susceptible to electrostatic
discharges (ESD). The chemical hazards may come from you- the oils on your
EE 737 Photonics Laboratory Manual Safety
4
skin, for example, can permanently ruin the coatings on a lens if you pick it up
without gloves.
Since people are harder to replace than lenses, we will begin with personal
safety.
lens
anterior
chamber
posterior chamber
ora serrataretina (inside surface)
cornea
optic nerve
iris
Figure 1. Structure of the eye.
RADIATION HAZARDS;
Eye DamageThe structure of the eye is shown in the crude drawing in Figure 1.
The eye has a thin eyelid made of skin (not shown), underneath which is the
cornea, a structure that is transparent to visible wavelengths (so you can see).
The cornea has no blood vessels, and has a refractive index of 1.376. [2] The
cornea absorbs radiation in the wavelength range of 200 nm-315 nm, and this
absoprtion can result in photokeratitis, an inflammation of the cornea. [3] To put
that wavelength range into perspective, the visible range is approximately
between 380 nm and 770 nm. Below 380 nm is the ultraviolet region. The cornea
is also susceptible to infared radiation in the range of 3µm to 1 mm. [3]
Behind the cornea is the anterior chamber, containing the aqueous humor,
a nutrient-bearing fluid that has an index of reaction of 1.336. Beyond this
chamber is the lens, which is also transparent to visible light, but is a different
EE 737 Photonics Laboratory Manual Safety
5
tissue type than the cornea. The lens has a refractive index of 1.41 (near the
center). [2] The lens is absorbing in the wavelength range of 315-400 nm [2] and
exposure to light at these wavelengths can produce cataracts. [3]
Behind the lens is the posterior chamber, filled with vitreous humor, a
gelatinous material of refractive index 1.336. It is in this chamber that small epics
of cellular material that are not transparent occasionally appear. These look like
squiggles or spots in your field of vision and are called vitreous floaters. If you are
familiar with the principles of Fourier optics, you will see diffraction patterns
around the edges of these (you'll see them anyway, but if you know your Fourier
you'll understand them).
At the back of the eye is the retina, which contains about 100 ! 106 rods
and 10 ! 106 cones. It is the retina that is primarily damaged by visible and near
infrared light, in the 400nm - 1400 nm range. [3] The lens and cornea pass
radiation in this range and focus it onto the retina. High power levels can
therefore result in burns to the retina that do not heal.
DAMAGE THRESHOLDS AND MPE's
It would be nice to give a number and say, "A laser more powerful than
this is dangerous." When you are about to work with a laser, you will want to
figure out the Maximum Permissible Exposure (MPE) for that laser.
Unfortunately, the thing is very complex. Damage thresholds depend on the
wavelength of the light, the time of exposure (how long the beam hits you and
whether or not it's pulsed), and the conditions under which it's viewed
(intrabeam versus extended source viewing).
"Intrabeam" viewing means viewing a laser beam by putting one's eye
directly into the beam. The same conditions can also be achieved by looking at a
specular reflection off an object such as a watchband or an optical post. The term
EE 737 Photonics Laboratory Manual Safety
6
"extended source" refers to conditions in which the beam is sufficiently divergent
(reflecting off a diffuse surface such as a piece of paper, or an uncollimated
beam). Figure 2 shows some typical situations that result in these types of
viewing.
a
r
!
eye
Figure 2a. Intrabeam viewing- primary beam. After [4]
mirror
eye
direct beam
indirect beam
laser
Figure 2b. Intrabeam viewing- specularly reflected secondary beam. After [4]
Figure 2 continued next page
EE 737 Photonics Laboratory Manual Safety
7
eye
laser
a
!
r
r1
"vDL
#
Figure 2c. Extended source viewing- normally diffuse reflection. After [4]
!
r
"
Figure 2 d. For a small, but highly divergent beam (such as that emitted from a
semiconductor laser), this is still intrabeam viewing, since it is the angle subtended by the eye , not the source, that is the issue.
The type of viewing one is exposed to is determined by the limiting angle
!min. When the angle is greater than !min, the MPE needed is that for extended
viewing, for angles less than !min, intrabeam viewing MPE's apply. Figure 3
shows the value of !min as a function of exposure time.
Figures 4, 5, and 6 gives the MPE values at the cornea for direct
(intrabeam) viewing, as a function of wavelength and time exposure. For
wavelengths between 0.7 µm and 1.4 µm, you use Figure 4, lower line, but you
must apply a correction factor from Figure 7. For example, you may be using a
940 nm laser diode, operating CW (continuous wave, meaning it's always on).
EE 737 Photonics Laboratory Manual Safety
8
Suppose you think it would not take you longer than 5 seconds to realize you are
being exposed and jerk your head away. (Radiation at 940 nm is in the infrared
region, so you wouldn't see it.) What is the MPE? Wavelengths between 0.4µm
and 1.4 µm are shown in Figure 4, from which the MPE is 1.8CAt3/4!10-3 J/cm2.
The factor CA must come from Figure 7, where it is found to be 2.9. Therefore the
Maximum Permissible Exposure is (1.8)(2.9)(53/4)(10-3) = 17.4 mJ/cm2.
For extended source (diffuse) viewing, Figure 8 must be used. Again,
correction factors for wavelengths between 0.7 and 1.4 µm must be applied from
Figure 7.
1
2
4
6
810
2
4
Subte
nse
angle
!m
in (
mra
d)
10-8
10-6
10-4
10-2
100
Exposure duration (sec)
Figure 3. Limiting angular subtense ("min), after ANSI. [5] Extended sources lie
above the line; apparent visual angles below the line are considered intrabeam
viewing.
EE 737 Photonics Laboratory Manual Safety
9
10-7
10-6
10-5
10-4
10-3
10-2
Radia
nt
Exposure
(J/
cm
2)
10-5
10-4
10-3
10-2
10-1
100
101
Exposure Duration (sec)
!=1.06 " 1.4 µm
!=0.4 " 0.7µm
Figure 4. MPE's for eye exposure to visible and near infrared (intrabeam
viewing), single exposure. After [5] Functional form of lower line: Radiant
exposure=1.8!10-3 t3/4 (J/cm2). For wavelengths between these two line (0.7
µm – 1.3 µm), functional form of lower line applies, with correction factors (Figure
7).
0.001
2
4
0.01
2
4
0.1
2
4
1
Radia
nt
Exposure
(J/
cm
2)
0.340.320.300.280.260.240.220.20
Wavelength (µm)
Figure 5. MPE for eye exposure to ultraviolet beams (intrabeam viewing), single
exposure. After [5] .
EE 737 Photonics Laboratory Manual Safety
10
0.01
2
4
6
80.1
2
4
6
81
Radia
nt
Exposure
(J/
cm
2)
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Exposure Duration (sec)
Figure 6. MPE's for eye exposure to far infrared beams (!-1.4µm – 1mm)
(intrabeam viewing), single exposure. After [5] Funtional form is Radiant
Exposure=0.56 t1/4 (J/cm2)
1
2x100
3
4
5
6
Corr
ecti
on F
acto
r C
A
1.41.31.21.11.00.90.80.7
Wavelength (µm)
Figure 7. Correction factors for wavelengths 0.7-1.4µm (CF8.5.2). After [5]
EE 737 Photonics Laboratory Manual Safety
11
0.001
0.01
0.1
1
10
100
Inte
gra
ted R
adia
nce (
J•cm
-2/
sr)
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Exposure Duration (sec)
!=1.06 " 1.4 µm
!=0.4 " 0.7 µm
Figure 8. Maximum Permissible Exposure (MPE) for viewing a diffuse reflection
of a laser beam or an extended source laser. After[5]
Skin Damage
Damage to the skin can also result from laser radiation. There is some
feeling in the laser community that laser damage to the skin is not as dangerous
since the skin can heal better than eyes can. Lasers can, however cause
"reddening, blistering, charring, and actual burned-out cavities" in the skin. [4]
Furthermore, the same lasers that can burn the skin can also damage materials
such as glass and plastic, [6] meaning a beam intense enough to burn your skin
can also burn your protective eyewear. Such a beam should be entirely enclosed
and inaccessible to the user. Still higher powers can set fire to clothing.
Gaussian and Elliptical beams: From Table 1, it is clear we will need to calculate
the power density (W/cm2) or energy density (J/cm2) for any lasers we might be
using. For a HeNe laser, which emits a circular Gaussian beam, the beam
diameter a is generally given. This number is actually the diameter of the 1/e
EE 737 Photonics Laboratory Manual Safety
12
Table 1. MPE's for Skin Exposure to a Laser Beam (Copied with permission from Laser Institute of America[5] Wavelength ! (µm) Exposure Duration t
(sec) Maximum Permissible Explosure (MPE)
Notes for Calculation and Measurement
Ultraviolet 0.200 to 0.302 10-9 to 3 " 104 3 " 10-3 J•cm-2
0.303 10-9 to 3 " 104 4 " 10-3 J•cm-2
0.304 10-9 to 3 " 104 6 " 10-3 J•cm-2
0.305 10-9 to 3 " 104 1.0 " 10 -2 J•cm-2 or 0.56 t1/4 J•cm-2 0.306 10-9 to 3 " 104 1.6 " 10 -2 J•cm-2 whichever is
0.307 10-9 to 3 " 104 2.5 " 10 -2 J•cm-2 lower
0.308 10-9 to 3 " 104 4.0 " 10 -2 J•cm-2
0.309 10-9 to 3 " 104 6.3 " 10 -2 J•cm-2 1 mm limiting
0.310 10-9 to 3 " 104 1.0 " 10 -1 J•cm-2 aperture
0.311 10-9 to 3 " 104 1.6 " 10 -1 J•cm-2
0.312 10-9 to 3 " 104 2.5 " 10 -1 J•cm-2
0.313 10-9 to 3 " 104 4.0 " 10 -1 J•cm-2
0.314 10-9 to 3 " 104 6.3 " 10 -1 J•cm-2
0.315 to 0.400 10-9 to 10 0.56 t1/4 J•cm-2
0.315 to 0.400 10 to 103 1 J•cm-2
0.315 to 0.400 103 to 3 " 104 1 " 10-3 W•cm-2
Visible and Near Infrared
0.400 to 1.400 10 -9 to 10 -7 2CA " 10-2 J•cm-2 1 mm limiting
10-7 to 10 1.1 CAt1/4 J•cm-2 aperture
10 to 3 " 104 0.2 CA W•cm-2
Far Infrared 1 mm limiting
1.4 to 103 10 -9 to 10 -7 10-2 J•cm-2 aperture for
10-7 to 10 0.56 t1/4 J•cm-2 1.4 to 100µm >10 0.1 W•cm-2 11mm limiting
aperture for 0.1 to 1mm
1.54 only 10-9 to 10-6 1.0 J•cm-2
points of the (circular) Gaussian energy profile. The peak power density of the
beam is then given by
E =4!
"a2
[1]
for a single transverse mode laser, where # is the total laser power.
EE 737 Photonics Laboratory Manual Safety
13
Semiconductor lasers, on the other hand, emit elliptical beams. You'll be
reading more about these if you are doing the lasers experiment, but in this case,
the peak power density is
E =1.27!
b + r"1( ) c + r"
2( ) [2]
where b is the length of the major axis of the elliptical beam cross section, c is the
minor axis, !1 is the beam divergence in the direction corresponding to the major
axis, and !2 is the beam divergence associated with the direction of the minor
axis. These parameters are shown in Figure 9.
cb
!2
!1
Figure 9. Definitions of terms in equation [2] for an elliptical beam.
Lasers used in this course
In this particular course, we will be using two basic types of lasers, HeNe
lasers that emit at 632.8 nm, at powers generally less than 5mW (which is still
dangerous), and semiconductor lasers, which may be visible, or in the infrared
region between 800 nm and 1.55µm. These are all dangerous lasers. How
dangerous? The American National Standards Institute classifies all lasers
according to their hazard potential. There are four categories, From Class 1 lasers
EE 737 Photonics Laboratory Manual Safety
14
which present essentially no hazard, to Class IV lasers, from which even a diffuse
reflection is hazardous. Class IV lasers can start fires and burn skin. These
classes are detailed in Table 2.
Table 2. Accesible Limits for Selected Continuous-Wave (>0.254 sec) Lasers and Laser
Systems, taken with permission from [5]
Wavelength Range
Emission Duration (sec)
Class I Class II (visible only)
Class 3 Class IV
Ultraviolet 3 ! 104 !0.8 ! 10-9 W to !8 ! 10-6
W depending on Wavelength
NA
>Class I but !0.5 W depending on wavelength
>0.5 W
Visible 0.4 -
0.550 µm 3 ! 104 <0.4 µW >Class I ,but <
1 mW >Class II but < 0.5 W
>0.5 W
Visible and Near Infrared
0.55 - 1.06 µm
3 ! 10-4 < 4 µW to
<200µW, wavelength dependent
NA
> Class I but < 0.5 W, wavelength dependent
>0.5 W
Near Infrared, 1.06 - 1.4 µm
3 ! 104 <200µW NA > 200µW but < 0.5 W
>0.5 W
Far Infrared,
1.4-100 µm
>10 <0.8 mW NA > Class I but < 0.5 W
>0.5 W
Safety Rules
Before you use any laser you will have to determine the type of hazards it
poses. Once that is known, you can establish what kinds of precautions are
necessary. These precautions are detailed in your supplemental laser safety text.
For example, when using a Class III laser, you should always wear goggles,
unless you completely encose the beam or use neutral density filters to reduce
the power of the beam immediately after the laser to levels classified as Class I.
Class I lasers require no special precautions. With a Class III laser, you should in
any case use a beam stop to prevent the laser beam from leaving your bench, so
EE 737 Photonics Laboratory Manual Safety
15
that a person walking by your table cannot accidently be exposed, and you
should avoid setups that put beams at or near eye level.
ELECTRICAL HAZARDS (to people)
The electrical hazards to people that will come up in this laboratory are
from the high voltage power supplies used to power the gas lasers, and AC line
voltages, particularly those experiments using a variable transformer (solar cells,
quantum wells).
Some good policies to follow for electrical safety are:
1. Assume all circuits are live until you have personally checked that they
are disabled.
2. Metallic or otherwise conductive rings, bracelets, watches, etc. should
not be worn when working with high voltage circuits, power supplies, etc. Also,
metallic pens, rulers, etc. should be avoided during work.
3. It only takes 200mA to kill a person (if it goes through the heart).
Current can go through the heart when travelling from a hand to a foot, or from
on hand to the other. Therefore:
a. Do not stand on a wet floor while working on circuitry. Assume
all floors are conductive and grounded unless special precautions have
been taken (insulating mats, etc.). Do not touch circuitry while hands, feet,
body are wet or perspiring.
b. Use only one hand whenever possible.
c. If you must touch an electrical device (to check for overheating,
for example), use the back of your hand. Electrical current makes the
muscles contract, causing your fingers to close into a fist. By using the
back of your hand, you ensure that your fingers close away from the wire,
naturally removing your contact, instead of causing your fingers to grip
EE 737 Photonics Laboratory Manual Safety
16
the wire, after which the flow of electricity will prevent you from letting
go.
d) Wear protective goggles if there is a possibility of sparks or
arcing.
If an electrical shock occurs, kill the circuit. Once that is done, it is safe to
touch to victim to administer first aid. This may include cardio-pulmonary
resuscitation to restart breathing and/or the heart.
CHEMICAL HAZARDS (to people)
The liquid crystal experiment requires you to use various chemicals,
including the liquid crystal material itself. The only potentially dangerous
chemical in this process is alcohol, which is flammable, and acetone, which is
carcinogenic. Therefore gloves must be worn to avoid skin exposure.
Furthermore, acetone qualifies as hazardous waste and must be disposed of
properly; it cannot be dumped down the drain. Used acetone must be kept in
special containers, and when the container is full, it will be collected by the
university and disposed of properly.
EQUIPMENT HAZARDS
In this section, we will be talking not about the hazards that the
equipment poses, but rather the ways in which you can damage the equipment
and devices through negligence or ignorance. The key issue in this section is
electro-static discharge (ESD), to which many of the devices are very sensitive.
That is, you can destroy some of the devices merely by picking them up.
Beyond ESD, there are certain rules for care and handling of optics (mirrors,
lenses, fibers) you are expected observe.
EE 737 Photonics Laboratory Manual Safety
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Electro-Static Discharge (ESD)
As you know from petting your cat on a fine winter's day in your living
room with the polyester carpet, you can carry around substantial amounts of
electricity. Your cat is a reasonably large structure and can dissipate the typical
amounts of charge you carry, but some of the semiconductor devices have very
small structures- on the order of microns- and these can be destroyed by
discharges so small you don't even feel them. You have these charges on you all
the time, just from walking and moving around.
For you to feel the shock of an electrostatic discharge, you must discharge
at least 4000V. [7] By contrast, Table 1 shows the damage thresholds for some
common semiconductor devices. You can see why the manufacturer of your
personal computer wants you to have a trained technician install the extra
memory! That technician is presumably taking the proper precautions to protect
those memory chips.
The charge that you carry is generated triboelectrically, tribo from the
Greek for "rub". Any two surfaces that rub together, such as your foot against
the floor, your hand across the table, the pen across the tablet, air blowing across
your work, can exchange charge, leaving both surfaces with a residual charge.
Even just two surfaces separating, without rubbing, can transfer charge from one
to the other. One insidious example is adhesive tape- when you take a peice off
the roll, you are forcibly separating one layer from another, leaving thousands of
volts on the tape. This charge is hard to dissipate since the tape is non-
conducting, but can be transferred to whatever you're taping.
During triboelectric generation, one material is generally left with net
positive charge, and the other with a net negative charge. These depend on what
two materials are involved- materials higher up in Table 2 tend to acquire
EE 737 Photonics Laboratory Manual Safety
18
positive charge when separated from materials lower down. Things like surface
finish and contamination will also affect relationships on this list.
TABLE 1. Representative Sample of Part Susceptibility Data [7] Device Number Technology Susceptibility
Level Pin Combination
VN98AK VMOS 110V (+) Gate to source 3N170 MOSFET 150V (-) Gate to source Custom IC CMOS 150V (-) Input to
ground 2N4416 JFET 220V (-) Gate to source Custom IC Bipolar op-amp 400V (-) Input to
ground 1N5711 Schottky diode 500V (-) Anode to
cathode MC1660 ECL 500V (+) Output to Vcc CD4001A CMOS 800V (-) Input to Vdd 54S04 Schottky TTL 1000V (+)Input to Vdd RNC50 Thin-film resistor 1000V Lead to lead 5404 TTL 1600V (+) Input to Vdd 54L04 Low-power TTL 3500V (-) Input to
ground 2N2222 Bipolar transistor 15,000V (+) Emitter to base
TABLE 2. Triboelectric Series [7] POSITIVE +
NEGATIVE (-)
Acetate Glass Nylon Wool Silk
Aluminum Polyester
Paper Cotton Steel
Nickel, copper, silver Zinc
Rubber Polyurethane foam
PVC (vinyl) Teflon
EE 737 Photonics Laboratory Manual Safety
19
A conductive material can easily be discharged by connecting it to
ground, while insulating materials are difficult to discharge. In general, the
surface resistivity of objects and people will affect how easily the charges can be
removed, and the capacitance combined with the resistance will affect the speed
with which the charge moves. Since there is no such thing as a perfect insulator,
the charge on any object will eventually bleed off. If the charge on object is
transferred instantly, as with a conductor, the instantaneous current can be very
high, and damage parts. On the other hand, insulators can more easily have
extremely high voltages on them to start with, and are therefore also potentially
dangerous. The charges on the insulator will cause en electric field, which can in
turn induce charges onto nearby objects. Nothing is innocuous.
ESD Precautions
In industry, precautions are taken during manufacture of sensitive items
(for example, computers). Depending on the degree of sensitivity, these
precautions may include conductive flooring, conductive smocks, ionized air
blowers, wrist straps, grounding chains hanging from carts, and conductive
packaging and bins for components.
The most dangerous object in the laboratory from the point of view of ESD
is a person, because people move around a great deal and touch everything. The
single most important precaution to take, therefore, is to ground the human. This
is most often done with a conductive wrist strap connected to a good ground.
These wrist straps generally have a 1M! resistor in series with the ground
connection, to prevent electrocution if the person accidentally touches 120V
while grounded.
Another effective and common strategy is to ground the work surface,
which must therefore be conductive. This is useless if the person walking up to
EE 737 Photonics Laboratory Manual Safety
20
the bench and picking up a board is not also grounded, meaning that wrist straps
are still necessary.
In this laboratory course, you will be using conductive work mats which
must be grounded and wearing a wrist strap which must be grounded when working
with sensitive devices. You should check that the mat is grounded every time
you use it, and check that it is clean. Excessive dust or contamination on the
surface will make it more resistive, therefore compromising its effectiveness.
The wrist strap must fit snugly enough to make good electrical contact
with your skin. It does no good to wear it over your sleeve. Those of us with
hairy arms may have to wear the strap such that the conductive button (some
styles) is on the inside of the wrist, where there is usually less hair.
The components themselves must be kept in a closed , conductive container
until a grounded work surface and grounded worker are available. Then and
only then may the part be removed from the packaging. The component must be
handled only by grounded personnel until it is safely installed in a circuit or
fixture with adequate grounding to protect the device.
In this laboratory course, the most ESD sensitive devices you will handle
are laser diodes. Not only can these devices be instantaneously destroyed by an
ESD event, they can be damaged in subtle ways that don't show up until some
later time. They can suddenly start to degrade faster, or stop lasing.
Transient Protection for Laser Diodes
The laser diodes can also be destroyed by transients that occur, for
example, when a power supply is turned on. For this reason, we have purchased
special laser power supplies with a soft turn-on. These prevent current to the
laser from spiking. When using any other type of power supply, you must 1)
disconnect the laser when plugging in and turning on the supply, then 2) with
EE 737 Photonics Laboratory Manual Safety
21
the current output set to zero, connect the laser, and only then 3) slowly increase
the current to the operating level.
Chemical Hazards to Equipment
The chemical hazards to the equipment come primarily from you. The
primary source of concern is skin oil and other organic contaminants, which can
ruin the optics, and mirrors and lenses are not cheap.
The mirror in your bathroom at home has a silvered back, with a layer of
glass on top. When the mirror is dirty, you wash it with Windex®. In the optics
laboratory, the mirrors are silvered (or aluminum'ed) on the front surface (why?).
If you touch that surface, your fingerprint, a messy pile of oil, will destroy the
reflectivity, and the mirrors are not much bigger than a fingerprint. Attempting
to wipe your print off may scratch the metallization. You must handle mirrors only
by the edges . As additional precaution, when mounting mirrors, you should
wear gloves. If the mirror is dusty, the preferred cleaning method is to blow the
dust off with N2 (not your breath! more chemicals!), or gently wipe with lens
tissue (not your sleeve).
The lenses and filters you will be using generally have some antireflection
coatings deposited on their surfaces, which can also be destroyed by skin oil. You
must handle lenses and filters only by the edges. You should also wear gloves when
mounting and aligning optics.
SUMMARY
You are the most dangerous component in the lab, since you a) move
around, b) look at things, c) touch things. Fortunately, you are also able to
control your behavior to eliminate all risks to yourself, your classmates, and the
equipment.
EE 737 Photonics Laboratory Manual Safety
22
Homework:
1) The acousto-optic experiment uses a HeNe laser that has a wavelength
of 632.8 nm (red), a minimum spot size of 0.59mm , a divergence (assuming no
additional optics) of 1.35 mrad, and a CW power of 1 mW.
a) What is the MPE? Assume a reaction time of a quarter of a second, sine
this is a visible laser. During typical use in the lab, you will be dealing with
intrabeam viewing situations. How close to the laser would you have to be for
the situation to be considered extended source viewing?
b) Classify this laser.
c) What part of your eye is the most susceptible to this laser, and what are
the possible effects of looking at this beam? Is there a potential for skin damage
(give numbers to defend your answer)?
d) How long can you safely look into this beam? Note you cannot use your
previous MPE value to compute this, since that was based on a specific time of
0.25 sec. You will have to use the functional form.
e) What safety precautions must you take?
2) The laser physics experiment uses a semiconductor laser that emits at
788nm (infrared), has a spot size of approximately 10µm by 1µm, a divergence of
10° parallel (HWHM) to the junction plane and 35° perpendicular to the junction
plane (the lit spot on the surface of the laser is an ellipse, and the emission is also
elliptical, figure next page), and a CW power of up to 50 mW.
a) What is the MPE? Assume it would not take you more than 10 seconds
to realize you are being exposed. You'll have to assume a distance from the laser-
how close are you likely to put your eye to the laser? 5 cm? 2 cm? Is this
intrabeam or extended source viewing?
EE 737 Photonics Laboratory Manual Safety
23
b) Classify this laser.
c) What part of your eye is the most susceptible to this laser, and what are
the possible effects of looking at this beam? Is there a potential for skin damage?
d) How long can you safely look into this beam?
e) What safety precautions must you take?
f) To use this laser, you will have to collimate the beam. Assuming you
use a 40X lens for this purpose, you will reduce the divergence to about .13°, and
increase the spot to a circle of radius about 0.2mm. Now what is the
classification?
g) Have the precautions needed changed? If so, how?
3. We have two kinds of goggles in the lab. Their attenuation curves are
given below. For each of the experiments above, decide which goggles to use.
Will they provide enough protection?
Optical Density (O.D.)=-log(Pout/Pin)
0
4
6
8
10
2
OPTIC
AL D
EN
SIT
Y
200 300 400 500 600 700 800
Wavelength (nm) Curve A.
EE 737 Photonics Laboratory Manual Safety
24
0
15
10
5
OPTIC
AL D
EN
SIT
Y
300 500 700
Wavelength (nm)
900 11001300 1500
Curve B.
LIBRARY PROBLEM:
This write-up addressed the safety issues with lasers, but what about
very bright incoherent light? Specifically, the light source in the solar cell and
quantum well experiments are 300-W quartz halogen, tungsten filament light
bulbs (ANSI designation ELH), with an approximately 1cm diameter emitting
area. They are incoherent sources, but still quite bright. Is there a danger to your
eyes from these?
Hints: You will need first to determine the energy spectrum of this source,
and find out how much energy is in the ultraviolet, visible, and infrared regions.
An incandescent filament acts as a blackbody radiator, and the temperature is a
function of the power of the bulb. Incandescent sources are also remarkably
inefficient; only about 10% of the electrical power into a standard incandescent
bulb is emitted as visible light.
REFERENCES
[1] C. D. Dekker, “Accident victim's view,” Laser Focus, August, p. 6,
1977. With permission.
EE 737 Photonics Laboratory Manual Safety
25
[2] F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics, Englewood
Cliffs: Prentice Hall, 1987.
[3] J. A. Smith, “Laser Safety Guide,” , 8th ed: Laser Institute of
America, 1992.
[4] A. Mallow and L. Chabot, Laser Safety Handbook. New York: Van
Nostrand Reinhold Company, 1978.
[5] The Laser Insitute of America, “American national standard for the
safe use of lasers ANSI Z136.1-1986,” , 1986.
[6] D. C. Winburn, Practical Laser Safety, vol. 11. New York: Marcel
Dekker, Inc., 1985.
[7] O. J. McAteer, Electrostatic Discharge Control. New York: McGraw-
Hill, 1990. With permission.
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!!""/0<$1#2,? @UB
C&808"$""*,"(&8".)934.0"D08V38)+1"'D"(&8"4*9&(F"23*%4&!*,"(&8"5'6/!6/71%-8!.)E"(&8
64.)8"C.H8"*,"60'6.9.(*)9".4')9"(&8"D*/80".(",'=8".)948"!"('"(&8"D*/80".>*,;"T8
.08"3,3.441"')41"+')+80)8E"./'3("(&8"4')9*(3E*).4"+'=6')8)("'D"(&8"C.H8H8+('0Q
!!""' ! $I2 +',! @#B
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 56(*+.4" 78),*)9
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k
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"
Figure 5. Relationship between " and k.
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A'=8?"<&*+&"60'6.9.(8,".4')9"(&8".>*,"0.(&80"(&.)".(",'A8".)948"('"*(B
!"#$%&'()*&'+#*'(,%-"()*&
5)8"<.1"('"+4.,,*@1"(168"'@"@*/80"*,"/1"(&8")3A/80"'@"A'=8,"*("+.00*8,C
,*)948"A'=8"@*/80"')41",366'0(,"')8"A'=8?".)="A34(*A'=8"@*/80",366'0(,"A.)1B
!.+&"A'=8"&.,"*(,"'<)"6.0(*+34.0""B
;8"&.D8",'"@.0"/88)"3,*)9"0.1,"('"=8,+0*/8"(&8"4*9&("60'6.9.(*')"*)".
@*/80E".)="(&.(F,"')8"<.1"('"=8,+0*/8"A'=8,B"G"6.0(*+34.0"A'=8"60'6.9.(8,".(".
6.0(*+34.0".)948?".)="&.,"."6.0(*+34.0""B""H)"08.4*(1"(&8"4*9&("*,".)"848+(0'EA.9)8(*+
@*84=?".)="(&8"0.1"*,".)".0(*@*+*.4"+'),(03+(*')"(&.("+'A8,"@0'A"+'))8+(8=".44"(&8
68068)=*+34.0,"('"(&8"<.D8@0')(B"H@"<8".,,3A8"(&.("(&8"<.D8,"*)"(&8"@*/80".08
64.)8"<.D8,?"(&8)"8.+&"A'=8"+'),*,(,"'@",",8("'@"64.)8"<.D8,"(0.D84*)9".(".
,68+*@*+".)948"('"(&8"@*/80".>*,?"I*9308"JB""!.+&"A'=8"(0.D84,".(",'A8"=*@@808)(
.)948B
Figure 5. Rays of a particular mode.
K.1,".08"3,8@34"('"@'0"98((*)9"."6*+(308"'@"(&8"60'6.9.(*')?"/3("(&8"0.1
.).4'91"='8,)F("08.441"&'4="36"*)",'A8"+.,8,B"L'"98("."08.441".++30.(8"6*+(308?"<8
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 56(*+.4" 78),*)9
$:
&.;8"('"9'"('"848+(0'<.9)8(*+,".)=",'4;8"(&8">.;8"8?3.(*')"*)"."+14*)=0*+.4
=*848+(0*+"<8=*3<",300'3)=8="/1".)'(&80"+14*)=0*+.4"=*848+(0*+"<8=*3<@"A&*,"*,
)')B(0*;*.4C".)=">8">*44"9*;8"D3,(",'<8"98)80.4"E.)=";801",3680F*+*.4G"08,34(,"&808@
H'0"8I.<648C"*F">8"8I.<*)8"(&8"F*84="'F"(&8"!(&"<'=8C">&808"!"*,",'<8
*)(8980C".,"."F3)+(*')"'F"0.=*3,"F0'<"(&8".I*,C"*("(30),"'3("(&8",'43(*')"*,"."J8,,84
F3)+(*')@"A&.("*,C"(&8"F*84="!!"#$%*,"9*;8)"/1K
%%""!! E#G ! &'! E(#G)*!") *E#+$%,G E+'08G LMN
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%%""(: ! -P
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+'08S"'3(,*=8"'F"(&8"+'08"(&8",'43(*')"*,"."J8,,84"F3)+(*')"'F"(&8",8+')="O*)=K
%%""!! E#G ! &4! E5#G)*!") *E#+$%,G E+4.==*)9G LPTN
U)"(&*,"8?3.(*')C
%%""5: ! % : $ -:
:.: LPPN
>&808"(&8"%"".)="."(80<,"&.;8"/88)"08;80,8="('"O886"(&8".093<8)(,"08.4@""A&8
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$$
1.0
0 .8
0 .6
0 .4
0 .2
0 .0
- 0 . 2
- 0 . 4
J!(x
)
1 21 086420
x
Bessel Function of the First Kind
J0
J1
J2
2 .0
1 .5
1 .0
0 .5
0 .0
K!(x
)
1 21 086420
x
Bessel Function of the Second Kind
K0
K1
K2
Figure 6. Bessel functions of the first and second kinds.
:'(*+8"(&.(!!"!!*,"+'),(0.*)8;"('",(.1"/8(<88)"!"#"".);"!$#"=",*)+8"#!+.)
)8>80"/8"908.(80"(&.)"#%"?08@A"!BA"C".);"!BA"$DA"E@""&!"#="(&8"F';8"*,"(&8
@3);.F8)(.4="'0".G*.4"F';8=".);""""+.))'("/8".)1"4.0980"(&.)"(&.(A"H'0"&*9&80
'0;80"F';8,=".,"(&8".)948"'@"(&8"F';8I,"0.1"98(,"&*9&80".);"&*9&80=""&#%'()#*
98(,",F.4480A"J'3"+.)",&'<"(&.("<&8)""&!$#="(&8"F';8"*,")'"4')980"93*;8;="'0
<8",.1"*("*,"%+,-'../
K&8,8"(<'"@3)+(*'),="(&8"L8,,84"@3)+(*')"'@"(&8"@*0,("M*);".);"(&8"L8,,84
@3)+(*')"'@"(&8",8+');"M*);="".08",8("8B3.4".("(&8"+'08"/'3);.01"('"'/(.*)"')8
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 56(*+.4" 78),*)9
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,;''(&<"+')(*)3'3,"+30=8">8,+0*/*)9"(&8">*,(0*/3(*')"'?"(&8"?*84>"*)"(&8"!!"";'>8<
@*9308"#A
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0.8
0.6
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0.0
a ar
Figure 7. The fundamental mode. Inside a, the function is a Bessel function of the first
kind, and outside a, in the cladding, it is a Bessel function of the second kind. The
vertical axis is field strength.
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;'>8,"."6.0(*+34.0"?*/80"+.00*8,D"F'"*)=8,(*9.(8"(&*,<"C8"C*44">8?*)8"C&.("*,"+.448>
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=.438<"*(">8(80;*)8,"(&8")3;/80"'?";'>8,"+.00*8>"/1"."6.0(*+34.0"?*/80A"F&*,"+.)
/8"08.>"'??"'?"."+&.0(<"?'0",;.44")3;/80,"'?";'>8,<"'0"+';63(8>"3,*)9"(&8
.660'L*;.(8"8M3.(*')N
$$"", !
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.664*+.(*'),"')8"3,8,"8*(&80",*)948";'>8"?*/80"'0"?*/80"(&.(",366'0(,"(&'3,.)>,"'?
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 56(*+.4" 78),*)9
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;'<8="">(",&'34<"/8")'(8<"(&.("(&8"?3)<.;8)(.4";'<8"*,")8@80"+3("'??=">(";.1")'(
60'6.9.(8"@801"!"##$"/3("*("*,".4A.1,",366'0(8<"/1"(&8"?*/80="B",*)948";'<8"?*/80
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7*)+8"(&8"%"6.0.;8(80"<8(80;*)8,"(&8")3;/80"'?";'<8,C".)<",*)+8"*(
<868)<,"')"(&8"+'08"0.<*3,".)<"(&8"*)<8D"<*??808)+8C"A8")'A"E)'A"&'A"('
<8,*9)"."?*/80"('"+.001"')41"')8";'<8="F&8"%&6.0.;8(80";3,("/8"48,,"(&.)"G=HI:C
'0".",8+')<";'<8"A*44",(.0("('"60'6.9.(8=">)(808,(*)941C"(&8"?3)<.;8)(.4";'<8"*,
)8@80"+3("'??"*)"'6(*+.4"?*/80,=""B"?*/80")88<,".",;.44"+'08"J(16*+.441":!;K"('"/8
,*)948";'<8=">)";',(",8),*)9".664*+.(*'),C";34(*;'<8"?*/80"*,"3,8<C".)<"(&*,"*,
<8,*9)8<"('"+.001"."#'("'?";'<8,="F&8,8"?*/80"&.@8"4.0980"+'08,"J(16*+.441":I"('"LII
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')8"(&*)9C".)<"8.,*80"('"+'3648"4*9&("*)('"?'0".)'(&80="F&808".08C"&'A8@80C""9''<
08.,'),"*)"(&8"(848+';;3)*+.(*'),"/3,*)8,,"('"3,8",*)948M;'<8"?*/80C"<*,680,*')
/8*)9"."E81"')8=
!"#$%&'(#)%*'(%+,*(#(%&'(#)%-&.#,
F&808".08"(A'"/.,*+"(168,"'?"?*/80",(03+(308,C"(&8",(86"*)<8D"?*/80".)<"(&8
90.<8<"*)<8D"?*/80="F&8",(86"*)<8D"?*/80"&.,"."+'08"'?"+'),(.)("08?0.+(*@8"*)<8D")*C
.)<"."+4.<<*)9"'?"+'),(.)("08?0.+(*@8"*)<8D")+=""F&808?'08C"(&8"64'("'?"08?0.+(*@8
*)<8D".9.*),("0.<*3,"4''E,"4*E8".",(86"?3)+(*')C"O*9308"P.=">)"90.<8<"*)<8DC"')"(&8
'(&80"&.)<C"(&8"*)<8D"60'?*48"*,"+')(*)3'3,41"@.01*)9".+0',,"(&8"+'08C"3,3.441
6.0./'4*+.441C".,"*)"O*9308"P/=
2a 2a
n1
n2
n1
n2
(0)
Figure 8. Fiber refractive index profiles: left, step index; right, graded index.
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 56(*+.4" 78),*)9
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&.@8")'("8=64*+*(41",(.(8<"*,"(&.("(&8"J'<8,"*)"(&8">*/80".08"%&$"'()*+I"(&.("*,?"*
+.)"')41"&.@8",68+*>*+"<*,+08(8"@.438,K"P&*,"+.)"/8",&'L)"<*08+(41"/1",'4@*)9"(&8
848+(0'J.9)8(*+,"60'/48J"J8)(*')8<"8.04*80?"/3("L8"L*44"9*@8"."0.1".093J8)(
&808">'0"(&8"6306',8,"'>"*443,(0.(*')K"";30(&80J'08?">'0"(&8",.Q8"'>"+4.0*(1?"L8"L*44
.,,3J8".",4./"L.@893*<8?"*)"L&*+&"(&8"+'08"*,"."64.)8"'>">*)*(8"(&*+Q)8,,"/3(
*)>*)*(8"8=(8)(?".)<"(&8"+4.<<*)9"'++36*8,"(&8"08,("'>",6.+8K";*9308"R",&'L,"."+0',,
,8+(*')"'>"(&8"L.@893*<8K
n2
n1
n2
A
BC
D
+
E
d
Figure 9. Congruent rays in a slab waveguide. After [1]
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 56(*+.4" 78),*)9
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:8";*44"<'44';"&808"(&8".093=8)(,"'<""2.0+3,8>"?@A""B&8"C.,&8C"4*)8,
08608,8)("(&8"6&.,8<0')(,"'<"."6.0(*+34.0"64.)8";.D8"*)"(&8";.D893*C8E"(&8"0.1"'<
;&*+&"*,"60'6.9.(*)9".(",'=8".)948"!"('"(&8"<*/80".F*,>"G'(8"(&.(".)1
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,'"(&.("(&8"0.1"8)C*)9".("!"*,""#$%&'($)*('"(&8"0.1"8)C*)9".("+,*"7*)+8"-".)C"!".08
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G';"48(H,"<'44';"(&8"6&.,8",&*<("'<"."0.1"60'6.9.(*)9"<0'="!"('"/I"')8
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.("(&*,".)948";'34C"/8"'3("'<"6&.,8";*(&"8.+&"'(&80E".)C"C8,(03+(*D841"*)(80<808E
=8.)*)9"(&808";'34C"/8")'"8)8091"*)"(&*,"='C8>
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)&<$3=(&3(*@<%$()4"*A?BCE".)C"(&8"6&.,8",&*<("')"08<48+(*')"*,"9*D8)"/1
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(&8""6&.,8"+&.)98"9'*)9"@0'A")"('"*"A3,("/8",'A8"*)(8980"A34(*648"'@
;+%C*@@808)("@0'A"(&8"6&.,8"+&.)98"9'*)9"@0'A"&!H/8@'08"08@48+(*')I"('"+""H.@(80
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Q'(8"(&.("(&808"*,")'"6',,*/48"J.1"@'0"#"('"8P+88C"$6%D"(&.("08608,8)(,"(&8".P*.4
0.1D"(0.E84*)9".4')9"(&8"@*/80"+'08F"R4,'")'(8"(&.("(&8"A'C8,".08"L3.)(*M8C"2$78"*)
(&8"089*')"!!""$;% & # & $<% F"2'C8,"&.E*)9"#9$:%""J*44")'("/8"93*C8CD".08")'(
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.)C"J*44")'("08.+&"(&8"C8(8+('0D".,,3A*)9"."4')9"8)'39&"@*/80"@'0"(&8,8"A'C8,"('
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continuum of
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!
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25 24 23 22 21 20 15 10 m=0
n1
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continuum of
radiation modes quantized guided modes
!
no modes
GRADED
m=0123456789101112131415161718
Figure 10. Mode spacing in !-space for step and index fibers.
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60'6.9.(*)9".(",'>8".)948""!B"/3(".<(80"(&8"<*/80"/8)="*,",3==8)41"60'6.9.(*)9".(
.")8D".)948"""?"E8",.1"(&.("(&8"8)8091"&.,"/88)"#$%&'()"('"(&8"'(&80">'=8?"F&8
<*9308",&'D,"."+.,8"D&808"4*9&("*,"+'3648="<0'>"')8"93*=8=">'=8"('".)'(&80
93*=8=">'=8G"*(">*9&("H3,(".,"4*I841"/8"+'3648="('"."0.=*.(*')">'=8"J.)="4',(K?
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 56(*+.4" 78),*)9
:;
!!
!"
Figure 11. Example of mode coupling, in this case due to a bend.
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Figure 12. Microbending sensor
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OBJECT 1 OBJECT 2
Figure 13. A proximity sensor.
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/8.<,64*((80"*)"+')A8)(*').4"H/34GI"'6(*+,?
Figure 14. Use of a directional coupler and a mirror in a proximity sensor
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reference arm
input
beam
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Figure 15. A Mach-Zehnder fiber interferometer
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Figure 17. Sensitivity and dynamic range of two sensors.
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)')O(0*B*.4J".)C"G6"G*44"<*B6"P3,(",':6"<6)60.4"Q.)C"B601",3860=*+*.4R"06,34(,"&606@
?",*)<46":'C6"=*/60"&.,"."=*64C"C*,(0*/3(*')"(&.("4''S,"4*S6"5*<306"T@"2',(J
/3(")'(".44J"'="(&6"6)60<1"80'8.<.(6,"*)"(&6"+.06"'="(&6"=*/60@"A&6"6)60<1"*)"(&6
(.*4,"*,".+(3.441"80'8.<.(*)<"*)"(&6"+4.CC*)<J"6B6)"(&'3<&"*("*,"8.0("'="(&6"<3*C6C
:'C6@"A&6")6D("=6G"&*<&60O'0C60":'C6,".06",&'G)"*)"5*<306"U@"?"(18*+.4
:34(*:'C6"=*/60"G'34C"+.001"(&'3,.)C,"'="(&6,6":'C6J".44",38608',6CJ",'"(&6
'B60.44"*)(6),*(1"G'34C"4''S"4*S6"5*<306"U@
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234(*:'C6"=*/60,"&.B6"+'06,"=0':"W;!:"')"38J"G*(&";X"!:".)C"EXX!:"/6*)<
(18*+.4"B.436,@"A&606='06J",*)<46":'C6"=*/60,".06":3+&":'06"C*==*+34("('"+'3846
4*<&("*)('J",*)+6"(&6*0"+'06,".06",'":3+&",:.4460@"A&6"+4.CC*)<"'="/'(&"(186,"'=
=*/60"(6)C"('"/6"./'3("(&6",.:6J"')"(&6"'0C60"'="EXX!:"('"E;X!:"'3(60"C*.:6(60@
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;<
0-a a
Inte
nsity
Radius
Figure 4. The field distribution of a single mode fiber.
0
Inte
nsity
-a a
first mode
second
mode
Figure 5. First and second modes (fundamental mode is zeroth mode, not shown)
Figure 6. Intensity distribution in a fiber supporting thousands of modes.
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;<
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.)A"."+4.AA*)C"'?"+'),(.)("06?0.+(*E6"*)A6B"!#DD""=&606?'06@"(&6"84'("'?"06?0.+(*E6
*)A6B".C.*),("0.A*3,"4''F,"4*F6".",(68"?3)+(*')@"5*C306"#G46?(HD""I)"C0.A6A"*)A6B
?*/60@"')"(&6"'(&60"&.)A@"(&6"*)A6B"80'?*46"*,"+')(*)3'3,41"E.01*)C".+0',,"(&6
+'06@"3,3.441"8.0./'4*+.441@".,"*)"5*C306"#G0*C&(HD
n2
n1
2a
corecladding cladding
2a
corecladding cladding
n (0)1
n2
Figure 7. Fiber refractive index profiles; left: step; right: graded.
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*)+06.,6"(&6"/.)A>*A(&"/1"06A3+*)C"A*,860,*')D"P&.("*,"A*,860,*')S"('".),>60
(&.("Q36,(*')@"+'),*A60"(&6"?'44'>*)C"?*/60"'8(*+"4*)FT"*("+'),*,(,"'?"."4*C&(",'30+6@
>&*+&"*,"/6*)C"834,6A"')".)A"'??".++'0A*)C"('"(&6"A*C*(.4"A.(.D"=&',6"4*C&("834,6,
(0.E64".4')C"(&6"?*/60@".)A".06"+')E60(6A"/.+F"*)('".)"646+(0*+.4",*C).4"/1"(&6
8&'('A6(6+('0".)A"06+6*E60D
U,"(&6"4*C&(",'30+6"*,"?4.,&*)C@".(".)1"(*:6"(&6"4*C&("*,"')"0.1,".06"/6*)C
6:*((6A"'E60"."0.)C6"'?".)C46,D"V':6"'?"(&6"0.1,".06"+'3846A"*)('"(&6"?*/60".)A
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;<
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,4*=&(41"B*??606)(".)=46,>"C&6",(66860"(&6".)=46"?'0"(&6"0.1A"(&6"=06.(60"(&6".+(3.4
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(&606?'06"(.@6"4')=60"('"06.+&"(&6"'3(83(A"+.3,*)="(&6"834,6"('",806.B"'3("*)
(*:6A".,",&'E)"*)"5*=306"F>"G,"1'3"+.)",66A"6D6)(3.441"(&6"834,6,"E*44",(.0("('
'D604.8A"*?"(&6"?*/60"*,"4')="6)'3=&".)B"/*("0.(6"*,"?.,("6)'3=&>"G)B"(&.(A"/'1,
.)B"=*04,A"*,"E&1"(&6"860?'0:.)+6"'?"."?*/60"*,"=*D6)".,"."B*,(.)+6H/.)BE*B(&
80'B3+(>
Figure 8. Dispersion.
!"#"$%"!&
-6(I,"(.@6"(*:6"'3("B*,+3,,*')"'?",*=).4,"?'0"."4*((46"E&*46".)B"(.4@"./'3(
?*/60"'8(*+"06+6*D60,>"C&6"06+6*D60"+')(.*),"(&6"B6(6+('0A"E&*+&"+')D60(,"(&6
'8(*+.4"8'E60"*)('".)"646+(0*+.4"+3006)(A".)B".)".:84*?*60>"50':"(&606"(&6",*=).4
='6,"('"."B6+*,*')"+*0+3*("J*)"B*=*(.4"+'::3)*+.(*'),KA"?'0"."6L.:846A".
+':8.0.('0>"5*=306"M",&'E,"."(18*+.4"06+6*D60"/4'+@"B*.=0.:>"C&6".:84*?*60"*,
'?(6)"."(0.),*:86B.)+6".:84*?*60A",'"(&6"/'',(6B",*=).4"*,"."D'4(.=6"0.(&60"(&.)".
+3006)(>
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;<
photodetector
amplifier
decision
circuit
outputlight
Figure 9. Receiver block diagram.
=&6"+':8.0.('0"+':8.06,"(&6"'3(83(",*>).4"('",':6"(&06,&'4?"@'4(.>6
+&',6)"/1"(&6"?6,*>)60"A*B6BC"1'3DC".)?"83(,"'3("."<"*E"(&6"@'4(.>6"*,"./'@6"(&6
(&06,&'4?C".)?"."F"*E"*("*,"/64'GB"=&6"+&'*+6"'E"(&*,"(&06,&'4?"46@64"G*44"?686)?"')
.")3:/60"'E"(&*)>,B""5'0"6H.:846C"E'0"(&6"*?6.4",*>).4"*)"5*>306"<F.C".)1"@'4(.>6
46@64"/6(G66)"F".)?"I"@'4(,"+'34?"/6"+&',6)B"5'0"(&6":'06"06.4*,(*+",*>).4",3+&".,
(&.(",&'G)"*)"5*>306"<F/C"G&*+&"&.,"3)?60>')6",':6"?*,860,*')C"*("/6+':6
+46.060"(&.("(&6"(&06,&'4?",&'34?"/6"+&',6)",':6G&606"*)"(&6":*??46B
a)
b)
c)
Figure 10. a) ideal pulse train; b) dispersed pulse train; c) dispersed pulse train with
noise.
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E*/60,"&.@6".((6)3.(*'),".,"4'G".,"FB<"?JKL:B"=&606".06"'(&60"'8(*+.4"4',,6,C"(''C
,3+&".,",84*+6"4',,6,C"+'384*)>"4',,6,C"/6)?*)>"4',,6,C"6(+B"=&6,6"('(.4"4',,6,"83(".
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E&6)"(&6",*>).4">6(,"B601"4'=@")'*,6"/6+':6,"*:8'0(.)(@"+.3,*)>"(&6
,*>).4"('")'*,6"0.(*'"FGHIJ"('"C6>0.C6D"K&606".06"B.0*'3,",'30+6,"'A")'*,6@"A'0
6L.:846"(&60:.4")'*,6"*)"(&6"06,*,('0,"'A"(&6"06+6*B60M,".:84*A*60D"K&606"*,",&'(
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#$%&'%(@".)C"(&6":'06"C*,860,*')".)C")'*,6"(&606"*,@"(&6":'06"+4',6C"(&6"U616U
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Figure 11. Eye diagram.
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,*)+6"(&6"(*:6"06R3*06<"('",6)<"6)'3?&"/*(,"('"80'<3+6"STT"600'0,"+.)"/6"D601
4')?="6,86+*.441"C&6)"600'0,".06"0.0641"<6(6+(6<@"A&6"M!F"(6,(60"&.,".)"*)(60).4
+4'+B"0.(6"'>"S2/*(W,6+@"A'"<6(6+("STT"600'0,">0':".",1,(6:"C*(&".""STUVM!F="STSS
/*(,":3,("/6",6)(="06R3*0*)?"STX",6+')<,="'0")6.041"YZ"&'30,@
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 5*/60"78(*+"9'::3)*+.(*'),"-./
#;
'LS198 'LS198
QA
QD
(2)QH
(10) (20)
(1)
(23)
S0S1
QA
Q
(20)
(23)
H
(1)S0S1
+5
(11) (11)(13)(13)
RST RST
RI(2)
RI
'86
(2)
(1)(3)
(1)
(2)'14
MOD
OUT
(2)
+
(4)
DS
Q
(11)
REC
IN(12)
DS
Q
+(10)
R
R
(1)
(3)
(3) (4)
'14
'LS74
'LS74
'86(5) (4)
(5)
(9)
CLK
(13)
RST(L)
'00
(6) (1)
(2)10K
RUN/STOP
+ '14
(3)
(11) (10)
ENABLE
'14 '14(6) (5)
(8)
(9)
10K
+
10uF+
-
RESET
5*<306"=>"?4'+@"A*.<0.:"'B"?!C"(6,(60""DA6,*<)E"F>"?066A*)<G
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 5*/60"78(*+"9'::3)*+.(*'),"-./
#;
0.1uF
'04(1) (2)
(3)
510 510
'04
100pF
(4)
1MHz
'04(5) (6)
CLK
+5V
0.1uF
'05
MOD
OUT
1K
70
TX(2,6,7)
HFBR-1412(3)
RX
HFBR-2412
Fiber
(2)
(3,7)0.1uF
115
+5
RECVR
IN
LED
CURRENT
ADJUST
5*<306"=>"94'+?@"'8(*+.4"*)(60A.+6"+*0+3*(,>"BC6,*<)D"E>"F066G*)<H
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 5*/60"78(*+"9'::3)*+.(*'),"-./
##
DIGIT 4 DIGIT 2DIGIT 3 DIGIT 1
a
b
c
d
e
f
g
74C947
Vcc
(1)
a4
35
a4
20
b4
b4
21
34
c4
c4
7
2
2 d4
d4
6
23
e4
e4 f4
f4
g4
g4
36375
2426
25
a3
a3
b3
b3
c3
c3
d3 e3
d3 e3
f3
f3
g3
g3
3029
1110
931
32
13 1514 16
17 1819
a2
a2 b2
b2
c2
c2
d2
d2
e2
e2
f2
f2
g2
g2
2524
15 1326
2714
67
89
10 1112
a1
a1
b1
b1 c1
c1
d1
d1
e1
e1
f1
f1
g1
g1
2120
1918
1722
23
3738
3940
24
3
backplane5
1com
com
P1
P1 P2
P2
P3
P3
P4
P4
P4
N.C
.
33
38
39
2
3
4
N.C
.N.C
.N.C
.N.C
.N.C
.
+
LZI
29
LZO
30
N.C
.
OSC
N.C
.
+
UP/DOWN
3
627
ENABLE
ENABLE
3
1
+
10K
HOLD
STORE
34
GND
35
'14
CLK
13 14
32
RST(L)
RESET
5*;306"$<"=*,84.1"*)(60>.+6"+*0+3*(<"?=6,*;)@"A<"B066C*);D
!"#$%&'()"!&%"!*
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 5*/60"78(*+"9'::3)*+.(*'),"-./
#;
<"=60)*60",+.46"*,".)"*)>6)*'3,"?6@*+6"3,6?"('"06.?":6.,306:6)(,"('"@601
&*>&".++30.+*6,A"B)"(&6"%&'(')*+,"-./C"1'3":.1"3,6"."=60)*60":*+0':6(60C"D'0
6E.:846C"('".?F3,("."8',*(*')"')"(0.),4.(*')",(.>6,A"B)",':6".884*+.(*')C"1'3")66?
')41"(30)"(&6"G)'/"3)(*4":.E*:3:"+'384*)>"*,"'/(.*)6?C"'0",':6(&*)>",*:*4.0C
.)?"(&6".:'3)("'D"(0.),4.(*')"*,")'("8.0(*+34.041"'D"*)(606,(A"B)"'(&60".884*+.(*'),C
&'H6@60C"1'3"H*44"H.)("('"G)'H"(&.("1'3":'@6?"(&6"ID*/60C"46),C"6(+AJ"6E.+(41
KAKK$L"::M"860&.8,"1'3N06"84'((*)>"+'384*)>"6DD*+*6)+1".,"."D3)+(*')"'D
?*,84.+6:6)(A
9'),*?60".":*+0':6(60",3+&".,"(&.("*)"5*>306"OA""<,"(&6"'3(60"G)'/"*,
(30)6?C"*(".?@.)+6,"'0"06+6?6,".4')>"(&6"&'0*P')(.4",+.46A"Q&6"D*0,("'0?60"'D
/3,*)6,,"*,"('"?6+*?6"H&6(&60"*("*,".":6(0*+"'0"!)>4*,&"=60)*60A"R'3"+.)"3,3.441"(644
/1",*:841":6.,30*)>"(&6"?*,(.)+6"/6(H66)":.F'0"(*+G,"')"(&6":.*)"I&'0*P')(.4"*)
(&6"D*>306J".E*,A"BD"(&6"?*,(.)+6"/6(H66)"K".)?"$"*,"$"+:C"(&6)"(&6"3)*(,".06
+6)(*:6(60A"BD"(&6"?*,(.)+6"/6(H66)"K".)?"OK"*,".)"*)+&C"(&6)"*(N,"80'/./41".)
!)>4*,&A"S'(6"(&.("(&6":.F'0"*)+06:6)(,":.1"/6"::C"'0"OTOKN,"'D".)"*)+&A
7)6"D344"06@'43(*')"'D"(&6"'3(60"G)'/".?@.)+6,"*(,"6?>6"/1"O":*)'0
*)+06:6)(A"-6(N,",388',6"(&.("(&6":.F'0"*)+06:6)(,".06"L"::".8.0(C":6.)*)>"(&6
:*)'0"(*+G,".06"KAL"::".8.0(A"U6"'/,60@6"(&.("(&6"6?>6"&.,"+0',,6?"/'(&"(&6"OV
::".)?"(&6"OVAL"::":.0GC",'"(&6"8',*(*')"*,".("46.,("OVAL::A
0 5 10 15
0
45
40
35
30
S6E("H6"4''G".("(&6":.0G*)>,"')"(&6"'3(60"G)'/A"B)"')6"06@'43(*')C"LK"'D
(&6,6":.0G,"H*44"8.,,"(&6":.*)".E*,C",'"LK"'D"(&6,6":.0G,"6W3.4,"KAL::C"'0"')6
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 5*/60"78(*+"9'::3)*+.(*'),"-./
#;
'3(60"<)'/"*)+06:6)("*,"6=3.4"('">?>@::?"A6")'B"&.C6"."06.D*)E"'F
@G?HI>?J@K@G?;@::?
7)",':6":*+0':6(60,L"(&606"*,".)".DD*(*').4",+.46"')"(&6"*))60",&.F(?"MF
(&606"*,L"1'3"+.)"E6("16(".)'(&60"D6+*:.4"84.+6?"M)"(&*,"+.,6L"1'3"4''<"F'0"(&6"4*)6
')"(&6"4.,("N60)*60"O&'0*P')(.4"*)"(&6"D0.B*)EQ"(&.("!"#$"4*)6,"38"B*(&"%&'""4*)6"')
(&6"'3(60"<)'/R,"6DE6?"M)"(&6"F*E306"/64'BL"*("4''<,"4*<6"(&6"4*)6,"4./646D"STS"4*)6,
38"/6,(L",'"(&6"F*).4"D*E*("*,"STS?"U&6"F*).4"06.D*)E"*,"(&606F'06"@G?;@T"::L"E*C*)E
1'3"."06,'43(*')"'F">?>>@"::"'0"@"!:?
30
25
20
15
10
24
6
8
0
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 5+'3,('678(*+,
9:
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*)(<0.+(*'),".)>"(&<",(3><)("*,".44'=<>"('"<G8<0*@<)("=*(&".,8<+(,"'B"'8(*+.4
/<.@"><B4<+(*')"(&.("+.)"/<"3,<>"B'0"<*(&<0",=*(+&*)DC",+.))*)D"'0",8<+(03@
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,(3><)(" @3,(" /<+'@<" B.@*4*.0" =*(&" (&<" @.))<0" *)" =&*+&" .+'3,(*+" =.?<,
80'8.D.(<E" " 5)" *)(0'>3+(*')" ('" .+'3,(*+" =.?<," *," 80<,<)(<>" *)" (&<" B'44'=*)D
,<+(*')E" " ;&<" *)(<0.+(*')" /<(=<<)" .+'3,(*+" .)>" '8(*+.4" =.?<," *," (18*+.441
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@'@<)(,E"";&*,"+4.,,*+.4"(&<'01"*,"80<,<)(<>")<G(E""H')(*)3*)D"=*(&"+4.,,*+.4"=.?<
(0<.(@<)(" 'B".+'3,(*+".)>"'8(*+.4"*)(<0.+(*'),C"><B4<+(*')"'B"(&<"'8(*+.4"/<.@" +.)
0<,34("/<+.3,<"(&<".+'3,(*+"/<.@" 0<80<,<)(,".">*BB0.+(*')"D0.(*)DE" ";&<0<B'0<C" .)
*)(0'>3+(*')" ('" >*BB0.+(*')"D0.(*)D," *," 80<,<)(<>" *)" (&<" (&*0>" ,<+(*')E" " I<G(C
/<+.3,<" (&<" *)(<0.+(*')" 'B" (&<" .+'3,(*+" .)>" '8(*+.4"=.?<," *," B3)>.@<)(.441" .
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+'),<0?.(*')"'B"<)<0D1".)>"@'@<)(3@"*,">*,+3,,<>E"""5"/0*<B"><,+0*8(*')"'B"&'=
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*)"*)"B*).4",<+(*')E
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B'4(.C<D";&<"+01,(.4"&.,".)".++'=8.)1*)C">0*B<0E"F&*+&"80'>3+<,".)"GH",*C).4"('
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F.B<"'?"&*C&"<)'3C&"?0<I3<)+1"(&.("(&<"><?4<+(*')"*,"'/,<0B./4<D"5,"1'3"J)'F
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.)>"(&<">*??0.+(*')".)C4<"*,"><(<0=*)<>"/1"(&<".+'3,(*+"?0<I3<)+1"K*)"(&*,"+.,<
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*)83(".)C4<"'?"(&<"/<.="B<01".++30.(<41D"H30(&<0='0<E"(&<"/<.="=3,("/<"F<44
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'0><0"('"J<<8"(&<"/<.="C'*)C"(&0'3C&"(&<"(F'".8<0(30<,".,"1'3"+&.)C<"(&<
.)C4<"'?"*)+*><)+<D
laser
crystal
RF signal in
;&<">0*B<0"&.,"(F'"*)83("8'0(,E"')+<"4./<4<>"Q='>*.)>"')<"4./<4<>"Q(0*K?'0
(3)*)C"(&<"?0<I3<)+1"'?"(&<"GH",*C).4OD"R("*,".)"*>*',1)+0.,1"'?"(&<"><B*+<"(&.("1'3
=3,("83(".("4<.,("SD#Q"')"(&<"='>34.(*')"*)83("Q='>"*)"'0><0"('",<<"><?4<+(*')
123*.#&*456!&65*&7!.*89*:2*9;:3<*;&<"='>34.(*')"*)83("*,"*)(<)><>"?'0
>*C*(.4"*)(<),*(1"='>34.(*')".884*+.(*'),E".)>"*("*,")<+<,,.01"('"<T+<<>"(&<
(&0<,&'4>"'?"(&<">*C*(.4"><B*+<,"*)"(&<">0*B<0"+*0+3*(01D"U'(<"(&.("(&*,"*,"."+3,('=
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+.)"/<";.0*<:"B0'="C"('"DEA",.B<41>
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acoustic wavefronts
Fig. 4. Acousto-optic switch (a) with beam deflection, (b) with no beam deflection.
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Figure 2. Distribution of electrons and holes in the semiconductor bands.
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(&*);H"I45.041":5"+.)"5FE5+(")'"5)50<1"J*75.441K"/54':"(&5"/.)7"<.E@"G&.(
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ENERGY
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Figure 3. The gain curve !#"! of a semiconductor
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.05"+'),50?57P"(&505D'05"<5"+.)"')41",(*;34.(5"5;*,,*')"/5(<55)"(<'"5)50F1
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Figure 4. E-k diagram of a direct-gap semiconductor, showing conservation of k.
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Figure 5. Pumping of a laser diode by injection under forward bias.
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Figure 6. Generic diode laser structure.
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Figure 7. Use of double heterostructure to confine carriers.
D*).441:",*)+5"(&5"(<'"@.(50*.4,"*)"(&5"&5(50',(03+(305"&.?5"7*>>505)(
*)7*+5,"'>"05>0.+(*'):"."+!%&,-$.&'*,"+05.(57".,",&'<)"*)"D*;305"EC"F5+.3,5"(&5
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*)(50>.+5,:"&54=*);"('"+')>*)5"(&5"4*;&("('"(&5";.*)"05;*')C"J&*,"*,"K)'<)".,"/2#$"!(
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Figure 8. Optical confinement in double heterostructure.
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+')(*)3'3,41B".)7"(&505<'05",'"*,"(&5"05<0.+(*?5"*)75@G"H&*,"&.,"(&5"5<<5+("'<
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*)75@"+&.)E5"<'0"5<<5+(*?5"*)75@"E3*7*)EG
n1
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Figure 9. Various lasers refractive index profiles (transverse): a) double
heterostructure, b) single quantum well, c) multiple quantum well, d) separate
confinement heterostructure, and e) GRINSCH
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+')C*)57"('"(&5".+(*<5"4.150"/1"."05C0.+(*<5"*)75@"+&.)>5"/3*4("*)('"(&5"75<*+5F"G44
7'3/45"&5(50',(03+(305"7*'75"4.,50,"H*44"&.<5"*)75@">3*7*)>"*)"(&5"(0.),<50,5
7*05+(*')F"I)"(&5")*+$,*)&"7*05+(*')E"&'H5<50E"(&505"*,"."H*75"<.0*5(1"'C",(03+(305,F
='0"[email protected]"+'),*750"(&5"4.,50",(03+(305"*)"=*>305"89.F";&*,"4.,50"&.,".)"*)(50).4
A5,."+')(.*)*)>"(&5".+(*<5"4.150E".)7"(&5".05.".0'3)7"(&5"A5,."*,"C*4457"*)"H*(&
H*75D/.)7">.B"J4'H"*)75@K"A.(50*.4F";&5"/5.A"B0'B.>.(*)>".4')>"(&5".+(*<5
4.150"*,"+')C*)57"*)"(&5"4.(50.4"7*05+(*')"*)"(&5",.A5"H.1".,"*("*,"*)"(&5"(0.),<50,5
7*05+(*')F""G"C5H"'(&50"*)75@">3*757",(03+(305,".05",&'H)"*)"=*>305"89".,"H544F
=*>305"89J+K",&'H,"."0*7>5"H.<5>3*75D")'(5"(&.(".4(&'3>&"(&5".+(*<5"4.150"*(,54C"*,
)'("/'3)757"4.(50.441"/1"."05C0.+(*<5"*)75@"+&.)>5E""(&5"4*>&("*)"(&5"4.,50"&.,
5<.)5,+5)("(.*4,"'3(,*75"(&5".+(*<5"4.150"*(,54CD".)7"(&5,5"5<.)5,+5)("(.*4,
5@B50*5)+5"(&5"*)75@">3*7*)>"*)"(&5"0*7>5F"L5.0"*)"A*)7"(&.("(&5"(50A"M*)75@
>3*7*)>M""'0"M>.*)">3*7*)>M"J7*,+3,,57")5@(K".4H.1,"05C50,"('"(&5")*+$,*)"7*05+(*')E
,*)+5"*("*,".,,3A57"(&505"*,"*)75@">3*7*)>"*)"(&5"(0.),<50,5"7*05+(*')F
metal contact
active layer
P
N
N N
n
oxide
(a)
P
N
(b)
P
N
(c)
Figure 10. Index guided laser structures: a) buried heterostructure, b) buried-
crescent, and c) ridge-waveguide.
G"'*!"-'(!#$#&4.,50E"&'H5<50E"*,",4*>&(41"7*CC505)(F";&5"4.(50.4">3*7*)>"*,",(*44
+.3,57"/1".)"*)75@"+&.)>5E"/3("(&5"*)75@"+&.)>5"*,)N("735"('"."+&.)>5"*)
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 -.,50"6*'75"%&1,*+,
889
:.(50*.4;"<("(30),"'3("(&.("."4.0=5")3:/50"'>"+.00*50,"*)"."05=*')"'>",5:*+')73+('0
+.)".>>5+("(&5"05>0.+(*?5"*)75@"4'+.441;"<)"."4.,50"7*'75A"&*=&"45?54,"'>"*)B5+(*')".05
)5+5,,.01"('"+05.(5"(&5"=.*)A".)7"(&5,5",.:5"*)B5+(*')"545+(0'),"C*44":.D5",4*=&(
+&.)=5,"*)"(&5"05>0.+(*?5"*)75@E":.D*)="*(",':5C&.("&*=&50;"F&505>'05A"4*=&("C*44
(5)7"('"/5":'05",(0')=41"+')>*)57"('"05=*'),"(&.(".05"5@G50*5)+*)="&*=&"45?54,"'>
5@+5,,"+.00*50,;"F&5"=5)50*+",(03+(305"*)"H*=305"I"*,"."+4.,,*+"*>"*)5>>*+*5)("5@.:G45
'>"."=.*)E=3*7*)=",(03+(305;"H*=305"88",&'C,".)'(&50"5@.:G45"'>"."=.*)E=3*7*)=
,(03+(305;"J'(5"(&.("*)"(&5"(0.),?50,5"7*05+(*')A"(&5"=3*7*)="*,",(*44"*)75@"(1G5;
%30541"=.*)E=3*7*)=",(03+(305,".05".?'*757"*)"G0.+(*+5"/5+.3,5"(&51"3,3.441"&.?5
*)>50*'0"/5.:"K3.4*(1A"/3(":.)1"4.,50"7*'75",(03+(305,"*)?'4?5"/'(&"*)75@E".)7
=.*)E=3*7*)=;
ion implant
(non-conductive)
Figure 11. Proton-implanted gain guiding laser diode.
!"#$%&%'($)'*+&,(+&-*'./('0"(1&2!.3)"!$'(+&-*'.
<)"(&5"<)(0'73+(*')A"C5":5)(*')57"(&.("4.,50"05K3*05"(C'"(&*)=,L"=.*)A".)7
.)"'G(*+.4"+.?*(1;"M5")'C"3)750,(.)7"(&.("'G(*+.4"=.*)"5@*,(,"*)"(&5
,5:*+')73+('0"C&5)"(&5"G0'/./*4*(1"'>",(*:34.(57"5:*,,*')"*,"&*=&50"(&.)"(&5
G0'/./*4*(1"'>"./,'0G(*')"'0",G')(.)5'3,"5:*,,*')A".)7"(&.("(&*,"+')7*(*')"*,
.+&*5?57"/1"G3:G*)="(&5"+.?*(1"('"*)?50("(&5"G'G34.(*');"N'"45(O,"(30)")'C"('"(&5
'G(*+.4"+.?*(1;"F&5"H./01E%50'("+.?*(1"*,",&'C)"*)"H*=305"8P;"<("+'),*,(,"'>"(C'
:*00'0,;"<>"C5"*:.=*)5""(&.("(&505"*,"."4*=&(">*547"*),*75"(&5"+.?*(1A"C5"+.)",55"(&.(
>'0"+50(.*)"C.?545)=(&,A"(&5">*547"/'3)+*)="/.+D".)7">'0(&"/5(C55)"(&5":*00'0,
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!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 -.,50"6*'75"%&1,*+,
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*)(50:505"75,(03+(*>541?"@&*,"A5.),"(&.("*:"1'3"<505"('"A5.,305"(&5"5)50B1"*)"(&5
+.>*(1"C1'3"<*44"7'"(&*,D".,".":3)+(*')"':"<.>545)B(&;"1'3"<'347",55"7*,+05(5
=5.9,"<&5)5>50
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&.4:"<.>545)B(&;"+'),(03+(*>5"*)(50:505)+5"<*44"05,34(?
d
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Figure 12. The Fabry-Perot cavity.
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)'("/5":*4457"<*(&".*0"/3("<*(&",'A5"A.(50*.4"C4*95;":'0"5J.A=45;",5A*+')73+('0LD?
M'"<5"&.>5"('"+')>50("!"N
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(&5"4.,50;",*)+5"*("+'A5,"'3("(&0'3B&".*0LD?
P++'07*)B"('"(&*,"7*,+3,,*');"(&5);"(&505",&'347"/5".)"*):*)*(5")3A/50"':
05,').)("<.>545)B(&,;",=.+57".=.0("/1
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n
8 75x10- 9
8 7 08 6 58 6 0
Wavelength (meters)
R1=R2=0.9
R1=R2=0.7
R1=R2=0.5
Figure 13. Transmission of Fabry-Perot Cavity
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05;45+(*<*(*5,C"E&505"*,".":5.,305"';"(&5"RO3.4*(1R"';"."P./01G%50'("+.<*(1>"+.4457
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.)7"/.+@"B.+5(,".05"A.0(*.441"05B45+(*)C>",'"(&5"/5.="5M*(,".("/'(&"5)7,G"<)"A0.+(*+5>
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+&.0.+(50*,(*+,"+&.);5"70.,(*+.441".)7",3775)41L"(&5"*)(5),*(1";'5,"@.1"3BC"735"('
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(&5"+.A*(1"@*44"5H+*(5"<'05"B&'('),"N3,("4*P5"*(>"Q'(*+5"(&.(",'<5"B&'('),",55
<'05";.*)"(&.)"'(&50,C"/5+.3,5"'D"(&5"5)50;1C",'"(&51".05"<'05"4*P541"('
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75D*)57"/1"5H(0.B'4.(*);"(&5"4*)5"*)"(&5"4.,*);"05;*')"/.+P"('"(&5".H*,>
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 -.,50"6*'75"%&1,*+,
889
P (mW)
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Figure 14. The power-current curve of a diode laser.
:5;("45(<,"4''=".("&'>"(&5"4.,*)?"@'75,"75A54'BC"D54'>"(&05,&'47E"(&505"*,
/0'.7/.)7",B')(.)5'3,"5@*,,*')E"F*?305"89.C"G&505"*,",'@5"(&05,&'47"+3005)(E
./'A5">&*+&"4.,*)?"/5?*),C"H,">5"0.*,5"(&5"+3005)("45A54E"(&5"+30A5"*)"F*?305"89.
@'A5,"3BE"@5.)*)?"(&505"*,"@'05"5)50?1".("5.+&">.A545)?(&C"D5+.3,5"'I"(&5
F./01J%50'("+.A*(1E"&'>5A50E"+50(.*)">.A545)?(&,".05"B05I50057"/1"(&5"+.A*(1C
H,"(&5,5"+.A*(1"@'75,",55"5)'3?&"?.*)"('"4.,5E"(&51"/5?*)"('"@34(*B41".("(&5
5;B5),5"'I"(&5")')J05,').)(">.A545)?(&,C"G&5"@'75",55*)?"(&5"&*?&5,("?.*)"K*)
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'(&50"@'75,">*44"?5("."+&.)+5"('"4.,5C"M)"B0.+(*+5E"&'>5A50E"7*'75"4.,50,"+.)
'B50.(5"*)"@'05"(&.)"')5"!"#$%&'(%#)!"@'75"K05,').)("@'75".4')?"(&5
4')?*(37*).4"7*05+(*')LE".)7"'I(5)"(&5"5)50?1"*,"/'3)+*)?".0'3)7"/5(>55)"(&5@
,'"I.,("1'3"7')<("75(5+("*(C"N&5)"1'3"@5.,305"(&5",B5+(03@"'I"(&5"4.,50E"1'3
@5.,305"(&5".A50.?5".@'3)("'I"B'>50"*)"(&5"A.0*'3,"4')?*(37*).4"@'75,"'A50
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thresholdgain
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wavelength
F-P
modes
wavelength
Figure 15. Evolution of lasing: a) below threshold, all emission is spontaneous, b)
above threshold, first mode sees gain, begins to lase c) well above threshold,
higher order modes can oscillate.
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Figure 16. Output pattern of a diode laser (edge-emitting).
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*)"')41"')5G"E3)7.F5)(.4"F'75M"O)"(&*,"+.,5"%56G".)7"(&5"*)(5),*(1"7*,(0*/3(*')"*,
."J305"K.3,,*.)":(&5"I50F*(5"J'41)'F*.4"*,"8;G"P*C305"8#".M""N&5"47%83*9:3%$4
;7:)1"'E"(&5"4.,50".05"(&5"P./01Q%50'("F'75,"B5"7*,+3,,57"5.04*50M
Mode 0 Mode 1 Mode 2
distance across beam
Figure 17. The Hermite-Gaussian intensity distributions for the first three spatial
modes.
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.("(&5"5)7,D";&5,5"B*00'0,".05")'("+45.>57"@.+5(,F"/3(".05"0.(&50"B*00'0",(.+K,A
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4.150,F"(&5,5"B*00'0,"+.)"/5"75,*C)57"@'0"&*C&"'0"4'J"05@45+(*>*(1D"M'(*+5"(&.("*)
(&*,"4.,50F"(&5"+.>*(1"*,"5L(05B541",&'0(A"(&*,"&.,"(&5"5@@5+("'@",I05.7*)C"(&5"G./01A
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&.>5">501"4'J"(&05,&'47,D
substrate
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Figure 18. The vertical-cavity surface emitting laser (VCSEL).
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'I(*+.4"@557/.+K"/1"05@45+(*)C"4*C&("/.+K".)7"@'0(&".4')C"(&5"+.>*(1F"."C0.(*)C
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5(+&*)C"."I50*'7*+",(03+(305"*)('"(&5",5B*+')73+('0F"(&5)"@*44*)C"*("*)"J*(&".)'(&50
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I50*'7*+">.0*.(*'),"*)"(&5"05@0.+(*>5"*)75LF".)7"05B5B/50"."+&.)C5"*)"*)75L
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Figure 19. Distributed feedback laser (top) and distributed Bragg reflector laser
(bottom).
lasing stripes (coherently coupled)
Figure 20. A phased laser array (evanescently coupled).
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O&.(",&'347"&.<<5)"*B"1'3"='D5"(&5"+.D*(1"=*00'0,"5*(&50"+4',50"('A5(&50
'0"B.0(&50".<.0(L""""6'".)"5;<50*=5)("('"75='),(0.(5"(&*,E""GP'C"+.)"1'3"+&.)A5
(&5"45)A(&"'B"(&5"+&*<LH""M,"(&5"'/,50D57"5BB5+("5)(*0541"5;<4.*)./45"/1"='D*)A"(&5
=*00'0,I"'0"*,",'=5(&*)A"54,5"A'*)A"')L""GP*)(:""O&.("<.0.=5(50"*)"Q*A305"?
=*A&(".4,'"&.D5"+&.)A57"730*)A"(&*,"5;<50*=5)(LH
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'<50.(*')I".)7"."75,+0*<(*')"'B"&'C"('"3,5"(&5"7.(.".+93*,*(*')"<0'A0.="('"(.S5
7.(.".3('=.(*+.441E
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8"=')'+&0'=.('0"@,'=5(*=5,"+.4457".",<5+(0'=5(50A"*,"."75B*+5"(&.(
=5.,305,"(&5",<5+(03="'C".)"'<(*+.4",'30+5D"E&5"/.,*+"*),(03=5)("*,",&'F)"*)
G*H305">D"I'=5"=')'+&0'=.('0,".05"4.*7"'3("7*CC505)(41J"F*(&"7*CC505)("'<(*+.4
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E&5"4*H&("*,"*)(0'73+57"*)('"(&5"=')'+&0'=.('0""B*.".)"5)(0.)+5",4*(D"E&5
/5.="*,"(&5)"*)+*75)("')"."H0.(*)HJ"."<50*'7*+",5("'C"H0''B5,"F&*+&"7*,<50,5"4*H&(
*)"."=.))50",*=*4.0"('"."<0*,=D""!.+&"F.B545)H(&"F*44"/5"05C0.+(57".(".",4*H&(41
7*CC505)(".)H45D"8,"(&5"H0.(*)H"0'(.(5,J"(&5"<.0(*+34.0"F.B545)H(&"*)+*75)("')"(&5
'3(<3(",4*("+&.)H5,D"K1"=5.,30*)H"(&5"'<(*+.4"<'F50"5=50H*)H"C0'="(&5
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F.B545)H(&"+.)"/5"=5.,3057D
detector
source mirrorsgrating
Figure 1. Structure of a monochromator
E&5"05,'43(*')"'C"(&5"=5.,305=5)("75<5)7,"')",5B50.4"(&*)H,D"G'0"')5
(&*)HJ"(&5"4')H50"(&5"'<(*+.4"<.(&".C(50"(&5"7*,<50,*B5"545=5)("@*)"(&*,"+.,5"(&5
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05,'43(*')"@F&.(L,"(&5"(0.75M'CCNAD"84,'J"(&5"5;*(",4*(",*O5"*,"*=<'0(.)(M"(&5
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,4*(".CC5+(,"05,'43(*')J",3CC*+5"*("('",.1"(&.("*("7'5,D"8"+'==')"<0.+(*+5"('".4F.1,
Q55<"(&5"(F'",4*(,"'<5)57"('"(&5",.=5",*O5D"E&505"*,"."(0.75M'CC"*)",4*("F*7(&,M"*C
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 -.,50"6*'75"%&1,*+,"-./'0.('01
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(&51".05").00'AB"1'3"C5("&*C&50"05,'43(*')"/3("45,,"(&0'3C&<3(D"EF"1'3".05"3,*)C
.)"*),(03=5)("A*(&".7G3,(./45",4*(,B"/5".A.05"(&.("!"#$%&'!%$()*$+#$,)-).#,$+/
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84*C)*)C"(&5"/5.="*)"=')'+&0'=.('0"*,")'("5.,1B".)7"=.1"/5"/5,("7')5
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=.J5",305"*("*,"&*((*)C"(&5"=*00'0,B"5(+D"K&.)C*)C"(&5"(*4(".)C45"'F"(&5"*)(50).4
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+'==.)7,"7*,+3,,57"&505I"<45.,5",55"1'30"4./'0.('01"*),(03+('0F
Figure 1. Main screen of data acquisition program.
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QUANTUM WELL DEVICES
INTRODUCTION
Quantum well devices have some very interesting and useful optical
properties. Aside from the fact that you can observe and believe in actual
quantum mechanics by measuring these devices, their importance in optical
modulation, particularly at high speeds, is increasing all the time.
We will review some basic quantum mechanics principles, then discuss
some specific devices. In the laboratory, you will be measuring electro-
absorption in some quantum well optical detectors. You will see the effects of
quantized energy levels, and the existence of excitons. You will believe.
REVIEW OF QUANTUM MECHANICS
You recall that quantum mechanics deals with very small objects, such as
electrons, atoms, photons, etc. Each of these can be thought of as either a particle
or a wave, depending on which is more convenient. For example, a photon,
which is a quantum of electromagnetic wave energy oscillating at some angular
frequency !, has energy E=h!, where "=!/2# is the frequency in Hz. Similarly, a
phonon is a quantum of acoustic energy (lattice vibration at angular frequency !,
whose energy is E=h!, or E = h! , where h is h/2".
We generally want to find out what is going on with electrons- what
energies can they have in a particular system, what is their average position, etc.
These things are found by solving Schrodinger's equation, which, to remind you,
is:
!
h2
2m"
2#(x ,y , z,t) + V(x , y, z)#(x, y ,z,t) = !
h
j
$#
$t [1]
where m is the mass. In semiconductors, of course, we'd use m* , the effective
mass. This equation is separable (for problems we'll encounter here) into time-
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dependent and time-independent parts. We'll only discuss non-time-varying
potential distributions V(x,y,z), so the two equations are then:
! 2"
!x2+
2m
h2
E # V(x)( )" = 0
!$
!t+
jE
h$ = 0
[2]
Notice that we have written these equations for a one-dimensional case.
The constant E is the energy associated with a particular !, or a particular state.
The time-independent wavefunction ! is a function only of space (in this case,
the one dimension x), and the time dependent part is ". Since the second
equation is always the same for cases we'll consider, the time-dependent solution
is always going to be
!(t) = e" j#t [3]
The solution to the time-independent Schrodinger's equation depends on
the actual system being analyzed, which is described by V(x). For example, in the
infinite potential well extending from x=0 to x=L, where V(x)=0 inside the well,
and goes to ! outside, the wavefunctions are sinusoidal, and have one hump for
!1, two humps for !2, etc. The solution to Schrodinger's equations (the
wavefunctions) are:
! n =2
Lsin
n"L
x#
$
%
& [4]
and the allowed energies (there is an infinite number of them) are given by
En =n
2!
2h
2
2mL2
(infinite potential well) [5]
where n is an integer naming the particular state, m is the mass of the electron,
and L is the width of the well. Figure 1 shows the first few solutions.
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x=0 x=L
! !
!1
!2
!3
! !
x=0 x=L
E 1
E
E
E4
3
2
Figure 1. The infinite potential well. On the left, the first three wavefunctions are
shown. On the right, the first four allowed energy levels are shown.
The wavefunction ! has no physical meaning, of course, but the quantity
!"! represents a probability density function. So for example, the most probable
location for the electron in state E1 is in the middle of the well, because that's
where the probability density function !*! is the highest for that state.
In a finite well, Figure 2, the potential energies outside the well are not
infinite. This results in wavefunctions that are mostly confined to the well, but not
completely. Since ! is non-zero outside the well, then the probability density
function !*! is also non-zero outside the well. This implies that the electron in
one of the confined states spends some fraction of its time actually outside the
well. Another way to look at this is that the electron, which is oscillating back
and forth in the well, actually penetrates the barriers a little bit. Note that there
are a finite number of solutions in a finite well; if the energy of the state gets
higher than the edge of the well, the electron won't be confined to the well, and
will appear to be a quasi-free electron. There is a quasi-continuum of states above
the top of the well.
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V
x=0x=L
!1
!2
V
x=0 x=L
"
"
"
1
2
3
continuum of states
Figure 2. The finite potential well. Left: first two wavefunctions. Note that they
extend outside of the well. Right: This well contains three discrete energy levels.
HETEROJUNCTIONS
All of the above should be familiar. If it isn't, go back to your notes from
EE331 and go through that material again. Meanwhile, a question that comes up
every quarter is, "So, where do these potential wells come from? How do you
make one?" The answer is "heterojunctions".
A heterojunction is a junction between two different materials. For
example, a simple silicon pn-junction diode is a homojunction because it has
silicon on both sides, even though the side may be doped differently. But a
junction between, say, GaAs and AlGaAs is a heterojunction. Each of these
materials has a different band gap, so at the junction some discontinuities occur
in the conduction band edge and valence band edge.
We will now develop the procedure for drawing the energy band diagram
for a heterojunction (or any junction, for that matter). For example, consider a
junction between a chunk of n-type material and a chunk of some different p-
type material, as shown in Figure 3.
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p n
Figure 3. A pn junction.
We'll need to know several material properties in order to proceed. The
work function, !, is the amount of energy needed to remove an electron from the
Fermi level to the vacuum level. In a semiconductor, however, there are usually
no states at the Fermi level, so perhaps the work function is not the most useful
number to know. In fact, the work function of a semiconductor will depend on
the doping since the location of the Fermi level depends on the doping. For
semiconductors, we use the electron affinity, ", which is the amount of energy
require to move an electron from the bottom of the conduction band to the vacuum level.
Consider a specific junction, of which we'll construct the energy band
diagram. Let the p-type material be GaAs, doped with NA = 1#1018. The electron
affinity " for GaAs is 4.07 eV, and the band gap is 1.43 eV. [1] We have to
calculate the location of the Fermi level; it is EF-EV=0.414 eV.
The n-type material we'll take to be Al.3Ga.7As, which is a material like
GaAs except that 30% of the atoms which would have been Ga have been
replaced with Al. This is a ternary material, meaning it has three different
elements in it. The electron affinity for Al.3Ga.7As is 3.74 eV, and the band gap is
1.8 eV. Let this material be doped with ND=1.5#1017cm-3, resulting in EC-
EF=0.41 eV. Figure 4 shows the individual energy band diagrams before the
materials are "joined".
Now, we are interested in the energy band diagram of the junction. To
construct an energy band diagram for anything, the following rules are applied:
1. The Fermi level is constant (flat) at equilibrium.
2. Evac is continuous.
3. Eg and " are constant in any given material.
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To draw the energy band diagram for the diode, we start with rule 1. The
Fermi level is flat, so start by drawing a straight line, Figure 5. Next, we
construct the energy band diagrams of each material, one on the left and one on
the right.
EF
!=4.07
vac
Eg
E
E
E
EV
C
F
!=3.74
EiEg
GaAs
=1.43=1.8
Figure 4. Energy band diagrams for GaAs and Al.3Ga.7As as described in text.
EF
!=4.07vac
Eg
E
E
E
EV
C
F
!=3.74
EiEgGaAs
=1.43
=1.8
Figure 5. Energy band diagram under construction.
Next, we invoke Rule 2: Evacis continuous. Therefore, the two Evac's must
be connected in some smooth, continuous fashion, as shown in Figure 5. Finally,
we invoke Rule 3: Eg and ! are constants of the material. Now, ! for each material
is a constant, but it is different for the two different materials. That means that EC
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is a constant distance from Evac on each side of the junction, but there will be a
discontinuity in EC at the junction. Similarly, Eg is a constant for each material,
and so EV will run parallel to EC in each material, but there will be a
discontinuity in EV at the junction. Figure 6 shows the completed energy band
diagram for this junction. In this particular example, the discontinuity in EV isn't
very obvious, but it's there. This procedure, which works for any junction
between any materials, is known as the electron affinity rule (EAR) or the
Anderson Model (really). Note that for different choices of materials or even
doping, the "dip" could appear in the valence band edge instead.
EF
!=4.07vac
Eg
E
E
E
EV
C
F
!=3.74
EiEgGaAs
=1.43
=1.8
Ei
AlGaAs Figure 6. Energy band diagram for this particular heterojunction.
There is a potential well in this figure These wells that form at the
junction are very narrow, and so they can actually be thin enough to be quantum
wells- the energy states in the well can be quantized (Figure 7). The number of
levels in the well depend on the well depth and width. The barrier on the right
side of the well can also be thin enough for tunneling to occur.
Figure 7. Quantum well results from heterojunction.
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There are other ways to make quantum wells, however, the most common
being the double heterostructure. In the double heterostructure, the narrow-gap
material is sandwiched between two layers of wide-gap material. In Figure 8, an
idealized sketch of the energy band diagram is shown. Note that at each junction,
there will be some band bending and these discontinuities in the conduction and
valence band edges, but we're leaving those out to show the big picture.
Nevertheless, you can see that if the layer of narrow-gap material is thin enough,
the depression in the conduction band edge could be narrow enough to be a
quantum well, usually less than about 100Å. Again, the narrower the well, the
fewer the number of confined states.
GaAsAlGaAs AlGaAs
Ec
Ev
E
distance Figure 8. Idealized energy band diagram of a double heterostructure.
Some devices are made with multiple quantum wells. Successive layers
are laid down by molecular beam epitaxy or metal-organic chemical vapor
deposition. These layers must be carefully controlled for composition and
thickness. Also note that any imperfection at the interfaces are likely to provide
recombination paths, hence leakage currents can be a problem.
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E D(E)
x
Energy Band Diagram Density of States
continuum
continuum
forbidden
xD(E)
n=1n=2n=3
n=1n=2n=3
xy
Physical Device
y
E
D(E)
E
D(E) Figure 9. Comparison of various structures and bulk material. From top to
bottom: 3-D (bulk), 2-D (quantum well), 1-D (quantum wire) and 0-D (quantum box).
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QUANTUM STRUCTURES
Let's compare quantum structures to bulk material. For example, a double
heterojunction material in which the potential well is wider than 100Å behaves
essentially like bulk material. Figure 9 shows some key differences between bulk
and quantum structures. Notice that if we confine the electron in one direction,
leaving it essentially two dimensions in which to travel, the density of states is a
step like structure. Also notice that there are no energy states at the bottom of the
well- the first state is always somewhere above the bottom of the conduction
band (for electrons) and somewhere below the top of the valence band (for
holes). When you get to this first energy, you suddenly add a whole plane of
states. That's why the density of states function looks the way it does.
One can also confine the electron in two dimensions, leaving it free to
travel in only one direction, resulting in quantum wires. One may even speak of
quantum boxes, which have confinement in all three directions. We'll restrict
ourselves to 2-dimensional quantum wells here, however.
EXCITONS
One of the important difference between quantum wells and bulk material
is the effect of excitons. An exciton is a sort of non-intuitive thing- the standard
line is that an exciton is an electron and a hole orbiting around each other. This is
a little hard to picture, so look at Figure 10. In the energy band diagram, we say
the electron is oppositely charged form the hole, so on the average, they'll be
slightly attracted to each and tend to stay in more or less the same physical area.
A look at the crystal picture shows that the electron and the hole are both
moving around, at some average distance from each other (about 140Å in bulk).
At these distances, there can be many atoms between them, effectively screening
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the charge. This means that excitons are extremely loosely bound, and it takes
very little energy (about 4.2meV) to separate the electron from the hole. In fact,
at room, temperature, you'll never observe the effects of excitons in bulk
materials.
EC
EV ++
Figure 10. Exciton in bulk material.
In a quantum well, however, the electron and the hole are artificially
confined by the well itself, so they have to stay closer to each other; remember
that a quantum well is less than 100Å wide. This is smaller than the bulk exciton
radius. Figure 11 shows the wavefunctions for an electron and a hole in the
quantum well.
The exciton has an interesting effect on the absorption of this material.
Normally, you'd expect to see no absorption at energies smaller than Eg+E1+E2.
The binding energy B of the exciton, however, reduces the actual energy needed
to move the electron from the valence band to the conduction band, which
creates this exciton. This results in an absorption peak associated with the
exciton, which appears at a photon energy
Eabsorbed= Eg+E1+E2.-B [6]
which can be seen on the right hand side of the figure.
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Eg
E1
E2
ideal
exciton
absorp
tion
Photon energy Figure 11. Wavefunctions of electron and hole confined to quantum well.
But even that's not the whole story. After all, there could be more than one
level in the wells, meaning there could be more than one exciton. That results in
an absorption spectrum that looks like Figure 12a. Furthermore, there are really
two different types of holes in the valence band, heavy holes and light holes.
Since effective mass depends on the curvature of the E-k diagram, there are two
lines on the E-k curve for the valence band. In bulk material these two are
degenerate at the top of the valence band, but the quantum well structure
destroys the degeneracy of the heavy hole and light hole bands, so it is possible
to sometimes observe double peaks, Figure 12 b.
ELECTRICALLY CONTROLLABLE OPTICAL EFFECTS IN
SEMICONDUCTORS
There are several electro-optical effects that can be exploited in
semiconductors, which fall into three general categories: electroabsorption,
electrorefraction, and the electro-optical effect. These are actually all the same
thing, and we'll show why.
Electroabsorption is the change in absorption under the influence of an
electric field. Electrorefraction is the change in refractive index with applied
field. The absorption coefficient ! of a material (at a particular wavelength) is
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abso
prt
ion
n=1
n=2n=3
photon energy
ab
sop
rtio
n
photon energy
lhhh
E
kBulk Quantum
Well
(a) (b) Figure 12. a) multiple excitons; b) heavy and light holes
actually interrelated with the refractive index no., by what are known as the
Kramers-Kronig relations. The complex index of refraction is given by:
˜ n = no + i!c
2"# [7]
where ̃ n is the complex index of refraction, c is the speed of light in vacuum, and
! is the frequency of the light. The point is, if something causes a change in " ,
there will also be a change in no; they are not independent. Therefore
electroabsorption and electrorefraction are different manifestations of the same
thing. You can change absorption to turn a beam on and off (amplitude
modulation), for example, or use the same material and the same effect to phase
modulate a beam by exploiting the change in refractive index. Electrorefraction
can be used for beam steering and guiding as well.
One way to change the refractive index in semiconductor is to inject
current. The presence of the carriers themselves changes the refractive index
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slightly[2, 3], an effect which is exploited a great deal in confining the beam in
lasers diodes to the active layer, where optical gain is present.
Many materials, not just semiconductors, exhibit the electro-optic effect,
meaning that the refractive index changes with applied electric field. The electro-
optic effect in this sense refers to a change in refractive index resulting from the
crystal structure shifting slightly under field. The idea is that in a particular
crystal, the atoms are arrange in such a way that they have some dipole
moments. When a field is applied, depending on the direction of the field, the
dipoles move a little, and since this changes the crystal structure, it changes the
refractive index. The refractive index is then written as a power series in the
applied field:
n(E) = no !
1
2rn
3E !
1
2sn
3E
2
[8]
where no is the refractive index with no field applied, r is called the linear electro-
optic coefficient, and s is called the non-linear electro-optic coefficient.
Sometimes the linear E-O effect is called the Pockels effect, and the nonlinear
effect is called the Kerr effect.
no field
Ec
Ev
Ev
Ec
!
field applied
E=h" ! E g
E=h" < Eg
Figure 13. The Franz-Keldysh effect.
It just so happens that these effects can be quite large in quantum well
structures- larger than they are in the same materials in bulk - in fact, the
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nonlinear term is two orders of magnitude large than bulk for InGaAsP! [4] That
makes InGaAsP a good candidate for monolithically integrated phase
modulators, for example.
Another way is to change the refractive index is, of course, to change the
absorption. Examples of electroabsorption effects are the Franz-Keldysh effect,
phase-space absorption quenching (bleaching of quantum wells), and the
quantum confined Stark effect. The Franz-Keldysh effect is shown in Figure 13.
Ideally, under no field, a photon needs energy as least as great as the band gap to
be absorbed. Under a very high field, however, the bands are extremely tilted.
Under these circumstances, the wavefunction of an electron in the conduction
band may extend somewhat out into the forbidden gap. It is therefore possible
(not likely, but possible) for an electron in the valence band to go up to this state
with a slightly smaller energy than the band gap, since the hole and electron
wavefunctions overlap at the same physical location. This results in a shift of the
absorption band edge to lower energies under applied field. The Franz-Keldysh
effect, however, is relatively weak and not used much.
Another example of electrically-controllable absorption is phase-space
absorption quenching,[5] or more simply, "bleaching". Because the quantum
wells are so narrow, there is a limited number of electrons you can put into a
state in a well. Once you fill, say, the lowest state, no more absorption can occur
at energies that would normally populate this state, for example E1 in Figure
14a. When the state is empty, absorption can occur. One can fill the well by
injecting current into it, or by applying a voltage to a structure whose energy
band diagram is shown in Figure 14b. When a voltage is applied, the Fermi level
is above the lowest well state; hence it is full. When the voltage is removed
(Figure 14c), the Fermi level is below the state, and it is empty, meaning
absorption can occur.
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By far the most promising electroabsorption effect, however, is the
quantum-confined Stark effect.
E1
E2
EC
EF
VE
EC
EF
VE
Figure 14. Phase space absorption quenching.
QUANTUM CONFINED STARK EFFECT.
The Stark effect is observed in many materials, and you know that the
energy levels are quantized in a potential well, and that the number of energy
states and the energy spacings depend on the depth and the width of the well. To
find the allowed energy states, we solve Schrodinger's equation for the V(x)
shown in Figure 15a. When an electric field is applied, however, the shapes of
the wells change. Since V(x) changes, you'd expect the solutions (allowed energy
levels) to change, too, and they do. This causes the absorption edge to move,
Figure 15b.
Notice that the edge didn't just move- it also changed its shape. Recall that
the exciton peak arises because the electron and hole are artificially kept close to
each other by the potential well. Under applied field, however, the wells become
asymmetric. The electron tends to be on the right hand side of the well, on the
average, statistically speaking, and the hole tends to be on the left. They are still
in the same neighborhood,, so there is still some excitonic effect, but they are also
slightly separated. This not only reduces the binding energy B in Equation [6],
but also tends to smear out the excitonic peak.
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Another point to notice is that the second curve is labeled "10V". This is
not a lot of voltage, but it is applied across a very thin structure-yielding high
electric fields, on the order of 100kV/cm.
a)
E1
new E1
Field appliedNo field
b)
0 V
10V
exciton
big change in absorption
ab
so
rptio
n
photon energy Figure 15. QCSE: a) shifting of the energy levels under applied field, and b)
change in absorption spectrum.
COLLECTION OF PHOTO-INDUCED CARRIERS
Once the photons are absorbed, in order to detect their presence we need
to collect the carriers that are produced (for example, electrons in the conduction
band). As a bias is applied across the well, the well is distorted and the electron
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sees a slightly lower barrier (refer back to Figure 15). Not only is the barrier
lower, but it is also narrower- under no field, the barrier is essentially infinitely
thick, but under an electric field, it is possible for the electron to tunnel through
the barrier. When it tunnels, it will no longer be confined to a well, and can be
swept out of the diode and collected as current.
HOW TO IMPROVE ABSORPTION IN AN EXPERIMENT
Finally, we come to the point of the absorption itself. You recall (or can
derive) that the absorption in a material is exponential in distance:
I(x) = Ioe
!"x [9]
where I(x) is the intensity at some distance x into the material, Io is the original
intensity, and ! is the absorption coefficient, in units of inverse length. Therefore,
to maximize absorption, you want to maximize the thickness of the material.
Therein lies a problem- a quantum well is very thin. Since the wide band gap
material on either side of the well is transparent to light that the well might
absorb, the actual absorption length is very very small, Figure 16a. This can be
gotten around by going to a multiple quantum well structure, Figure 16b. Each
well is of the exact width and depth needed for the particular absorption needed,
and they are place just far enough apart that they don't interfere with each other.
(a) (b) Figure 16. Single and multiple quantum wells.
Notice that we have assumed that the light comes in perpendicular to the
quantum well layers. This is the typical configuration for photodetectors, in
which light is either shone onto the surface of the chip or sometimes brought in
through the transparent substrate.
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There is another configuration, however: the waveguide configuration,
Figure 17. It is a fortunate coincidence that the same materials that have narrow
band gaps have higher refractive indices. That means the quantum well layers
act like a dielectric waveguide (like a planar optical fiber, if you will). You can
inject light into the edge of the chip, and it will propagate along the layers and
experience a large interaction length, and hence good absorption. This
configuration is ideal for optoelectronic integrated circuits, where the light to be
absorbed (modulated or detected) is coming from another device on the same
chip, such as a laser or passive waveguide. This approach is difficult for discrete
detectors, however, because it is very difficult to couple light efficiently into
those narrow layers from an external source.
QUANTUM WELL DETECTORS
We've seen how quantum wells can be intensity or phase modulators, but
what about detectors? It turns out there are some advantages to going to a
multiple quantum well structure for photodiodes, as well.
As you know, a photodiode is a device whose output current (under
reverse bias) is proportional to the light intensity being shone on it. Figure 18
shows the current-voltage characteristic of a diode under no bias and reverse
bias. First of note that if you measure the voltage across the diode, it does change
with light but not linearly- it has a logarithmic dependence. To use a detector,
you want to operate it under reverse bias, and monitor the current. Even in the
dark, of course, there is the reverse leakage current of any diode- this is known as
the dark current. When light is incident on the detector, the current increases in a
manner linearly proportional to the light intensity.
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Figure 17. Perpendicular (top), or parallel (waveguide) configuration (bottom).
The arrows represent the incoming beam; the curve in the lower figure is the electric field strength of the guided light.
I
V
no light
light
dark current
Figure 18. I-V characteristic of a photodiode
To maximize the responsivity (the amount of current produced per watt
of incident light), you want to maximize how much light is absorbed. However,
once the electron-holes pairs are produced, you also need to collect them. In a
p.n. junction diode, one photons absorbed near the junction will produce useful
current- other e-h pairs will wander around and recombine, whereas those near
the junction will be swept across the junction and collected. Therefore, one wants
to increase the width of the junction as much as possible, so as much light as
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possible is absorbed there. One approach is to use a p-i-n structure, Figure 19.
The electric field across the intrinsic region sweeps the electrons and holes across
the junction for collection.
pn junctionp-i-n diode
Figure 19. Different junction styles have differing junctions widths; the pin structure produces more collectable current.
Quantum wells are another way to increase the responsivity by increasing
the probability that a particular electron will be collected. Under high field
(reverse bias), the barriers between the wells look thin near the tops, as shown in
Figure 20. Electrons excited up to states near the top of the well can tunnel
through this barrier. As they travel to the right, they are high above the well
energies, and so are not easily captured by any of the other wells. If they were
captured, they wouldn't easily escape, and eventually they'd recombine and the
information would be lost. With a quantum well structure, however, the
probability of collecting the photocurrent is increased, thus increasing the
efficiency.
Finally, one interesting use for quantum well photodetectors are for very
short wavelengths (by semiconductor standards). By intersubband detection, that
is, from one level to another in the same well, Figure 21, one can obtain detector
wavelengths previously not possible in semiconductors.
EE 737 Photonics Laboratory Manual Quantum Well Devices
157
No field
Field applied
Figure 20. Increased responsivity in quantum well detectors due to difficulty in
recapturing electrons.
Figure 21. Intersubband detection for high energy photons.
HOMEWORK:
1. Draw the energy band diagram for junctions between the following
materials. Use graph paper and a ruler to make sure you keep ! and Eg constant.
a) n-type Si, EC-EF=0.2 eV; !=1.39 eV for Si, Eg =1.11 eV
p-type Si, EF-EV=0.1eV
b) n-type Si, EC-EF=0.2 eV
intrinsic Si, EF=Ei
c) n-type GaAs, EC-EF=0.1 eV; !=4.07 eV, Eg =1.43 eV
EE 737 Photonics Laboratory Manual Quantum Well Devices
158
p-type Al0.3Ga0.7As, EF-EV=0.2eV; !=3.74 eV, Eg =1.8 eV
2. Describe in words what you think will happen when several wells are
brought close together, like maybe 10 or 20Å. Will electrons still be confined?
3. We described how the photocurrent produced by absorption of the light
is collected when the diode is under bias. Why is current collected when no bias
is applied, as shown in Figure 15 b? Hint: in a typical MQW detector, the doping
is p-i-n, where the wells are intrinsic, and are between the n and p regions. You
may wish to consult your notes from EE432 for pn junctions.
Library Problems:
Choose one of the following:
1. Find an example of a quantum well modulator that operates at 1GHz or
more in the literature. Write a page or so explaining its structure and capabilities,
and what principle it operates by (i.e. electrorefraction or electro- absorption, or
other)
2. Find an example of a quantum well detector reported inthe literature.
Write a page or so describing its fabrication, operation, capabilities, etc.
REFERENCES
1. S. Adachi, "GaAs, AlAs, AlxGa1-xAs: Material parameters for use in
research and device applications," J Appl Phys, 58, p. 62-89 (1985).
2. S. Shin and C. B. Su, "The sublinear relationship between index change
and carrier density in 1.5 and 1.3µm semiconductor lasers," IEEE Phot Tech Lett,
4, p. 534-537 (1992).
EE 737 Photonics Laboratory Manual Quantum Well Devices
159
3. J. Manning, R. Olshansky, and C. B. Su, "The Carrier-Induced Index
Change in AlGaAs and 1.3µm InGaAsP Diode Lasers," IEEE J. Quant. Elect.,
QE-19, p. 1525-1530 (1983).
4. J. E. Zucker, I. Bar-Joseph, B. I. Miller, U. Koren, and D. S. Chemla,
"Quaternary Quantum Wells for Electro-Optic Intensity and Phase Modulation at
1.3 and 1.55µm," Appl. Phys. Lett., 54, p. 10-12 (1988).
5. D. S. Chemla, I. B. Joseph, C. Klinshirn, D. A. B. Miller, J. M. Kuo, and T.
Y. Chang, "Optical reading of field-effect transistors by phase-space absoprtion
quenching in a single InGaAs quantum well conducting channel," Appl. Phys.
Lett., 50, p. 585-587 (1987).
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EE 737 Photonics Laboratory Manual Liquid Crystals
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LIQUID CRYSTALS
INTRODUCTION
Liquid crystals find application in many areas, from thermometers to
displays and back. In this lab we will learn a bit about liquid crystals in general
and then construct a simple liquid crystal cell and use it to investigate some of
properties of liquid crystal displays.
GENERAL CONSIDERATIONS
Liquid crystals are liquids which possess some of the regularity of crystals
and a delightful interaction with light. These properties occur because liquid
crystal molecules have a long skinny molecular shape. Molecular drawings of a
few representative liquid crystal molecules are shown in Fig. 1. There the C's
represent carbon atoms, the H's represent hydrogen, the O's represent oxygen
and the N's represent nitrogen. The long skinny shape is quite evident. We note
that not all the molecules shown there have axial symmetry. The lack of
symmetry can well contribute to specific properties. The chemical names are
given along with the shapes.
The long skinny shape affects the material properties by restricting some
of the usual freedom of the molecules. In an ordinary liquid the molecules
generally have six degrees of freedom, three translational degrees and three
rotational degrees. The molecules flow back and forth (translation), roll over
each other (rotation). Intermolecular forces then keep the molecules from flying
off like a gas.
In a liquid crystal the shape of the molecules in conjunction with
intermolecular forces suppresses the rotational motion while leaving the
molecules free to translate. The intermolecular forces cause the molecules to line
up in a regular pattern. For some liquid crystals all molecules are parallel to each
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174
other, for other liquid crystals other configurations such as helical arrangement
are found. To see the effect of molecular shape and also the effect of
temperature, one might draw an analogy with a box of pencils. If one were to
shake the box long enough in the right way the pencils at the bottom would find
that they could fit better if they were parallel. Then the pencils above them
would drop in place, etc. In this way they would slowly start lining up. If we
shake too hard with too much energy they would come misoriented. If we shake
weakly they stay aligned.
H-C-C-C-C-O-C
H
H H H H
HHHC=C
C-C
C-C
H
N-C
H H
H H
C=C
C-C
H H
H H
C-C-C-C-C-C-C-C-H
H HHH H HH
H H H H H H H
Butoxybenzylidene octylanilene
H HHH H H
H H H H H H H
H-C-C-C-C-C-C-C-C-C-O-C
H H H
H H
C=C
C-C
H H
H H
C-C
C=C
C-C
H H
H H
C-C N
Octyloxy-
cyanobiphenyl
Figure 1. Typical liquid crystal molecules.
Similar things happen with liquid crystals. At high temperatures (heavy
shaking) the molecules shake enough that they loose their orientation so that the
liquid crystal acts like any other isotropic liquid. At lower temperatures the
molecules settle down and we find the delightful liquid crystal properties.
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175
Liquid crystals are divided into three main types, nematic, smectic, and
cholosteric as shown in Fig. 2. There we see a typical cube of molecules that
might be taken from a large bottle without any interaction with the container and
in the absence of any applied fields.
z
yx
Nematic
Cholesteric
A
B
C
Figure 2. Liquid crystal types.
At the upper left is a cube filled with a nematic liquid crystal. The
molecules are all parallel to each other. The ends don't necessarily line up as they
will in some other types . Mixtures of nematic liquid crystals are often used in
standard liquid crystal displays.
At the right we see three cubes of smectic liquid crystals. They have the
property that the molecules are arranged in layers as well as being all parallel. In
smectic A liquid crystals the molecules in any plane parallel to the layers are
oriented perpendicular to the planes and are located at random positions. In
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176
smectic B liquid crystals the molecules are also perpendicular to the planes but
are arranged in lines in planes parallel to the layers. In smectic C liquid crystal
the molecules are tipped with respect to the molecular planes.
At the lower left is a cube containing cholosteric liquid crystals. We see
planes parallel to the top and bottom of the cube. In any plane the molecules are
in the plane and are parallel to each other, as shown. The defining feature is that
in going from one plane to the plane above it the direction of the molecules
twists. Along any vertical line the tips of the molecules exhibit a helical
behavior. The pitch of cholosteric molecules, i.e. the vertical distance required
for a single end-to-end rotation is often of the order of magnitude of the
wavelength of light so a given cholosteric molecule will diffract light at of a
particular color. This property will often be temperature dependent so
cholosteric liquid crystals are sometimes used in thermometers.
A given liquid crystal may change from one type to another depending on
temperature. At high temperatures the liquid will be isotropic and none of the
special properties will be observed. The energy associated with intermolecular
forces is less than kT where K is Boltzmann's constant and T is the absolute
temperature. When the temperature is below a critical value nematic behavior
will be observed. For some liquid crystals lowering the temperature still further
may cause transitions to one or more smectic phases.
A very simple conceptual mechanical model showing the orientational
properties of a liquid crystal is shown in Fig. 3. There at the left we see the long
skinny liquid crystal molecules. Connecting them at each end are tiny springs
representing the intermolecular forces. When the springs are not stretched or
compressed the molecules are parallel. If some external force is applied the
relative orientation of the molecules changes and the springs are stretched
adding energy to the system as shown in the figure at the right. At equilibrium a
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177
configuration is obtained where each spring is stretched a bit and the energy
added in stretching the springs in minimized.
Figure 3. Simple mechanical model of a liquid crystal.
LIQUID CRYSTAL CELL
In useful applications liquid crystals are found in liquid crystal cells. A
simple liquid crystal cell is shown in Fig. 4. In the figure we see two parallel
glass substrates with nematic liquid crystal between them. These are glass plates
which in practice are separated by a distance of 3 to 15 microns. The surfaces of
the substrates are specially prepared to make the liquid crystal molecules at the
surface all line up in a given direction, the y direction in the figure. With no
other forces the intermolecular forces then align the rest of the molecules parallel
to those at the surface.
The inside of both glass substrates is coated with a transparent conductor,
indium-tin-oxide (ITO) to which a D.C. voltage may be applied. With no voltage
applied all the molecules are parallel to each other and to the surface in (y)
direction as shown in the drawing at the left.
When a D.C. voltage is applied to the transparent electrodes the
molecules sense the resulting D.C. electric field and develop an induced dipole
moment. That dipole moment causes them to feel a torque and to rotate to try
and align themselves with the electric field. This is resisted by the
intermolecular forces and an equilibrium configuration is attained where the
molecules are partially tipped as shown in the cross-sectional drawing at the
upper right in Fig. 4. If the applied voltage is sufficiently high all the molecules
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178
except those very close to the surface will all be tipped so as to be parallel to the
applied electric field as in the cross-sectional drawing at the lower right in Fig. 4.
x
y
z
substrate
ITO liquid crystal
ITOsubstrate
E
Eno field applied large field applied
small field applied
Figure 4. Typical nematic liquid crystal cell
LIQUID CRYSTAL INTERACTION WITH LIGHT
Liquid crystals can interact with light in two ways, scattering and
polarization modification. The interaction of light with liquid crystals that is
most widely used involves polarization and the change of refractive index with
molecular orientation. Since the liquid crystal molecules are long and thin they
have a larger optical dipole moment along the long axis than they do
perpendicular to it. This results in a larger refractive index for light polarized in
the long direction of the molecules than the short direction. The refractive index
also depends on the direction of propagation since the light can be polarized
only perpendicular to the propagation direction. From a different point of view,
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179
for a given direction of propagation the refractive index depends on the
orientation of the molecules with respect to the polarization. Now let's examine
some of these.
To see how the interaction of light with liquid crystals depends on the
orientation of the molecules and polarization direction for a given direction of
propagation consider Fig. 5. There we see three different cases. In each there is a
plane wave propagating in the z direction through liquid crystal molecules. In
the case at the left the molecules are oriented along the x direction and in the case
at the at the right they are oriented in the z direction. In the figure at the left light
with electric field (polarization) along the x direction will experience a large
refractive index while light polarized in the y direction will find a smaller one.
In the figure at the right light polarized in both the x and y directions will see the
same small refractive index.
xy
z
EE
xE
xyEE
y
xx
y y
x
Figure 5. Molecular refractive index, direction of propagation, and polarization of
the light.
Now consider the figure in the center where light is polarized in the x
direction. If the molecule were to start with orientation in the x direction and
slowly rotate towards the z axis the refractive index would slowly decrease until
we get to the situation in the figure at the right. Light polarized in the y
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180
direction would see the skinny dimension and small refractive index
independent of the molecular tip.
If the light is polarized other than along the x or y directions as shown in
Fig. 5 then we resolve the optical E field into components and treat the
components in the x and y directions independently. Thus to reiterate in Fig. 5 as
the molecules rotate from being parallel to the x axis to being parallel to the z axis
then the y polarization component sees a constant index and the x component
sees a decreasing index.
LIGHT INTERACTION WITH LIQUID CRYSTAL CELLS
Now let's apply what we know about refractive index variation and
polarization to our liquid crystal cell. There are various configurations we might
consider. In one simple configuration we let the light be polarized only in the x
direction. We then apply a voltage to the cell which makes the molecules rotate
in the x-z plane as in the middle case in Fig. 5. This produces a phase shifter!
Phase Shifter
To see the action of the phase shifter quantitatively we can write down a
few equations. As the molecules slowly rotate towards the z axis the refractive
index decreases causing a change in the phase of the light leaving the cell. This is
seen in the equations relating the refractive index, n, and the wavelength !, in the
liquid crystal, !", the wavelength in free space and velocities in the material, v
and in free space, c..
! =
v
f [1]
! o =
c
f [2]
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181
n(z) =
c
v(z) [3]
Eliminating v, and c we have
! (z) =
!o
n(z) [4]
In Eq. 1 we have written n(z) just to remind ourselves that the refractive
index depends on the angle of tip of the molecules and that the tip angle and
therefore the refractive index varies along the z axis.
Eq. 4 shows us physically why the phase of the cell in the center drawing
in Fig. 4 acts as a phase shifter. Eq. 4 says that the wavelength of the light
changes as the refractive index changes. Thus as the wavelength changes the
number of cycles of phase shift of the light that we can fit between the ends of the
cell changes, changing the phase of what comes out.
To see this in mathematical terms, this consider the expression for !, the
phase of the light leaving the cell. The incremental phase change in a distance dz
in the liquid crystal is d! =
2"
#dz and the phase change in a full cell is
!x= 2"
dz
# (z)o
L
$ [5]
where L is the cell thickness and we have put a subscript x on ! to remind
ourselves that the light is polarized in the x direction.
Eliminating " with Eq. 4 gives our most useful expression.
!x=
2"#
n(z)dz
0
L
$ [6]
We note that if the light had been polarized in the y direction there would
have been no change because the refractive index for that case remained the
same and the integral corresponding to Eq. 6 would have been
EE 737 Photonics Laboratory Manual Liquid Crystals
182
!y =
2"#o
nyodzo
L
$ [7]
=
2!
" o
nyoL [8]
where nyo is the constant index for light polarized in the direction of the small
dimension of the molecules.
Intensity modulator
Now consider a cell with a different configuration of liquid crystal and
polarization. Imagine that the light is polarized at 45° to the x and y axis and
there is a polarizer (usually called an analyzer because it is transmitting output
light) which passes light polarized at 135° to the x axis and perpendicular to the
incident polarization as shown in Fig. 6. There we see the cell and analyzer and
two sets of axis drawn at the input and the output to the cell. Looking at the axes
at the cell input we see the electric field components drawn. We see that the x
and y components are nicely in phase so that they will line up to give a field in
the 45° direction as expected.
To the right of the cell we see the corresponding set of axes and the two
electric field components. There is a difference, however! The electric field
component in the x direction is lagging in phase behind that in the y direction
because the refractive index is larger and the velocity is slower.
After the light has passed through the cell the electric field component
polarized in the y direction lags in phase because it has experienced a larger
refractive index and has traveled more slowly. If we arrange things just right, by
having a voltage applied to the cell that produces just the right amount of liquid
crystal tip then the y electric field component will lag by exactly 180°. The result
is that the y field component reverses its direction and recombines with the x
EE 737 Photonics Laboratory Manual Liquid Crystals
183
field component so that the combination is perpendicular to the incident light as
shown in Fig. 6. The result is that the electric field components of the exiting
light are then polarized at 135° to the x axis and pass through the analyzer.
x
y
z
E
x
y
E
analyzer
x
y
liquid
crystal cell
incoming light
polarized at 45°
Figure 6. Liquid crystal cell configured as intensity modulator
If we were to plot the light transmitted by the analyzer as a function of
voltage we would get a curve similar to that in Fig. 7. At high voltages the
output light is a minimum. That is because essentially all the liquid crystal
molecules are lined up perpendicular to the cell walls. In that case both
polarizations components experience the same refractive index and have the
same phase delay. After leaving the cell the polarization components recombine
to give the same direction as when entering it. The field is stopped by the
polarizer and the output intensity is zero. As the voltage is decreased the phase
delay between the two components increases from zero, the light becomes
elliptically polarized and a small portion is transmitted by the analyzer. This
increases until there is exactly 180° phase difference and the optical field
components combine so as to be all transmitted by the polarizer. With
continuing voltage decrease the phase difference increases and the output
EE 737 Photonics Laboratory Manual Liquid Crystals
184
intensity oscillates as shown. The output intensity for zero voltage depends on
the refractive indices of the liquid crystal and the thickness of the cell.
1
0
voltage
I/I 0
Figure 7. Intensity of light transmitted by analyzer as a function of voltage applied
to the nematic liquid crystal cell.
We can now express the preceding in analytical terms. The optical electric
field entering the cell is given by
E = Eo( ˆ x + ˆ y )ej!t [9]
The field leaving the cell is
E = Eo( ˆ x ej!x + ˆ y e
j!y )ej"t [10]
The field transmitted by the polarizer Eout is that in the direction of the unit
vector
1
2( ˆ x + ˆ y )
. Thus the field of the light leaving the analyzer, Eout will be
Eout =
1
2( ˆ x ! ˆ y ) "E [11]
=
1
2( ˆ x ! ˆ y ) "Eo
ˆ x ej# x + ˆ y e
j# y( )e j$t [12]
=
Eo
2e
j!x " ej!y( )e j#t [13]
The intensity of the light leaving the analyzer is then
EE 737 Photonics Laboratory Manual Liquid Crystals
185
I =1
2Z0
E2
[14]
=
Eo2
2Z0
ej! x " e
j! y( ) e" j! x " e
" j! y( ) [15]
=
Eo2
2Z0
2 ! 2cos "x ! "y( ){ }( ) [16]
=
Eo2
Z0
sin2 1
2!x " !y( )#
$ %
& ' (
[17]
Substituting in Eq. 17 for !x and !y using Eqs. 6 and 8 we have
I =Eo
2
Z0
sin2 !" o
n(x)dz0
L
# $ nyoL%
&
' (
)
* +
,
-
.
/
0
[18]
In the particular situation where the applied voltage is zero none of the
molecules are tipped and the refractive index for light polarized in the direction
is nx from the back to the front of the cell, n(z)=nxo , Eq. 18 can be further
simplified.
I = Io sin2 !
"o
nxo # nyo( )L$ % &
' ( )
[19]
where
Io=
Eo
2
Zo
[20]
The quantity Io is the maximum intensity. Eq. 19 can be used in the laboratory to
predict the results of measurements made using a parallel cell. The procedure is
to plot the output of a parallel cell and polarizer as set up in Fig. 6. The thickness
of the mylar spacer gives the thickness, L, and the refractive indices nxo and nyo
are provided along with the liquid crystal. Then the quantity
!
" o
nx0# nyo( )L
gives the number of cycles of fluctuation of the intensity. In addition the
EE 737 Photonics Laboratory Manual Liquid Crystals
186
quantity
sin2 !
" o
nxo # nyo( )L$ % &
' ( )
gives the relative intensity at zero voltage. These
quantities can both be observed.
COMMERCIAL LIQUID CRYSTAL DISPLAYS
Commercial liquid crystal cells will often have a somewhat different
construction than the simple cell we have been considering. The liquid crystal
will have the twisted arrangement shown in Fig. 8. A nematic liquid crystal will
still be used with a small amount of cholosteric liquid crystal to help with the
twist. For a twist cell the treated substrates are oriented so that the liquid crystal
molecules at the cell walls are at a 45° angle rather than parallel as in the cell we
have been considering. For this case with no applied voltage the molecules
undergo a slow twist from one wall to the other as shown in Fig. 8. There is a
mirror on one side of the cell so that the light passes through the cell, is reflected
off the mirror and passes back through the cell. The polarization of the light is
rotated 45° on the first pass through the cell, and then unrotated back to its
original orientation on the return pass.
The twisted nematic cell has a intensity-voltage characteristic curve which
is more useful in many circumstances. It is shown in Fig. 9. Instead of oscillating
as in Fig. 7 it decreases monotonically from maximum to zero. This is more
useful if one wants only the one range from minimum to maximum, or if one
wants a binary output as in many liquid crystal displays. It also has the
advantage that it is easier to mass produce so that there is the desired
polarization rotation with no applied voltage.
When a voltage is applied, most of the molecules are aligned
perpendicular to the substrate. Near the ITO-coated substrate, however, there is
a thin layer whose molecules are still oriented along their original alignment. The
EE 737 Photonics Laboratory Manual Liquid Crystals
187
liquid crystal material in these regions is birefringent, and causes a total phase
retardation between the x and y polarizations of 90° during one round trip. In
this case, the return light does not pass through the front polarizer, and the
display appears dark.
mirror twist cell polarizer
Figure 8 Twisted nematic liquid crystal cell
1
o
I/I0
voltage
Figure 9 Voltage characteristic curve of a twisted nematic cell.
EE 737 Photonics Laboratory Manual Liquid Crystals
188
In practice the transparent electrode on one side is divided into separate
areas with independently controlled voltages so as to form voltage controlled
patterns. A typical pattern might be a seven segment display used to display
numbers.
HOMEWORK
1. Consider a transmissive (as opposed to reflective) nematic liquid crystal
display, such as that shown below, which is meant to be similar to that of Figure
6 in the lab manual. The voltage, when applied, produces exactly 180° of phase
lag between the x and y polarizations. There are electrodes on the front and back
glasses, and wherever they overlap it is possible to generate a field and rotate the
liquid crystals in that region. Assume the crystals are aligned vertically at both
glasses.
For each of the following display configurations, describe the appearance
you expect when a voltage is applied to some segments, and when no voltage is
applied. Do both displays work?
polarizer
electrodes LC cell
x
y
z polarizer
electrodes LC cell
2. Consider the phase shifter described in the chapter. Suppose you
wished to convey binary information on the phase of an optical wave (as in
EE 737 Photonics Laboratory Manual Liquid Crystals
189
phase-shift keying). Also suppose the input beam is purely x polarized. Describe
how you would imprint the data onto the phase of the lightwave (shift the phase
back an forth between two states that you define). Sketch the system. How
would you detect the phase at the other end? (Hint: how do you detect the phase
of any wave? Compare it to something of known phase, such as another wave
whose phase is not being modulated. Hint 2: Interfere two beams together to find
the phase relationship.) Sketch that system too.
LIBRARY PROBLEM:
Choose one of the following:
1. Find an example of a use of liquid crystals that is not a display. Write a
page or so describing it operation and application.
2. How do they make color liquid crystal displays? (There is more than
one way, find at least one.) Describe in a page or so how these work.
EE 737 Photonics Laboratory Manual Liquid Crystal Laboratory
190
LIQUID CRYSTALS LABORATORY EXPERIMENT
As explained in your course notes, a liquid crystal cell operates by altering
the polarization state of the light passing though it. To acquaint yourself with a
simple polarization effect, design and construct an experiment with linear
polarizers to demonstrate Malus’ Law. If you use a white light source, you
might want to include an ultraviolet light blocking filter and an infrared-blocking
filter zZwithin your experiment. (Any ideas why?) Compare your results with
what you would expect theoretically and explain any discrepancies between the
two.
Next, you will actually make a simple liquid crystal display and
investigate some of its properties such as the contrast ratio of the display and the
effects of viewing angle.
To construct your liquid crystal display, you will sandwich the liquid
crystals between two pieces of ndium-tin-oxide (ITO) coated glass. (The ITO
coating is only on one side of the glass) ITO is a conductor that is transparent to
visible light. To obtain an optimally functioning display, it is important to begin
with the ITO glass as clean as possible. To this end, the following cleaning steps
have been developed to remove any contamination from the glass. In this
cleaning process, chemicals which are to varying degrees harmful to your skin
will be used, so USE CAUTION AT ALL TIMES. While handling the glass
plates, to protect yourself as well as the materials, you must wear gloves or
finger cots.
You will be given two 1” square pieces of ITO glass along with a special
holder which you will mount the glass in for cleaning. Make sure you know
which surface of the glass has the conductive coating on it before you put them
into the holders. You will need three 100 ml beakers in which to put the various
EE 737 Photonics Laboratory Manual Liquid Crystal Laboratory
191
cleaning materials. Please use only enough cleaning liquid to just cover the
glass. The cleaning procedure is detailed below.
CLEANING PROCEDURE FOR ITO COATED GLASS
1.) In a 100 ml beaker, pour in the detergent Liqui-Nox. Insert the glass
holder into the detergent and put the beaker in the ultrasonic cleaner for
approximately 5 minutes.
2.) Remove the glass from the Liqui-Nox and rinse thoroughly with DI
water.
3.) Submerge glass in acetone from about 2 minutes.
4.) Blow dry the glass with nitrogen gas.
5.) Submerge glass in isopropyl alcohol to dissolve any excess acetone.
6.) Rinse thoroughly with DI and blow dry.
Once the glass substrates have been thoroughly cleaned, you must treat
the conductive surface so that the liquid crystals align in a specific direction. You
will be making a parallel cell in this experiment. That is, with no applied
voltage, the orientation of the liquid crystal molecules will be the same between
the front and back pieces of glass. To achieve this, set the glass substrate ITO
side up on a piece of tissue paper. Using your thumb and another piece of tissue
EE 737 Photonics Laboratory Manual Liquid Crystal Laboratory
192
paper, slowly swipe the tissue paper along the ITO surface applying a fair
amount of pressure (but not enough to break the glass). BE SURE TO SWIPE
ONLY ONCE AND ONLY IN ONE DIRECTION, KEEPING NOTE OF THE
DIRECTION. Do this to both substrates.
To contain the liquid crystal material, you will need to cut out a spacer of
0.5mil mylar as shown in Figure 1.
1"
0.75"
Figure 1 Mylar Spacer
Take the mylar cut-out and place it between the two glass plates with the
conductive side on the inside such that the directions in which you swiped the
ITO surface are parallel. Note that the glass plates should not be placed directly
on top of one another, but rather, should be offset about 1/8” horizontally and
lined up vertically (this is so that you can apply electrical contacts to the ITO
coating). This is shown in Figure 2. Use binder clips to hold the cell together.
Using a micropipet, insert a small amount (half a pipette or so) of the liquid
crystal material between the plates. Capillary action will cause the liquid to fill
the well which you have created. You now should have a functioning liquid
crystal cell.
EE 737 Photonics Laboratory Manual Liquid Crystal Laboratory
193
Figure 2 Position of Glass Plates
The liquid crystal cell you have built is a transmissive device. Design and
construct an experiment to measure the contrast ratio that is achievable by your
cell.
Note that a DC voltage will quickly deplate the thin electrodes. Therefore,
you should applied an AC signal that has zero volts average value (DC offset).
Keep the applied RMS voltage below 12 volts.
A commercially made liquid crystal display is also available for your use.
This display is a reflective device. Measure the contrast ratio of this display (also
applying only AC voltage) and compare this with the value obtained for your
cell. How do they compare?
Finally, you will note that depending upon the angle at which you view
the commercial display, the characters may be more or less brilliant. Design an
experiment to measure this viewing angle effect. Can you explain the origin of
this effect?
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Figure1. Spectral distribution of sunlight. Shown are AM0, AM1.5 and the radiation distribution of
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Figure 2.Simple diagram of photon absorption by direct gap semiconductor wherethe photon energy is greater than the bandgap.
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*)+*<7)(",'4.0"7)70A1"+.)"/7"+')J70(7<"('"747+(0*+.4"7)70A1".)<"*,"4',(".,"&7.(?""M'(7"(&.(
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,<.+7",(.(7,".07".A.*4./47".,"(&7"+01,(.4">'>7)(3>"*)+07.,7,"B07+.44"<"C"&?DE
Figure 3. Absorption processes for indirect gap semiconductors.
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Figure 4. Optical absorption coefficients of various single crystal semiconductors. [1]
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Figure 5. Percentage of light reflected as a function of wavelength from Si with andwithout AR coatings having the refractive indices shown. [1]
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Figure 6. Geometry of a typical solar cell structure, based on the GaAs "heteroface" design.
Only the p-type emitter and n-type base are considered in the I-V derivation.
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Figure 7. Common solar cell configurations. In cases a,b and c, the contribution from thelarge bandgap material to the photocurrent is negligible.
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Figure 8. Terminal I-V characteristics of an illuminated pn junction solar cell. Maximum powerpoint is indicated and the dark I-V is shown for comparison.
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Figure 9.Solar cell efficiency limits as a function solar cell bandgap for different solar spectra.[1]
Figure 10. Major features of a simple solar cell.
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Figure 11. Equivalent circuit of an ideal solar cell.
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Figure 12.. Effect of parasitic resistances on cell I-V characteristics for (a) Rs and (b) Rsh.
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4.170,"')"('A"'?"(&7"7>*((70@".07".44"+0*(*+.441"*>A'0(.)("('".+&*7=7"&*<&"7??*+*7)+1",'4.0
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(&7"+744@".+0',,"(&7"A)"I3)+(*')F""J'E7=70"('"7H(0.+("(&7"+3007)("=*."+')(.+(,"(&.("')41
A.0(*.441"+'=70"(&7",30?.+7"K07>7>/70@"(&*,"*,"('".44'E"4*<&("*)('
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89:
Figure 13. Various approaches to top contact designs.
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+'447+(*')"*)",'4.0"+744,@
!!"#$#"%&'(')*+,"-./'0.('01"2.)3.4 5'4.0" 6744,
89#
Figure 14. Band diagrams of cell structures for enhanced carrier collection.
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*<@'0(.)(B"A*(&"3)*='0<"*)(7),*(1".+0',,"(&7"+744E""K'(7"(&.(",[email protected].(7"+')(.+("47.>,"='0
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*)+07.,7,E
Figure 15. DOE/NASA solar cell testing methods.
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89:
Figure 16 Simple solar simulator.
Figure 17. Experimental configuration for measuring cell characteristics. (a) Efficiency and (b)spectral response.
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226
PART II: Introduction to several optical technologies not coveredexplicitly in the lab:
Review of OpticsPolarization
Radiometry and PhotometryUseful Optical Devices
Things Statistical(or, How to treat mesurement errors in the lab)
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@8.),"(&.(")'"+48.0"*@.B8"*,"F'0@8CD"F'0"')8"(&*)BD".)C"(&.("(&8"08F48+(8C"4*B&("*,
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Figure 5. Specular vs. diffuse reflections
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Angle of incidence (degrees)
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Figure 6.
1.0
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Angle of incidence (degrees)
Fresnel Reflection, glass to air
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Figure 7.
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Figure 9. Absorption and emission
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Figure 1. Linearly polarized light. Top: x polarization, center: y polarization; bottom:
linearly polarized light at +45°.
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Figure 2. Linear polarization at +135°.
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Figure 3. Circularly polarized light, with &=+90°.
T&<";.1"(&<":*>30<"*,"B0.;)A"&""*,"WJFGH""9:";<"?3("'30"+&.*0".("*):*)*(1".)B
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RADIOMETRY AND PHOTOMETRY
INTRODUCTION
When dealing with light sources, a common question is “How bright is
it?” It turns out that “bright” can means lots of different things- high intensity,
high luminance, high radiance.... It’s important to be precise, since two sources
emitting the same number of photons may have, for example, different
intensities. To characterize a detector, one can express the incident light in a
variety of units as well, depending on the detector and the application.
Furthermore, different systems of units are sometimes used for visible
light than for invisible light. The photometric system of units is corrected for
human eye response, and therefore only applies to visible light. In this system,
since the eye is more sensitive to green light than purple light, a green source
will be “brighter” (whatever that means) than a purple source of equal energy.
The radiometric units are the same regardless of wavelength, and so apply to all
regions of the electromagnetic spectrum.
RADIOMETRIC UNITS OF EMISSION
A light source emits energy in the form of electromagnetic waves (or
photons, depending on the day of the week). The basic measurement of light
might therefore be considered to be the radiant energy Q, measured in Joules.
Qe=radiant energy (Joules) [1]
The amount of energy delivered within a certain time (or the rate of
energy delivery) would then be the radiant power, or radiant flux:
! e =
dQe
dt (Watts) [2]
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If the source is an extended source (for example an electroluminescent panel
such as those used in cockpits), the total optical power is perhaps not so
important as they energy emitted per unit area, the radiant fluence:
Fe =
dQe
dA (J/m2) [3]
The volumetric radiant density is the energy per unit area, symbolized by W:
We =
dQe
dV (J/m3) [4]
Even a very weak source can emit a sizable amount of energy over a
sufficiently long time, so perhaps it would be more useful to consider the rate of
energy flow than the total energy. This leads to a series of power-based units.
An extended source may emit a given amount of energy per unit area per
unit time, which would be its radiant excitance:
Me=d!
e
dA (W/m2) [5]
The most intuitive quantity is perhaps the radiant intensity, however, at
least from a perceptual point of view. From a mathematical point of view,
radiant intensity is
I
e=
d!e
d" (W/sr) [6]
where ! is a solid angle, measured in units of steradians (sr). A steradian is a unit
of solid angle, Figure 1, the same way a radian is a unit of flat angle. There are 4"
sr in a sphere:
d! = 4"sphere
# [7]
and
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d! =
da
r2
[8]
x
y
z
d!
r
da
Figure 1. Solid angle.
For an extended source with directional emission, the radiance may the
most useful unit:
L
e=
d2!
e
d"dA cos#=
dIe
dAcos# (W/sr-m2) [9]
PHOTOMETRIC UNITS OF EMISSION
As opposed to radiometric units, photometric units have been corrected
for the response of the human eye. The eye responds differently in daylight than
when it is night-adapted. These two curves are shown in Figure 2. The daylight
curves is the photopic response, while the night-adapted curve is the scotopic
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curve. When the eye is daylight-adapted, the average human vision peaks at
about 555 nm, in the yellow-green part of the visible spectrum. If we call the
photopic curve K(!), then for each radiometric unit Xe there is a corresponding
photometric unit Xv such that
Xv=K(!)Xe (monochromatic light) [10]
We have specified here that the light is monochromatic for Eq. 10 to apply.
In general the source will have some spectral spread, so one must integrate the
radiometric unit over the entire applicable spectrum:
!e= !
e(" )d"
"1
"2
# [11]
where the range !1 through !2 is limited to the visible spectrum. Energy outside
this range is not detectable by the eye an therefore does not contribute to the
photometric quantity.
The peak of the photopic curve has a value of Kmax=673 lm/W, where lm
stands for “lumens”, the photometric analog to power. The conversion factor will
be different at each visible wavelength, and it is handy to use the normalized
response curve, in which
V(! ) =
K(!)
Kmax
(photopic) [12]
and
V' (! ) =
K(! )
K 'max
(scotopic) [13]
where K’max =1725 lm/W, and occurs at 510 nm.
If luminous flux corresponds to radiant flux or power, then we should
call it
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!v=luminous flux (lumens) [14]
where the subscript v suggests “vision”. The analog to radiant energy is
luminous energy, measured in Talbots:
Qv=luminous energy (Talbots, or lm-s)[15]
and
!v =
dQv
dt (lumens) [16]
Similarly, the luminous energy density is given by
Wv =
dQv
dV (lm-s/m3) [17]
The photometric unit corresponding to radiant excitance is, predictably,
the luminous excitance
M
v=
d!v
dA (lm/m2 or lux) [18]
Now, the luminous intensity is given by
I
v=
d!v
d" (lm/sr or candela) [19]
The SI unit of luminous intensity is the candela (cd), and when you are
purchasing, for example, light-emitting diodes, they are typically specified in
miilicandela (mcd), since it that is the unit most relevant to our perception of the
lamps’ brightness.
Finally, the luminance corresponds to apparent brightness of an extended
source, taking into account the directionality of the source:
L
v=
dIv
da cos! (lm/m2/sr, or cd/m2) [20]
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Figure 2. Effect of viewing angle on radiance, intensity of a Lambertian source.
This last business of luminance (or radiance) bears some further
investigation. For example, consider a 1 cm2 flat square that emits light
uniformly over its surface and isotropically (evenly in all directions). A source
that radiates evenly in all directions (radiance or luminance = constant) is called
a Lambertian source. The intensity, however, is not a constant with viewing angle,
as may be seen from Figure 2. The intensity is the power per unit solid angle.
When the viewer (or measurement device) is directly in front of the emitting
surface, some maximum reading is obtained. As the viewer goes off axis,
however, the source plane is tilted with respect to the eye, so a cos! projection
factor must be included, which reduces the number or rays that will go through
the detection aperture. The solid angle remains the same, but the effective area
from which rays can be detected has been reduced by the projection factor. A
good example of a Lambertian source is an LED chip. It’s flat, and light is only
emitting from the top planar surface. When you look at the edge, you will detect
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essentially zero light, even though the photons are emitting evenly in all
directions.
The radiance is also invariant with distance from a source, whereas the
intensity is not.
UNITS OF INCIDENT LIGHT
The quantities discussed up till now have all been applied to emitting
sources. We now need to consider how to measure the amount of light landing
on an object or surface. It turns out, fortuitously, that most of the units can be
used in the same way. The total energy incident on a surface is the power
striking the surface integrated over time. The intensity of light striking, say, your
eye can be computer from the solid angle your eye subtends based on your
position relative to the source.
There is one unit that is different, however- the radiant flux crossing a unit
area. If the area in question lies on the source, this quantity is termed “radiant (or
luminous) excitance” (M), as defined earlier. If the flux being measured is
crossing a unit of area on a detecting surface, the convention is to call it
irradiance for radiometry:
E
e=
d!e
dA (W/m2) [22]
and illuminance for photometry:
E
v=
d!v
dA (lm/m2) [23]
Table 1. Radiometric Units
Quantity Symbol Units How to find
Energy Qe Joules (J) fundamental
Fluence Fe J/m2 dQe/dA
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Energy Density We J/m3 dQe/dV
Power (or radiant flux) !" Watts (W) dQe/dt
Intensity Ie W/sr d!"/d#
Excitance (emitters) Me W/m2 d!"/dA
Irradiance (detectors) Ee W/m2 d!"/dA
Radiance Le W/sr-m2 d2!"/d#dAcos$
Table 2. Photometric Units
Quantity Symbol Units How to find
Luminous energy Qv Talbot (lm-s) fundamental
Luminous energy density Wv lm-s/m3 dQv/dV
Luminous flux !% Lumens (lm) dQv/dt
Luminous intensity Iv lm/sr d!%/d#
Luminous excitance Mv lux (lm/m2) d!%/dA
Illuminance Vv lux d!%/dA
Luminance Lv lm/sr-m2 d2!%/d#dAcos$
OTHER SYSTEMS OF UNITS
The units of Watts, Joules, meters, and lumens are all part of the SI system
of units. You will occasionally encounter terms such as “stilb” and “footcandles”
from the cgs and English systems. For example, the SI unit of luminous excitance
or illuminance is the Lux (lumens/m2). In cgs units, the unit is the Phot
(lumens/cm2), and in English units the Footcandle (lm/ft2) was used. Table 3
summarizes the conversions between these systems.
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The basic unit of luminance in the SI systems is the candela/m2,
sometimes called the nit. The cgs equivalent is the stilb (candela/m2), and the
English system has the candela/ft2, no special name.
Finally, there are some units that apply only to Lambertian, diffuse
surfaces, having one lumen per unit area excitance, and these are the Apostilb
(SI), Lambert (cgs), and Footlambert (English). For more discussion of these see
the Handbook of Optics.
Table 3. Conversion factors for units of illuminance (after [1])
Footcandle Lux Phot Milliphot
1 footcandle 1 10.76 1.08E-3 1.076
1 lux 0.0929 1 100E-6 0.1
1 phot 929 10E3 1 1E3
1 milliphot 0.929 10 1E-3 1
Bibiliography
Introduction to Optics, Frank Pedrotti and Leno Pedrotti, Prentice-Hall,
1987.
Electro-optics Handbook, Ronald Waynant and Marwood Ediger, McGraw-
Hill, 1994.
HOMEWORK:
Prove Equation 7.
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REFERENCES
1. The Laser Institute of America, "American national standard for the
safe use of lasers ANSI Z136.1-1986","1986).
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USEFUL OPTICAL COMPONENTS
INTRODUCTION
When you are design a circuit, you can probably design one to do perform
a given function even with a limited vocabulary of components. If you didn’t
know that op amps existed, though, it’d be much more difficult and time
consuming. If you didn’t know transistors existed, there’d be functions you
couldn’t perform at all. The same is true in optics- there are lots of interesting
optical components out there that you may not know about. The purpose of this
section is to tell you about the existence of some of the optical components that
can make your life easier when designing an experiment. We are not trying to
imply we have all of these components at your disposal, but it is useful to know
they exist.
MIRRORS
Of course you already know about lenses and mirrors. But there are some
specialized types of both out there. Let’s start with mirrors.
The mirrors in your bathroom consists of a piece of glass with a silvered
back. If the silvering is good and the surface flat, as shown in Fgure 1a, you’ll get
a decent reflected image, with the travelling the paths shownin the top part of
Figure 1. If the back surface is curved, however, you get a fun-house mirror that
distorts your image. We can use curved mirrors intentionally to focus light,
however. Consider the spherical mirror in the bottom of Figure 1. Parallel rays
striking the surface are reflected back but not along their incoming paths. Note
that this is only true in paraxial approximation; that is, the rays are very close to
the optical axis over their entire lengths. This mirror acts just like a lens, and
even has a focal length.
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Figure 1. Top: a flat mirror. Bottom: a curved mirror focuses like a lens.
The focal length of a spherical mirror is given by
f = !
R
2 { 1}
where R is the radius of curvature of the mirror. The negative sign here reflects
the fact that the image appears on the same side of the mirror as the incoming
beam. One common use of a curved mirror is in a laser resonator cavity- the
mirrors provide focusing and optical feedback at the same time. Also, it is
sometimes necessary to use mirrors to fold an optical path back and forth to
obtain a long path on a short table. In this case, some curved mirrors are often
used to prevent the beam from expanding too much over the length of the path.
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The mirrors used in optics laboratories are front surface mirrors, meaning
that unlike your bathroom mirror, the reflecting surface is not protected by a
layer a glass. The reason for this construction is that Fresnel reflections off the
front surface of the glass would occur in addition to the expected reflection off
the metallization on the back, producing a spurious image. By putting the
reflecting surface on the front, this extra reflection is avoided. The down side of
front surface mirrors is that they are easily damaged since the metallization is
exposed, and therefore the reflecting surface should never be touched with
anything, particularly not your fingers.
You should also be aware that the effectiveness of a mirror depends on the
material it is coated with, and must be matched to the intensity of your source.
You may be surprised to learn that the standard aluminum mirror is only about
90% reflective over the visible and near-IR range, where it is most commonly
used. This can be improved with dielectric coatings (at additional cost). The
energy tolerance of the mirror before it is damaged also depends on the coating.
For mid- and far infrared light, gold mirrors are commonly used because of their
high reflectance in these wavelengths.
Other specialized mirrors in the curved mirror family are parabolic and
elliptical mirrors, used for illumination applications and concentration of light.
For example, consider a light bulb such as that you might find inside an
overhead projector. (Check out the bulb in the solar simulator or quantum well
experiment next time you are down in the lab.) It emits rays in all directions, but
the desired direction of propagation is through the optics of the projector and
onto the screen. If the bulb is placed at the focus of a parabolic reflector, then all
rays will be reflected off the parabola such that they are parallel, Figure 2.
Parabolic reflectors are also used in antennas- all rays entering the parabola are
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reflected through the focus (the incoming rays are assume parallel since the
source of the rays is assumed to be a great distance).
focus
Figure 2. Top: a parabolic reflector. Bottom; an elliptical reflector.
The elliptical reflector is useful for collecting light from a source and
focusing it at another point, as seen in the bottom of Figure 2. If the source is at
one of the foci, the rays will all converge at the other focus. Note that both the
parabolic and ellipsoidal reflector are not good for imaging, but rather for
concentration or collection of light.
Another type of reflector used in the laboratory is the retroreflector, or
corner cube. This device contains three mirrors, all mutually orthogonal. Any ray
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entering the corner cube is reflected off some or all of the surfaces, and the exit
ray will always be parallel to the incoming ray, Figure 3.
Figure 3. A retroreflector.
The utility of such a device can be understood by considering the
following engineering problem: It is desired to measure the distance to the moon
by shining a laser beam to the moon, reflecting it off a shiny object left there by
astronauts, and measuring the time it take the beam to make the round trip. Note
that implicit is the assumption that the reflected beam will return to the same
spot from which it is launched. This has actually been done, and the distance to
the moon measured to within a centimeter. It may strike you as difficult to hit the
corner cube from earth, but even the most directional laser beam spreads out to a
radius of several km by the time it gets to the moon, so aim is not as big an issue
as it first appears. Quiz question: when you look into a corner cube, what will
you see?*
Another device used to reflect a beam back parallel to its incoming path is
a roof prism, Figure 4. This works similarly to the mirror-based corner cube,
except that the optical path is no longer entirely in air, and the reflections are
Answer: your own eye. Can't be anything else.
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now total internal reflections. A solid glass trihedral retroreflector can be used to
retroreflect in three dimensions.
Figure 4. Roof Prism.
LENSES
You already know about thin lenses, which have spherical or planar
surfaces. These can be double convex (both surfaces bow out), plano-convex,
double concave, or plano concave. You have also used compound lenses- the last
time you looked through a microscope. Compound lenses are composed of
several different lenses, as in a microscope objective lens.
Although the focal length of a compound lens is often specified, this
number is not terribly useful in the lab, because it is measured with respect to an
imaginary surface corresponding to the equivalent surface in the equivalent thin
lens. For example, suppose a compound lens has a focal length of 20 mm. A ray
entering the lens parallel to the optical axis will emerge from the lens and cross
the optical axis at the focal point. If we extend the incoming ray and the outgoing
ray until they meet, that defines the principal surface from which the focal length
is measured. For a thin lens, this plane is in the center of the lens. Figure 5
illustrates these two examples. Not that for a compound lens, the location of the
principal planes (there are two, a front one and a back one) can be somewhere
inside the lens, and not necessarily in the center.
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A more useful quantity for a compound lens is its working distance,
which is the distance from the physical front of the lens to the focal point. That is
a distance you can physically measure in the lab. For a thin lens, the working
distance is essentially the same as the focal length since the lens is thin.
front
principal
plane
back
principal
plane
ff
f
working
distance
principal plane
Figure 5. Focal length, principal planes, and working distance. Top: compound
lens. Bottom: thin lens.
Cylindrical lenses are used to focus light along one direction, and leave it
unchanged along another. Such a lens might be used to create a line of light
instead of a point. Figure 6 illustrates the principle behind this type of lens.
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incoming beam
Figure 6. Cylindrical lenses focuses light along one direction only.
A lens (any lens discussed so far) operates by Snell’s law. The curved
surface causes the angle of incidence to change for the rays of a collimated beam
as one goes away from the optical axis. This changing angle of incidence also
changes the angle of refraction, so that the rays are bent according to how far
from the optical axis they lie.
An alternative way to achieve this varying bending of the rays is to
change the refractive index of the glass gradually rather than the angle of
incidence. Such a lens is called a graded index, or GRIN lens. In order to provide
enough bending, the lenses have to be reasonably "thick." Graded index lenses
are generally shaped like rods, hence the term GRIN rod, Figure 6. Usually the
grading is parabolic, that is, goes as r2 where ris the radial position. As a result,
the optical rays are continually bent. The GRIN rod in Figure 6 is cut to a length
that achieves focusing, but if it were longer, the rays would all follow sinusoidal
paths, and all the rays intersect at the nodes, or zeros of the sine function. The
length of one complete sine wave is known as the pitch of the GRIN rod.
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incoming beam
GRIN ROD
GRIN RODcollimated output
Figure 6. A graded index lens (GRIN rod). Top: focusing. Bottom: collimating.
GRIN lenses are generally quite small, and so are useful in applications
where compact size is needed, such as in coupling optical fibers to laser diodes.
They are often used in pairs, to collimate a beam for transmission across a gap,
and to collect the light and focus it onto a fiber or detector.
There is another type of lens that is pretty interesting – the Fresnel lens.
This lens, shown in Figure 7, is usually cast in acrylic and looks a little like a
regular convex lens that has been collapsed. These are often used in overhead
projectors to magnify the light coming from the light source in the base. There is
another Fresnel device called a Fresnel zone plate, which consists of a series of
transparent and opaque rings, also on a flat surface. To explain how these work,
we'd have to go into diffraction theory, which is beyond the scope of this
manual. Fresnel zone plates (not shone) act as spherical lens with multiple focal
lengths. Another key difference between Fresnel lenses and Fresnel zone plates
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in that in the zone plate, the concentric circles get closer and closer away from
the center.
Figure 7. Fresnel Lens. Top: Cross section. Bottom: top view.
OPTICAL FLATS
These are pieces of glass or other optical material, on which one or both
sides is polished to a very high degree of smoothness. They are used to measure
surface quality- the surface under test is brought very close to the polished
surface of the flat. When monochromatic light is shone through the flat, the
reflections from the flat surface interfere with the reflections from the surface
under test. For every 2! phase difference between the two surfaces results in a
fringe, so the fringes give a direct map of the contours of the surface being tested.
BEAMSPLITTERS
Beamsplitters are devices that divide an optical beam into two paths.
These can be power beamsplitters, in which some specified percentage of the
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power is deflected, while the rest continues along its original path (i.e. 50-50, 90-
10, etc.). Figure 8 shows a cube beamsplitter and how it’s used.
primary
reflection
ghost image
Figure 8. Beamsplitters Left: cube type. Right: plate beamsplitter.
The beamsplitting agent itself is a dielectric film. In the cube case, the film
is deposited onto the hypotenuse side of one of the triangular pieces, and the
pieces are cemented together. In the plate case, the film is on one of the surfaces.
In the plate case, a second, Fresnel reflection will occur at the other surface, and
antireflection coatings must be applied to avoid ghost images. In the cube case,
the Fresnel reflection goes back along the incoming path and does not create
ghosts. Because of the optical cement between the two glass pieces, however,
these cannot withstand as high of optical powers as the plate type.
There is a third type of beamsplitter- the pellicle beamsplitter. This one is
made of a stretched membrane, which is so thin that ghost images are essentially
not a problem. These are delicate, however, and can be deformed easily, ruining
them. These are also available with coatings controlling the ratio of reflected to
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transmitted power. Note that as membranes they are sensitive to acoustic
vibrations in the laboratory.
A beamsplitter can also be used as a beam combiner, as in the
interferometer of Figure 9. Two incoming rays are combined such that they travel
along the same path. In the interferometer, one beamsplitter is used to separate
the beam into two components of equal power. One beam stays unchanged, and
the other passed through some device or material that delays it. When the beams
are combined at the second beamsplitter, the phase difference acquired by the
second beam with respect to the first is measured by measuring the degree of
interference. (There is an additional fixed delay due to extra path to and from the
fixed mirrors.)
Object
under test
Figure 9. Use of beamsplitters in an interferometer.
Note that a quick and cheap way to pick a small amount of energy off a
beam, to check the signal or whatever, is to use the 4% Fresnel reflection off the
surface of a piece of glass- microscope slides are mighty handy for that. This
technique is useful for checking that a “signal” one is seeing is actually due to the
experiment, and not fluctuations in the laser source itself-, for example. The
picked off signal could also be used in a feedback loop to subtract out source
fluctuations.
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There are also polarizing beamsplitters, which send each polarization in a
different direction, as will be discussed under prisms, mext.
POLARIZERS
Polarizers are made in a variety of ways. The cheap ones are made by
stretching a polymeric plastic materials (polyvinyl alcohol, or PVA). The
stretching causes the long molecules to line up. The molecules are then dyed
with iodine, and they act as polarizers, absorbing light polarized parallel to the
molecules. The PVA is laminated between two pieces of glass or plastic for
protection. Better quality polarizers are made by embedding a grid of fine wires
in a material. Light whose electric field is polarized parallel to the wires is
transmitted
birefringent
material
Figure 10. A birefringent material can be used to separate the polarizations.
Birefringent materials are those in which the refractive index of the
material depends on the polarization of the light. When light comes into the
material, Figure 10, the refraction angle will be different for the two
polarizations, and therefore the polarizations will be separated. A classic
birefringent material is calcite.
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Separation of polarizations can also be achieved by reflection. Since
Fresnel reflection coefficient is in general different for the two polarizations, the
light after a Fresnel reflection will be partially polarized. If a number of reflecting
surfaces are stacked, then after multiple Fresnel reflections the polarization
purity of the reflected beam can be quite high. Multiple stacks are laminated
between the two pieces of glass in a polarizing beamsplitter cube, for example.
Several types of polarizing prisms are based on birefringent materials. For
example, a Wollaston prism uses birefringence to separate the two polarizations
spatially as shown in Figure 11. There is a fairly wide separation angle for this
geometry, and neither of the emerging rays propagates in the same direction as
the incoming wave, which can complicate alignment of large systems. One
normally puts a stop in front of the unwanted polarization to keep it from
propagating. On the right of Figure 11, the Rochon prism operates similarly to
the Wollaston, but has the feature that part of the beam being “kept” propagates
in the same direction as the incoming beam, the trade-off being smaller angular
separation. The output directions are controlled by proper choice of orientation
of the birefringent crystal and the angle of the interface.
The Glan-Thompson prism uses total internal reflection of one
polarization to deflect that polarization to an absorber, while the unreflected
component is allowed to progress, Figure 12.
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Figure 11. Left: Wollaston prism. Right: Rochon prism. (After [1])
absorber
Figure 12. The Glan-Thompson prism.
WAVE PLATES
In addition to linear polarizers, there are devices to create and analyze
circular and elliptical polarizations. These are the wave plates, which, in
conjunction with linear polarizers, are immensely useful in the lab.
For example, consider a quarter-wave plate. This device is birefringent, so
that the two polarizations propagate at different velocities. If the incident ray is
normal to the surface, there is no angular deviation- both polarizations continue
along the same path, but one travels faster than the other. The result is that the
two electric field components, one polarized perpendicular to the other, are out
of phase at the exit surface. If the thickness of the plate is chosen such that the
phase difference is !/4, it is called a quarter wave plate. If the incoming light has
equal field strengths in the two polarizations (i.e. is linearly polarized at 45°), the
EE 737 Photonics Laboratory Manual Useful Optical Components
274
output will be circularly polarized. Whether the output is RCP or LCP depends
on which polarization is retarded and which is not.
45°
linear
polarizer
circularly
polarized light
quarter
wave
plate
Figure 13. Use of a linear polarizer plus a quarter-wave retarder plate to create
circularly polarized light.
Here is a typical use of a quarter-wave plate. One big problem in working
with lasers is that any back reflection off of optical components can interfere with
proper operation of the laser, depending on the strength of the reflections.
Unfortunately, reflections are unavoidable, since there will be a Fresnel reflection
off every lens and other piece of glass in the optical system. Using a quarter-
wave plate (QWP) can prevent these reflections from entering the laser cavity in
the following manner: the laser light is either already polarized, or passed
through a linear polarizer. On passing through the QWP, the light becomes, let
us say, RCP (right circularly polarized). Upon reflection, however, the returning
beam is LCP (left circularly polarized). When the LCP light passes through the
QWP, the two polarization components are retarded again by a difference of 90°.
The resultant leaving the QWP is now linearly polarized again, but at -45°,
whereas the incoming light was polarized at +45°. The polarizer in Figure 13
stops the back reflection, making an optical isolator. The QWP must be oriented
EE 737 Photonics Laboratory Manual Useful Optical Components
275
properly, however, that is, with its optical axis at 45° to the incoming
polarization.
A half-wave plate can be used to rotate the polarization of a linearly
polarized beam to any arbitrary angle. If the HWP’s optical axis is set at some
angle ! to the incoming light’s polarization (assumed linear) then the output will
also be linearly polarized, but at an angle of 2! to the original polarization.
These wave plates discussed so far are based on optical retardation- one
(linear) polarization travels at a different velocity than the other. All
polarizations (except unpolarized light) can be broken up into linear polarized
components, and the effect of the retardation plate can be worked out.
Just as you can resolve a vector into its Cartesian components, or
equivalently into polar coordinates, so can you resolve any polarization into
either two linear polarizations, or a sum of LCP and RCP components. There are
materials (such as quartz) that are optically active- that is, they slow one of the
circularly polarized components more than the other. This also results in a
change in the output polarization state. For a more detailed discussion, take
EE833.
REFERENCES
1. R. Guenther, Modern Optics, John Wiley and Sons, New York
(1990).
EE 737 Photonics Laboratory Manual Things Statistical
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THINGS STATISTICAL
(or How To Treat Measurement Errors In The Lab)
We are considering taking measurements– analog measurements– in the
laboratory. Suppose, for example, you measure the length of your pencil using a
meter stick marked in mm. You might get the results 15 mm, 15 mm, and 15
mm. Now suppose you try to measure to within 0.1mm. In this case you might
get 15.1, 15.2, and 14.9. This is because you pushing the limits of the resolution of
the meter stick. There is some scatter to your data.
Suppose you try to locate the position of an optical image position
precisely on an optical bench. You have to judge by eye the point at which the
image is perfectly focused. Because there is a limit to how well you can do this,
you may have considerable scatter in your data. Some approach is needed to
handle these variations. In electronic measurements there will always be Johnson
noise caused by random motion of electrons in resistors. If we go to the ultimate
in precision we have to start counting electrons. If we turn up the volume on a
radio we can hear a hiss. That is noise and it will provide a limit on how small a
signal can be and still be detected. In any case, if you try to make precise
measurements your data will be random and unreproducible. This can be a
problem. We would be tempted to start pulling our hair out in response.
Average values
The solution is to consider averages! They are reproducible. There are
several averages of interest. We need to develop background on what they are
and how they will be used.
Let's consider for a moment some random data, N of 'em. Call the
quantity being measured x (for lack of imagination) and the measured values xi,
EE 737 Photonics Laboratory Manual Things Statistical
277
where i= 1, 2, 3, .... N. In general, many values will be duplicated. If one plots the
values of x measured against the number of times that particular value is
measured, one might get a curve such as that shown in Figure 1. This curve is
the Gaussian distribution, sometimes called the bell curve or normal distribution,
and there is a theorem (called the Central Limit Theorem) that says that for truly
random data, if you take enough points they will describe the Gaussian curve.
0
x
Figure 1. Number of times a particular measurement occurs.
If you normalize this curve such that the area under it is 1, the curve is
then a probability density curve, P(x). It is centered around the mean, x , and
reflects the relative probability that on your next measurement you will obtain
some particular value x. The average value, or expectation value, x is given by
x
Nxi
i
N
=-
Â1
1
[1]
Now, group the like values. Let there be M different values, and let j be
the index that counts through the different ones; that is, j=1....M. Let the jth
value be repeated Nj times. Then we can rewrite our summation as
x
NN xj j
i
M
=-
Â1
1
[2]
The curve in Figure 1 would be a plot of Nj versus xj (for a large number
of data, not the example data). Next, define the probability density function P(xj):
P x
N
Nj
j( ) =[3]
EE 737 Photonics Laboratory Manual Things Statistical
278
so that we have
x P xj j
j
M
==
Â1 [4]
For example, consider the following set of data:
xj Nj Pj
1.0 1 0.01
1.1 1 0.01
1.2 3 0.03
1.3 7 0.07
1.4 12 0.12
1.5 17 0.17
1.6 18 0.18
1.7 17 0.17
1.8 12 0.12
1.9 7 0.07
2.0 3 0.03
2.1 1 0.01
2.2 1 0.01
For this data, N=100 (there are 100 data points), and M=13 (there are 13
different values that occur). We see from the table that the number 1.0 occurred
only once, the number 1.1 occurred once, the number 1.2 occurred three times,
etc. The number 1.6 occurred the most often, namely 18 times.
Let's compute the average:
x x Nj j
j
= ==
Â1
100 1
13
EE 737 Photonics Laboratory Manual Things Statistical
279
1
100
1 0 1 1 1 2 3 1 2 7 1 4 12 1 5 17
1 6 18 1 7 17 1 8 12 1 9 7 2 0 3 2 1 2 2
. . . . . .
. . . . . . .
+ + ¥ + ¥ + ¥ + ¥ +¥ + ¥ + ¥ + ¥ + ¥ + +
ÊËÁ
ˆ¯̃
=1.6
These data fall nicely on a bell-shaped curve, as shown in Figure 2. For
some cases, e.g., small values, the probability density curve may not be
symmetric. The Poisson distribution, for example, is skewed to the left. It was
initially checked out by Poisson to predict the number of soldiers in the Prussian
army kicked to death by mules.
1 8
1 6
1 4
1 2
1 0
8
6
4
2
Nu
mb
er
or
occu
ren
ce
s
2 .22 .01 .81 .61 .41 .21 .0
Value
Figure 2. The data from the table fall on a bell-shaped curve.
Even if the measurements are not consistent and reproducible the
averages are. We can now stop pulling our hair out.
Variance and standard deviation
Given that our measurements are random, we might like to know how
close they are. This leads to the ideas of variance and standard deviation. Let us
develop each of these. First, for each measurement xj, let's find out how far that
measurement is from the average: x xj - . Then, we'll square that difference, so
that positive and negative differences have the same effect (we're just interested
EE 737 Photonics Laboratory Manual Things Statistical
280
in how far from the average the measurements generally are): x xj -( )2
. Then
we'll sum the squares- this results in the variance,
variance = x x Pj
j
M
j-( )=Â
2
1
[5]
where Pj is the probability of the jth value occurring.
The standard deviation, call it s, is the square root of the variance:
s = -Â( )x x Pj j2 [6]
which makes it the root mean square of the difference from average of the data.
The standard deviation is nice because it has the same units as the quantity being
measured, x.
When the number of measurements is very large, the discrete
measurements xj go to a continuous variable x. The probability density function
becomes a continuous distribution P(x), which is still normalized:
P x dx( )
-•
•
Ú = 1 [7]
The equation for P(x) for a Gaussian distribution then becomes
P x
x x( ) exp
( )=
- -ÊËÁ
ˆ¯̃
1
2 2
2
2s p s[8]
There are the comparable definitions.
x xP x dx=
-•
•
Ú ( ) [9]
and
s 2 2 2= -( ) = -
-•
•
Úx x x x P x dx( ) ( ) [10]
These can be easily established using two integrals from an integral table:
e dxx-
-•
•
Ú =2
p [11]
x e dxx2 2
2-
-•
•
Ú =p
[12]
and of course the appropriate substitutions.
EE 737 Photonics Laboratory Manual Things Statistical
281
Looking at Eq. 8 we see that P(x) is indeed the maximum when x= x . We
also see that P(x) drops off to e-1 of its maximum value when ( )x x- = 2s . We
call the half width 2s . Thus the more scattered the measurements, the wider
the curve.
We can use the area under the curve to tell what fraction of the
measurements will satisfy certain criteria. For example
P x dx( )
-Ús
s
tells the fraction of measurements falling between minus one standard deviation
and plus one standard deviation. This turns out to be 68%. Between the two
sigma points it is 95%, and for the three sigma points 99.7% of the data will fall
between those values.
This is used by manufacturers. If for example their widget is advertised to
have 3 grams of quantity "X" then the manufacturer may decide that he wants at
least 97.7 of his widgets to contain at least 3 grams, and he designs his
manufacturing process to put three sigma more of "X" into the widgets. That
means he is providing many customers with more than their share of "X". If s is
large, he is giving away more "X" than if s is small, so he wants to improve his
manufacturing technique to make s as small as possible. If he is selling a liquid
and he wants to make sure that everyone gets he advertised amount then the
average he puts in his containers has to be greater than the advertised value by
two or three sigma in order to make sure people will get their due. The extra
added above the advertised value represents waste to the manufacturer. If he
can get instruments that are more precise and can reduce s, he saves money. The
point is that the standard deviation is an integral part of his vocabulary.
Subsidiary measurements and calculated values
EE 737 Photonics Laboratory Manual Things Statistical
282
When you are making measurements in the laboratory, you often
measure several quantities, and then compute some other quantity from your
measurements. If you original data has some randomness to it, how does that
affect your final results? We will develop some equations to find the standard
deviation of calculated values.
For example, suppose you are to calculate some quantity z based on your
measurements of x and y . Then z is given by the formula z=f(x,y). We assume
that x , y, and z are random uncorrelated variables. That means that if x x- is
negative, then y y- is still just as likely to be positive as negative. Therefore we
can write:
x x y y-( ) -( ) ªÂ 0 [13]
We make a large number of set of x , y, and z . We find x , y , and z and
substitute x and y into the formula and calculate z, call it zcalc. We then ask it
zcalc is close enough to z , whatever "close enough" means.
To get a measure of "close enough:, we derive a general formula. Imagine
we take the total derivative of f(x,y).
dz
f
xdx
f
ydy= +
!
!
!
![14]
We evaluate the partial derivative using average values of x and y. We
also replace dx by x x- and similarly for dy and dz, giving
z zf
xx x
f
yy yj
x y
j
x y
j- = - + -!
!
!
!, ,
( ) ( ) [15]
We square Eq. 15 and sum over all the measurements giving
z zf
dxx x
f
dyy y
f
dx
f
dyx x y y
jx y
j
x y
j
x y x y
j j
-( ) =Ê
ËÁˆ
¯̃-( ) +
Ê
ËÁÁ
ˆ
¯˜̃ -( ) +
Ê
ËÁˆ
¯̃
Ê
ËÁÁ
ˆ
¯˜̃ -( ) -( )
ÂÂ Â
Â
22
2
2
2
2
! !
! !
, ,
, ,
[16]
EE 737 Photonics Laboratory Manual Things Statistical
283
We notice that because the x and y measurements are uncorrelated, the
last term sums to zero. We then replace the sums of the squares in terms of the
standard deviations.
s!
s!
sfx y
x
x y
y
f
dx
f
dy2
2
2
2
2=Ê
ËÁˆ
¯̃+Ê
ËÁÁ
ˆ
¯˜̃
, ,
[17]
We substitute the measured values of sx, sy and calculate sf. If the
calculated value of z falls within z f±s then we can feel that the expression
z=f(x,y) is verified. If it is outside that range, then the formula is highly
improbable.
As an example, suppose we want to verify the formula for the focal
lengths of a lens:
1 1 1
s d f+ = [18]
We take a lens of known focal length, set the object at a particular
distance, and measure the object and image distances a large number of times.
We find the average values s and d and use them in Eq. [18] to calculate a
predicted focal length. We then calculate the standard deviations, ss and sd. We
need to find sf to compare. We find
!
!
f
s and
!
!
f
d.
f
sd
s d=
+[19]
!
!
f
s
d
s d
sd
s d=
+-
+( )2[20]
!
!
f
d
s
s d
sd
s d=
+-
+( )2[21]
We evaluate these with the average values of s and d and plug them into
the expression for sf along with the values of ss and sd.
s
!
!s
!
!sf s d
f
s
f
d2
22
22= Ê
ËÁˆ¯̃
+ ÊËÁ
ˆ¯̃
[22]
There are two general rules. If we have a sum of terms z=Ax+By, then we
add the squares of the uncertainties.
EE 737 Photonics Laboratory Manual Things Statistical
284
s s sz x yA B2 2 2 2 2= + [23]
If we have a product, such as z=Cxy, then we add the squares of the
percentage uncertainties in x and y to get the square of the percentage
uncertainty in z.
s s sz x y
z x y
ÊË
ˆ¯
= ÊË
ˆ¯
+Ê
ËÁˆ
¯̃
2 2 2
[24]
We can use the preceding procedure in a more qualitative way to get a
quick estimate of uncertainties by replacing the standard deviations by an
estimate of the maximum possible error. For example in many cases a
measurement with a meter stick cannot be off by more than a millimeter, or
perhaps a half a millimeter if one is looking closely. One would replace s
calculated as the root mean square difference by one millimeter. That will
provide a quick approximate estimate of the uncertainty before one is ready to
do the final statistical calculation.
Rejection of data
There may be cases where we have a series of random measurements,
and one datum differs considerably from the rest. We might be tempted to reject
it. If we have taken all the data in a consistent manner, we must have good
reason to throw it out. Intuition will not suffice. We can find some rationale by
using the standard variation and looking at the probability density function. Let
the measurement under question be xi, the average be x and the variance be s2.
Then the expression
exp- -( )Ê
ËÁˆ
¯̃
È
ÎÍÍ
x xi
2
2
s[25]
EE 737 Photonics Laboratory Manual Things Statistical
285
gives the probability of the particular number occurring. If the probability of an
outrageous datum occurring is sufficiently small then one might have reason to
neglect it.
Summary
Precise data will be random and unreproducible. We must use average
values; they are reproducible. The average is the most basic quantity. The
probability density is important, and the variance, and standard deviation are all
well used quantities.
We can use the uncertainties in measured values to find the uncertainties
in calculated from measured values.
HOMEWORK:
1. Derive Equation [23].
2. Derive Equation [24].
EE 737 Photonics Laboratory Index
287
A
absorption 98, 153, 199, 239absorption coefficient 201absorption, effect of excitons 146absorption, indirect 200acoustic waves 81active layer, 107air mass zero 196American National Standards Institute 13amorphous silicon 194amplitude sensors 41anterior chamber 5antireflection coatings 203Apostilb 257atmospheric absorption 196atmospheric effects 217attenuation (fiber) 62Auger recombination 202
B
back surface field (BSF) region 203band gap 101band structure 198beam steering 90beamsplitter 269beamsplitter as beam combiner 270beamsplitter, cube 269beamsplitter, pellicle 270beamsplitters, polarizing 271BER (see bit-error-rate) 63Bessel function 32birefringence 271bit error rate 63bit error rate, measurement of 63Bragg angle 88Bragg condition 88Bragg regime of acousto-optic interactions 88Brewster angle 236Brewster windows 236bulk material 145bus lines 215Butoxybenzylidene octylanilene 174
C
cadmium telluride 194calcite 272candela 253carrier confinement 106chemical hazards (to people) 16
EE 737 Photonics Laboratory Index
288
chemical hazards to equipment 21cholosteric 175chopper 161, 163circularly polarized light 274cladding 27, 55cladding, 107CLEANING PROCEDURE FOR ITO COATED GLASS 191clock recovery 63collimate a beam 233combiner, beam 270Commercial liquid crystal cells 186compound lenses 264connectors (fiber) 65contact designs 215core (fiber) 27, 55cornea 4corner cube 262critical angle 27, 54current, short circuit 208current-voltage (I-V) characteristics 203cylindrical lens 266
D
dark current. 155dB 64dBm 64DBR (distributed Bragg reflector 120defect state 202degrees of freedom 173density of states function 145depletion width 206dielectric mirrors 238diffraction 117diffraction efficiency 89diffraction order. 87Diffuse reflection 234diffuse source, MPE's for 11diffusion current density 206diffusion equation 204diffusion length 202direct bandgap 198, 200direct recombination 202directional coupler 42dispersion 60distance-bandwidth product 53, 60distributed Bragg reflector 120distributed feedback laser 120divergence (of a laser diode beam) 117Doppler effect 90
EE 737 Photonics Laboratory Index
289
double-heterostructure 106dynamic range 46
E
effective mass 147efficiency loss 212Einstein coefficients 103elastomer splice 68electric dipole moment 84electric field vector 228electric flux density 228electric polarization field 84electric susceptibility 85electrical hazards (to people) 15electro-optic effect 149electro-static discharge 16electroabsorption 147, 150electron affinity 140electrorefraction 148elliptical polarization 247energy band diagram, rules for drawing 140energy bands, semiconductor 101equipment hazards 16errors, measurement 276ESD Precautions 19ESD see electro-static discharge 16excitons 145extended source viewing 5extended source viewing, MPE's for 8external reflection, defined 236eye 4Eye Damage 4eye diagram 62eye, human 252
F
Fabry-Perot cavity 110far field, 117far-field output pattern of a phased array 120Fermi level 140fill factor 209fill factor loss 212finesse, 113flats, optical 268footcandles 257Footlambert 257Fourier optics 5Fourier transform 91Franz-Keldysh effect 150
EE 737 Photonics Laboratory Index
290
free spectral range 113Fresnel reflection 269, 270Fresnel reflections 235, 261fusion splicer 67
G
gain curve 114gain guiding 110gallium arsenide 194, 200Glan-Thompson prism 272Goos-Hänchen shift 37graded index fiber 35graded index lens 266graded-index fiber 59grating 127grid lines 215GRIN (graded index) lens 266GRIN rod 266GRINSCH 108guiding of light 28
H
H2O absorption 217half-wave plate 275heavy holes 147HeNe laser 13Hermite polynomial 118Hermite-Gaussian modes 118heteroface 216heterojunction 139high voltage 15homojunction 216homojunction laser 106
I
I-V characteristics 203illuminance 255index guiding 109index of refraction 85, 229index of refraction, complex 148indium phosphide 194indium-tin-oxide 177infrared booster 210intensity modulator, liquid crystal 182interferometric sensors 42internal reflection, defined 236intersubband detection 157intersymbol interference 62intrabeam viewing 5
EE 737 Photonics Laboratory Index
291
intrabeam viewing, MPE's for 7irradiance 255ITO (indium tin oxide) 177
Jjacket (fiber) 27, 55jitter 64
KKerr effect 149Kramers-Kronig relations 148
LLabView 129, 165Lambert 257Lambertian source 254lasing threshold 114lattice vibrations 200LED (light emitting diode) 114lens (of the eye) 5lens law 232lens, compound 264lens, concave 264lens, convex 264lens, cylindrical 266lens, graded index 266lens, microescope objective 264lenses 264lifetime, minority carrier 202light emitting diode 114light holes 147linear polarization 241linearity 45link budget 64liquid crystal cell 177lnear field, 117longitudinal mode 115longitudinal modes of a laser 119loss, open circuit voltage 212loss, short circuit current 212loss,fill factor 212lumens 252luminance 254luminous energy 253luminous energy density 253luminous excitance 253luminous flux 253luminous intensity 253Lux 257
EE 737 Photonics Laboratory Index
292
M
Mach-Zehnder interferometer 43macrobending loss 28macrobending sensor 41magnetic field vector 228magnetic flux density 228magnetization density 228magnification M of a lens 233Malus' Law 244Malus’ Law 190materials, solar cell 194Maximum Permissible Exposure 5Maxwell's equations 227microbending 40mirror stacks 119mirror, curved 259mirror, front surface 261mirror, spherical 259mirrors 259mirrors, elliptical 261mirrors, gold 261mirrors, parabolic 261misalignment, angular (fiber) 67misalignment, lateral (fiber) 66misalignment, longitudinal (fiber) 66mode 28, 56mode coupling 39monochromator 127MPE see maximum permissible exposure 5multimode (fiber) 57multimode (laser) 115multiple quantum wells 153
N
NA see numerical aperture 56ndium-tin-oxide 190nematic 175noise 62NRZ(non-return-to-zero) 62numerical aperture 30, 56
O
Octyloxy-cyanobiphenyl 174open circuit voltage 208open circuit voltage loss 212optical activity 275optical cavity 110optical confinement. 107
EE 737 Photonics Laboratory Index
293
optical flats 268optical gain- 100optical path length 230optical retardation 275optical scanner 91optical switching 90
P
parasitic resistances 213Part Susceptibility Data 18particle theory of light 230pellicle beamsplitter 270permeability 228permittivity 228phase grating 86phase locking 120phase sensitive detector 163phase sensors 42phase shifter, liquid crystal 180phase-space absorption quenching 150phased laser array 120phonon 199Phot 257photodetector 195photodiode 154photometric units 256photometry 252photons 230photopic response 252photovoltage 201plane wave 229Pockels effect 149Poisson distribution 280polarization density 228polarization, circular 274polarization, elliptical 247polarization, linear 241polarizers, plastic 271polarizing beamsplitters 271polyvinyl alcohol 271population inversion 104posterior chamber 5potential well 142potential well, finite 138potential well, infinite 137power-current curve (of a laser) 114Poynting Vector 228pressure waves 81principal planes 264
EE 737 Photonics Laboratory Index
294
principal surface 264prism, Rochon 272prism, roof 263prism, Wollaston 272propagation constant b 30pumping 105PVA (polyvinyl alcohol) 271
Qquadrature. 45quantization of modes 36quantum mechanics 136quantum noise 62quantum structures 145quantum well detectors 154quarter-wave plate 273quasi-continuum 138
Rradiance 251radiant density 250radiant energy 249radiant excitance 250radiant fluence 250radiant flux 249radiant intensity 250radiant power 249radiation damage 194radiative recombination 202radiometric units 256radiometry 249Raman-Nath regime of acousto-optic interactions 87Rayleigh scattering 217real image 232receiver 60recombination processes 202reflection coefficient 113reflection.surface 203reflective device liquid crystal display 193reflector, elliptical 262refractive index 229repeatability 47resistance, lateral 214resistance, sheet 215resolution 47responsivity 156retina 5retinal damage 2retroreflector 262, 264
EE 737 Photonics Laboratory Index
295
Rochon prism 272roof prism 263
S
safety 2safety rules 14saturation current density 206Schrodinger's equation 136scotopic response 252sensitivity 46separate confinement 108seven segment display 188shear waves 81Shockley-Read-Hall (SRH) recombination 202short circuit current 208short circuit current loss 212shot noise 62signal to noise ratio 62silicon 194, 200single mode 57slab waveguide 36slit widths, monochromator 132, 162, 169slits widths, tradeoffs 128slits, monochromator 128smectic 175Snell's law 26, 54, 107, 232Snell’s law 266SNR see signal to noise ratio 62solar cell 194solar constant 196solar radiation 196solid angle 250specular reflection 234SPLICE LOSSES 65splices (fiber) 65spontaneous emission 99, 239standard deviation 280Stark effect 151step index fiber 35, 59steradians 250stilb 256stimulated emission 100, 240strain 82stress, 82sunglasses 236supermode 120surface reflection 203surface-emitting laser 119
EE 737 Photonics Laboratory Index
296
T
Talbots 253texturing (of solar cell surfaces) 216the Fresnel reflection 113thermal noise 62thermalization 198threshold level, choice of 61threshold, lasing 114timing jitter 64total internal reflection 26, 54, 107transient protection for laser diodes 21transimpedance amplifier 61transmissive liquid crystal display 193Triboelectric Series 19trunk lines 53tungsten filament projector lamp 218
V
V parameter 34vacuum level 140variance 280velocity, surface recombination 207Vernier micrometer 78vertical cavity laser 119virtual image 232vitreous humor 5voltage, open circuit 208
W
wave equation 228wave plates 273waveguide, laser 107wire 53Wollaston prism 272working distance 265wrist strap 20