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Photonics and Optical Communication, Spring 2007, Dr. D. Knipp
1Waveguides
Photonics and Optical Communication
(Course Number 300352)
Spring 2007
Waveguides
Dr. Dietmar Knipp
Assistant Professor of Electrical Engineering
http://www.faculty.iu-bremen.de/dknipp/
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2Waveguides
Photonics and Optical Communication
3 Waveguides3.1 Introduction
3.2 Reflection and Refraction at the Boundary between two Media3.3 Total internal reflection
3.3.1 Light propagation in an optical fiber
3.3.2 Acceptance angle
3.4 Planar Waveguide
3.4.1 Planar Mirror Waveguide3.4.2 Planar Dielectric Waveguide
3.5 Modes in Waveguides
3.5.1 Transverse Electric Waves
3.5.2 Transverse Magnetic Waves
3.5.3 Transverse Electro Magnetic Waves3.5.4 Calculating Modes in a planar wave guide
3.5.5 The effective refractive index
3.5.6 The Mode chart
3.5.7 Designing a planar wave guide
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3Waveguides
3.5.8 TE versus TM Modes
3.5.9 Types of Modes
3.5.10 Numbering of modes
3.6 Coupling between Waveguide
References
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4Waveguides
3.1 Introduction
Light can be confined by an optical waveguide. The waveguide is formed by a
medium which is embedded by an another medium of lower refractive index.
The medium of higher refractive index acts as a light trap. Light is confined in
the waveguide by multiple total internal reflections. By doing so light can betransported from one location to another location. Waveguides can be
distinguished in terms of slabs, strips and fibers. The most widely applied
waveguide structure is the optical fiber, which is made out of two concentrically
cylinders of low-loss glass with slightly different refractive index.
Waveguides can be distinguished in terms of a slab, a strip or a fiber.
Ref: Saleh & Teich, Fundamentals of Photonics
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5Waveguides
3.1 Introduction
If a lot of optical component like wave guides, light sources and light receivers
are integrated together on a substrate (chip) we speak about integrated
optics. The goal is to miniaturize optics like electronics to improve performance
and reduce cost.
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6Waveguides
3.1 Introduction
The optical fiber in its existing form (the fiber consists of a core and a cladding)
was invented 40 years ago. The first fibers were used in the near infrared
wavelength region at around 800nm-900nm. As technology of fibers and light
sources evolved the optical transmission window was shifted to 1310nm in themid 1980s and 1550nm in the 1990s.
Internal reflection is a requirement for the guidance or confinement of waves in
a waveguide. Total internal reflection can only be achieved if the refractive
index of the core is larger than the refractive index of the cladding. In the
following, we will briefly repeat the related ray optics.
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7Waveguides
3.2 Reflection and Refraction at the Boundary between two Media
The reflection and refraction of light at an interface can be described by Snells
law. The angle of incidence is given by 1 which is related to the angle ofrefraction 2.
Reflection of rays at an interface. (a) From a high to a low refractive medium, (b)
The critical angle, (c) Total internal reflection.
Ref: J.M. Senior, Optical Fiber Communication
2211 sinsin = nn
Snells law.
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8Waveguides
3.3 Total internal reflection
With increasing angle of incidence 1 the angle of refraction 2 also increases.
If n1 > n2, there comes a point when 2 =/2 radians. This happens when1=sin
-1(n2 / n1). For larger values of 1, there is no refracted ray, and all the
energy from the incident ray is reflected. This phenomena is called totalinternal reflection. The smallest angle for which we get total internal reflection
is called the critical angle and 2equals /2 radians.
The total internal reflection is an requirement for the guidance of light in an
optical fiber.
1
2sinn
nc = Critical angle
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3.3.1 Light propagation in an optical fiber
An optical fiber can be described by an cylindrical core surrounded by a
cladding. Usually (at least for optical communication) the fiber core and the
cladding are made of silica (SiO2). The refractive index of the core is slightly
higher than the refractive index of the cladding so that the light is guided in thefiber.
Transmission of a light ray in a perfect optical fiber.
Ref: J.M. Senior, Optical Fiber Communication
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10Waveguides
3.3.2 Acceptance angleTotal internal reflection is required to guide light in an optical fiber. We know
that only light under sufficient shallow angles (angle greater than the critical
angle) can propagate in the fiber. The question is now under what angle a ray
can enter a fiber? It is clear that not all rays entering the fiber core will continue
to be propagated along the fiber. Only rays that enter the fiber within aacceptance cone (acceptance angle) will propagate along the fiber, whereas
rays outside of the cone will not be guided.
Coupling of a ray into a fiber. The ray can only be coupled into the fiber
when the angle of incident is within the acceptance cone.
Ref: J.M. Senior, Optical Fiber Communication
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11Waveguides
3.3.2 Acceptance angle
In the following we will derive an expression for the the acceptance angle
from the refractive indices of the three media involved, namely the core of the
fiber (n1), the cladding of the fiber (n2) and the air (n0).
In order to enter the fiber Snells law has to be fulfilled.
The angle 2can now be
described by
So that the Snells law can
be modified to
2110 sinsin = nn
=2
2
cossin 110 = nn
Coupling of a ray into a fiber. The ray can onlybe coupled in the fiber when the angle of
incident is within the acceptance cone.
Ref: J.M. Senior, Optical Fiber
Communication
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12Waveguides
3.3.2 Acceptance angleIf we consider now the trigonometrically relationship
The expression can be modified to
Now the equation can be combined with the equation for the critical angle
Leading to the relationship for the numerical aperture
The acceptance angle can now be calculated by
( ) ( ) 1cossin 22 =+
2110 sin1sin = nn
( )121sin nn=
22
2110 sin nnnNA == Numerical aperture
Acceptance angle
=<
0
22
211
1 sinn
nna
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14Waveguides
3.4.1 Planar Mirror waveguideA planar mirror waveguide is shown on this
slide. The cladding of the waveguide is formed
by a conducting material, which can be a
mirror.
As a consequence of the conducting cladding
the tangential components of the electric and
the magnetic field is zero at the interface
between the core and the cladding.
Therefore, the waves can not extend in the
cladding of the waveguide. Subsequently the
modes of propagation are defined by thedimensions of the core of the waveguide.
Cross section of a mirror
waveguide
Ref: Back to Basics in Optical
Communications, Tutorial
Agilent Technologies
0=TE 0=TB
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15Waveguides
3.4.2 Planar Dielectric waveguideA planar dielectric waveguide is shown
on this slide. The cladding of the
waveguide is formed by a dielectric
medium of lower refractive index.
The waves extend in the cladding of
the waveguides. The wave propagating
the waveguide can be described by
The complex amplitude of the wave
corresponds to the transverse standing
wave perpendicular to the direction of
propagation. Due to the fact that wavesextend in the cladding the wavelengths
that can propagate are larger than 2
times the diameter of the core. is apropagation constant.
Cross section of a dielectric
waveguide.
Ref: Back to Basics in Optical
Communications, Tutorial Agilent
Technologies
( )( )ztjrEtrE = exp,
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16Waveguides
3.5 Modes in Waveguides
The planar waveguide is the simplest form of an optical waveguide. The
waveguide can be realized by a simple sandwich structure which consists of a
slab embedded between two regions of lower refractive index.
The optical ray within the waveguide can be described by a transverseelectromagnetic wave, which can be a TE, TM or TEM wave.
Propagation of a wave in a planar
waveguide. We can impose self-consistency condition which
requires that the wave reproduces
itself. Fields that satisfy this
conditions are called Eigenmodes
(modes) of the waveguide.
Ref: Saleh and Teich,
Fundamentals of Photonics
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17Waveguides
3.5 Modes in a Waveguides
As a consequence of the superposition of planar waves we get an interference
pattern, which is formed in the waveguide (the z-direction is the propagation
direction of the wave). If the total phase change upon two successive reflections
is equal to 2m constructive interference is observed, where m is a positive
integer. The phase shift has two contributions. The reflection of the plane wave atthe interface leads to a phase shift, which depends on the angle and thedistance traveled. Later on we will derive an expression for the modes of
propagation in such a structure.Formation of modes in a planar
dielectric waveguide. (a) planewave propagation in a waveguide
and corresponding electric field
distribution in the optical fiber.
The interference of the plane
waves in the waveguide isforming the lowest order mode
(m=0).
Ref: J.M. Senior, Optical Fiber
Communication
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18Waveguides
3.5 Modes in a Waveguides
In the figure on the previous slide the lowest order mode (m=0) is shown. A mode
of propagation is only observed when the angle between the propagation vector
and the interface (boundary of the cladding and the core) has particular values.
For all modes of propagation a standing wave is formed in the waveguide.
Depending on the mode of propagation an electric field distribution is formed. Forthe lowest order mode the electric field is maximized in the center of the core.
The electric field decays towards the boundaries. For all modes of propagation
the self-consistency condition has to be satisfied which means that the wave in
the waveguide reproducing itself.
Before discussing the modes of propagation mathematically we will discuss the
propagation of waves in a waveguide phenomenologically .
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19Waveguides
3.5 Modes in WaveguidesAgain we assume a plane wave which
propagates in the z-direction. We
observe constructive interference
across the waveguide as a
consequence of the superposition of the
propagating waves. In the examples
shown on this slide the propagation
modes are m=1, 2 and 3. The number
of modes corresponds to the number of
zeros in the transverse electric field
pattern.
How do we determine the self-
consistency conditions. In order to
achieve total internal reflection theangle of incidence has to be smaller
than
Propagation of waves in a waveguide
and the corresponding transverseelectric (TE) field pattern of three
lower order models m=1, 2, 3.
Ref: J.M. Senior, Optical Fiber
Communication
=
<
1
21
1
21 cossin
2 n
n
n
nC
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20Waveguides
For self-consistency the wave
reproduces itself and the phase
shift between the two waves has
to be zero or a multiple of 2.We can assume that the field in
the slab is in the form of a
monochromatic plane wave
bouncing back and forth at an
angle smaller than the criticalangle C. A round trip can bedescribed by:
For self-consistency the phaseshift between the two waves has
to be zero or a multiple of 2.
Planar dielectric waveguide.
Ref: Saleh and Teich, Fundamentals of
Photonics
3.5 Modes in Waveguides
sin2dABAC =
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21Waveguides
3.5 Modes inWaveguides
There is a phase r introduced by each internal reflection at the boundary. Thereflection phase is a function of the angle and it depends on the polarization ofthe incidence wave (TE, TM or TEM wave), which is described by the complex
reflection and transmission coefficients (see Review of optics). The complex
reflection and transmission coefficients can be separated in an amplitude and a
phase, where the phase depends on the angle of incidence.
,2,1,02sin22
== mformd r
Mode Equation
Reflection coefficient and phase shift on
reflection for a transverse electric wave
as a function of the angle of incidencefor a glass/air interface.
Ref: J.M. Senior, Optical Fiber
Communication
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22Waveguides
3.5.1 Transverse Electric WavesIn the case of a transverse electric field (TE mode) the electric field is
perpendicular to the direction of propagation of the wave (z-direction). As we
are dealing with electro-magnetic waves each wave consists of a periodically
varying electric and magnetic field which is again perpendicular to each other.
In the case of a TE transverse wave the electric is perpendicular to thedirection of propagation (Ez=0) and the magnetic field has a (small) component
to the direction of propagation. This is due to the fact that the traveling wave is
not propagating in a straight line in the wave guide, meaning the ray is
propagation on a zigzag path.
Propagation of a TE wave in a slab
waveguide.
Ref: J.C. Palais, Fiber Optic
Communication
z
x
B
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23Waveguides
3.5.2 Transverse Magnetic WavesIn the case of transverse TM modes the magnetic field is perpendicular to the
direction of propagation and the electric field has a (small) component to the z-
direction of propagation. Again the traveling wave is propagating on a zigzag path
rather than a straight line in the wave guide.
Propagation of a TM wave in a slab
waveguide.
Ref: J.C. Palais, Fiber Optic
Communication
z
x
E
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24Waveguides
3.5.3 Transverse Electro Magnetic Waves
In the case of a TEM transverse wave (TEM modes) both the electric and the
magnetic field are perpendicular to the direction of propagation, which means
that the rays propagate straight in the fiber. Such cases occurs only for single
mode fibers.
3.5.4 Calculating Modes in a planar wave guideThe mode equation and the equation for the phase shift have to be merged,
which leads to a transcendental equation. The transcendental equation has to
be solved to get the modes which propagate in a given waveguide structure. A
detailed mathematical description is given by Saleh and Teich in their book
Fundamentals of Photonics.
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3.5.5 The effective refractive indexThe effective refractive index is defined by:
The effective refractive index is a key parameter for waveguides like the
refractive index is a key parameter for the free space propagation of waves.
The effective refractive index changes the wavelength in the same way that abulk refractive index does. The idea of the effective refractive index gets clear
by simply looking at the structure of a waveguide. The effective refractive
index is a corrected refractive index which simply assumes that the wave
propagates in a straight line the media (in our case in the core of the
waveguide structure.)
sin1nneff =
Plane wave propagating in a
waveguide. The effective refractive
index considers that the plane wavepropagates by following a zigzag path.
Ref: J.C. Palais, Fiber Optic
Communicationz
x
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3.5.6 The Mode chartThe mode chart for a waveguide structure is an absolutely essential graph to
study the propagation of modes in a given waveguide structure. The
thickness/diameter (d) of the core of the waveguide is usually normalized by
the wavelength of the incident light. The different modes of propagation are
plotted for the propagation angle and the effective refractive index.
Mode chart for a
symmetric slab. The
following refractive indices
were assumed for the core
n1=3.6 and the cladding
n2
=3.55 (AlGaAs
structure).
Ref: J.C. Palais, Fiber
Optic Communication
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3.5.6 The Mode chartFor the particular mode chart shown on the previous slide an AlGaAs waveguide
structure (used for a laser diode) is assumed The critical angle for the structure
is
Therefore, the range of angles for which the ray is trapped in the waveguide is
then 80.4 90. As a consequence the effective refractive index ranges from
3.55 to 3.6.
From the mode chart we can draw the following conclusions:
When the core thickness is very small in comparison to the wavelength of the
propagating light the wave travels very close to the critical angle and the
effective index is close to the refractive index of the refractive index of the
cladding. The wave penetrates deeply into the outer layers, because the rays are
near the critical angle.With increasing thickness of the core the ray travels at larger angles. The ray
travels more parallel to the waveguide axis. For thick films (thickness is large in
comparison with the wavelength of the propagating light) the effective index is
very close to the refractive index of the film itself.
( ) == 4.80sin 121 nnc
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3.5.7 Designing a planar wave guide
For any given propagation angle there is a set of film thicknesses that will allow
rays to propagate. The following equation has to be satisfied for the higher
modes, where m is a positive integer.
In order to change the mode the normalized thickness has to change by:
cos2 10 n
mdd
m+=
( )
cos2
1
1n
d =
Table of TEm modes in a gallium arsenide waveguide.
Ref: J.C. Palais, Fiber Optic Communication
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3.5.7 Designing a planar wave guide
The following equation can be applied to calculate the number of TE modes
supported by the dielectric waveguide, where m is increased to the nearest
integer.
3.5.8 TE versus TM Modes
So far we discussed only the propagation of TE modes. However, TM modes
exhibit almost identical propagation behavior. This is why we will not distinguish
between TM and TE modes. Therefore, the curve in the mode chart were labeled
as both TE and TM modes. This is mostly true since the difference in the
refractive index for the core and the cladding are very small (in the range of a fewpercent). Even with increasing difference the cutoff modes are identical. For each
TE mode there will be always a TM mode. The number of total modes is
therefore twice the number of TE modes. The electric field distribution for the
different mode is shown in the following.
( )21
22
21
0
2
nnNA
NAd
m
=
=
Number of TE Modes
Numerical Aperture
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30Waveguides
3.5.9 Types of ModesIn the case of a transverse TE (TE modes) wave the electric field is
perpendicular to the direction of propagation and the magnetic field has a
(small) component that is in the direction of the propagation.
In the case of a transverse TM (TM modes) wave the magnetic field isperpendicular to the direction of propagation and the electric field has a (small)
component that is in the direction of the propagation.
In the case of a TEM transverse wave (TEM modes) both the electric and the
magnetic field are perpendicular to the direction of propagation, which means
that the rays propagate straight in the fiber. Such cases occurs only for single
mode fibers.
Furthermore, helical modes (HE or EH) modes exist. Under such conditions
the ray travels in a circular path in the fiber and electric and magnetic field have
components in the z-direction. These modes can be realized either as a HE or aEH mode depending on which field contributes most to the z-direction.
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3.5.10 Numbering of Modes
It turns out that the difference in refractive index between the cladding and the core
is usually very small. The modes for TE, TM, HE and EH modes are very similar.
Therefore, we can simplify the way we look at modes in waveguides and fibers. The
listed modes can be summarized and explained using only a single set of LP (linear
polarized) modes.The TE and the TM modes were numbered based on the number of zeros in their
electric field pattern across the waveguide. Therefore a TE0 mode would be a
continuous distribution with only a single maxima but no zeros. A TE00 mode would
be a mode for a 2-dimensional waveguide structure and the electric field distribution
would correspond to a single spot. Obliviously a waveguide structure does not haveto be symmetric. A TE21 would be now a pattern with 2 zeros in one direction and a
third zero in the perpendicular direction.
Electric field distribution for some
symmetric and asymmetric slabwaveguides. The numbering for TE,
TM and TEM mode is identical.
Ref: H. Dutton, Understanding Optical
Communication
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3.5.10 Numbering of modes
Numbering of linear polarized modes is different from numbering TE and TM
modes, but LP numbers are only used for fibers (circular waveguides). LP
modes are described by LPlm where m is the number of maxima along the
radius of the fiber and l is half of the number of maxima around the
circumference.
Correspondence between the linear
polarized modes and the traditional
exact modes in a cylindrical fiber.
Ref: J.M. Senior, Optical Fiber
Communication
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3.6 Coupling between waveguidesIf waveguides are sufficiently close to each other light can couple from one
waveguide to the other. Coupling occurs if the electric fields of the two
waveguides overlap. This effect can be used to build couplers and switches. The
two waveguides are formed by two slabs of higher refractive index similar to single
waveguide structures. Maxwell equations can be used to describe the coupling of
modes from one waveguide to the other waveguide. The problem can be
described by two coupled differential equations (Coupled-Mode Equations).
Coupling between two parallelplanar waveguides.
At z1 most of the light is guided in
waveguide 1,
at z2 the light is equally divided
amongst the two waveguides,at z3 most of the light is guided in
waveguide 2.
Ref.: B.E.A. Saleh, M.C Teich,
Fundamentals of Photonics
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3.6 Coupling between waveguidesLets assume that light propagates in waveguide 1. No light (at least at this
point) propagates in waveguide 2. If the waveguides are close enough to each
other and the electric fields of the waveguides overlap, the wave (in waveguide
1) couples in the waveguide 2. Based on intuition it could be expected that half
of the light is coupled in the waveguide 2. However, this is not the case. Almost
all the light is coupled from waveguide 1 in waveguide 2. The length after which
all the light is coupled form waveguide 1 to waveguide 2 is called the coupling
length L0.
If the two waveguides are close to each other for more than the coupling lengththe light starts to couple back in the waveguide 1. Depending on the length of
the coupling structure the waves couple back and forth between the
waveguides.
Ph t i d O ti l C i ti S i 2007 D D K i
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3.6 Coupling between waveguidesThis effect can be used to build switches and 3dB couplers. If the two
waveguides are close to each other for a distance which is equal to the
coupling length the entire optical power is coupled from one waveguide to the
other waveguide. The structure can be used as switch. If the two waveguides
are close to each other for only half of the coupling length the incoming opticalpower is divided into two equal intensities. Such a structure can be used as an
3dB coupler. In discrete optics a beam splitter would be used to separate a
beam into two equal beams.
Waveguide based optical coupler, (left) switching of the power from one
waveguide to the other, (right) a 3dB coupler.
Ref.: B.E.A. Saleh, M.C Teich, Fundamentals of Photonics
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3.6 Coupling between waveguidesSimilar coupling behavior is observed for single mode fibers. Two single-mode
fibers have to be placed close and parallel to each other to accomplish
coupling. In this case we are of course speaking about the cores of the single-
mode fibers, which have to be placed close to each other. Like already
described for the planar waveguides the electric field of the two fiber cores hasto overlap so that waves can couple from one fiber in the other. Again, we can
define a coupling length. If the fiber cores are close for longer than the coupling
length we observe an oscillation of the intensity from one fiber to the other fiber.
Coupling between
single mode fibers.
Ref.: H. J.R. Dutton,
Understanding optical
communications
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3.6 Coupling between waveguidesThe coupling length strongly dependent on the separation of the two single
mode fiber cores. The further apart they are the greater the coupling length.
Furthermore, the coupling length is strongly wavelength dependent. For
different wavelengths the coupling length changes.
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References:Stamatios V. Kartalopoulos, DWDM, Networks, Devices and Technology,
IEEE press and Wiley Interscience, 2003.
Eugene Hecht, Optics, Addison Wesly, 4th edition, 2002
Fawwalz T. Ulaby, Fundamentals of Applied Electromagnetics, PrenticeHall, 2001.
John M. Senior, Optical Fiber Communications, Prentice Hall Series in
Optoelectonics, 2nd edition, 1992.
Bahaa E.A. Saleh, Malvin Carl Teich, Fundamentals of Photonics,Wiley-Interscience (1991)
Harry J. R. Dutton, Understanding Optical Communications,
Prentice Hall Series in Networking, 1998. (Formerly freely available as a red
book on the IBM red book server.
Joseph C. Palais, Fiber Optic Communications,
Prentice Hall Series, 1998. 4th edition.
Recommended