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Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Optical Telecommunications
Chapter 2 - Optics Refresher
Author : Kevin Heggarty
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Lecture plan
Light as an Electromagnetic wave – orders of magnitude . Quantum nature - «photon» Polarisation Interaction with matter, reflection, refraction, TIR. Dioptre, paraxial approximation Simple lenses, imaging, 2 et 4f setups Lens performance, resolution Aberrations, ray tracing GRIN lenses Gaussians beams.
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Bibliography
• Optics, E. Hecht, Addison-Wesley (Second edition
1984).
• Principals of Optics, M.Born and E.Wolf,
Pergamon.
• The practical application of light, Melles Griot
Catalogue: www.mellesgriot.com
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Light as an electro-magnetic wave Light is an electro-magnetic wave A wave solution of Maxwell’s equations
∇ . E=0 ∇ . H=0 ∇∧H= D
t∇∧E= −
B
t
Progressive wave: E x ,t =E0cos t−kx
Speed: c=1/ ~ 3x108 m/s in vacuum
Light polarisation = polarisation of the electrical field
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
EM Spectum refresher
Visible spectrum~400nm (violet) à 700nm (red) Ultraviolet (UV) ~50-400nm, Infrared (IR) ~ 700nm-1mm
Optical « Telecoms » : 850nm, 1300nm, 1550nm
Frequencies : c= 1550nm : ~1015 Hz
Huge potential bandwidth !
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Quantum behaviour Wave-particle duality. Simultaneously a EM wave and a particle =
«photon» Photon Energy E=h
In the visible spectrum, photon energy ~10-19 J Photonic behaviour is rarely visible in the IR, more in the l’UV
Rarely need to allow for quantum nature of light in optics. Exceptions:
Light emission by sources, particularly semi-conductor sources,
LED, Diode Laser ... Detection, photosensitive materials, photodiodes ... In the fundamental limit of «statistical» noise in a detector
(“quantum noise”).
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Polarisation Defined by the electrical field (the field that interacts with matter) In general a propagating wave is separable in x and y
The phase and amplitude relationship between these two components determines
the state of polarisation : Linear (equal amplitude, in phase) Circular (equal amplitude, phase difference of /2, 3/2) Elliptical (unequal amplitude and/or unrelated phase)
In general – light is not perfectly polarised. We talk of the “degree of polarisation”
E z , t =Excos kz− t E
ycoskz− t
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Polarisation : birefringence and dichroics
The interaction between a light wave and matter often depends on polarisation:Reflection – dielectrics can reflect some polarisations preferentiallyDichroics – absorb one polarisation preferentially (polarisers)Birefringents – have different refractive indices (speed of light propagation) for
different polarisations (waveplates).
DichroïqueRéflexion Biréfringent
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Geometrical OpticsLight is an EM wave
wave solution
speed
Spherical, planar ... wavefronts
Huygens’ principle (remission of a wavefront)
Light “rays”
Conceptual construction (ex. Laser “beam”)
Corresponds to the direct of energy flow
Orthogonal to the wavefront
Parallel to the wavevector, k
E x , t =E0cos t−kx
c = 1 /o o = 1 / = c /n
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Light-matter interaction The EM field acts on charges (atoms, electrons) in
matter.
Principally interaction with electrical field
Several models can be used:
− EM – continuity of the E field.
− Fermat - « shortest path »
− Huygens (re-emission
− Stokes - conservation of energy
− Quantum mechanics – conservation of k
In general reflected and transmitted components
Exact behaviour depends on type of matter (metal,
dielectric), on polarisation, wavelength ...
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Reflection and refraction
Reflection:
Refraction: Snel/Decartes law
Angles are measured between the ray and te
normal to the surface.
r=
i
n t s int = n i s in i
i
r
t
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Critical angle and total internal reflection Light ray passes into a lower index medium (eg. glass-to-air)
We increase i
When t > 90o no longer a transmitted wave – all light is reflected
Total internal reflection (TIR). Waveguide – optical fibre
A more complete treatment of light guiding leads to propagation “modes”
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Fresnel coefficients
Ray treatment ignores polarisation
and energy considerations of
reflected and transmitted rays
A more complete EM analysis
leads to the Fresnel Coefficients :
Amplitude coefficients, energy coefficients : |t|2
itti
ii
ttii
ii
itti
tiit
ttii
ttii
θn+θn
θn=t
θn+θn
θn=t
θn+θn
θnθn=r
θn+θn
θnθn=r
coscos
cos2
coscos
cos2
coscos
coscos
coscos
coscos
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Some consequences of Fresnel coefficients
Reflection losses
eg. Normal incidence, air/glass interface
(n=1.0/1.5)
R = 0,2
|r|2 = 0,04 ~ 4% reflected
Multiple interfaces (0,96)n
AR coatings, multilayer ...
Brewster Angle: one polarisation transmitted:
photography (remove reflections)
Laser mirrors – lower losses
i
t=90o , tan
i=n
t/n
i, r∥ = 0
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
DiopterRefraction at a spherical, dielectric interface
Snel and basic geometry gives usn
1
lo
n
2
li
=1
R n2si
li
−n
2so
lo
So
SiS
o
lo
li
i
r
n2n
1
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Paraxial Approximation
Consider rays close to the optical axis.
Small angles ().
Hence : and we obtain
When
We can define «focalisation »
lo≈s
oet l
i≈ s
i R
nn=
s
n+
s
n
io
1221
R
nn=
s
n,s
ii
121
Rnn
n=fi
12
2
0sin753
sin753
θ!
θ
!
θ+
!
θθ=θ
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Thin lens approximation
Two surfaces (R1 and R
2), at least one spherical
We assume Thin lens (d << f ) In air (n
i = 1)
This gives
and the “imaging”
equation
1
f= n
l−1
1
R1
−1
R2
1
so
1
si
=1
f
(NB Other types of lenses exist, eg. GRIN lenses: flat surfaces, varying index, n.
nl
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Simple imaging
So Si
f f
Objet Image
Graphical technique:Centre ray : undeviatedParallel ray : through focal pointFocal ray : parallel to axis.
Mathematical technique
1
So
1
Si
=1
f⇒ S
i=
f So
So− f
Transverse magnification
Object before f - real image (eg. Photographic lens)
Object at f - image at infinity (ex. collimation of a light
source)
Object after f - virtual image (eg. Magnifying glass)
MT=
hi
ho
=si
so
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Basic lens arrangements
Collimation 1:1 imaging, “2f” setup
Coherent imaging, “4f” setup
2f 2f
f f2f
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Characteristics of simple (singlet) lenses
Focal length, f
Diameter, D
F number (f#) = f/D
Numerical aperture (N.A.) :
N.A. = sin = D/2f
N.A. et f# are measures of the optical « power » or curvature of a lens
(N.A. Also used for other components, eg. fibre).
D
f
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Resolution and diffraction limit
Resolving power
Distinguish two points of an object : Rayleigh criterion
Transfer function – impulse response
Limited by the aperture (diameter), , of the lens
Diffraction theory gives :
“Diffraction limited”
φ
λf=d 2,44
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Resolution – test targets
Evaluated visibility of a test target.
Examples test targets :
Binary amplitude grating (B&W)
USAF test target
Pros :
Quick measurement
Cons : Disadvantages
Qualitative measurement only - difficult to compare lenses
Tends to over-estimate resolving power (aided by 3rd order harmonics)
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Modulation transfer function (MTF) An object with a sinusoidally varying optical density (grey)
Measure the contrast, C, of the image of the grating
Plot contrast against grating spatial frequency
A quantitative criterion – probably best found so far
Can be used for all imaging systems (2f, at infinity, intermediate, on-
axis, off-axis ...)
C=T
max−T
min
TmaxT
min
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Aberrations Paraxial theory assumes small
True close to optical axis, low curvature lenses, long focal lengths ...
However, remains an approximation
Deviations due to this approximation = « aberrations »
Several types of aberration : (monochromatic ...)
1st order aberrations (3rd order theory, Seidel ...)
spherical
coma
astigmatism
field curvature
distortion
sin ≈ , cos≈1
s in =− 3 /3 !
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Aberrations - 2
Spherical Coma
Off-axis object : different rays
image at different points.
External rays focus at different points
(typically closer to the lens)
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Aberrations – 3
Astigmatism Field curvature
Off-axis object : asymmetry, different
focal lengths in different directions.
In reality, image forms on a sphere –
deviation from true position = “distortion”
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Aberrations - 4
Distortion Magnification, M
T varies with off-axis
Chromatic aberrations Refractive index, n, depends on , n() Different colours refracted by different angles Focus (and so image) at different points
Lateral Colour
For similar reasons, MT varies with
Plan
focale
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Lens shape
Aberrations depend strongly on how a lens is used. For each “conjugate” there is a optimal lens shape. “Optimal” when the rays are deviated equally at each surface. Simple rule : “most curved surface toward the infinite conjugate”
2f (1:1) - Symmetrical Biconvex Infinite conjugate - plano-convex
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Doublets, triplets ... acromats Different types of glass, different refractive indices, n
(eg. “crown” and “flint”, n= 1,51 – 1,72) Aberrations depend on n
(ray deviation too strong or weak for = sin
SOLUTION = a “lens” made of different glasses (« doublet, triplet ...)
(errors of one glass correct those of the other)
H1
H2
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Examples of classic “objectives »
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Advantages of « acromats »
Correction of chromatic aberration (b, r)
Correction of spherical aberration
(improved performance even with
monochromatic light)
Improved performance for varying
conjugates.
Crown glass more durable
Large series production : cost reduction.
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Ray tracing Aberrations eliminated or reduces with two glasses Impossible to eliminate all aberrations – use more than 2 glasses ?
Laws of refraction known – calculate deviation at each surface
Ray path through an optical system can be calculated
The calculation is straightforward (matrices) but long
COMPUTER !!
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Ray tracing - 2 Trace several rays through the optical system (often 1D simulation – use
rotational symmetry).
Gradually build up the image/focal spot by summation.
Can allow for : indices, lens shape, aperture stops, chromatics, dispersion ...
Can optimise (computer chooses) lens shape, indices ...
Can use supplier libraries of standard lenses/glasses
Can calculate/compare expected performance : (MTF), throughput, cost ...
Examples de commercial ray-tracing software : Code V, OSLO, ZEMAX....
Tools for experts !!
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Lecture plan
Light as an Electromagnetic wave – orders of magnitude . Quantum nature - «photon» Polarisation Interaction with matter, reflection, refraction, TIR. Dioptre, paraxial approximation Simple lenses, imaging, 2 et 4f setups Lens performance, resolution Aberrations, ray tracing GRIN lenses Gaussians beams.
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
GRIN lenses
GRadient d'INdex lens Deflection by variation of refractive index Flat surfaces, no curvature Similar operation to gradient index fibres Optimal index profile is parabolic
nr=nO1−Ar2/2
n(r)
r
n(r)
Continuous deviation Snel’s law and differential calculus
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
GRIN lenses – fabricationFabrication
Glass doped to exhange its refractive index
Ag+ or Tl+ ions replace Na+ ions in glass
Refractive index depends on ion concentration
Ion concentration is controlled to control index
“Rod” lenses (individual)
- doped preform and “pulling” (as for fibres)
Linear and planar lenslet arrays
- photolithographic masking followed by doping
Masking
UV
Doping
Ag+
Lenses
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Applications of GRIN lensesApplications
Imaging large areas - photocopiers, scanners, fax
Advantages Good optical quality/cost tradeoff Assembly/alignments greatly simplified - groves et gluing to fibre end
Choice of lens Required length depends on and the application Lens “pitch” = P
L=P /4
Collimation
L= P /2
Imaging – inverted
L=P
Imaging - upright
GRIN lens transformation of Gaussian beams
- Yuan, Riza, Applied Optics Vol 38, p3214, 1999
P =2 A
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Lecture plan
Light as an Electromagnetic wave – orders of magnitude . Quantum nature - «photon» Polarisation Interaction with matter, reflection, refraction, TIR. Dioptre, paraxial approximation Simple lenses, imaging, 2 et 4f setups Lens performance, resolution Aberrations, ray tracing GRIN lenses Gaussians beams.
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Gaussian beams
Why important ? We meet them often – naturally occurring :
At the output of a single mode optical fibre Too laser beam Many other wavefronts are close to or tend to Gaussian
A propagating Gaussian beam remains Gaussian : stable
TF(Gauss) = Gauss
A Gaussian beam can be modelled well (analytically) It is a physical solution of Maxwell’s equations cf a spherical or planar wave
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Gaussian beams - definitions
Gaussian profile of the electrical field and irradiance (energy/area)
2220
22
0// weI=rIweE=rE rr
The width, 2w, defines the “waist” of the beam when I(r) falls to 1/e2 of the
maximum (or to 1/e for the electrical field E=1/e E0)
The waist is the only parameter required to characterise the beam profile
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Gaussian beams - propagation Gaussian beam remains Gaussian as it propagates. But it diverges : the waist varies with z.
wz = w0[1 z w
0
2 2]
1/2
R z = z [1w0
2
z 2 ]
Divergence results simply from diffraction - unavoidable A flat wavefront (phase) becomes naturally curved on propagating, the
radius of curvature is given by R(z) The “waist”, w
0 , is the radius where the beam is narrowest
Phase is flat at beam waist: z=0, R(z) infinite.
Asymptote : z∞ , w z z
w0
,
≈ wzz= w
0
, R z ∞ flat
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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Gaussian beams - transformation
Gaussian beams do not obey the usual imaging formulae One possible approach « Self’s Law
Input waist = objet, output waist = image
Consequences: Maximum and minimal image distances (not infinity or at the lens) Maximum image distance (collimation) à s
o=f+Z
R not s
o=f
Special case : so/f = s
ii/f = 1 (focus to focus imaging)
1
Soz
R
2 /So− f
1
si
=1
fmagnificationm =
wi
wo
=1
[1−So/ f ]2 zR/ f 2
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Gaussian beams - truncation A true (mathematical) Gaussian beam is of infinite extent
Any practical optical system will truncate the beam
Power loss :
Infinite T, Dt ~ 0, so illumination uniform ... Airy disc pattern
T=1 (Dt= “waist”) 14% of energy lost – allow safety margin ~ 1,5*beam waist
T =D
g
Dt
=Diamètre faisceau Gaussien à 1 /e2
Diamètre de troncature de la lentille
P L = e−2
D t
D g
2
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
Conclusion – what you need to remember
• Laws of refraction (Snell/Descartes) and reflection
• Critical angle and normal incidence Fresnel coefficients.
• Laws and schema for simple imaging
• Resolution limits
• Know about the existence of simple lenses, acromats
aberrations and ray tracing ... beyond that is “specialist”.
• GRIN lenses – much used in optical telecoms
• Gaussian beams : slightly unusual behaviour, transformation
– care with truncation.
Revision :
Veronique MoeyaertOptical Transmission, VI – Storage Area Networks
References
1.Principals of Optics, M.Born and E.Wolf, Pergamon.
2.The practical application of light, Melles Griot
Catalogue: www.mellesgriot.com
3.http://www.electro-optical.com (Octobre 2005) –
educational site on electro-magnetisme
4.Optics, E. Hecht, Addison-Wesley (Second edition 1984).