ECEN 4616/5616
4/1/2013
Polarization
The Zemax we have can do polarization calculations. Any use of anti-reflection
(or other) coatings or analysis of energy loss due to reflections or absorption
requires polarization analysis.
Read Chapter 20 in the Zemax Manual: “Polarization Analysis”.
A very useful Zemax surface for polarization analysis is the Jones Matrix
surface:
Excerpts from chapter 20:
Defining polarizing components
Any boundary between two media can polarize a beam. However, ZEMAX supports an
idealized model for a general polarizing device. The model is implemented as a special
"Jones Matrix" surface type for sequential ray tracing, and a "Jones Matrix" object type
for non-sequential ray tracing. The Jones matrix modifies a Jones vector (which
describes the electric field) according to
𝐸𝑥′
𝐴
=[
𝐸𝑦′
𝐶
𝐵 𝐸𝑥
]
𝐶 𝐸𝑦
,
where A, B, C, D, Ex, and Ey are all complex numbers. In the lens data and in the nonsequential components editor, ZEMAX provides cells for defining A real, A imag, etc.
(See print-out next page for examples)
pg. 1
ECEN 4616/5616
4/1/2013
Polarization Example:
Air-Spaced doublet (from homework):
Choose “Analysis / Polarization / Polarization Pupil Map”:
Note that input polarization is not changed.
pg. 2
ECEN 4616/5616
4/1/2013
Now, view the report “Analysis / Polarization / Transmission:
Note that there is a 20% loss of light through the system. (Transmission by
surface can be read by scrolling down in the window.)
Now, add Zemax’s standard Anti-Reflection coating to each lens surface (λ/4
thickness of MGF2, n = 1.3778):
pg. 3
ECEN 4616/5616
4/1/2013
Check the Transmission data again:
A significant improvement in transmission (and reduction of stray light).
Why does the ¼ wave coating work? Light experiences a 180 deg phase change
when they reflect from a medium of higher index, and no phase change when
they reflect from a medium of lower index. If the coating material has an
intermediate index between air and the glass, then the reflections from each side
will have the same phase change. The ¼ wave thickness insures that these
reflections will then be out of phase and destructively interfere.
𝑛 −𝑛 2
From the Fresnel reflection coefficient (for normal incidence), 𝑅 = |𝑛1 +𝑛2 | , you
1
2
can calculate that the reflections from each side of the film will be equal (and
hence cancel completely) when the film index is the geometric mean of the
indices on either side; i.e. 𝑛𝑓𝑖𝑙𝑚 = √𝑛1 𝑛2 . Since MGF2 does not exactly meet
this criteria, the reflection cancelation is not complete.
pg. 4
ECEN 4616/5616
4/1/2013
If we define the input polarization as diagonal, the output is also diagonal:
Now, we add a Jones Matrix following the last lens surface,
and define it as a Quarter-Wave Plate in the X-direction:
pg. 5
ECEN 4616/5616
4/1/2013
Now the polarization output map looks like this:
The QWP has converted the diagonal linear polarization into right-circular
polarization.
pg. 6
ECEN 4616/5616
4/1/2013
Appendix A
Small n Approximation for Uniaxial Retarders
(A simple way to calculate the effect of retarders on off-axis rays.)
(These equations not readily available elsewhere.)
Rotated Index Ellipsoid:
For a uniaxial retarder with the extraordinary axis along Z, the index ellipsoid is:
X 2 Y2 Z2
(eq. 1)


1.
no2 no2 ne2
To find new axes, x, y, z  , rotated  degrees about the Y axis, we use the
following transformations:
 x  cos  0  sin   X 
 y   0
1
0    Y 
  
 z   sin  0 cos    Z 
-and X   cos  0 sin    x 
Y    0
1
0    y 
  
 Z   sin  0 cos   z 
Using the second transformation above to substitute into equation 1, we get the
equation of the index ellipsoid in the new axes:
x cos   z sin 2  y 2   z cos   x sin 2  1 .
no2
no2
ne2
For a ray propagating along the (new) z -axis, the effective indices will be given
by the major and minor axes of the ellipse obtained by cutting the above ellipsoid
with the z  0 plane. Setting z  0 in the above equation and rearranging gives
the ellipse:
x2
y2

 1.

 no2
no2 ne2
 2

2
2
2
n
cos


n
sin

o
 e

Hence, the effective extraordinary index for this ray is:
ne no
n e 
(eq. 2)
n e2 cos 2   n o2 sin 2 
pg. 7
ECEN 4616/5616
4/1/2013
Expand the right hand side of equation 2 and apply the approximation,
ne  no  n 1:
1
A)
ne2 cos 2   no2 sin 2   ne2  no2   ne2  no2 cos 2
2


ne2  no2  (no  n) 2  no2
 2no2  2no n  n 2
 2no2  2no n,
B)
n  1
 2no (no  n)
 2 no ne
ne2  no2  no  n   no2
2
 2no n  n 2
C)
 2 n o  n,
n  1
Putting expression A, and approximations B & C into equation 2:
ne no
ne 
ne no  no n cos 2
ne no

1
n
cos 2
ne
no 1 

1
n
no
n
cos 2
ne
Expanding the radicals in truncated Taylor Series:


n 
n
1 
ne  no 1 
cos 2 ,
 2no  2ne

n  1
n n  no 
n 2


 no 

cos 2 
cos 2
2
2  ne 
4ne
n n

cos 2,
2
2
 no  n sin 2 
 no 
n  1
n  ne  no   n sin 2 
Hence,
(eq. 3)
Where  is the angle between the ray and optic axis of the retarder.
pg. 8
ECEN 4616/5616
4/1/2013
Retardance of Plate:
Consider a ray incident on a plane, plate retarder with angle of incidence  i . In
general, there will be two refracted rays in the plate, related by Snell’s law:
sin  i  ne sin  e  no sin  o .
Under the assumption that n  1, we can say that:
sin  e  sin(  o  )
 sin  o cos   cos  o sin 
 sin  o   cos  o ,
since   1
Substituting the above result into Snell’s law and solving for  we get:
 
n 
tan  o .
ne
(eq. 4)
Also, expanding cos  e ;
cos  e  cos o   
 cos  o cos   sin  o sin 
 cos  o   sin  o ,
  1.
Substituting  from equation 4 above:
cos  e  cos  o 
n
tan  o sin  o
ne
(eq. 5)
The retardance of a birefringent plane/parallel plate is:1
  t ne cos e  no cos o ,
where t is plate thickness. Substituting the approximation for cos  e from eq. 5:
  t ne cos  o  n tan  o sin  o  no cos  o 
 tncos  o  tan  o sin  o 

tn
.
cos  o
Hence,

1
tn
cos  o
(eq. 6)
See Principles of Optics, Born & Wolf, p.697, for example.
pg. 9
ECEN 4616/5616
4/1/2013
Summary:
Equations 3 & 6 give the retardance of an arbitrary uniaxial retarder, given that
(ne  no )  1. Summarizing the relevant relations:
n  n sin 2 
1.
2.
 
tn
cos 
where:
 is the angle between a refracted ray and the optic axis,
and,
 is the angle between the refracted ray and the normal to the plate.
In one equation:
sin 2 
(, )  0
,
0  t ne  no 
cos 
pg. 10
ECEN 4616/5616
4/1/2013
Appendix B
(This is a general scheme for using two intensity holograms to recover the phase
of the initial signal. It works for optical frequencies as well. For example, if two
holograms, as described, as taken of a microscope image, the wave field can be
recreated and re-imaged at any desired depth, without the artifacts inherent in
optical hologram reconstruction.)
Microwave Holography
General Scheme:
Far-field antenna patterns are easily determined by near-field scanning. When
amplitude and phase are both recorded in the near field, the far field pattern can
be determined by simply propagating the recorded field. In millimeter
wavebands, however, it is difficult to measure phase accurately. (At optical
frequencies, it is impossible) Intensities are easily measured, and the intensity of
the antenna signal mixed with a reference signal is, in fact, a hologram. Opticalstyle holography with radiated reference waves and standard reconstruction
suffers from the same types of spurious images and background noise that
plague optical holography.
With microwave technology it is easy to replace the reference wave with a
constant signal mixed with the output of the probe antenna. This is a method of
reconstructing the fields of the antenna under test (AUT) from near-field
holographic recordings that eliminates spurious images by making use of the
flexibility of this method of recording the hologram.
Symbolism:
The reference signal is
R  Er ( x, y )ei
r
( x, y )
 Er ei
 Er cos( r )  iEr sin( r )
r
 Err  iEri ,
and the AUT field is
A  Ea ( x , y )ei
a
( x, y )
 Ea ei
 Ea cos( a )  iEa sin( a )
a
 Ear  iEai .
The recorded hologram is
pg. 11
ECEN 4616/5616
4/1/2013
H  R A
2


 R  A  RA  R A.
2
2
(1)
Direct solution for the AUT Field
This technique makes use of the ability to shift the phase ( r ) of the reference
wave by an arbitrary angle,  . Let two holograms be recorded, identical except
for a phase difference of  in the reference waves; i.e.:
R(1)  Er ei
r
 Err  iEri
(1)
R( 2 )  Er ei (
(1)
r
 )
 Err  iEri
(2)
(2)
and the holograms are
H(1)  R(1)  A
2
2
H( 2 )  R( 2 )  A .
The solution proceeds as follows:
2
1. Measure A when the reference signal is turned off.
2
2. Measure R when the AUT is turned off. In the case of an injected
2
reference signal, R is simply a constant scalar.
2
2
3. Subtract A and R from the two measured holograms. Referring to
equation (1), we see that:
H  H  R  A
2
2
 RA  R  A
 2( Ear Err  Eai Eri ).
4. From the expressions for the two holograms,
H(1)  2( Ear Err  Eai Eri )
(1)
(1)
H(2 )  2( Ear Err  Eai Eri ),
(2)
(2)
pg. 12
ECEN 4616/5616
4/1/2013
solve for the AUT field components:
Ear
i
i
1 H(1) Er  H(2 ) Er

2 Eri Err  Err Eri
(2)
(2)
(1)
(1)
(2)
(1)
and
 Eai
r
r
1 H(1) Er  H(2 ) Er
.

2 Eri Err  Err Eri
(2)
(2)
(1)
(1)
(2)
(1)
Thus, the AUT field is recovered, artifact free except for measurement errors,
from the measured holograms and the measured (or otherwise known) reference
source.
The denominator in the above expressions, in terms of the reference amplitude,
E r , and phase shift between the two holograms,  , is  Er  sin( ) . Thus,
2
choosing    / 2 allows the denominator to be replaced with R , measured
as described in step (2) above.
2
pg. 13
ECEN 4616/5616
4/1/2013
Appendix C
Improved Algorithm for Reconstructing Digital
Holograms
(This is a general scheme for recreating the image from a digitally recorded
hologram, without many of the artifacts in optical resconstruction.)
Holograms
A standard representation for a (monochromatic) optical wave field generally
traveling near the z direction is, (in complex notation):
O  Ao ( x , y )eik o ( x , y )
where Ao ( x , y ) is the (non-negative) amplitude, and  o ( x , y ) is the (arbitrary)
phase of the field. (In the standard notation, z dependence is left implicit, since it
i t
can be deduced from the x and y dependence, and the time dependence, e
, is left off, since it would simply cancel from both sides of the equations.)
With this notation in mind, let's define O to be the light scattered from an object
we wish to make a hologram of, and R is a (nearly) plane-wave reference beam.
The hologram is made by allowing O and R to interfere at the site of a detector
(a CCD camera, say). At the detector, then, the total wave field is   ( R  O)
and the detected intensity is proportional to:
H
2
  R  O
2
  R  O R  O

(1)
 R  O  RO   R  O,
2
2

where indicates the operation of complex conjugation. This expression
describes the detected hologram.
Optical Reconstruction of Holographic Images
If the reference beam, R , is a good approximation of a plane wave, then the
amplitude of R: AR ( x , y )  constant and the phase of R: R ( x , y )  constant ,
and:
R  AR ( x , y )e i R ( x , y )
 RC
pg. 14
ECEN 4616/5616
4/1/2013
where RC is a constant which can be made real by choosing the origin such that
 R  0 . Then, the 4th term in (1) (ie.: R  O ) is proportional to the original
object wave, O . Given this approximation, we can recover the image of the
object by passing a plane wave through a mask (film) with transparency equal to
H. This wave will then be modulated so that it is proportional to H , and the



object-wave term R O  RC O will produce an image at the original z-distance
away from the hologram. Equivalently, we can simulate the propagation of a
wave field equal to H digitally.
Of course, R is never exactly a uniform plane wave, so distortion and noise are
thereby introduced. The other terms in (1) that are not proportional to the object
wave also introduce noise into the image. In addition, if the wave used in
reconstructing the image optically is not a uniform plane wave, then even more
distortion and noise are present in the image.
The meaning of all of the terms in (1) are:
2
R : This is the background reference illumination; it is what the detector would
see if no object wave were present.
2
O : This is the diffraction pattern of the object; it is what the detector would
see if no reference wave were present. Generally, O  R .
2
2
RO  : This is the conjugate object wave, distorted by any non-uniformity's in the
reference wave. The conjugate object wave is the same as the object
wave, except it focuses in the opposite direction; thus the conjugate wave
will be out-of-focus at the point where the object wave forms an image and
vice-versa.
RO: This is the object wave, distorted by non-uniformity's in the reference
wave. A distorted reference wave will distort the final image.
Special Digital Reconstruction of Holographic Images
One of the advantages of having access to a digitized hologram is that we are no
longer bound to the limitations of optical reconstruction. Since the detected
2
reference wave, R , can be recorded when no objects are in the beam, the first
H, is to subtract out the background:
2
H  H  R
step in improving the hologram,
 O  RO   R O.
2
Our first improved hologram, H  , has one less noise term than the standard
`optical' hologram. For further improvement, we would like to remove the
pg. 15
ECEN 4616/5616
4/1/2013
distorting effect of multiplying the object wave,

O , by R . Unfortunately, we do
not know the amplitude function R directly, as we only have access to the
2
detected intensity of the reference wave, R .
At this point, we make some reasonable approximations. Remember, by
i ( x , y )
. In general, AR ( x , y )  constant , since
definition, R  AR ( x , y )e R
significant variations in the intensity of the background is commonly seen in
 R ( x , y ) of R , however, would be directly
related to components of R propagating in different directions. Since R is a wellholograms. Variations in the phase,
collimated laser beam that propagates in (nearly) one direction only, evidently
 R ( x, y )  constant i. This, plus the fact that AR ( x , y )  0 (by definition),
means that:
R  R 
2
R .
The next improvement in our digital hologram is therefore:
H  
H
R
2
O  RO  RO
2

R
2
2
O

 O  O.
R
This modified hologram, H  , is much improved over the original detected
hologram, H . The background variations are removed, the object wave is
2
distortion-free, and the (already small) diffraction pattern overlay, O , is reduced
by division with the (generally larger) background. Virtually the only noise term of

note left is the out-of-focus conjugate object wave, O .
Images produced by digitally simulating the propagation of H  have 10  20
times the Signal/Noise of optically reconstructed hologram images (and are even
better than many in-focus images, made with coherent light). They are a much
better target for automatic image analysis techniques than images from optically
reconstructed holograms.
There is a further, iterative, technique whereby even the last two noise terms of
H  might be removed, leaving only the object wave. In H  itself, we have no

way to distinguish between O, O , and.
2
O
R
. However, if the object that
O
comes from is compact (that is, it forms a small image), then when we propagate
pg. 16
ECEN 4616/5616
4/1/2013
H  over distance z to form an image of the object, O and O (as well as
O
R
2
)
become separated geometrically. Then OZ  z is the image, which is compact,
real, and localized, while the other two terms are spread over a much larger
region. We physically separate OZ  z from the other terms by selecting a small
area around the image. We can then apply any apriori knowledge we have about
the image to improve it (for example, should it be a complete shadow image?),
back-propagate it to the hologram plane (ie.; calculate OZ 0 ) from which we can

calculate the complex representation of O and the diffraction pattern (
O  OO ). These two non-object wave terms would then be subtracted from
H  to yield a new estimate for O: H  , which could then be propagated
2
forward to find the next approximation to the image. We have not tried this yet,
since the current algorithm works very well, but it might prove useful for
applications where high-precision images are required.
i
 R x, y  is an actual constant only for the case of in-line or Gabor holograms – for off-axis reference
holograms, perhaps a linear phase function can be used as an approximation, depending on the angle of the
reference beam. This has not been experimentally attempted, due to the difficulty of doing off-axis
reference holography with limited resolution CCD cameras.
pg. 17