Read Chapter 20 in the Zemax Manual: Polarization Analysis

ECEN 4616/56164/1/2013

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

where A, B, C, D, Ex, and Ey are all complex numbers. In the lens data and in the non-sequential components editor, ZEMAX provides cells for defining A real, A imag, etc.

(See print-out next page for examples)

Polarization Example:

Air-Spaced doublet (from homework):

Choose “Analysis / Polarization / Polarization Pupil Map”:

Note that input polarization is not changed.

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):

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.

From the Fresnel reflection coefficient (for normal incidence), , you 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. . Since MGF2 does not exactly meet this criteria, the reflection cancelation is not complete.

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:

Now the polarization output map looks like this:

The QWP has converted the diagonal linear polarization into right-circular polarization.

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:

. (eq. 1)

To find new axes,, rotated degrees about the axis, we use the following transformations:

-and-

Using the second transformation above to substitute into equation 1, we get the equation of the index ellipsoid in the new axes:

For a ray propagating along the (new) -axis, the effective indices will be given by the major and minor axes of the ellipse obtained by cutting the above ellipsoid with the plane. Setting in the above equation and rearranging gives the ellipse:

Hence, the effective extraordinary index for this ray is:

(eq. 2)

Expand the right hand side of equation 2 and apply the approximation, 1:

Putting expression A, and approximations BC into equation 2:

Expanding the radicals in truncated Taylor Series:

Hence, (eq. 3) Where is the angle between the ray and optic axis of the retarder.

Retardance of Plate:

Consider a ray incident on a plane, plate retarder with angle of incidence . In general, there will be two refracted rays in the plate, related by Snell’s law:

Under the assumption that , we can say that:

Substituting the above result into Snell’s law and solving for we get:

(eq. 4)

Also, expanding ;

Substituting from equation 4 above:

(eq. 5)

The retardance of a birefringent plane/parallel plate is:[1]

wheret is plate thickness. Substituting the approximation for from eq. 5:

Hence,

(eq. 6)

Summary:

Equations 36 give the retardance of an arbitrary uniaxial retarder, given that Summarizing the relevant relations:

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:

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. Optical-style 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

and the AUT field is

The recorded hologram is

(1)

Direct solution for the AUT Field

This technique makes use of the ability to shift the phase 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.:

and the holograms are

The solution proceeds as follows:

Measure when the reference signal is turned off.

Measure when the AUT is turned off. In the case of an injected reference signal, is simply a constant scalar.

Subtract and from the two measured holograms. Referring to equation (1), we see that:

4. From the expressions for the two holograms,

solve for the AUT field components:

and

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, , and phase shift between the two holograms, , is . Thus, choosing allows the denominator to be replaced with , measured as described in step (2) above.

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 direction is, (in complex notation):

where is the (non-negative) amplitude, and is the (arbitrary) phase of the field. (In the standard notation, dependence is left implicit, since it can be deduced from the and dependence, and the time dependence, , is left off, since it would simply cancel from both sides of the equations.)

With this notation in mind, let's define to be the light scattered from an object we wish to make a hologram of, and is a (nearly) plane-wave reference beam. The hologram is made by allowing and to interfere at the site of a detector (a CCD camera, say). At the detector, then, the total wave field is and the detected intensity is proportional to:

(1)

where indicates the operation of complex conjugation. This expression describes the detected hologram.

Optical Reconstruction of Holographic Images

If the reference beam, , is a good approximation of a plane wave, then the amplitude of and the phase of , and:

where is a constant which can be made real by choosing the origin such that . Then, the 4th term in (1) (ie.: ) is proportional to the original object wave, . Given this approximation, we can recover the image of the object by passing a plane wave through a mask (film) with transparency equal to This wave will then be modulated so that it is proportional to , and the object-wave term 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 digitally.

Of course, 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:

This is the background reference illumination; it is what the detector would see if no object wave were present.

This is the diffraction pattern of the object; it is what the detector would see if no reference wave were present. Generally, .

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.

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 reference wave, can be recorded when no objects are in the beam, the first step in improving the hologram, is to subtract out the background:

Our first improved hologram, , has one less noise term than the standard `optical' hologram. For further improvement, we would like to remove the distorting effect of multiplying the object wave, , by . Unfortunately, we do not know the amplitude function directly, as we only have access to the detected intensity of the reference wave, .

At this point, we make some reasonable approximations. Remember, by definition, In general, , since significant variations in the intensity of the background is commonly seen in holograms. Variations in the phase, of , however, would be directly related to components of propagating in different directions. Since is a well-collimated laser beam that propagates in (nearly) one direction only, evidently [i]. This, plus the fact that (by definition), means that:

The next improvement in our digital hologram is therefore:

This modified hologram, , is much improved over the original detected hologram, . The background variations are removed, the object wave is distortion-free, and the (already small) diffraction pattern overlay, , 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, .

Images produced by digitally simulating the propagation of have 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 might be removed, leaving only the object wave. In itself, we have no way to distinguish between . However, if the object that comes from is compact (that is, it forms a small image), then when we propagate over distance to form an image of the object, and (as well as ) become separated geometrically. Then is the image, which is compact, real, and localized, while the other two terms are spread over a much larger region. We physically separate 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 ) from which we can calculate the complex representation of and the diffraction pattern (). These two non-object wave terms would then be subtracted from to yield a new estimate for , which could then be propagated 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.

pg. 1

[1] See Principles of Optics, Born & Wolf, p.697, for example.

[i]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.