# ATLAS TCAD: Simulation of Frequency Response from Light Impulse

Recently a project was posted for a high speed photodetector. Part of that project was to develop a program that takes the frequency response of a light impulse. My thought is to create a program that can perform these tasks, including an impulse response for any structure.

Generic Light Frequency Response Simulator Program in ATLAS TCAD

The first part of the program should include all the particulars of the structure that is being simulated:

go atlas

[define mesh]

[define structure]

[define electrodes]

[define materials]

Then, the beam is defined. x.origin and y.origin describes from where the beam is originating on the 2D x-y plane. The angle shown of 270 degrees means that the beam will be facing upwards. One may think of this angle as starting on the right hand sixe of the x-y coordinate plane and moves clockwise. The wavelength is the optical wavelength of the beam and the window defines how wide the beam will be.

beam num=1 x.origin=0 y.origin=5 angle=270 wavelength=1550 min.window=-15 max.window=15

The program now should run an initial solution and set the conditions (such as if a voltage is applied to a contact) for the frequency response.

METHOD HALFIMPL

solve init
outf = lightpulse_frequencyresponse.str
LOG lightpulse_frequencyresponse.log

[simulation conditions such as applied voltage]

LOG off

Now the optical pulse is is simulated as follows:

LOG outf=transient.log
SOLVE B1=1.0 RAMPTIME=1E-9 TSTOP=1E-9 TSTEP=1E-12
SOLVE B1=0.0 RAMPTIME=1E-9 TSTOP=20E-9 TSTEP=1E-12

tonyplot transient.log

outf=lightpulse_frequencyresponse.str master onefile
log off

The optical pulse “transient.log” is simulated using Tonyplot at the end of the program. It is a good idea to separate transient plots from frequency plots to ensure that these parameters may be chosen in Tonyplot. Tonyplot does not give the option to use a parameter if it is not the object that is being solved before saving the .log file.

log outf=frequencyplot.log
FOURIER INFILE=transient.log OUTFILE=frequencyplot.log T.START=0 T.STOP=20E-9 INTERPOLATE
tonyplot frequencyplot.log
log off

output band.param ramptime TRANS.ANALY photogen opt.intens con.band val.band e.mobility h.mobility band.param photogen opt.intens recomb u.srh u.aug u.rad flowlines

save outf=lightpulse_frequencyresponse.str
tonyplot lightpulse_frequencyresponse.str

quit

Now you can focus on the structure and mesh for a light impulse frequency response. Note that adjustments may be warranted on the light impulse and beam.

And so, here is a structure simulation that could be done easily using the process above. # High Speed UTC Photodetector Simulation with Frequency Response in TCAD

The following is a TCAD simulation of a high speed UTC photodetector. An I-V curve is simulated for the photodetector, forward and reverse. A light beam is simulated to enter the photodetector. The photo-current response to a light impulse is simulated, followed by a frequency response in TCAD.

Structure: I-V Curve Beam Simulation Entering Photodetector: Light Impulse: Frequency Response in ATLAS: The full project (pdf) is here: ece530_final_mbenker

# Sinusoidal and Exponential Sequences, Periodicity of Sequences

Continuing our discussion on discrete-time sequences, we now come to define exponential and sinusoidal sequences. The general formula for a discrete-time exponential sequence is as follows:

x[n] = Aα^n.

This exponential behaves differently according to the value of α. If the sequence starts at n=0, the formula is as follows:

x[n] = Aα^n * u[n]. If α is a complex number, the exponential function exhibits newer characteristics. The envelope of the exponential is |α|. If |α| < 1, the system is decaying. If |α|> 1, the system is growing. When α is complex, the sequence may be analyzed as follows, using the definition of Euler’s formula to express a complex relationship as a magnitude and phase difference. Where ω0 is the frequency and φ is the phase, for n number of samples, a complex exponential sequence of form Ae^jw0n may be considered as a sinusoidal sequence for a set of frequencies in an interval of 2π.

A sinusoidal sequence is defined as follows:

x[n] = Acos(ω0*n + φ), for all n, and A, φ are real constants.

Periodicity for discrete-time signals means that the sequence will repeat itself for a certain delay, N.

x[n] = x[n+N] : system is periodic.

t = (-5:1:15)’;

impulse = t==0;
unitstep = t>=0;
Alpha1 = -0.5;
Alpha2 = 0.5;
Alpha3 = 2.5;
Alpha4 = -2.5;
cAlpha1 = -0.5 – 0.5i;
cAlpha2 = 0.5 + 0.5i;
cAlpha3 = 2.5 -2.5i;
cAlpha4 = -2.5 + 2.5i;
A = 1;

Exp1 = A.*unitstep.*Alpha1.^t;
Exp2 = A.*unitstep.*Alpha2.^t;
Exp3 = A.*unitstep.*Alpha3.^t;
Exp4 = A.*unitstep.*Alpha4.^t;

cExp1 = A.*unitstep.*cAlpha1.^t;
cExp2 = A.*unitstep.*cAlpha2.^t;
cExp3 = A.*unitstep.*cAlpha3.^t;
cExp4 = A.*unitstep.*cAlpha4.^t;

%%
figure(1)
subplot(2,1,1)
stem(t, impulse)
xlabel(‘x’)
ylabel(‘y’)
title(‘Impulse’)

subplot(2,1,2)
stem(t, unitstep)
xlabel(‘x’)
ylabel(‘y’)
title(‘Unit Step’)
%%
figure(2)
subplot(2,2,1)
stem(t, cExp1)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = -0.5 – 0.5i’)

subplot(2,2,2)
stem(t, cExp2)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = 0.5 + 0.5i’)

subplot(2,2,3)
stem(t, cExp3)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = 2.5 -2.5i’)

subplot(2,2,4)
stem(t, cExp4)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = -2.5 + 2.5i’)
%%
figure(3)
subplot(2,2,1)
stem(t, Exp1)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = -0.5’)

subplot(2,2,2)
stem(t, Exp2)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = 0.5’)

subplot(2,2,3)
stem(t, Exp3)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = 2.5’)

subplot(2,2,4)
stem(t, Exp4)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = -2.5’)

# Mathematical Formulation for Antennas: Radiation Integrals and Auxiliary Potentials

This short paper will attempt to clarify some useful mathematical tools for antenna analysis that seem overly “mathematical” but can aid in understanding antenna theory. A solid background in Maxwell’s equations and vector calculus would be helpful.

Two sources will be introduced: The Electric and Magnetic sources (E and M respectively). These will be integrated to obtain either an electric and magnetic field directly or integrated to obtain a Vector potential, which is then differentiated to obtain the E and H fields. We will use A for magnetic vector potential and F for electric vector potential.

Using Gauss’ laws (first two equations) for a source free region: And also the identity: It can be shown that: In the case of the magnetic field in response to the magnetic vector potential (A). This is done by equating the divergence of B with the divergence of the curl of A, which both equal zero. The same can be done from Gauss Law of electricity (1st equation) and the divergence of the curl of F.

Using Maxwell’s equations (not necessary to know how) the following can be derived: For total fields, the two auxiliary potentials can be summed. In the case of the Electric field this leads to: The following integrals can be used to solve for the vector potentials, if the current densities are known: For some cases, the volume integral is reduced to a surface or line integral.

An important note: most antenna calculations and also the above integrals are independent of distance, and therefore are done in the far field (region greater than 2D^2/λ, where D is the largest dimension of the antenna).

The familiar duality theorem from Fourier Transform properties can be applied in a similar way to Maxwell’s equations, as shown. In the chart, Faraday’s Law, Ampere’s Law, Helmholtz equations and the above mentioned integrals are shown. To be perfectly honest, I think the top right equation is wrong. I believe is should have permittivity rather than permeability.

Another important antenna property is reciprocity… that is the receive and transmit radiation patterns are the same , given that the medium of propagation is linear and isotropic. This can be compared to the reciprocity theorem of circuits, meaning that a volt meter and source can be interchanged if a constant current or voltage source is used and the circuit components are linear, bilateral and discrete elements.

# Discrete-Time Impulse and Unit Step Functions

Discrete-Time Signals are understood as a set or sequence of numbers. These sequences possess magnitudes or values at a given index.

One mark of Discrete-Time Signals is that the index value is an integer. Thus, the sequence will have a magnitude or value for a whole number index such as -5, -4, 0, 6, 10000, etc.

A discrete-time signal represented as a sequence of numbers takes the following form:

x[n] = {x[n]},          -∞ < n < ∞,

where n is any real integer (the index).

An analog representation describes values of a signal at time nT, where T is the sampling period. The sampling frequency is the inverse of the sampling period.

x[n] = X_a(nT),      -∞ < n < ∞.

Common Sequences

Both a very simple and important sequence is the unit sample sequence, “discrete time impulse” or simply “impulse,” equal to 1 only at zero and equal to zero otherwise. The discrete time impulse is used to describe an entire system using a delayed impulse. An entire sequence may also be shifted or delayed using the following relation:

y[n] = x[n – n0],

where n0 is an integer (which is the increment of indices by which the system is delayed. The impulse function delayed to any index and multiplied by the value of the system at that index can describe any discrete-time system. The general formula for this relationship is, The unit step sequence is related to the unit impulse. The unit step sequence is a set of numbers that is equal to zero for all numbers less than zero and equal to one for numbers equal and greater than zero. The unit step sequence is therefore equal to a sequence of delta impulses with a zero and greater delay.

u[n] = δ[n] + δ[n-1] + δ[n-2] + . . . The unit impulse can also be represented by unit step functions:

δ[n] = u[n] – u[n-1].

Below I’ve plotted both the impulse and unit step function in matlab. ```t = (-10:1:10)';

impulse = t==0;
unitstep = t>=0;

figure(1)
subplot(2,1,1)
stem(t, impulse)
xlabel('x')
ylabel('y')
title('Impulse')
figure(1)
subplot(2,1,2)
stem(t, unitstep)
xlabel('x')
ylabel('y')
title('Unit Step')```

# Image Resolution

Consider that we are interested in building an optical sensor. This sensor contains a number of pixels, which is dependent on the size of the sensor. The sensor has two dimensions, horizontal and vertical. Knowing the size of the pixels, we will be able to find the total number of pixels on this sensor.

The horizontal field of view, HFOV is the total angle of view normal from the sensor. The effective focal length, EFL of the sensor is then:

Effective Focal Length: EFL = V / (tan(HFOV/2)),

where V is the vertical sensor size in (in meters, not in number of pixels) and HFOV is the horizontal field of view. Horizontal field of view as an angled is halved to account that HFOV extends to both sizes of the normal of the sensor.

The system resolution using the Kell Factor: R = 1000 * KellFactor * (1 / (PixelSize)),

where the Pixel size is typically given and the Kell factor, less than 1 will approximate a best real case result and accounts for aberrations and other potential issues.

Angular resolution: AR = R * EFL / 1000,

where R is the resolution using the Kell factor and EFL is the effective focal length. It is possible to compute the angular resolution using either pixels per millimeter or cycles per millimeter, however one would need to be consistent with units.

Minimum field of view: Δl = 1.22 * f * λ / D,

which was used previously for the calculation of the spatial resolution of a microscope. The minimum field of view is exactly a different wording for the minimum spatial resolution, or minimum size resolvable.

Below is a MATLAB program that computed these parameters, while sweeping the diameter of the lens aperture. The wavelength admittedly may not be appropriate for a microscope, but let’s say that you are looking for something in the infrared spectrum. Maybe you are trying to view some tiny laser beams that will be used in the telecom industry at 1550 nanometer.

Pixel size: 3 um. HFOV: 4 degrees. Sensor size: 8.9mm x 11.84mm.  # Spatial Resolution of a Microscope

Angular resolution describes the smallest angle between two objects that are able to be resolved.

θ = 1.22 * λ / D,

where λ is the wavelength of the light and D is the diameter of the lens aperture.

Spatial resolution on the other hand describes the smallest object that a lens can resolve. While angular resolution was employed for the telescope, the following formula for spatial resolution is applied to microscopes.

Spatial resolution: Δl = θf = 1.22 * f * λ / D,

where θ is the angular resolution, f is the focal length (assumed to be distance to object from lens as well), λ is the wavelength and D is the diameter of the lens aperture. The Numerical Aperture (NA) is a measure of the the ability to of the lens to gather light and resolve fine detail. In the case of fiber optics, the numerical aperture applies to the maximum acceptance angle of light entering a fiber. The angle by the lens at its focus is θ = 2α. α is shown in the first diagram.

Numerical Aperture for a lens: NA = n * sin(α),

where n is the index of refraction of the medium between the lens and the object. Further,

sin(α) = D / (2d).

The resolving power of a microscope is related.

Resolving power: x = 1.22 * d * λ / D,

where d is the distance from the lens aperture to the region of focus. Using the definition of NA,

Resolving power: x = 1.22 * d * λ / D = 1.22 * λ / (2sin(α)) = 0.61 * λ / NA.

# Telescope Resolution & Distance Between Stars using the Rayleigh Limit

Previously, the Rayleigh Criterion and the concept of maximum resolution was explained. As mentioned, Rayleigh found this formula performing an experiment with telescopes and stars, exploring the concept of resolution. This formula may be used to determine the distance between two stars.

θ = 1.22 * λ / D.

Consider a telescope of lens diameter of 2.4 meters for a star of visible white light at approximately 550 nanometer wavelength. The distance between the two stars in lightyears may be calculated as follows. The stars are approximately 2.6 million lightyears away from the lens.

θ = 1.22 * (550*10^(-9)m)/(2.4m)

Distance between two objects (s) at a distance away (r), separated by angle (θ): s = rθ

s = rθ = (2.0*10^(6) ly)*(2.80*10^(-7)) = 0.56 ly.

This means that the maximum resolution for the lens size, star distance from the lens and wavelength would be that two stars would need to be separated at least 0.56 lightyears for the two stars to be distinguishable. # Diffraction, Resolution and the Rayleigh Criterion

The wave theory of light includes the understanding that light diffracts as it moves through space, bending around obstacles and interfering with itself constructively and destructively. Diffraction grating disperses light according to wavelength. The intensity pattern of monochromatic light going through a small, circular aperture will produce a pattern of a central maximum and other local minima and maxima. The wave nature of light and the diffraction pattern of light plays an interesting role in another subject: resolution. The light which comes through the hole, as demonstrated by the concept of diffraction, will not appear as a small circle with sharply defined edges. There will appear some amount of fuzziness to the perimeter of the light circle.

Consider if there are two sources of light that are near to each other. In this case, the light circles will overlap each other. Move them even closer together and they may appear as one light source. This means that they cannot be resolved, that the resolution is not high enough for the two to be distinguished from another. Considering diffraction through a circular aperture the angular resolution is as follows:

Angular resolution: θ = 1.22 * λ/D,

where λ is the wavelength of light, D is the diameter of the lens aperture and the factor 1.22 corresponds to the resolution limit formulated and empirically tested using experiments performed using telescopes and astronomical measurements by John William Strutt, a.k.a. Rayleigh for the “Rayleigh Criterion.” This factor describes what would be the minimum angle for two objects to be distinguishable.

# Optical Polarizers in Series

The following problems deal with polarizers, which is a device used to alter the polarization of an optical wave.

1. ### Unpolarized light of intensity I is incident on an ideal linear polarizer (no absorption). What is the transmitted intensity?

Unpolarized light contains all possible angles to the linear polarizer. On a two dimensional plane, the linear polarizer will emit only that amount of light intensity that is found in the axis of polarization. Therefore, the Intensity of light emitted from a linear polarizer from incident unpolarized light will be half the intensity of the incident light.

### c) Is it possible to reduce the intensity of transmitted light to zero by removing a polarizer(s)?

a) Using Malus’s Law, the intensity of light from a polarizer is equal to the incident intensity multiplied by the cosine squared of the angle between the incident light and the polarizer. This formula is used in subsequent calculations (below). The intensity of light from the last polarizer is 19.8% of the incident light intensity.

b) My removing polarizer three, the total intensity is reduced to 0.0516 times the incident intensity.

c) In order to achieve an intensity of zero on the output of the polarizer, there will need to exist an angle difference of 90 degrees between two of the polarizers. This is not achievable by removing only one of the polarizers, however it would be possible by removing both the second and third polarizer, leaving a difference of 90 degrees between two polarizers. # Jones Vector: Polarization Modes

The Jones Vector is a method of describing the direction of polarization of light. It uses a two element matrix for the complex amplitude of the polarized wave. The polarization of a light wave can be described in a two dimensional plane as the cross section of the light wave. The two elements in the Jones Vector are a function of the angle that the wave makes in the two dimensional cross section plane of the wave as well as the amplitude of the wave. The amplitude may be separated from the ‘mode’ of the vector. The mode of the vector describes only the direction of polarization. Below is a first example with a linear polarization in the y direction. Using the Jones Vector the mode can be calculated for any angle. See calculations below: The phase differences of the Jones Vector are plotted for a visual representation of the mode. If both components of the differ in phase, the plot depict a circular or oval pattern that intersects both components of the mode on a two dimensional plot. The simplest of plots to understand is a polarization of 90 degree phase difference. In this case, both magnitudes of the components of the mode will be 1 and a full circle is drawn to connect these points of the mode. In the case of a zero phase difference, this is demonstrated at 45 degrees where both sin(45deg) and cos(45deg) equal 0.707. In this case, the phase difference is plotted as a straight line, indicating that polarization is of equal phase from each axis of the phase difference plot. # Acoustics and Sound: The Vocal Apparatus

The study of modulation of signals for wireless transmission can, to some extent, be applied to the human body, In the RF wireless world, a “carrier” signal of a high frequency has a “message” encoded on it (message signal) in some form or fashion. This is then transmitted through a medium (generally air) as a radio frequency electromagnetic wave.

In a similar way, the vocal apparatus of the human body performs a similar function. The lungs forcibly expel air in a steady stream comparable to a carrier wave.  This steady stream gets encoded with information by periodically varying its velocity and pressure into two forms of sound: voiced and unvoiced. Voiced sounds produce vowels and are modulated by the larynx and vocal cords. The vocal chords are bands which have a narrow slit in between them which are flexed in certain ways to produce sounds. The tightening of the cords produces a higher pitch and loosening or relaxing produces a lower pitch. In general, thicker vocal cords will produce deeper voices. The relaxation oscillation produced by this effect converts a steady air flow into a periodic pressure wave. Unvoiced sounds do not use the vocal chords.

The tightness of the vocal cords produces a fundamental frequency which characterizes the tone of voice. In addition, resonating cavities above and below the larynx have certain resonant frequencies which also contribute to the tone of voice through inharmonic frequencies, as these are not necessarily spaced evenly.

Although the lowest frequency is the fundamental and most recognizable tone within the human voice, higher frequencies tend to be of a greater amplitude. Different sounds produced will of course have different spectrum characteristics. This is demonstrated in the subsequent image. The “oo” sound appears to contain a prominent 3rd harmonic, for example. In none of these sounds is the fundamental of highest amplitude. The image also shows how varying the position of the tongue as well as the constriction or release of the larynx contributes to the spectrum.

It is interesting to note the difference between male and female voices: male voices contain more harmonic content. This is because lower multiples of the fundamentals are more represented in the male voice and are spaced closed to one another in the frequency domain.

# The Cavity Magnetron

The operation of a cavity magnetron is comparable to a vacuum tube: a nonlinear device that was mostly replaced by the transistor. The vacuum tube operated using thermionic emission, when a material with a high melting point is heated and expels electrons. When the work function of a material is overcome through thermal energy transfer to electrons, these particles can escape the material.

Magnetrons are comprised of two main elements: the cathode and anode. The cathode is at the center and contains the filament which is heated to create the thermionic emission effect. The outside part of the anode acts as a one-turn inductor to provide a magnetic field to bend the movement of the electrons in a circular manner. If not for the magnetic field, the electrons would simple be expelled outward. The magnetic field sweeps the electrons around, exciting the resonant cavities of the anode block.

The resonant cavities behave much like a passive LC filter circuit which resonate a certain frequency. In fact, the tipped end of each resonant cavity looks much like a capacitor storing charge between two plates, and the back wall acts an inductor. It is well known that a parallel resonant circuit has a high voltage output at one particular frequency (the resonant frequency) depending on the reactance of the capacitor and inductor. This can be contrasted with a series resonant circuit, which has a current peak at the resonant frequency where the two devices act as a low impedance short circuit. The resonant cavities in question are parallel resonant.

Just like the soundhole of a guitar, the resonant cavity of the magnetron’s resonance frequency is determined by the size of the cavity. Therefore, the magnetron should be designed to have a resonant frequency that makes sense for the application. For a microwaves oven, the frequency should be around 2.4GHz for optimum cooking. For an X-band RADAR, this should be closer to 10GHz or around this level. An interesting aspect of the magnetron is when a cavity is excited, another sequential cavity is also excited out of phase by 180 degrees.

The magnetron generally produces wavelength around several centimeters (roughly 10 cm in a microwave oven). It is known as a “crossed field” device, because the electrons are under the influence of both electric and magnetic fields, which are in orthogonal directions. An antenna is attached to the dipole for the radiation to be expelled. In a microwaves oven, the microwaves are guided using a metallic waveguide into the cooking chamber. # Optical Polarization, Malus’s Law, Brewster’s Angle

In the theory of wave optics, light may be considered as a transverse electromagnetic wave. Polarization describes the orientation of an electric field on a 3D axis. If the electric field exists completely on the x-axis plane for example, light is considered to be polarized in this state.

Non-polarized light, such as natural light may change angular position randomly or rapidly. The process of polarizing light uses the property of anisotropy and the physical mechanisms of dichroism or selective absorption, reflection or scattering. A polarizer is a device that utilizes these properties. Light exiting a polarizer that is linearly polarized will be parallel to the transmission axis of the polarizer. Malus’s law states that the transmitted intensity after an ideal polarizer is

I(θ)=I(0)〖cos〗^2 (θ),

where the angle refers to the angle difference between the incident wave and the transmission axis of the polarizer.

Brewster’s Angle, an extension of the Fresnel Equation is a theory which states that the difference between a transmitted ray or wave into a material comes at a 90 degree angle to the reflected wave or ray along the surface. This situation is true only at the condition of the Brewster’s Angle. In the scenario where the Brewster’s Angle condition is met, the angle between the incident ray or wave and the normal, the reflected ray or wave and the surface normal and the transmitted ray or wave and the surface normal are all equal. If the Brewster’s Angle is met, the reflected ray will be completely polarized. This is also termed the polarization angle. The polarization angle is a function of the two surfaces. # Fourth Generation Optics: Thin-Film Voltage-Controlled Polarization

###### Michael Benker ECE591 Fundamentals of Optics & Photonics April 20,2020

Introduction

Dr. Nelson Tabiryan of BEAM Engineering for Advanced Measurements Co. delivered a lecture to explain some of the latest advances in the field of optics. The fourth generation of optics, in short includes the use of applied voltages to liquid crystal technology to alter the polarization effects of micrometer thin film lenses. Both the theory behind this type of technology as well as the fabrication process were discussed.

First Three Generations of Optics

A summary of the four generation of optics is of value to understanding the advancements of the current age. Optics is understood by many as one of the oldest branches of science. Categorized by applications of phenomena observable by the human eye, geometrical optics or refractive optics uses shape and refractive index to direct and control light.

The second generation of optics included the use of graded index optical components and metasurfaces. This solved the issue of needing to use exceedingly bulky components although it would be limited to narrowband applications. One application is the use of graded index optical fibers, which could allow for a selected frequency to reflect through the fiber, while other frequencies will pass through.

Anisotropic materials gave rise to the third generation of optics, which produced technologies that made use of birefringence modulation. Applications included liquid crystal displays, electro-optic modulators and other technologies that could control material properties to alter behavior of light.

Fourth Generation Optics

To advance technology related to optics, there are several key features needed for output performance. A modernized optics should be broadband, allowing many frequencies of light to pass. It should be highly efficient, micrometer thin and it should also be switchable. This technology is currently present.

Molecule alignment in liquid crystalline materials is essential to the theory of fourth generation optics. Polarization characteristics of the lens is determined by molecule alignment. As such, one can build a crystal or lens that has twice the refractive index for light which is polarized in one direction. This device is termed the half wave plate, which polarizes light waves parallel and perpendicular to the optical axis of the crystal. Essentially, for one direction of polarization, a full period sinusoid wave is transmitted through the half wave plate, but with a reversed sign exit angle, while the other direction of polarization is allowed only half a period is allowed through. As a result of the ability to differentiate a sign of the input angle to the polarization axis (full sinusoid polarized wave), the result is an ability to alter the output polarization and direction of the outgoing wave as a function of the circular direction of polarization of the incident wave.

The arrangement of molecules on these micrometer-thin lenses are not only able to alter the direction according to polarization, but also able to allow the lens to act as a converging lens or diverging lens. The output wave, a result of the arrangement of molecules in the liquid crystal lens has practically an endless number of uses and can align itself to behave as any graded index lens one might imagine. An applied voltage controls the molecular alignment.

How does the lens choose which molecular alignment to use when switching the lens? The answer is that, during the fabrication process, all molecular alignments are prepared that the user plans on employing or switching to at some point. These are termed diffraction wave plates.

## Problem 1. The second lens is equivalent to the first (left) lens, rotated 180 degrees. In the case of a polarization-controlled birefringence application, one would expect lens 2 to exhibit opposite output direction for the same input wave polarization as lens 1. For lens 1 (left), clockwise circularly polarized light will exit with an angle towards the right, while counterclockwise circularly polarized light exits and an angle to the left. This is reversed for lens 2.

## Problem 2. There are as many states as there are diffractive waveplates. If there are six waveplates, then there will be 6 states to choose from.