# Calculating Bandwidth for RF/Photonic Components based on Velocity mismatch

The bandwidth of a device such as a modulator or photodetector is an important figure. When designing a modulator or photodetector for high frequencies, much attention is paid to matching the velocity of the optical waves and the RF waves.

By finding the propagation time difference between the optical and RF waves, we model this in the time domain as a rect function. Note that for the rect function, the difference in propagation time is the tau variable. Performing the Fourier transform on the rect function will give us a sinc function. The 3dB cutoff point of this sinc function in the frequency domain gives us the device bandwidth. Note the MATLAB algorithm used below. The 3dB bandwidth is calculated using a simple manipulation of the frequency vector indices.

v_optical = ; %simulated optical velocity [define]

v_RF = ; %simulated RF velocity [define]

l_device = ; %device length [define]

f_max = ; %max frequency of vector (should be higher than bandwidth) [define]

f_num = ; %number of frequencies in vector [define]

tau = abs((l_device/v_optical)-(l_device/v_RF)) ; %propagation time difference

W = linspace(0,f_max,f_num); %frequency vector

S = tau*sinc(W*tau/2); %sinc function in frequency domain

Qs = find(20*log10(S)<=(20*log10(S(1))-3)); %intermediate calculation for index of 3db cutoff

BW_3dB= f_max*(Qs(1))/f_num %This is the result

# I.                   Optical Isolators

## Introduction

An optical isolator is a device that allows light to travel in only one direction. Isolators have two ports and are made for free-space and optical fiber applications. Lasers benefit from isolators by preventing backscatter into the laser, which is detrimental to their performance. Other applications include fiber optic communication systems, such as CATV and RF over fiber, and gyroscopes. Isolators are magneto-optic devices that use Faraday rotators and polarizers to achieve optical isolation. The magneto-optic effect was discovered by Michael Faraday, when he observed that polarized light rotates when propagating through a material with a magnetic charge.

## Types of Isolators

Two categories of optical isolators are free-space and fiber isolators. Isolators in both these categories see use in a wide range of applications and for many wavelengths from ultraviolet to long-wavelength infrared. Isolators can be fixed for isolation at a single wavelength, tunable for multiple wavelengths, or wideband. Adjustable isolators come with a tuning ring to adjust the Faraday rotator’s position and effect in the isolator. Adjustable isolators can be narrowband for specific wavelengths or broadband adjustable.

Polarization-dependent isolators and polarization-independent isolators are two different operation concepts that are used to achieve isolation. Polarization-dependent isolators use polarizers and faraday rotators, while polarization-independent isolators use a Faraday rotator, a have-wave plate, and birefringent beam displacers. In either case, polarization is used in both systems to achieve isolation, exploiting the magneto-optic effect using the Faraday rotator.

## Key Parameters of Isolators

An isolator’s performance is measured by its transmission loss, insertion loss, isolation, and return loss. The transmission loss or S(1,2) should be low, meaning that there is no loss in optical power from the direction of port 1 to port 2. Optical power should not be transmitted from port 2 to port 1. The reduction in optical power from port 2 to port 1 is termed insertion or S(2,1) and should be high. Optical isolators can have 50 dB or higher isolation  . Insertion loss is reflected optical power from port 1 and should be reduced.

Figure 2. Optical isolator as 2-port Element

Pulse dispersion is a relevant parameter in the design of isolators for specific pulsed laser uses, such as an ultrafast laser. This is measured as a ratio between pulse time width before the isolator and the pulse time width after leaving the isolator .

For fiber isolators, the type of fiber will need to be considered for its application. Size is a consideration for many applications. The design of the magnet in the isolator often a major limiting factor in size reduction. Operating temperature and accepted optical power are two other considerations to ensure the isolator is in acceptable operating conditions and is not damaged. This information can be found in datasheets for common isolators.

## Concept of Operation

Polarization is an important concept in the operation of an optical isolator. Polarization refers to the orientation of waves transverse to the direction of propagation. Polarization can be written as a vector sum of components in two directions perpendicular to the direction of propagation.

Figure 3. Wave polarized in the x-direction and propagating in the -z-direction

A polarizer changes the polarization of an incident wave to that of the polarizer. However, light is only allowed to pass through the polarizer to the extent that the incident wave shares a level of polarization in the direction of the polarizer. The output intensity from the polarizer is defined using Malus’ Law, where is the angle between the polarization of the incident wave and the polarizer’s direction:

For example, a wave polarized in the x-direction propagating in the negative z-direction may enter a polarizer positioned in the x-direction. In this case, the angle difference between the polarizer and the incident wave is zero, meaning that full optical power is transmitted. If an angle is introduced between the polarizer and incident wave, the output intensity is reduced. Two polarizers are used in an optical isolator, as well as a Faraday rotator.

Figure 4. The angle between polarizers and incident waves

When light propagates through a magnetic material, the plane of polarization is rotated. This is termed the Faraday effect or the magneto-optic effect . The Verdet constant measures the strength of the Faraday Effect in a material. The Verdet effect units are radians per Tesla per meter, and it is a function of the wavelength, electron charge and mass, dispersion, and speed of light.

Figure 5. Rotation of Polarization using Faraday Effect

Figure 6. Optical Isolator in Forward Direction

Figure 7. Optical Isolator in Reverse Direction

Using the Faraday rotator and two polarizers, a non-reciprocal isolator is made. The forward and reverse directions for an isolator are demonstrated below, made of a Faraday rotator and two polarizers. The Faraday rotator is made to rotate the polarization 45 degrees in a clockwise direction.

Since the clockwise rotation is different respective to the direction of light in the Faraday rotator, a non-reciprocal effect is expected. The two polarizers are positioned at 45 degrees from one another. In the forward direction, the direction of polarization of the optical wave matches that of the second polarizer, meaning that full optical power is present at the isolator’s output. In the reverse direction, the 45 degrees difference from the Faraday rotator and 45 degrees angle difference from the polarizers are applied constructive, producing a 90-degree difference in polarization direction for the second polarizer. Using Malus’s law, the resultant intensity for the outgoing light wave is zero. Since the Verdet constant is dependent on wavelength, there is a variety of materials that can be used to demonstrate the Faraday effect. For 1550 nm light, widely used in telecommunications, Yttrium Iron Garnet has a strong effect and sees use in optical isolators .

## Designing an Isolator

To design an isolator, we must first consider the wavelength that we are using, the material that we are using for Faraday rotation, and the electromagnet output B field. Recall that the Verdet constant is a function of the wavelength. For Yttrium Iron Garnet, the Verrdet constant is 304 radians/Tesla/meter for 1550 nm-wavelength light . The rotation of the polarization plane for linearly polarized light is:

where L is the length of the magneto-optic material and B is the applied magnetic field to the magneto-optic material. Given that we are looking for a Faraday rotation of 45 degrees, we can either solve for the required magnet field strength B given a certain length L, or we can determine how long (L

the Faraday rotator should be given an applied magnetic field strength, B. An optical isolator can be described using the following formations. The first polarizer is for polarization in the x direction.

The Faraday rotator rotates the polarization by radians and then goes through the second polarizer, which is 45 degrees to the first polarizer.

The forward and reverse polarization can be calculated as follows:

The components of the resulting Jones’ Matrices for forward and reverse directions are plotted, while varying the magnetic field strength to find the optimal strength B for reverse isolation.

Figure 8. Forward Direction, Polarization from Isolator vs. Magnetic Field Strength

The Faraday rotator is 1 mm long, rotates polarization by pi/4 radians, and is made of Ytterbium Iron Garnet with a Verdet constant of 304 radians/Tesla/meter, I can find what strength magnetic field should be applied. The result shows that the optimal field strength is 2.584 Tesla to achieve full isolation.

Figure 9. Backward Direction, Polarization from Isolator vs. Magnetic Field Strength

The output matricies for the forward, reverse directions and the Faraday rotation matrix are then found to be:

N that the dimensions of the isolator have been calculated, these numbers can be loaded into an optical simulation for the permittivity tensor of the material. Simulation as an approach to designing an isolator is also useful to measure at which length the polarization has rotated the desired amount.

## Other Magneto-Optic Devices

The optical isolator is a magneto-optic component. Closely related is the circulator. The circulator is a non-reciprocal component with multiple ports and is commonly used for transceivers and radar receivers. When light enters a four-port circulator, the output port is dependent on the input port. A simple diagram and S-matrix are shown below for a four-port circulator, where the columns denote where light entered, and the row denotes which port light exits. Like the isolator, the circulator also uses Faraday rotation.

Figure 11. Circulator Concept and S-Matrix

Other magneto-optic devices include beam-deflectors, multiplexers, displays, magneto-optic modulators . Magneto-optic memory devices, including disks, tapes, and films, were commonplace but have been largely replaced by solid-state memory. Thin-film magneto-optic waveguides have also been demonstrated . Magnetic-tunable optical lenses allow for a dynamically tunable focal length .

# II.                Photonic Integrated Isolators

## Introduction

Integrated photonics is a technology that, like integrated electronics, allows for many components to be made on a single semiconductor chip, allowing for more complex systems with reduces size and improved reliability. The photonic IC market was about \$190M in 2013 and is estimated between \$1.3B and \$1.8B in 2022 . An integrated isolator is a highly sought technology, currently in the beginning stages of commercial availability and still in research and development. Photonic integrated circuits are usually developed on Silicon, Indium Phosphide and Gallium Arsenide; each having advantages. Indium Phosphide is of particular importance for the telecommunications wavelength (C-band at 1550nm) because this platform is used for lasers as well as photodetectors. Other methods to include active components such as lasers made on Indium Phosphode onto a Silicon wafer have been realized with limited success. Given the benefit of an isolator to a laser, an integrated isolator, if designed well could improve the performance of the semiconductor laser on Indium Phoshide.

Figure 12. Photonic Integrated Circuit

As mentioned previously, the circulator is based on the same non-reciprical Faraday Effect. When conducting research on integrated isolators, it is of interest to consider that advancements in integrated isolator technology can be applied to design an integrated circulator. However, the circulator is also used for a very different purpose, which may be more compatible to passive integrated photonics on a platform such as silicon. Integrated circulators can enable the realization of miniaturized receivers and a wide range of applications.

Integrated isolators can improve the performance of semiconductor lasers by reducing backscattering. Laser backscattering can reduce the laser linewidth and relative intensity noise (RIN), a limiting factor for next-generation high dynamic-range microwave photonic systems. Improving the signal to noise ratio by reducing RIN noise in a microwave photonic link can therefor allow for an overall signal-to-noise ratio (SNR).

Figure 13. Noise Limits in an RF Photonic Link

Benchtop laser units typically come packaged with isolating components as needed to improve performance, but to design an entire high dynamic range microwave photonic system on a chip, the integrated lasers will need sufficient isolation to ensure that the system works properly. For a fully integrated system, an integrated isolator therefore can reduce the noise floor and enable high dynamic range system operation.

Figure 14. High Dynamic Range Integrated Microwave Photonic System

Because of the potential of integrated isolators, the US Air Force has taken an interest in this technology and sought to introduce this technology to the AIM Photonics Foundry in Albany, New York . AIM Photonics currently includes an integrated isolator in its process design kit.

The first challenge in making an integrated isolator is the question of making an integrated Faraday rotator. Since the magneto-optic effect is dependent on the length of the Faraday Rotator, this may cause an issue with the size of the component. Semiconductor platforms do not exhibit a magneto-optic effect. One main material used for the magneto-optic effect at the 1550 nm wavelength is Yttrium Iron Garnet (YIG). For fiber isolators and free-space isolators, light propagates through the magneto-optic material. Two solutions were made for the issue of using YIG on a photonic IC: YIG waveguides  and layering YIG on top of the optical waveguides , each including the use of an electromagnet for the magnetic field. When layering YIG on top of the optical waveguides, the magneto-optic effect is applied to the evanescent waves outside of the waveguide, producing a weaker effect. The advantage of layering YIG rather than using YIG waveguides is its relative ease of fabrication.

## Challenges: Fabrication

Fabrication with garnets and semiconductors is one challenge for integrated isolators. One issue is that garnets are not typically used in semiconductor fabrication, presenting several unique challenges specific to their material properties. It has found to be unreliable especially in the deposition process . When growing semiconductor materials, the wafer is exposed to very high temperatures. Thermal expansion mismatch between garnets and semiconductors makes growing YIG on semiconductor difficult without cracking. To avoid thermal expansion mismatch, alternate methods using lower temperatures are used such as rapid annealing (RTA) . While direct bonding techniques are preferred over YIG waveguides, YIG waveguides on deposited films are made using a H3PO4 wet etch .

Polarizers in integrated photonics are achieved using waveguide polarizers. Waveguide polarizers have been realized using a variety of approaches, including metal-cladding and birefrinfence waveguides , photonic crystal slab waveguides . Polarizers have been fabricated using ion beam lithography  .

Figure 15. Integrated Waveguide Polarizer Example

Another question related to the exploitation of Faraday rotation on an integrated isolator is the design of the optical waveguide structure and phase shift or use of polarizers to reduce optical power in the reverse direction. Two designs are a microring resonator and a Mach-Zehnder interferometer. Due to the challenges of desigon developing integrated isolators that do not use the magneto-optic effect  . A non-magneto-optic isolator was designed as a Mach-Zehnder Interferometer, providing some backwards isolation .

Figure 16. Travelling-Wave MZ Modulator as Isolator

## Challenges: TE and TM Polarization

One issue with integrated isolators is that they require light to be TM polarized for operation, making them incompatible with TE polarized lasers. To circumvent this issue, some isolator designs are being optimized for TE polarization, or include polarization rotators between the laser and isolator . The polarization rotator between the laser and isolator would then need to have low loss and a high polarization extinction ratio. Below is a model for an integrated waveguide polarization converter from TE1 to TM0 modes , which would follow after a TE0 to TE1 mode coupler:

Figure 17. TE1 to TM0 Converter for Integrated Isolators

## Challenges: Performance

The performance of current integrated isolator designs is a major drawback. Integrates isolators should provide wide band isolation across the C-band, have high isolation, and low insertion loss. Wideband operation ensures that the isolator will prevent backscattering from all wavelengths from a C-band laser. Achieving low insertion loss is needed to prevent optical loss. Finally, isolation measures how much loss is provided in the reverse direction, which for an isolator should be high.

Discrete component isolators can offer up to 60 dB isolation with <1dB insertion loss. Integrated isolators have been shown with much lower isolation and often large insertion loss, while being too narrowband for some applications. Improving their performance is a research topic still being explored. Large optical loss hass been theorized to be due to the loss from scattering of the YIG layered above the waveguide, the interface of the bond and losses in YIG material. Other methods for improving on previous designs include electromagnet design and waveguide design, especially for coupling of the electromagnet and a microring resonator insolator design .

Figure 18. Integrated Isolator Performance

## Design on Indium Phosphide

Design of integrated isolators on the Indium Phosphide platform has the advantage of being integrated directly after a 1550 nm wavelength laser, providing numerous benefits to the integrated laser. There remains an interest in integrating isolators on this platform with the laser rather than a separate, discrete component. One design uses Yttrium Iron Garnet as the magneto-optic material with a Mach-Zehnder interferometer structure and a reciprocal phase shifter  . A design on the indium phosphide platform was proposed in 2007, utilizing the magneto-optic effect on an MZI design and achieving greater than 25 dB isolation over the telecommunications wavelength C-band . Another was proposed and developed in 2008 as an interferometric isolator on the indium phosphide platform based on non-reciprocal phase shifts in the modulator’s arms . The YIG magneto-optic material was bonded using a surface activated direct bonding technique .

Figure 19. Integrated Isolator Design on InP with DFB Laser

The above design also features a 3×2 MMI coupler, so that the backward light wave is radiated out of the sides of the coupler, avoiding backscatter to the laser.

## Design on Silicon

Microring resonator isolators and Mach-Zehnder Interferometers (MZI) utilizing the magneto-optic effect have been developed on heterogeneous silicon integration platforms. Integrated micro-ring resonator isolators provide isolation with low insertion loss, but the isolation is too narrow for most applications . An advantage to the MZI integrated isolator is its high isolation. However, this is largely offset by its large optical insertion loss, making them impractical for most RF Photonics applications .

Figure 20. Integrated MZI and Micro-ring Isolator

## Integrated Circulator

Designing an integrated circulator has several similarities to an integrated isolator, since they are both based on the non-reciprical magneto-optic effect.

Figure 21. Integrated Isolator Design using Mircoring Resonator

## Conclusion

In conclusion, isolators, and particularly integrated isolators, show a promising future for enabling photonics technology in the future. There is a clear demand for isolators to enable narrow-linewidth lasers with low RIN noise and considerable effort to apply new approaches to integration technology to realize an isolator that is wideband with high isolation and low insertion loss.

# Bibliography

## MATLAB Code

%Michael Benker

%ECE591 Photonic Devices

clf;

A_P2 = pi/4 %Polarization shift of P2

P1 = [1,0;0,0] %P1 Matrix

P2=[0.7071,0;0.7071,0] %P2 Matrix

B=2.584; %Magnetic Field

L = 0.001; %Length

V = 304; %Verdet Constant

beta = B*V*L; %Polarization shift

Forward = P2*FR*P1

Backward = P1*FR*P2

h=101; %Number of points on the plot

for x=1:h

B=1.5+(x-1)*(3-1.5)/h;

Bvect(x) = B;

beta = B*V*L;

FR= [cos(beta),-sin(beta);sin(beta),cos(beta)];

Forward = P2*FR*P1;

Backward = P1*FR*P2;

Forw11(x) = Forward(1,1);

Forw21(x) = Forward(2,1);

Forw12(x) = Forward(1,2);

Forw22(x) = Forward(2,2);

Back11(x) = Backward(1,1);

Back21(x) = Backward(2,1);

Back12(x) = Backward(1,2);

Back22(x) = Backward(2,2);

end

figure(1)

plot(Bvect,Forw11)

hold on

plot(Bvect,Forw12)

plot(Bvect,Forw21)

plot(Bvect,Forw22)

legend([‘F(1,1)’;’F(1,2)’;’F(2,1)’;’F(2,2)’])

title([‘Isolator: Forward Direction vs. B field (L =1mm,V=304)’])

ylabel(‘Matrix component value’)

xlabel(‘Magnetic Field Strength B’)

figure(2)

plot(Bvect,Back11)

hold on

plot(Bvect,Back12)

plot(Bvect,Back21)

plot(Bvect,Back22)

legend([‘B(1,1)’;’B(1,2)’;’B(2,1)’;’B(2,2)’])

title([‘Isolator: Backwards Direction vs. B field (L =1mm,V=304)’])

ylabel(‘Matrix component value’)

xlabel(‘Magnetic Field Strength B’)

# Photolithography for Device Fabrication

Introduction

Photolithography is a technique used in semiconductor device fabrication. A light sensitive layer is added to a semiconductor wafer. Light is applied to the parts of the light-sensitive layer to remove the layer where needed. After this is done, etching can be performed exclusively to the parts of the wafer without a layer of the photo-sensitive layer.

Photoresist

The light-sensitive layer added to the wafer is called the photoresist. To apply photoresist to a wafer, first clean the wafer. The photoresist should cover most of the wafer. This should be done while the wafer is sitting in the spinner. The spinner rotates the wafer so that the photoresist has an even coating. Then the spun wafer is placed on a heating plate. The RPM and temperature of the hot plate are important. The photoresist data sheet can indicate the required spin rate and temperature.

When the photoresist layer is uneven, an interference pattern can be seen on the wafer, as shown below. The interference pattern shown is less than ideal. It is normal however that there will be more photoresist build-up at the edges of the wafer. The excess photoresist at the edges of the wafer can be removed using acetone and a q-tips or swabs.

There is also the question of whether to cut the wafer before applying photoresist or afterwards. The advantage of cutting the wafer afterwards is that the built up layer of photoresist at the edges will not be used, since the cut wafer will be taken from the middle. A disadvantage of cutting the wafer afterwards is that cutting the wafer can cause damage to the photoresist layer. Also, the issue of built up photoresist at the edges of the cut wafer will remain. If the cut wafer is not round, there will be more build-up at corners of the cut wafer. If using a silicon wafer, cutting a clean square will be more difficult than when using a III-V semiconductor. Generally, one can create a square or rectangle by making a notch in the side of the wafer. The lattice of the material will cause a break to be a straight line.

The mask is what is used to select which parts of the photoresist layer will be removed and which will stay. Masks are ordered from a company such as Photronics with a .gds file for the die. The mask and the cut wafer are placed in a mask aligner. Here, a vacuum press is used. UV light is directed at the wafer from above the mask aligner.

Developing

The UV light from the mask aligner breaks the bond of the photoresist where it was applied. Now the broken-bond photoresist needs to be removed using a developer solution. The amount of time that the wafer is rinsed in the developer solution is critical. Too much time can cause the photoresist to be removed further than needed. This is especially important for small features, such as a waveguide.

Next, we can view the wafer using a microscope. The combination of the thickness of the photoresist, how even the photoresist layer is, the type of photoresist, how fast it was spun, how hot it was baked on the hot plate, the mask, the developer solution, how long it was soaked in the developer solution and how much care was given to the wafer during the process, including the presence of dust will all contribute to the overall result of the wafer. Below, curved waveguides are shown on the microscope. These are layers of photoresist. For a higher magnification, an electron microscope can be used.

# Photonic Components: Multimode Interference Waveguides

Multimode Interference Waveguides, also termed MMI Couplers, are used to split light from one waveguide into two or more paths. MMI couplers are designed to match the power at each output port. The length, width and positioning of the output ports are critcal to the design of the MMI coupler. The MMI coupler is also difficult to build in device fabrication due to the sensitivity of the width of the multimode waveguide to the performance.

Below are two MMI couplers, designed in Rsoft. The 3dB Coupler has two output ports of half the input power. The approach for both couplers is to monitor the optical power at each output port in the simulation. Initially, we design the length of the multimode section to be longer than estimated. The length of the multimode waveguide section is reduced to the length at which the optical power in each of the output paths is equal.

3dB Coupler

MMI Coupler

# Designing a Waveguide Photodetector in Rsoft

The following images depict the first stages of a waveguide photodetector design in Rsoft., The input waveguide is 2 microns, followed by a tapered section to a 10 micron wide photodetector region. Three tapering typologies are used. Following these initial simulations come optimization of the photodetector region and electrical simulations.

First, the layer view. This section is at the input waveguide.

The InGaAs layers above the waveguide serve to absorb the optical power in the photodetector region:

Three different input tapers are used:

Exponential Taper:

Absorption in the photodetector region is in the range of 95%.

Here is the optical power remaining in the waveguide region:

Linear Taper:

# IMD3: Third Order Intermodulation Distortion

We’ll begin a discussion on the topic of analog system quality. How do we measure how well an analog system works? One over-simplistic answer is to say that power gain determines how well a system operates. This is not sufficient. Instead, we must analyze the system to determine how well it works as intended, which may include the gain of the fundamental signal. Whether it is an audio amplifier, acoustic transducers, a wireless communication system or optical link, the desired signal (either transmitted or received) needs to be distinguishable from the system noise. Noise, although situationally problematic can usually be averaged out. The presence of other signals are not however. This begs the question, which other signals could we be speaking of, if there is supposed to be only one signal? The answer is that the fundamental signal also comes with second order, third order, fourth order and higher order distortion harmonic and intermodulation signals, which may not be averaged from noise. Consider the following plot:

We usually talk about Third Order Intermodulation Distortion or IMD3 in such systems primarily. Unlike the second and fourth order, the Third Order Intermodulation products are found in the same spectral region as the first order fundamental signals. Second and fourth order distortion can be filtered out using a bandpass filter for the in-band region. Note that the fifth order intermodulation distortion and seventh order intermodulation distortion can also cause an issue in-band, although these signals are usually much weaker.

Consider the use of a radar system. If a return signal is expected in a certain band, we need to be able to distinguish between the actual return and differentiate this from IMD3, else we may not be able to trust our result. We will discuss next how IMD3 is avoided.

# Mode Converters and Spot Size Converters

Spot size converters are important for photonic integrated circuits where a coupling is done between two different waveguide sizes or shapes. The most obvious place to find a spot size converter is between a waveguide of a PIC and a fiber coupling lens.

Spot size converters feature tapered layers on top of a ridge waveguide for instance, to gradually change the mode while preventing coupling loss.

The below RSoft example shows how an optical path is converted from a more narrow path (such as a waveguide) to a wider path (which could be for a fiber).

While the following simulation is designed in Silicon, similar structures are realized in other platforms such as InP or GaAs/AlGaAs.

RSoft Beamprop simulation, demonstrating conversion between two mode sizes. Optical power loss is calculated in the simulation for the structure. This is the 3D structure. Notice the red section present carries the more narrower optical path and this section is tapered to a wider path. The material layers are shown: Structure profile: # Discrete-Time Signals and System Properties

First, a comparison between Discrete-Time and Digital signals:

Discrete-Time and Digital signal examples are shown below:

Discrete-Time Systems and Digital Systems are defined by their inputs and outputs being both either Discrete-Time Signals or Digital Signals.

### Discrete-Time Signals

Discrete-Time Signal x[x] is sequence for all integers n.

Unit Sample Sequence:
𝜹[n]: 1 at n=0, 0 otherwise.
Unit Step:
u[n] = 1 at n>=0, 0 otherwise.

Or,

Any sequence: x[n] = a1* 𝜹[n-1] + a2* 𝜹[n-2]+…
where a1, a2 are magnitude at integer n.
or,

### Exponential & Sinusoidal Sequences

Exponential sequence: x[n] = A 𝞪n
where 𝞪 is complex, x[n] = |A|ej𝜙 |𝞪|e0n=|A||𝞪|n ej(ω0n+𝜙)
= |A||𝞪|n(cos(ω0n+𝜙)+j sin(ω0n+𝜙))
Complex and sinusoidal: -𝝅< ω0< 𝝅 or 0< ω0< 2𝝅.

Exponential sequences for given 𝞪 (complex 𝞪 left, real 𝞪 right):

Periodicity:        x[n] = x[n+N],  for all n. (definition). Period = N.
Sinusoid: x[n] = A cos(ω0n+𝜙) = A cos (ω0n+ ω0N+ 𝜙)
Test: ω0N = 2𝝅k,                            (k is integer)

Exponential: x[n] = e0(n+N) = e0n,
Test: ω0N = 2𝝅k,                            (k is integer)

### System Properties

System: Applied transformation y[n] = T{x[n]}

Memoryless Systems:

Output y[nx] is only dependent on input x[nx] where the same index nx is used for both (no time delay or advance).

Additive property:         Where y1[n] = T{x1[n]} and y2[n] = T{x2[n]},
y2[n] + y1[n] = T{x1[n]+ x2[n]}.

Scaling property:            T{a.x[n]} = a.y[n]

Time-Invariant Systems:

Time shift of input causes equal time shift of output. T{x[n-M]} = y[n-M]

Causality:

The system is causal if output y[n] is only dependent on x[n+M] where M<=0.

Stability:

Input x[n] and Output y[n] of system reach a maximum of a number less than infinity. Must hold for all values of n.

### Linear Time-Invariant Systems

Two Properties: Linear & Time-Invariant follows:

“Response” hk[n] describes how system behaves to impulse 𝜹[n-k] occurring at n = k.

• Convolution Sum: y[n] = x[n]*h[n].

Performing Discrete-Time convolution sum:

1. Identify bounds of x[k] (where x[k] is non-zero) asN1 and N2.
2. Determine expression for x[k]h[n-k].
3. Solve for

General solution for exponential (else use tables):

Graphical solution: superposition of responses hk[n] for corresponding input x[n].

### LTI System Properties

As LTI systems are described by convolution…

LTI is commutative: x[n]*h[n] = h[n]*x[n].

… is additive: x[n]*(h1[n]+h2[n]) = x[n]*h1[n] + x[n]*h2[n].

… is associative: (x[n]*h1[n])*h2[n] = x[n]*(h1[n]*h2[n])

LTI is stable if the sum of impulse responses

… is causal if h[n] = 0 for n<0                  (causality definition).

Finite-duration Impulse response (FIR) systems:

Impulse response h[n] has limited non-zero samples. Simple to determine stability (above).

Infinite-duration impulse response (IIR) systems:

Example: Bh=

If a<1, Bh is stable and (using geom. series) = Delay on impulse response: h[n] = sequence*delay = (𝜹[n+1]- 𝜹[n])* 𝜹[n-1] = 𝜹[n] – 𝜹[n-1].

______________________________________________________

Continued:

# Arrayed Waveguide Grating for Wavelength Division Multiplexing

Arrayed Waveguide Grating (or AWG) is a method for wavelength division multiplexing or demultiplexing. The approach for multiplexing is to use unequal path lengths to generate a phase delay and constructive interference for each wavelength at an output port of the AWG. Demultiplexing is done with the same process, but reversed.

Arrayed Waveguide Gratings are commonly used in photonic integrated circuits. While Ring Resonators are also used for WDM, ring resonators see other uses, such tunable or static filters. Further, a ring resonator selects a single wavelength to be removed from the input. In the case of AWGs, light is separated according to wavelength. For many applications, this is a more superior WDM, as it offers great capability for encoding and modulating a large amount of information according to a wavelength.

Both the design of the star coupler and the path length difference according to the designed wavelength division make up the significant amount of complexity of this component. RSoft by Synopsys includes an AWG Utility for designing arrayed waveguide gratings.

Using this utility, a star coupler is created below:

# Methods of Optical Coupling

An optical coupler is necessary for transferring optical energy into or out of a waveguide. Optical couplers are used for both free-space to waveguide optical energy transmission as well as a transmission from one waveguide to another waveguide, although the methods of coupling for these scenarios are different. Some couplers selectively couple energy to a specific waveguide mode and others are multimode. For the PIC designer, both the coupling efficiency and the mode selectivity are important to consider for optical couplers.

Where the coupling efficiency η is equal to the power transmitted into the waveguide divided by the total incident power, the coupling loss (units: dB) is equal to
L = 10*log(1/η).

Methods of optical coupling include:

• Direct Focusing
• End-Butt Coupling
• Prism Coupling
• Grating Coupling
• Tapered Coupling (and Tapered Mode Size Converters)
• Fiber to Waveguide Butt Coupling

### Direct Focusing for Optical Coupling

Direct focusing of a beam to a waveguide using a lens in free space is termed direct focusing. The beam is angled parallel with the waveguide. This is also one type of transverse coupling. This method is generally deemed impractical outside of precision laboratory application. This is also sometimes referred to as end-fire coupling.

### End-Butt Coupling

A prime example of end-butt coupling is for a case where a laser is fixated to a waveguide. The waveguide is placed in front of the laser at the light-emitting layer.

### Prism Couplers

Prism coupling is used to direct a beam onto a waveguide when the beam is at an oblique incidence. A prism is used to match the phase velocities of the incident beam and the waveguide.

### Grating Couplers

Similar to the prism coupler, the grating coupler also functions to produce a phase match between a waveguide mode and an oblique incident beam. Gratings perturb the waveguide modes in the region below the grating, producing a set of spatial harmonics. It is through gratings that an incident beam can be coupled into the waveguide with a selective mode.

### Tapered Couplers

Explained in one way, a tapered coupler intentionally disturbs the conditions of total internal reflection by tapering or narrowing the waveguide. Light thereby leaves the waveguide in a predictable manner, based on the tapering of the waveguide.

### Tapered Mode Size Converters

Mode size converters exist to transfer light from one waveguide to another with a different cross-sectional dimension.

### Butt Coupling

The procedure of placing the waveguide region of a fiber directly to a waveguide is termed butt coupling.

# Programs for PIC (photonic Integrated Circuit) Design

For building PICs or Photonic Integrated Circuits, there are a number of platforms that are used in industry today. Lumerical Suite is a major player for instance with built in simulators. Cadence has a platform that can simulate both photonic and electronic circuits together, which for certain applications provides a major advantage. There are two platforms that I’ve become familiar with, which are the Synopsys PIC Design Suite (available for students with an agreement, underwritten by a professor at your university to ensure it’s use is for only educational purposes) and Klayout using Nazca Design packages.

Synopsys is another great company with advanced programs for photonic simulation and PIC design. Synopsys Photonic Design Suite can include components that are designed using Rsoft. OptoDesigner is the program in the PIC design suite where PICs are designed, yet the learning curve may not be what you were hoping. The 3,000+ page manual let’s the user dive into the scripting language PheoniX, which is necessary to learn for PIC design using Synopsys. Using a scripting language means that designing your PIC can be automated, thereby eliminating repetitive designing. There also comes other advantages to this such as being able to fine tune one’s design without needing to click and drag components. Coding for PIC design might sound tedious, but if you start to use it, I think you’ll realize that it’s really not and that it’s a very powerful way of designing PICs. If you’d like to use PheoniX scripting language using the Synopsys PIC design suite, note that the scripting language is similar to C.

One of the greatest aspects of OptoDesigner and the PIC Design Suite is the simulation capabilities. Much like the simulations that can be run in Rsoft, these are available in OptoDesigner.

The downside of Synopsys PIC Design Suite is in the difficulty of obtaining a legal copy that can be used for any and all purposes, even commercial. I mentioned that I obtained a student version. This is great for learning the software, to a certain extent. The learning stops when I would like to build something that could be sent out to a foundry for manufacture. Let’s be honest though, there is a lot to learn before getting to that point. Still, if we would even like to use a Process Design Kit (PDK) which contains the real component models for a real foundry so that you can submit your design to be built on a wafer, you will need to convince Synopsys that the PDK is only used for educational purposes and not only for learning, but as part of an education curriculum. If your university let’s you get your hands on a PDK with Synopsys Student version, you will essentially have free range to design PICs to your hearts content. If you have a student version, you’ll still have to buy a professional version if you want to design a PIC using a foundy PDK, submit it for a wafer run and sell it. I’ll let you look up the cost for that. The best way to use Synopsys is to work for a company that has already paid for the profession version, in conclusion.

Now, if you find yourself in the situation where all the simulation benefits of using OptoDesigner are outweighed by the issue of needing to perform a wafer run, you might just want to use Klayout with Nazca Design photonic integrated circuit design packages. These are both open source. Game changer? Possibly. Suddenly, you picture yourself working as an independent contractor for PIC design someday and you’ll have Klayout to thank.

Klayout and the Nazca Design packages are based on the very popular Python programming language. Coding can be done in Spyder, Notepad or even Command Prompt (lol?). If you aren’t familiar with how Python works, PIC design might move you to learn. Python takes the place of PheoniX scripting language as is used in OptoDesigner, so you still have the automation and big brain possibilities that a scripting language gives you for designing PICs. As for simulations, you’ll have to go with your gut, but you could use discrete components to design your circuit and evaluate that.

Klayout doesn’t come with a 3,000+ page manual, but you’ll likely find that it is a simpler to use than OptoDesigner. Below is a Python script, which generates a .gds file and then the file opened in Klayout.

# Ring Resonators for Wavelength Division Multiplexing

The ring resonator is a rather simple passive photonic component, however the uses of it are quite broad.

The basic concept of the ring resonator is that for a certain resonance frequency, those frequencies entering port 1 on the diagram below will be trapped in the ring of the ring resonator and exit out of port 3. Frequencies that are not of the resonance frequency will pass through to port 2. Ring resonators can be used for Wavelength Division Multiplexing (WDM). WDM allows for the transmission of information allocated to different wavelengths simultaneously without interference. There are other methods for WDM, such as an Asymmetric Mach Zehnder Modulator.

Here I present one scheme that will utilize four ring resonators to perform wavelength division multiplexing. The fifth output will transmit the remaining wavelengths after removing the chosen wavelengths dependent on the resonating frequency (and actually, the radius) of the ring resonators. # Quantum Well: InP-InGaAsP-InP

Quantum wells are widely used in optoelectronic and photonic components and for a variety of purposes. Two materials that are often used together are InP and InGaAsP. Two different models will be presented here with simulations of these structures. The first is an InP pn-junction with a 10 nm InGaAsP (unintentionally doped) layer between. The second is an InP pn-junction with 10 nm InGaAsP quantum wells positioned in both the positive and negative doped regions.

Quantum Well between pn-junction The conduction band and valence band energies are depicted below for the biased case: The conduction current vector lines: ATLAS program:

go atlas
Title Quantum Wells
# Define the mesh
mesh auto
x.m l = -2 Spac=0.1
x.m l = -1 Spac=0.05
x.m l = 1 Spac=0.05
x.m l = 2 Spac =0.1
#TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.5 material = InP NY = 10 acceptor = 1e18
region num=3 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393  y.comp = 0.3048
region num=2 bottom thick = 0.5 material = InP NY = 10 donor = 1e18
# Electrode specification
elec       num=1  name=anode  x.min=-1.0 x.max=1.0 top
elec       num=2  name=cathode   x.min=-1.0 x.max=1.0 bottom

#Gate Metal Work Function
contact num=2 work=4.77
models region=1 print conmob fldmob srh optr
models region=2 srh optr
material region=2

#SOLVE AND PLOT
solve    init outf=diode_mb1.str master
output con.band val.band e.mobility h.mobility band.param photogen opt.intens recomb u.srh u.aug u.rad flowlines
tonyplot diode_mb1.str
method newton autonr trap  maxtrap=6 climit=1e-6
solve vanode = 2 name=anode
save outfile=diode_mb2.str
tonyplot diode_mb2.str
quit
Quantum Well layers inside both p and n doped regions of the pn-junction
Structure: Simulation results: #TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.25 material = InP NY = 10 acceptor = 1e18
region num=3 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393  y.comp = 0.3048
region num=4 bottom thick = 0.25 material = InP NY = 10 acceptor = 1e18
region num=2 bottom thick = 0.25 material = InP NY = 10 donor = 1e18
region num=6 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393  y.comp = 0.3048
region num=2 bottom thick = 0.25 material = InP NY = 10 donor = 1e18

# Capacitance and Parallel Plate Capacitors

Capacitance relates two fundamental electric concepts: charge and electric potential. The formula that relates the two is Capacitance = charge / electric_potential.

The term equipotential surface refers to how a charge, if moved along a particular path or surface, the work done on the field is equal to zero. If there are many charges along the surface of a conductor (along an equipotential surface), then the potential energy of the charged conductor will be equal to 1/2 multiplied by the electric potential φ and the integral of all charges along this surface.

Ue = ½ φ ∫ dq.

Given a scenario in which both charge and electric potential are related, we may introduce capacitance. The following formula proves important for calculating the energy of a charged conductor:

Ue = ½ φ q = ½ φ2 C = q2 / (2C).

A parallel plate capacitor is a system of metal plates separated by a a dielectric. One plate of the capacitor will be positively charged, while the other is negatively charged. The potential difference and charge on the capacitor places causes a storage of energy between the two plates in an electric field. # Electric Potential and Electric Potential Energy

Electric potential can be summarized as the work done by an electric force to move a charge from one point to another. The units are in Volts. Electric potential is not dependent on the shape of the path that the work is applied. Being a conservative system, the amount of energy required to move a charge in a full circle, to return it back to where it started will be equal to zero.

The work of an electrostatic field takes the formula

W12 = keqQ(1/r1 – 1/r2),

which is found by integrating the the charge q times the electric field. The work of an electrostatic field also contains both the electric potential and electric potential energy. Electric potential energy, U is equal to the electric potential φ multiplied by the charge q. Electric potential energy is a difference of potentials, while electric potential uses the exact level of electric potential in the given case. To calculate electric potential energy, it is convenient to assume that the potential energy is zero at a distance of infinity (and surely it should be). In this case, we can write the electric potential energy as equal to the work needed to move a charge from point 1 to infinity. We’ll consider a quick application related to both the dipole moment and the electric potential. The dipole potential takes the formula in the figure below. Dipole potential decreases faster with distance r than it would for a point charge. 