# The Sommerfeld Model – Metals

Following the Drude model describing the movement of electrons in metals, Sommerfeld developed yet another model for electrons in metals in 1927. This new model would account for electron energy distributions in metals, Pauli’s exclusion principle, and Fermi-Dirac statistics of electrons. This model factors in quantum mechanics and the Schrödinger Equation.

The Somerfeld model’s view of electrons in metals can be taken as an example of a large volume of metal with electrons confined in the volume. We’ll call this a potential well. Inside this volume or potential well, electrons are ‘free’ with zero potential. Outside the potential well, the potential is infinite. The electron states inside this box are governed by the Schroedinger equation.

The quantum state of an electron is generally described by the Schroedinger equation as shown below.

The result of the Schroedinger equation, applying the boundary conditions of the problem with the potential being zero at the boundary, the solution of the wavefunction is below. This solution introduces a concept called the k-space, a 3D grid of allowed quantum states.

The density of grid points in the k-space is related to Lx, Ly, and Lz of the solution. The number of grid points per unit volume or density (i.e. density of states) will be V/(2*pi)^3, where the spacing of points in the 3D k-space are defined as 2pi/Lx, 2pi/Ly, and 2pi/Lz.

# Drude Model – Metals

In 1900, Paul Drude formed a theory of conduction in metals using the newly discovered concept of the electron. The theory states:

1. Metals have a high density of free electrons.
2. Electrons in metals move according to Newton’s laws
3. Electrons in metals scatter when encountering defects, ions and the momentum of the electron is random after this scattering event.

In short, the Drude Model explains how electrons can be expected to move in metals, which is fundamental to the operation of many devices.

Applied Electric Field

In the presence of an electric field, electron motion on average can be described with the following momentum p(t), when tau is the scattering time and 1/tau is the scattering rate:

Now consider several cases for electron motion in metals.

No applied electric field, E(t) = 0
In this case, electrons move randomly and the electron path momentum averages zero.

Constant Uniform Electric Field
When a uniform electric field is present, electron movement averages in the opposite direction of the electric field.

Relating momentum to velocity, we can find the electron drift velocity which is the rate at which the electron travels in an average direction caused by the field. The electron mobility is the relationship between drift velocity and the electric field.

Electron current density is related to the number of electrons, an electron’s charge, the mobility, and the electric field. The factor between electron current density and electric field is the conductivity.

# Nanofabrication Processes

Due to the size of the structures made during nanofabrication, the semiconductor wafers are highly vulnerable to foreign objects, such as dust. The waveguide structures being fabricated see a width of 2 microns. Interference by a piece of dust of similar width results in device fabrication failure if left on the wafer before depositing an oxide layer. For this reason, cleanrooms and their procedures are designed to eliminate dust particles and organic material that may invade the device fabrication process. Still, caution must be taken at each step to ensure that there is no dust on one’s sample. The wafer should be observed under a microscope before each etching and depositing procedure and cleaned using the appropriate method, depending on the previous and next steps.

If the wafer does not have a layer of photoresist on it, the cleaning procedure is to dip the wafer into a beaker of acetone, then isopropanol and then DI water. The typical duration of this is for about 1 minute each. If there is a layer of old photoresist on the wafer, this may need to be increased to five minutes each. If old photoresist remains, a flood UV exposure using the mask aligner and development may proceed a second solvent wash. The RIE machine is also used for removing photoresist on wafers if used on an O2 descum process.

Masks are templates used in the mask aligner to create patterns on wafers. These plates must also be cleaned after using, since they will make contact with wafers that have photoresist on them. This is especially critical for masks that have small patterns (<20 microns). After finishing use, soak in acetone, isopropanol and DI water for 5 minutes or longer. If after inspecting the mask on a microscope there is still photoresist remaining, an ultrasonic bath for 20 minutes or more or a flood exposure on the contact side of the mask can be attempted, however it likely will not be needed if it is cleaned continually after use for each day.

If the wafer being used requires a custom epitaxial structure, the first step in the fabrication process is to grow these layers. A common method for research applications is molecular beam epitaxy (MBE). MBE machines are used primarily for research due to their accuracy but slow growth rate. MOCVD, on the other hand is used for simpler epitaxial structures and mass production.

PECVD is used to deposit oxides such as SiO2, SiNx and others. If using small wafers, using a larger carrier wafer is common practice. When creating a PECVD deposit recipe, the gas mixture and temperature are selected.

The E-beam evaporator is used to deposit metals such as gold, aluminum, chrome, platinum and other materials such as germanium.

F E-Beam Evaporator

Reactive Ion Etching (RIE) is used for etching oxides and deposits on wafers using a chemical plasma that is charged with an electromagnetic field and under a strong vacuum. This is termed a dry etching process. For RIE etching recipes, the pressure, chemical and RF power are chosen. The RIE is also used to remove photoresist and organic material using an O2 clean process.

The ICP (or Inductively Coupled Plasma) tool can etch many materials including SiO2, SiNx, Cr, GaAs and AlGaAs. ICP etch recipes are designed using a selected pressure, RF and ICP power, etchant gas and temperature. When using the ICP, run a chamber cleaning process with O2 and Argon with a dummy wafer loaded. After a cleaning run, the desired etch process should be run with a dummy wafer first before loading the desired wafer. For smaller wafers, thermal conducting paste can be applied between the wafer and a larger carrier wafer. For deep wafer etches on semiconductor material such as GaAs, the edges of the wafer will be etched more. To avoid this, other wafers can be placed aside it.

Acid Etching is a wet etch process, unlike RIE and ICP. This is performed at a bench using a blend of chemicals to etch the semiconductor wafer itself, typically. Heavier protective gear is worn during this process to prevent contact with some of the most dangerous chemicals used in a nanofabrication facility. Hydroflouric acid, a deadly neurotoxin is one chemical that is used frequently in a nanofabrication facility [37]. One use of an acid etch recipe used is an AlGaAs-selective etch.

Photolithography is a technique used in semiconductor device fabrication. First, a light-sensitive layer called photoresist is added to a semiconductor wafer. Depending on the type of resist (positive or negative), this layer can be removed using developer after applying UV light. A mask is template used to apply UV light only to a desired region or shape on the wafer. After this is done, etching can be performed exclusively to the parts of the wafer without a photo-sensitive layer.

The spinner is the machine that is used to apply photoresist to a wafer. The wafer is first held on a vacuum arm and photoresist is applied. The vacuum arm is then spun at a desired spin rate and duration. This creates a uniform film of photoresist on the wafer. After running on the spinner, the wafer should sit on a hot plate for a specified time and temperature.

Particularly for non-circular wafers, photoresist can build up along the edges, creating an uneven surface. This is problematic for the following steps, so the photoresist is removed from the edges and underside of the wafer using a swab and acetone.

If there is already a pattern on the wafer, the wafer position in the mask aligner can be adjusted to ensure alignment. It is recommended to include an alignment feature on the mask die, such as a veneer mark, especially if alignment is critical to that fabrication step. After aligning as needed, the UV light exposure time is selected and applied to the wafer. The wafer is dipped in developer for a specified time, then in DI water, and gently blow dried with a nitrogen gun.

Lift-off photoresist is used when creating a metal feature on a wafer. In this process, normal photoresist can be applied over the lift-off resist (before putting on a hot plate). After running the wafer in the spinner, lift-off resist needs to be removed from the edges using tweezers. Lift-off resist will react differently to acetone, so for edge bead removal, a separate solution needs to be used. After performing photolithography and metal depositions, the lift-off resist may need to be developed using yet another type of solution. For LOR 20-3, it is recommended that the wafer sit on a hot plate at 80 degrees C in the lift-off developer solution for 12 hours. It also recommends a wash in cool lift-off developer and isopropanol after sitting on the hot plate. Refer to data sheets for specific instructions for chemicals.

Electron beam Lithography (EBL) performs the same role as the mask aligner, but with much higher precision due to the smaller electron wavelength. Photoresist still needs to be applied before using an EBL machine. Instead of using a mask, it follows the pattern on a GDSII file. An EBL can be used for all lithography steps in fact, though it is much slower, so it is used only for steps with a narrow feature that is not achievable on a mask aligner or stepper. An EBL machine also contains an SEM, scanning electron microscope and can be used for that function as well.

The SEM or Scanning Electron Microscope is used for examining structures that are too small for an optical microscope. This is achieved using an electron beam, which excites conductive materials. SEM is necessary for inspecting etch qualities and for making accurate measurements of component sizes. The SEM operates by directing an electron gun at the sample. The charged atoms release electrons, producing the signal that is detected for the image.

One tool that is useful for measuring the height profile on a wafer is the Dektak tool. This is often used after an etching process. This gives the height profile along a line on the wafer.

The ellipsometer is used for measuring the thickness of a film deposit. Unlike the Dektak, the ellipsometer is able to measure multiple layers. However, it is not useful for precise positioning or height profiles over a distance on the wafer. The ellipsometer is typically only used for uniform deposits on wafers that have not been exposed to etching or photolithography, unless the shapes are very large.

# Optical Loss in Optical Waveguides and Free Carrier Absorption

Sources of loss in optical waveguides include free carrier absorption, band edge absorption, surface roughness, bending loss, and two photon absorption. Optical loss can be determined from the imaginary index of refraction.

Band edge absorption is a wavelength-dependent absorption based on material properties. For wavelengths above the bandgap wavelength (approx. 1 micron), the band edge absorption and free-carrier absorption of GaAs is greatly reduced. Free-carrier absorption caused by doping is still a concern for optical waveguide loss, however.

Free carrier absorption is loss in optical waveguides due to interaction of photons and charge carriers. The effects of free carrier absorption can be calculated using the free carrier coefficients of electrons and holes for the material and the doping concentration. Since doping is used to create a PIN structure, it is therefore wiser based on free carrier absorption to have the regions surrounding the intrinsic waveguide core to be lightly doped. The imaginary dielectric constant due to free carrier absorption, based on doping levels is calculated as follows. The doping concentration for electrons and holes are n and p respectively, the bulk refractive index is n0, the wavenumber is k, and FCN and FCP are the free carrier coefficients of electrons and holes respectively.

# Converting from normalized SFDR (dBHz^(2/3)) to real SFDR (dB)

SFDR is frequently written in the units of dBHz^(2/3), particularly for fiber optic links. Fiber optic links can often have such high bandwidth, that assuming a bandwidth in SFDR is unhelpful or misleading. Normalizing to 1Hz therefore became a standard practice. The units of SFDR for a real system with a bandwidth are dB.

Now consider that the real system has a specific bandwidth. The real SFDR can be calculated using the following formulas:
SFDR_real = SFDR_1Hz – (2/3)*10*log10(BW)

Here are a few examples.

# Noise Figure in a Microwave Photonic Link

The standard definition for noise figure (NF) is the degradation of signal to noise ratio (SNR). That is, if the output noise power of the system is increased more than the output signal power, then this implies a significant noise figure and a degredation of SNR.

For an RF photonic link, there are a couple assumptions that result in a slightly altered definition and calculation for noise figure. One assumption is that the input noise is the thermal noise (kT), such as would be detected from an antenna receiver. It is also the case that RF photonic links may be employed in a case where the input signal power level is not defined. In simple telecommunications aplications, it is standard to expect a certain input power level, but as a communications system at a radar front end for instance, the input signal is not known. We can use the gain of the link as a relationship between output signal and input signal instead of a known input and output signal power.

It is a goal of the link designer in those cases to ensure that all true signals can be distinguished from noise. For these reasons, we may also think of noise figure in the following definition:

Noise figure (NF) is the difference between the total equivalent input noise and thermal background noise.

The equivalent input noise is the output noise without considering the gain of the link.

For the noise figure calculation, we have then:

NF = 10*log_10( EIN / GkT ),

where EIN is the equivalent input noise, G is the link gain, k is Boltzmann’s constant, and T is the temperature in Kelvin.

Equivalent input noise (EIN) is as follows:

EIN = GkT + <I^2>*Z_PD,

where <I^2> is the current noise power spectral density at the output of the link and Z_PD is the photodetector termination impedance.

These together, we have noise figure:

NF = 10*log_10(1+(<I^2*Z_PD)/GkT)

# Noise Sources in RF Photonic Links

Identifying the noise sources in an RF Photonic link allows us to determine the performance of the link and helps us to identify critical components to link and device design to develop a high performance link. Below is an intensity modulated optical link. Other modulation schemes in RF photonic links may be discussed at a later point.

Since the output of the RF photonic link is the photocurrent generated by the photodetector, the noise sources are a current noise power spectral density.

Noise sources from the laser:

Laser RIN (relative intensity noise) is the fluctuation of optical power. Relative intensity noise is the noise of the optical power divided by the average optical power in a laser. RIN noise originates from spontaneous radiative carrier recombination and photon generation.

Noise sources from the modulator:

Noise in a modulator is due to thermal noise of electrode termination and ohmic loss in the electrodes.

Noise sources from the photodetector:

Shot noise occurs as a result of the quantization of discrete charges or photons. Noise is also generated by the photodetector termination.

Total current noise power spectral density of the RF photonic link:

RF Photonic links (also called Microwave Photonic Links) are systems that transport radiofrequency signals over optical fiber. The essential components of an RF photonic link are the laser as a continuous-wave (CW) carrier, a modulator as a transmitter and the photodetector as a receiver. A low-noise amplifier is often used before the modulator.

Optical fiber boasts much lower loss over longer distances compared to coaxial cable, and this flexibility of optical fiber is one advantage over conventional microwave links. Another advantage of RF photonic links are their immunity to electromagnetic interference, which plays a more significant role in electronic warfare (EW) applications. RF Photonic links are employed in telecommunications, electronic warfare, and quantum information processing applications, although the performance requirement in each of these situations vary. In telecommunications, a high bandwidth is required, while in EW applications having high spurious-free dynamic range (SFDR) and a low noise figure (NF) is critical. In quantum information processing applications, a low insertion loss is critical.

In EW scenarios, unlike in telecommunications, the expected signal frequency and signal power is unknown. This is because typically, an RF photonic link is found as a radar receiver. In a system with high SFDR and low NF, distortion is minimized, the radar has stronger reliability and range, and smaller signals can be registered. Here is a demonstration of two scenarios with different SFDR and NF:

Low SFDR, High NF:

High SFDR, Low NF:

# Thermal Background Noise

Any object with a temperature above absolute zero (Kelvin) radiates electromagnetic energy, or thermal noise. Noise is generated by the earth and cosmos, and this is background thermal noise, which is received by an antenna.

Thermal background noise is the starting point for system performance. A signal of strength below the thermal background noise will be indistinguishable from noise.

The thermal background noise power is proportional to the temperature (P = kTB, k being Boltzmann’s constant, T the temperature in Kelvin, and B the bandwidth in hertz). The thermal background noise power spectral density is the fundamental noise minimum at -174 dBm/Hz at 300K.

The gain of the device or system further amplifies the thermal background noise. RF Photonic links most often use a low noise amplifier (LNA) directly before the modulator, amplifying the thermal background noise.

The definition of thermal noise applied to electronics is the movement of charge carriers caused by temperature in a conductor.

# Mean Squared Noise Power

What does it mean when people say “mean squared”?

The average value of a noise waveform is zero. The square of the waveform mean is also equal to zero. The square of the noise signal and the mean of the square are non-zero. This is because the negative values associated with the zero-mean noise waveform are made positive by squaring, and the entire waveform is positive. Taking the root of the averaged square of the waveform yields the RMS.

The mean of the squared (“mean square”) noise waveform is the noise power with respect to a 1 Ohm resistor (units: V2/Ω=W, “power” if noise signal is a voltage signal, and units I2/Ω=W, also “power” if noise waveform is current).

The power spectral density is the power of the signal in a unit bandwidth.

What is a current noise power spectral density?

The correct definition of current noise spectral density is the mean of the squared current per hertz, <i2>. The units are A2/Hz.

The square of the mean is equal to zero, because the mean of the noise waveform is zero and squaring that number remains zero. The mean of the square is a non-zero number. Taking the square of a noise current results in a positive valued current waveform. Taking the average of the square is a non-zero number used for the spectral density.

# What are the frequencies of the second-order and third-order distortion tones given two frequency peaks?

In general, the third order distortion tones are understood to exist as in-band distortion at frequencies 2ω21 and 2ω12 in a two tone intermodulation test. Third order distortion also exists at frequencies ω1 and ω2. Second order distortion tones are found outside of a narrowband system at 2ω2, 2ω1, and ω12.

Consider the two-tone input of a non-linear system with frequencies ω1 and ω2:

Vin = A[cos(ω1t)+cos(ω2t)]

The second order and third order distortion tones are calculated on the following page. In summary, the tones are shown in the table below. This shows that third order distortion tones are found not only in the positions mentioned above, but also contribute to the fundamental tone frequencies.  In a spurious-free system, all third order tones will be below the noise floor. This is verified in MATLAB with ω1, ω2 at 500kHz, 501kHz.

# What does the term “Spurious-free” mean in Spurious-free Dynamic Range (SFDR)?

In the term spurious-free dynamic range (SFDR), spurious-free means that non-linear distortion is below the noise floor for given input levels. The system is spurious when non-linear distortion is present above the noise floor. The system is spurious-free when non-linear distortion is below the noise floor. SFDR therefore is the range of output levels whereby the system is undisturbed by non-linear distortion or spurs.

SFDR contrasts with compression dynamic range (or linear dynamic range (LDR)) which is the range of output levels whereby the fundamental tone is proportional to the input, irrespective of distortion tone levels. The fundamental tone is no longer considered to be linear beyond the 1dB compression point, after which the output fundamental tones do not increase at the same rate as the input fundamental tones.

Spurs are non-linear distortion tones generated by non-linearities of a system. The output of a non-linear system can be modeled as a Fourier series.

The first term a0 is a DC component generated by the non-linear system. The second term a1Vin is the fundamental tone with some level of gain a1. The third term a2Vin2 is a second order non-linear distortion tone. The fourth term a3Vin3 is the third-order non-linear distortion tone. Further expansion of the Fourier series generates more harmonic and distortion tones. Even order harmonic distortion tones are usually outside of the band of interest, unless the system is very wideband. Odd order distortion tones however are found much closer to the fundamental tone in the frequency domain. SFDR is usually taken with respect to the third order intermodulation distortion, however it may also occasionally be taken for the fifth order (or seventh).

# Where do the units of SFDR “dB·Hz^(2/3)” come from?

The units of spurious-free dynamic range (SFDR) are dB·Hz^(2/3). The units can be a source of confusion. The short answer is that it is a product of ratios between power levels (dBm) and noise power spectral density (dBm/Hz). The units of dBHz^(2/3) are for SFDR normalized to a 1Hz bandwidth. For the real SFDR of a system, the units are in dB.

If we look at a plot of the equivalent input noise (EIN), the fundamental tone, OIP3 (output intercept point of the third order distortion), and IMD3 (intermodulation distortion of the third order), a ratio of 2/3 exists between OIP3 and SFDR. This can be recognized from the basic geometry, given that the slop of the fundamental is 1 and the slope of IMD3 is 3.

Now, we need to look at the units of both OIP3 and EIN. The units of OIP3 are dBm and the units of the equivalent input noise (a noise power spectral density) are dBm/Hz.

SFDR = (2/3)*(OIP3 – EIN)

[SFDR] = (2/3) * ( [dBm] – [dBm/Hz] )

Now, remember that in logarithmic operations, division is equal to subtracting the denominator from the numerator. and therefore:

[dBm/Hz] = [dBm] – 10*log_10([Hz])

Note that the [Hz] term is still in logarithmic scale. We can use dBHz to denote the logarithmic scale in Hertz.

[dBm/Hz] = [dBm] – [dBHz]

Substituting this into the SFDR unit calculation:

[SFDR] = (2/3) * ( [dBm] – ( [dBm] – [dBHz] )

This simplifies to:

[SFDR] = (2/3) * ( [dBm] – [dBm] + [dBHz] )

Remember that the difference between two power levels is [dB].

[SFDR] = (2/3) * ( [dB] + [dBHz] )_

The units of [dB] + [dBHz] is [dBHz], as we know from the same logarithmic relation used above for [dBm] and [dB].

[SFDR] = (2/3) * [dBHz]

Now, remember that this is a lkogarithmic operation, and a number multiplying a logarithm can be taken as an exponent in the inside of the logarithm.Therefore, we can express Hz again explicitly in logarithm scale, and move the (2/3) into the logarithm.

(2/3) * [dBHz] = (2/3) * 10*log_10([Hz]) = 10*log_10([Hz]^(2/3))

We can return our units back to the dB scale now, giving us the true units for SFDR: dBHz^(2/3):

[SFDR] = [dBHz^(2/3)]

# 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  [1]. 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 [1].

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 [2]. 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 [3].

## 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 [4]. 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 [5]. 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 [6]. Magnetic-tunable optical lenses allow for a dynamically tunable focal length [7].

# 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 [8]. 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 [9]. 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 [10] and layering YIG on top of the optical waveguides [11], 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 [11]. 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) [12]. While direct bonding techniques are preferred over YIG waveguides, YIG waveguides on deposited films are made using a H3PO4 wet etch [12].

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 [13], photonic crystal slab waveguides [14]. Polarizers have been fabricated using ion beam lithography [10] [15].

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 [16] [17]. A non-magneto-optic isolator was designed as a Mach-Zehnder Interferometer, providing some backwards isolation [16].

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 [11]. 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 [11], 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 [11].

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 [18] [11]. 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 [19]. 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 [20]. The YIG magneto-optic material was bonded using a surface activated direct bonding technique [20].

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 [9]. 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 [9].

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’)