# Quantum Theory of Solids

Classical mechanics have long been proven to be useful for predicting the motion of large objects. Newton’s laws however prove to be highly inaccurate for measurements involving electrons and high frequency electromagnetic waves. Semiconductor physics, for example requires that a new model be adopted. The quantum mechanical model proves to be appropriate in these cases. Quantum mechanics allows for the calculation of the response of an electron in a crystallized structure to an external source such as an electric field, for instance. The movement of an electron in a lattice will differ from it’s movement in free space and quantum mechanics is used to relate classical Newtonian mechanics to such circumstances.

The photoelectric effect is one example of a circumstance that is not describable using classical mechanics. Planck devised a theory of energy quanta in a formula that states that the energy E is equal to the frequency of the radiation multiplied by h, Planck’s constant (h = 6.625 x 10^(-34) J*s). Einstein later interpreted this theory to conclude that a photon is a particle-like pack of energy, also modeled by the same equation, E = hv. With sufficient energy can remove an electron for the surface of a material. The minimum energy required to remove an electron is called the work function of the material. Excess photon energy is is converted to kinetic energy in the moved electron.

Hertz discovered the photoelectric effect in 1887. He found that polished plates irrradiated may emit electrons. This was termed the photoelectric effect. It was found that there was a minimum frequency threshold required to produce a current. The minimum frequency threshold was a function of the type of metal and configuration of atoms at the surface. The magnitude of the current emitted is proportional to the light intensity. The energy of the photo-electrons (electrons emitted by photons) was independent of the intensity of light, however the energy emitted increased linearly with the frequency of light.

Einstein in 1905 explained that light is composed of quanta (photons) with energy E = h*ν, where h is Planck’s constant and ν is the frequency. The work function specifies how much energy is needed to release electrons from a metal. The energy of the electron then is equal to the energy of the photon minus the work function. The remainder energy of the photon is transmitted as kinetic energy. An experimental verification of Einstein’s prediction came 10 years later.

The following is an example problem for photoelectric effect calculations:

The wave-particle duality principle was presented by de Broglie to suggest that, since waves exhibit particle-like behavior, particles also should show wave-like properties. The momentum of a photon was then proposed to be equal to Planck’s constant devided by the wavelength. The ultimate conclusion to de Broglie’s hypothesis was that in some cases, electromagnetic waves behave as photons or particles and sometimes particles behave as waves. This is an important principle used in quantum mechanics.

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously describe with absolute accuracy the momentum and the position of a particle. This may also include angular position and angular momentum. The principle also states that it is impossible to describe with absolute accuracy the energy of a particle and the instant of time that the particle is energized. Rather than determining the exact position of an electron for instance, a probability density function is developed to determine the likelihood that an electron is in a particular location or has a certain amount of energy.

# Optics, Optoelectronics, Electro-Optics and Photonics

Optics vs. Photonics

What is the difference between Optics and Photonics? These words are sometimes used interchangeably. A distinction may be made however. Optics, on one hand is a very old subject, whereas Photonics is a term that has only recently been used. Photonics is a word which refers to devices that primarily involve the flow of photons as opposed to electronics, which deals with the flow of electrons. The main inventions that lead to the use of the word Photonics are the laser, fabrication of low-loss optical fibers and semiconductor optical devices. Other terms that are often used to refer to these inventions and their various applications are electro-optics, optoelectronics, quantum electronics, quantum optics and lightwave technology. Many of these terms may be used interchangeably, although some of them refer to specific technologies.

The first figure below may be viewed as an optical system, while the second figure may be referred to as a photonic system. The first figure features a light beam that is modulated, reflected and reflacted through a medium. The second figure is of a photonic integrated circuit device.

Electro-Optics is term for devices that incorporate both an optical and electrical properties, however are primarily optical devices. Examples of electro-optical devices are lasers and electro-optic modulators and switches.

Optoelectronics refers on the other hand to devices that are primarily electronic, but involve light, such as light-emitting diodes, photodetectors or liquid-crystal display devices.

Quantum Optics refers to the study of the quantum mechanical and coherence properties of light. Lightwave Technology typically is used to describe optical communications and optical signal processing devices and systems. Quantum Electronics is the study of technology concerned with the interaction of light and matter, such as lasers, optical amplifiers and optical wave mixing devices.

# Branches of Optics

Optics is a broad subject that has been studied and renovated over a long period of time. The following is a quick list of the different branches, theories and studies of optics. This list does not cover all concepts related to Optics by any means, however when discussing a topic related to Optics, it will be important to know which type of Optics is being discussed.

Optics

Ray Optics

Geometrical Optics

Wave Optics

Electromagnetic Optics

Quantum Optics

Paraxial Optics

Matrix Optics

Classical Optics

Beam Optics

Gaussian Optics

Fourier Optics

Physical Optics

Polarization Optics

Magneto-Optics

Electro-Optics

Metal Optics

Metamaterial Optics

Transformation Optics

Fiber Optics

Micro-Optics

Nano-Optics

Guided-Wave Optics

Resonator Optics

Photonic-Crystal Optics

Statistical Optics

Photon Optics

Spatial Optics

Acousto-Optics

Semiconductor Optics

Nonlinear Optics

Ultrafast Optics

Atom Optics

# Negative Resistance

RF/Photonics Lab
November 2019
Jared Alves

Negative Resistance

Arguably the most fundamental equation in electrical engineering is Ohm’s Law (V = I*R) which states that voltage is proportional to the product of current and resistance. From this equation, it is apparent that increasing a voltage across an element will increase the current through that element assuming the resistance is fixed. With a resistor, electrical energy is dissipated in the form of thermal energy (heat) due to the voltage drop between the terminals of the device. This is in direct contrast to the concept of negative resistance, which causes electrical power to be produced instead of dissipated.

Generally, negative resistance refers to negative differential resistance, as negative static resistance is not typically used.  Static resistance is the standard V/I ratio while differential resistance takes the derivative dV/dI. The following image shows an I-V curve with several slopes. The inverse of B yields a static resistance, and the inverse of line C is differential resistance (both evaluated at the point A). If the differential curve has a negative slope, this indicates negative differential resistance.

Even when differential resistance is negative, static resistance remains positive. This is because only the AC component of the current flows in the reverse direction. A device would consume DC power but dissipate AC power. This is because the current decreases as the voltage increases, leading to

A tunnel diode is a semiconductor device that exhibits negative resistance due to a quantum mechanical effect called “tunneling”.

# Doppler Effect

RF/Photonics Lab
November 2019
Michael Benker

Doppler Effect

The Doppler Effect is an important principle in communications, optics, RADAR systems and other systems that deal with the propagation of signals through space. The Doppler Effect can be summarized as the resultant change to a signal’s propagation due to movement either by the source or receiving end of the signal. As the distance between two objects changes, so does the frequency. If, for instance, a signal is being propagated towards an object that is moving towards the source, the returning signal will be of a higher frequency.

The Doppler Effect is also applied to rotation of an object in optics and RADAR backscatter scenarios. A rotating target of a radar or optical system will return a set of frequencies which reflect the distances of each point on the target. If one side of the target is moving closer while the other side is moving away, there will be both a higher and lower frequency component to the return signal.

# Skin Effect

RF/Photonics Lab
Jared Alves
November 2019

Skin Effect

The skin effect is an important characteristic of alternating current within conductors. With direct current, charges are distributed evenly when flowing through a conductor. However, due to the Skin Effect as the frequency of the conduction current is increased, the charges distribute in greater quantities towards the surface of the conductor. In other words, the current density (J) decreases with greater depth in the conductor.   As shown, the current density is per area.

The skin depth of the conductor is the length from the surface of the conductor inward in which the majority of the charge is contained at frequencies higher than DC.

As shown in the equation above, skin depth is inversely proportional to frequency so at higher frequency values, the effective resistance of the conductor increases which reduces the cross-sectional area, as shown below.

The figure demonstrates that the conductor becomes more “hollow” at higher frequencies as the electric charges avoids traveling through the center. This is because the back EMF is strongest towards the center of the conductor. Maxwell’s equations explain that magnetic field strength is proportional to current and therefore as current intensity changes, so does magnetic field strength. The changing magnetic field creates an electric field opposing this change in intensity which causes the counter EMF effect. This creates an almost “Faraday cage” effect with the electrons at the center of conductor as the electric field cannot penetrate as deep into the conductor with increasing frequency.

The skin depth is technically defined as the length from the surface to the inside of a conductor in which J (current density) decays to 1/e of Js (current density at the surface). The imaginary part of the above equation shows that for each skin depth of penetration, the current density phase is delayed by 1 radian.

# Interferometry – Introduction

RF/Photonics Lab
Jared Alves
November 2019

Interferometry – Introduction

Interferometry is a family of techniques in which waves are superimposed for measurement purposes. These waves tend to be radio, sound or optical waves. Various measurements can be obtained using interferometry that portray characteristics of the medium through which the waves propagate or properties of the waves themselves. In terms of optics, two light beams can be split to create an interference pattern when the waves combine (superimpose). This superposition can lead to a diminished wave, an increased wave or a wave completely reduced in amplitude. In an easily realizable physical sense, tossing a stone into a pond creates concentric waves that radiate away from where the stone was tossed. If two stones are thrown near each other, their waves would interfere with each other creating the same effect described previously. Constructive interference is the superposition of waves that results in a larger amplitude whereas destructive interference diminishes the resultant amplitude. Normally, the interference is either partially constructive or partially destructive, unless the waves are perfectly out of phase. The following image displays total constructive and destructive interference.

A simple way to explain the operation of an interferometer is that it converts a phase difference to an intensity. When two waves of the same frequency are added together, the result depends only on the phase difference between them, as explained previously.

The image above shows a Michelson interferometer which uses two beams of light to measure small displacements, refractive index changes and surface irregularities.  The beams are split using a mirror that is not completely reflective and angled so that one beam is reflected, and one is not. The two beams travel in separate paths which combine to produce interference. Whether the waves combine destructively or constructively depends on distancing between the mirrors. Because the device shows the difference in path lengths, it is a differential device. Generally, one leg length is kept constant for control purposes.

# IP3 Distortion and Linearity

RF/Photonics Lab
November 2019
Michael Benker

IP3 Distortion & Linearity

Linearity is the measure of a system’s performance as an output signal being proportional to the input signal level as characterized by Ohm’s Law, V = I*R. Not every system can be expected to perform ideally and thus linearly. Devices such as diodes and transistors are examples of non-linear systems.

The intercept point of the third order, IP3 is a measure of the linearity of a system. IP3 is the third order of a Taylor series expansion of the input signal’s presence in the frequency domain. Being third order, this term in a Taylor series expansion is understood as distortion since it is different from the sought output signal. In contrast to the second order harmonics, which fall outside of the frequency band of the first order signal, the third order is found in the same frequency band as the original or first order signal. Similarly, consecutive even orders (4, 6, 8, etc) are found outside of the frequency band of the first order signal. Consecutive odd orders beyond the third order such as IP5 and IP7 also cause distortion but are not of primary focus since the amplitude of these order signals are weaker after consequent exponentiation.

The meaning of an intercept point of an nth order (IPn) on a dBm-dBm axis is the point at which the first-order and nth-order powers would be equal for a given input power. In the case of IP3, this indicates the power level needed for a third-order power to potentially drown out the first-order signal with distortion. The 1 dB compression point defines the range of linear operation for a system.

# Scattering Parameters

RF/Photonics Lab
Jared Alves
November 2019

Scattering Parameters

After the mid-1900s, high frequency networks became increasingly prevalent. When analyzing low frequency circuits parameters such as voltages and currents are easily realized. From these signals, Y and Z (admittance and impedance) parameters can be used to describe a network. For the Radio Frequency and Microwave range, S parameters are much more applicable when studying a network of a single port or multiple ports. Each S parameter can be placed in an NxN square matrix where N is the number of ports. For a single port network, only the parameter S11 (also known as ᴦ (gamma or voltage reflection coefficient)) can be realized. The S parameters are unitless because they are ratios of voltages. The parameters can be viewed as both reflection and transmission coefficients for multi-port networks. S parameters with subscripts of the same number are reflection coefficients, as they describe the ratio of voltage waves at a single port (reflected to incident).

For a two-port network, only parameters S11, S12, S21, S22 exist. For a simple network like this, S11 represents return loss or reflection at port 1. S22 is the output reflection coefficient.  S12 and S21 are transmission coefficients where the first subscript is the responding port and the second the incident port. For example, S21 would be the “forward gain” at port 2 incident from port 1. The following diagram shows an abstracted view of a two-port network, where each “a” and “b” are normalized by the system’s characteristic impedance. Each S parameter can be calculated by terminating a port with a matched load equal to the characteristic impedance. For example, when calculated return loss for a two-port network, port 2 should be terminated by a matched load reducing a2 to zero.  For calculating S1,2 or S2,2 port 1 would be terminated with a matching load to reduce a1 to zero. Each “a” is an incident wave and “b” a reflected wave. Having a matched load at a port results none of the incident wave being reflected due to impedance mismatching.

This leads to the following voltage ratios:

An amplitude with a negative superscript indicates a reflected wave, and an amplitude with a positive superscript indicates a forward propagating wave.

SParameters

# Smith Chart

RF/Photonics Lab
Jared Alves
November 2019

Smith Chart

The Smith Chart, named after laboratories engineer Phillip Smith, is a graphical tool for solving RF transmission line problems. There are many specific uses for a Smith Chart, but it is most commonly used to visually represent impedance matching problems. Although paper Smith Charts are outdated, RF equipment such as Network Analyzers display information using the chart as well.

The Smith Chart is a unit circle (radius of one) plotted on the complex plane of the voltage reflection coefficient (ᴦ – gamma). As with any complex plane, the vertical axis is the imaginary and the horizontal axis the real. The Smith Chart can be used as an admittance or impedance chart or both. For a load impedance to be plotted on the chart, it must be normalized (divided by) the characteristic impedance of the system (Zo) which is the center of the chart. With this information in mind, it is apparent that a matched load condition would result in traveling to the center of the chart (where ZL=Zo). Along the circumference of the chart, there are two scales: wavelength and degrees. The degrees scale can be used to find the angle of the complex reflection coefficient. Since the plot is the polar representation of the reflection coefficient, if a line is drawn from the load impedance point to the center of the chart this would be considered the magnitude of the reflection coefficient. By extending the line to the circumference of the circle, the angle (in degrees) can be found. The wavelength scale shows distance across a transmission line in meters. A clockwise rotation represents moving towards the generator whereas a counter-clockwise rotation represents moving towards the load side.

It is important to note that a Smith Chart can only be used at one specific frequency and one moment in time. This is because waves are functions of both space and time as shown by the equations:

VF is the forward propagating voltage wave and VR is the reverse propagating voltage wave. If a transmission line system is not impedance matched, a reflected wave will exist on the line which will cause partial or fully standing waves to occur on the line (the reflected wave will add to the incident wave). For the matched condition the reflected wave is zero. Because the Smith Chart can only be used at a specific instant in time and at one frequency the first exponential term in each equation drops out. Because the reflection coefficient is the ratio of the reflected wave to the forward propagating wave, the reflection coefficient becomes:

Where C is the ratio of the amplitudes of both waves. For a passive load, the reflection coefficient must be equal to one or less because the reflected wave cannot be greater in amplitude than the incident wave.

Many transmission lines can be approximated as lossless and therefore have zero attenuation. This leads to:

The propagation constant is a complex number that describes how a wave changes as it propagates down a transmission line. The real part is attenuation constant (Nepers/meter) and the imaginary part is the phase constant or wave number (radians/meter).

For the lossless condition the attenuation is zero, as stated previously.

On the Smith Chart, the wavelength λ = 720. This is because the reflected wave must travel the roundtrip distance moved (it must propagate forward and then back again). Using the piece of information, a half wavelength distance is one complete revolution on the chart. This leads to the conclusion that a transmission line that is a half wavelength long does not transform impedance.

The following image shows common points on the Smith Chart.

The left-hand side of the chart (lying on the real axis) represents a short circuit load. This makes intuitive sense because the reflection coefficient must be real and negative for a short circuit. This is because short circuits have a voltage drop of zero across them which would require a same-amplitude wave with a 180-degree phase shift to cancel the forward propagating wave. The right-hand part of the real axis represents the open circuit load, where the reflection coefficient is purely real but has no phase shift. For an open circuit, the current wave would have to be phase shifted by 180-degrees, but since the reflection coefficient is a voltage reflection coefficient it is not necessary for it to be phase shifted. As shown in the image, the upper half plane is inductive (positive reactance) and the lower half is capacitive (negative reactance).

Smith Chart

# Angle Modulation

RF/Photonics Lab UMASS Dartmouth
November 2019
Michael Benker

Angle Modulation

In comparison to Amplitude Modulation, which varies the magnitude of the sinusoidal carrier wave, Angle Modulation varies the phase of the carrier wave. The two most common forms of angle modulation are phase modulation (PM) and frequency modulation (FM). Phase modulation varies the instantaneous angle linearly with the message signal, while frequency modulation varies the instantaneous frequency with the message signal. The signals on the right are understood (from top to bottom) as the carrier frequency,the modulating wave and the result signal of amplitude modulation, phase modulated and frequency modulation. Due to phase modulated and frequency modulated waves having constant amplitude AC, noise is expected to be lower, although the transmission bandwidth is increased.Rates of distortion are reduced with a reduced possibility of a polarity shift. The average power for angle modulated wave is Pave=(1/2)*(AC)2.The table below summarizes the relationship between phase-modulated and frequency-modulated waves. An FM wave can be seen as a PM wave with a substitution of the integral of the message signal for the message signal. Further, an FM wave can be represented as having gone through an integrator while a PM wave is represented as having gone through a differentiator.

The benefits of conserving bandwidth lead to the development of the narrow-band frequency modulation scheme. To achieve this, several parameters are defined. The frequency deviation, or the maximum departure of the instantaneous frequency from the carrier frequency is defined as Δf = kfAm, where kf (as mentioned in Table 4.1) is the frequency sensitivity factor.The modulation index, β is the ratio of the frequency deviation to the modulation frequency: β = Δf/fm. The angle of the FM wave and the FM wave itself are described as: The following block diagram depicts a method for generating a narrow-band FM wave:Carson’s rule defines an approximate relation for the transmission bandwidth of an FM wave generated by a single-tone modulating wave. From the following expression(Carson’s rule), it is understood that large values of the modulation index β the bandwidth is slightly greater than the twice the frequency deviation Δf and for small values of the modulation index, the spectrum is limited to the carrier frequency and a pair of side-frequencies at fc± fm, in which case the bandwidth approached 2*fm.

anglemodulation

# Fiber Optics (Introduction)

RF/Photonics Lab at UMASS Dartmouth
November 2019
Michael Benker

Fiber Optics

When the frequency of a signal is increased, so does the transfer rate. On the electromagnetic spectrum, light waves occupy frequency ranges of several hundred Terahertz. Fiber optics and photonics take advantage of the speed of light waves to allow for a different approach to data communications. When using light waves instead of electrical charges, this drastically alters the normal characteristics of electrical information transfer. A light wave being sent through glass in a fiber optic wire is no longer restricted to Ohm’s law for example, since a light wave will move through a resistor without any loss. Although light waves are susceptible to quantum noise, they are immune to noise caused by heat (in many cases, this means they are virtually noise-less). Fiber optics, due to their high data rates, flexibility and immunity to noise offer an extraordinary opportunity for scientific and engineering progress.

fiberoptics

# NanoVNA – Handheld Vector Network Analyzer 50kHz-900MHz

A Network Analyzer for \$60 on Amazon. Looking forward to owning my own and spending more time with Network Analyzers, like the one’s in the RF/Photonics Lab.

This post outlines the steps needed to create a schematic using OrCAD and then prepare it for manufacture as a PCB board. Writing this post helps me to learn OrCAD better and this will serve as a guide for review later. I will be using the free version, OrCAD Lite.

## Opening a new project

1. First, start a new project.

2. Give the project a name and create the folder that you want for the project files. Select PSpice Analog or Mixed A/D.

3. Select “Create a blank project” if starting from scratch.

## Building the Schematic

4. Select the “Place Part” button or press P to open the parts menu.

5. This next part requires a bit of knowledge about where which libraries the components are found under. Here, I want to place a resistor, so I typed R and selected the library “Analog”. If the libraries are not added, you can find them in the OrCAD folder on the PC and add them using the “Add libraries” button shown on the screen.

6. Double-click on the part in the menu to place it on the schematic page. I also added a VDC, which is found in the Source library. Finish placing parts.

7. In this case, I will add an LED, but I am unable to find it in the Place Part menu. To find what I am looking for, I chose “Place”, “PSpice Component…” and “Search…” to open a new menu shown below. Further components can be found here if you are unable to find what you need. Under part name and description, select one from the list to add it to the schematic.

8. Press G on the keyboard to add a ground. I chose “0/CAPSYM”. Now select the “Place Wire” button or W to put down the wires.

9. Double-click on the voltage and resistor values to change them as necessary.

## Simulation

10. To run a simulation, on the drop down menu, select “PSpice”, “New Simulation Profile”. Give this simulation a name.

11. Define the parameters for this simulation, click apply and Ok.

12. Select the voltage probe and add it to the circuit.

13. Select “Run” and open the new simulation window to view results.

## PCB Design

14. First, a folder will be created for the PCB. Rename the schematic folder in the main project folder., then rename the default “PAGE1” page name.

15. Right-click on the main project folder and create a new schematic. This will be for the PCB board. Now, copy the schematic from the schematic folder and paste it to the new PCB folder. Rename the copied schematic to indicate it is for PCB and not the schematic.

16. Make the PCB folder the root folder. Click and open the PCB schematic file.

17. For the PCB board, the DC voltage needs to be replaced with connectors. Select the VDC and delete it. Select “Place Part” and choose to add a new library. The connectors are found in the library folder shown below.

18. Select CON1 from the part list and place the parts where the VDC was connected. Remember to save.

19. Select all the components and go to “Edit”, “Properties”.  Select the “Parts” tab on the lower left.

20. Scroll to the right to view the PCB Footprint tab. The footprint names here are then changed to the footprint names found in the libraries. Save.

21. Now, open the OrCAD PCB Designer program and create a new drawing in the main project folder. I put it in a separate folder inside the project folder. Selecting the board wizard will take you through a series of prompts.

22. Continue through the wizard (in this case using only default settings until Spacing Constraints). At Spacing Constraints, change the Minimum line width from the default 0 (this default setting can be problematic). Then select the default via padstack. I chose “Via”. Select ok. Continue through the wizard. In this case, choose a rectangular board. After finishing the wizard, you will see an empty square. Now, save and close the PCB designer.

23. Go back to Capture CIS, open the project tab and select the PCB schematic file. Select “Tools”, “Create Netlist…” to begin transferring the schematic to a printed circuit board. Select Create of Update PCB Editor Board and choose the file created using OrCAD PCB Designer. For the output file, I chose to output the board to the same file, since I won’t be needing the empty board file. Now, select Open Board in OrCAD PCB Editor and select OK.

24. Now, go to “Place”, “Components Manually…” to add the parts from the schematic to the PCB. Select the components you need to place (or select all if you will place all of them) and hide the menu.

25. Place the parts on the board. Parts may need to be rearranged to fit nicely. When satisfied,, right-click and select “Done”. Save and “overwrite”.

26. Now select in the menu, “Route”, “Connect” to place wires connecting the components. When finished, right-click and select “Done”. Once again, save and “overwrite”.

# References

[1]B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

[3]Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

[4]R. Hunsperger, Integrated optics. Berlin: Springer, 2002.

[5]K. Ng, Complete guide to semiconductor devices. New York: Wiley-Interscience, 2002.

[6]J. Wilson and J. Hawkes, Optoelectronics. Prentice Hall, 1998.

[7]D. Pozar, Microwave engineering. .

[8]D. Neamen, Semiconductor physics and devices. New York: McGraw-Hill, 2012.

[9]S. Haykin and M. Moher, Introduction to analog and digital communications. Hoboken, N.J.: J. Wiley & Sons, 2007.

[10]SILVACO, ATLAS User’s Manual. 1998.

[11]2.3.1 III-V Semiconductors and Optoelectronics. (n.d.). Retrieved March 08, 2020, from https://www.tf.uni-kiel.de/matwis/amat/semitech_en/kap_2/backbone/r2_3_1.html