# Linearity of Quantum Mechanics, Schödinger’s Equation

Linearity

A linear function follows two properties:

1. If a function is a solution and each variable is scaled or multiplied by the same number, then this is also a solution.
2. If two solutions to a function are found, then a third solution of the function is the summation of each variable in the function.

Given a linear operator L and an unknown variable u, the following properties apply:

Here is an example:

Linearity as related to Quantum Mechanics

A linear system is far less complicated than a non-linear system.

Maxwell’s equation is linear, for instance. Newton’s equations are not linear.

Consider the example below that explains a particular scenario in which Newton’s equations are shwon to be non-linear:

Quantum Mechanics is linear. Schroedinger’s Equation, devised in 1925 for a dynamic variable, ψ termed the wavefunction.

H_hat is the Hamiltonian, a linear operator as was L in the previous example for Linearity [link]. This means that in Quantum Mechanics, solutions can easily be scaled and added together. Thus, it is proven that Quantum Mechanics is actually simpler than classical mechanics. i is the complex number operator equal to the square root of negative one and h_bar is Planck’s constant.

You might ask what the wavefunction is about, if there are any units, for instance. Interestingly, Schroedinger was not sure what the wavefunction referred to exactly. Max Born later proposed that it had to do with probability.

Complex Numbers in Quantum Physics

The complex operator i at the front of the formula notes a significant departure from classical mechanics in which almost all systems are primarily real. In the case of Quantum Mechanics, complex numbers are essential.

Euler’s formula, e^(i*x) = cos(x) + i*sin(x) also proves useful in Quantum Mechanics.

Below is some review relevant to Quantum Mechanics:

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.

# Object Oriented Programming in C# – Constructors

Object oriented programming is an entirely new perspective on programming from procedural programming. It focuses on the manipulation of objects rather than a “top-down” approach. This in general makes object oriented programs easier to modify than procedural based programs. A class is a category of objects which defines common properties of all the objects that belong to it. An object is a self contained entity that consists of data and procedures to manipulate the data.

In the following snippet of a program, a class “Person” is initialized after the class “Program”. The variables defined within the class are hidden from the parent class unless they are made public. The “perOne” is in instance of a class which is an object.

The “Firstname”, “LastName”, etc are properties of each object which are set to different strings.

A “constructor” is a method with the same name as the class, as shown by the line “Person perOne = new Person()”. The following shows the constructor for the class “Person”

If a property for “firstname” is not set, when a new object is created that property will default to “unknown” due to the constructor.

# Keysight ADS – Microstrip Line Design

The goal of the project is to design a 50 ohm microstrip line at an operating frequency of 10 GHz and phase delay of 145 degrees.

The following ADS simulation will be composed of four major parts:

a) Designing the microstrip lines using two models (I.J. Bahl and D.K. Trivedi model and E. Hammerstad and Jensen model). The insertion loss (S(2,1)) will be plotted over the range of 10 MHz to 30 GHz.

b) Assuming reasonable dielectric losses, results should be compared to part a

c) Creation of ideal transmission lines with same parameters compared to part a and b

d) Showing dispersion on the lossless microstrip line. This is compared to the ideal line.

The LineCalc tool (which uses the Hammerstad and Jensen model) within ADS is used to design the second line with the correct specifications. The first circuit is designed using hand calculated values.

The following shows using the LineCalc tool to get the values for the second schematic.

The simulation is shown below.

A new substrate is created with a loss tangent of .0002 for the second schematic. The simulation results in:

An ideal transmission line circuit is created and compared with both the lossy and lossless lines.

In order to demonstrate dispersion, the phase velocity must be calculated. As shown by the values compared from 0 GHz to 30 GHz, the phase velocity does not change for the ideal line, but does for the microstrip line.

# Optical Waveguides

Just as a metallic strip connects the various components of an electrical integrated circuit, optical waveguides connects components and devices of an optical integrated circuit. However, optical waveguides differ from the flow of current in that the optical waves travel through the a waveguide in a spatial distribution of optical energy, or mode. In contrast to bulk optics, which guide optical waves through air, optical waveguides guide light through dielectric conduits.

Bulk Optical Circuit:

Optical Waveguides:

The use of waveguides allows for the creation of optical integrated circuits or photonic integrated circuits (PIC). Take for example, the following optical transmit and receive module:

Planar Waveguides

A planar waveguide is a structure that limits mobility in only one direction. If we consider the planar waveguide to be on the x axis, then the waveguide may limit the travel of light between two values on the x axis. In the y and z directions, light may travel infinitely. The planar waveguide does not serve many practical uses, however it’s concept is the basis for other tpyes of waveguides. Planar waveguides are also referred to as slab waveguides.Planar waveguides can be made out of mirrors or using a dielectric with a high refractive index slab. See also, Planar Boundaries, Total Internal Reflection, Beamsplitters.

Rectangular Waveguides

Rectangular waveguides can also be built either from mirrors or using a high refractive index rectangular waveguide.

The following are useful waveguide geometries:

Various combinations of waveguides may produce different and useful configurations of waveguides:

# Optoelectronic Integrated Circuit Substrate Materials

The substrate material used on an optical integrated circuit (OIC) is dependent primarily on the function performed by the circuit. An optical integrated circuit may consist of sources, modulators, detectors, etc and no one substrate will be optimal for all components, which means that a compromise is needed when building an integrated circuit. There are two main approaches that taken to deciding on a solution to this compromise: hybrid and monolithic approaches.

Hybrid Approach

The hybrid approach attempts to bond more than one substrate together to obtain an optimization for each device in the integrated circuit. This approach allows for a more optimized design for each component in theory, however the process of bolding the various elements together is prone to misalignment and damage from vibration and thermal expansion. For this reason, although the hybrid approach is a theoretically more otpimized approach, it is more common to use the monolithic approach for OIC.

Monolithic Approach

The monolithic OIC uses a single substrate for all devices. There is one complication in this approach which is that most OIC will require a light source, which can only be fabricated in optically active materials, such as a semiconductor. Passive materials, such as Quartz and Lithium Niobate are effective as substrates, however an external light source would need to be coupled to the substrate to use it.

Optically Passive and Active Materials

Optically active materials are capable of light generation. The following are examples of optically passive materials:

• Quartz
• Lithium Niobate
• Lithium Tantalate
• Tantalum Pentoxide
• Niobium Pentoxide
• Silicon
• Polymers

The following are optically active materials:

• Gallium Arsenide
• Gallium Aluminum Arsenide
• Gallium Arsenide Phosphide
• Gallium Indium Arsenide
• Other III-V and II-VI semiconductors

Losses in Substrate due to Absorption

Monolithic OICs are generally limited to the active substrates above. Semiconductors emit light at a wavelength corresponding to their bandgap energy. They also absorb light at a wavelength equal to or less than their bandgap wavelength. It follows then, for example, if a light emitter, a waveguide and a detector are all fabricated in a single semiconductor, there is a considerable issue of light being absorbed into the substrate, meaning that not enough light will be present for the detector. Thus, reducing losses due to absorbtion is one of the main concerns in substrate materials.

# Keysight ADS – Frequency Dependence of Microstrip Lines

The following ADS simulation will demonstrate how the characteristic impedance and effective dielectric constant change with frequency.  In the simulation, a quarter wave section of multi-layer microstrip line is used to demonstrate frequency effects. The result are expected to show that the dielectric constant and the characteristic impedance are inversely related. When the frequency of the electric field increases, the permittivity decreases because the electric dipoles cannot react as quickly. The multi-layer component is used in place of an ideal component because frequency dependence must be demonstrated. An “MLSUBSTRATE2” component is used with the updated dielectric constant and Dielectric loss tangent.

For S parameter analysis, two terminated grounds are required. The effective dielectric constant must be solved for by unwrapping the phase of S(2,1). The results show the characteristic impedance (both real and imaginary parts) increasing with frequency and the dielectric constant decreasing.

# Gas Laser and Semiconductor Lasers

The Gas Laser

In laboratory settings, gas lasers (shown right) are often used to eveluate waveguides and other interated optical devices. Essentially, an electric charge is pumped through a gas in a tube as shown to produce a laser output. Gasses used will determine the wavelength and efficiency of the laser. Common choices include Helium, Neon, Argon ion, carbon dioxide, carbon monoxide, Excimer, Nitrogen and Hydrogen. The gas laser was first invented in 1960. Although gas lasers are still frequently used in lab setting sfor testing, they are not practical choices to encorperate into optical integrated circuits. The only practical light sources for optical integrated circuits are semiconductor lasers and light-emitting diodes.

The Laser Diode

The p-n junction laser diode is a strong choice for optical integrated circuits and in fiber-optic communications due to it’s small size, high reliability nd ease of construction. The laser diode is made of a p-type epitaxial growth layer on an n-type substrate. Parallel end faces may functions as mirrors to provide the system with optical feedback.

The Tunnel-Injection Laser

The tunnel-injection laser enjoys many of the best features of the p-n junction laser in it’s size, simplicity and low voltage supply. The tunnel-injection laser however does not make use of a junction and is instead made in a single crystal of uniformly-doped semiconductor material. The hole-electron pairs instead are injected into the semiconductor by tunneling and diffusion. If a p-type semiconductor is used, electrons are injected through the insulator by tunneling and if the semiconductor is n-type, then holes are tunneled through the insulator.

# Keysight ADS – Transient Propagation

The following ADS simulation will demonstrate the effects of transients on a transmission line. A rectangular pulse of duration .5 microseconds will be generated and a net voltage vs time will be plotted for a period of .7 microseconds. The circuit has a mismatched load, producing reflections. A time domain reflectometry analysis will prove that the propagating signal voltage steadily increases after the initial time and as time increases, the reflections will eventually die out and leave a steady state response. This is shown with transient analysis.

The schematic above contains two circuits for the two parts of the rectangular pulse (one with and one without a time delay). The simulated results are shown below.

A bounce diagram can also be used to convey Time domain reflectometry analysis, as shown below. This diagram is a plot of the voltage/current at the source or load side after each reflection. This is a general diagram and does not apply to the problem.

# The Quantum Well

What is a Quantum Well

Optical Integraded devices are normally built with the consideration that the device size will be large compared to the wavelength of the beams in the system. When however, the device size is reduced to a size of the same order of magnitude as the wavelength of light in the system, unique properties can be observed. The class of device that operates under the unique properties of this arrangement is the “quantum well.”

Uses of Quantum Wells

Quantum wells may be integrated to other optical and opt-electronic integrated circuits. Uses of quantum wells include improved lasers, photodiodes, modulators and switches.

Building a Quantum Well

A quantum well structure features one or more very thin layers of narrow bandgap semiconductor material, interleaved with layers of wider bandgap semiconductors. The thickness of the layers in a quantum well are typically 100 Angstroms or smaller. Quantum wells with many layers are termed a “Multiple Quantum Well” (MQW) structure and quantum wells with only one layer are termed a “Single Quantum Well (SQW) structure. A typical MQW structure may have around 100 layers. The GaAs-AlAs material system or GaInAsP are common choices for materials in quantum well structures.

Superlattice Structure

A superlattice structure is a term for a case in whic a multiple quantum well structure is built with barrier wals that are thin enough that electrons are able to tunnel through the structure.

The Quantum Well and Quantum Dot

The quantum well reduces the separation between an electron and hole in a semiconductor, altering the wavefunction and allowing a strong exciton bonding effect at room temperature. The semiconductor laser results from this process. Wave functions in the well are shown to the right.

When a field is applied across the well, this can result in the tilting of the wells. This can reduce the effective band gap of the material. The process of tilting the wells the alter the band gap is called the Quantum Confined Stark Effect.

Quantum wells are generally understood in two dimensions. The conduction band is forced to be closer the valence band. When this is done in three dimensions to create a small box, where this squeezing effect can be emulated in all dimensions, this is termed a Quantum Dot. A Quantum Dot it turns out is highly effective at producing a high level of energy and as a result there is a high probability that it works as a coherent light source (laser). Quantum dots are readily used today, however since the process of fabrication employs the use of defects in a material to create a quantum dot, the coherency of the light produced is not perfect. Quantum dots are used in data centers for light transmission at a distance of meters. Quantum dots remain a low cost and reasonably efficient light transmission source for small distances. One reason for the low cost of quantum dots is that they can be grown on silicon wafers. A quantum well is not easily (highly unreliably, but perhaps not impossible) grown on Silicon wafers. The issue that arises with quantum wells when being grown on silicon wafers is that the size of atoms in the wafers and thereby the lattice constant is not readily compatible.

# Acoustics and Sound: Beating

Beating is a very important concept in musical instruments. This tremolo-like variation in sound intensity occurs when two pure tones of slightly different frequencies are sounded simultaneously. An experiment can be performed with two tuning forks (one regular and one wrapped with tight rubber bands on the prongs) struck at the same time. The resultant sound intensity will vary, rising and falling periodically. When the sound wave arrives at the ear, the waves initially appear out of phase (destructive interference) then appear in phase (constructive interference). The superposition of the two waves is shown to have a pulsation effect. The frequency of the pulsations is determined by the beat frequency, which is the difference in frequency of the pure tones.

The pitch perceived by the ear is the average of the two pure tones. A demonstration can be done with two pipes in an organ (one adjustable pipe). The adjustable pipe is varied within a certain range of the other pipe’s frequency. If the second pure tone is greater than approximately 15 Hz away from the first tone, beating is no longer heard.

# Keysight ADS – Conjugate Matching

This project will use conjugate matching to match a capacitive load of 50-j40 to a generator of impedance 25+j30. Since the generator impedance is complex, conjugate matching is required to match the network, as opposed to in situations of low frequency where the reactive components are negligible. In the example, an L network is used to match the generator to the load. Theoretically, differentiating the power and setting this equal to zero proves that maximum power is transferred when the resistance of the source and load are equal and the reactive portions are equal and opposite phase shift/sign.

The first step is to use the impedances given to calculate the actual lumped inductor and capacitor values to use for the network to work at 2 GHz. 25+j30 corresponds to a 25 ohm resistor in series with a 2.387732 nH inductor and 50-j40 corresponds to a 50 ohm resistor in series with a 1.98944 pF capacitor.

The following shows the schematic with the source, matching network and load.

Running the simulation with Data Display equations yields….

This shows maximum power transfer at the correct frequency of 2 GHz. The next step is to use the Smith Chart tool. A shunt inductor and series capacitor is used to form the L Network. Exact values can be typed in for these to get the impedance value Z = 0.5 +j0.6 which is the normalized equivalent source impedance (divided by 50).

With the capacitor and inductor values recorded, these values can be loaded into a separate schematic and compared to the original schematic results.

Conjugate matching is not achieved with this Smith Chart configuration so there is no peak at 2 GHz.

Alternatively, the Smith Chart tool can be used from the palette. From this point with the chart icon selected, the network can be created by selecting “Update Smart Component” from the Smith Chart tool window. These results show that it is important to select the proper design network for the specifications for optimal results.

# Keysight ADS – Quarter Wave Transformer Matching

In ADS, a batch simulation can be implemented to run different load impedance simulations. This function will be used to simulate a quarter wave transformer matching system for various loads (25, 50, 75, 100, 125 and 150 ohms),  The system is used to match a 50 ohm line with an electrical length of 60 degrees at 1 GHz.

The simulation will demonstrate that an unmatched load will generate a constant VSWR at all frequencies. With the implementation of the matching network, the VSWR varies because it is only designed to match the network at a specific frequency. A previous post derived the relationship to find the impedance of a quarter wave matching transformer.

The VSWR can be plotted by adding equations into the data display window and manually adding equations into the plot window to plot VSWR against frequency for both the matched circuit and the unmatched circuit. The mismatched circuits appears constant over frequency with a very high SWR, as it does not have the matching transformer. The quarter wave transformer is shown to provide excellent matching at specific frequencies.

For batch simulations, a slider tool can be implemented to show only specific impedances. Clicking on the axes and changing the names to include the index will update the plot with the specific impedances one at a time. The plot is updated to match the slider value for the load impedance.

With the axes correctly updated, sliding the slider tool will change the plot automatically. Also in the data display window, tables can be added to view specific values at different frequencies.

# Hermite-Gaussian, Laguerre-Gaussian and Bessel Beam

The Gaussian Beam [link] is not the only available solution to the Helmholtz equation [link]. The Hermite-Gaussian Beam also satisfies the Helmholtz equation and it shares the same wavefronts (shape) of the Gaussian Beam. Where it differs is in the distribution of intensity in the beam. The Hermite-Gaussian Beam distribution is a modulated Gaussian distribution in the x and y directions which can be seen as a number of functions in superposition. The below figures depict the cross-sections of ascending order intensity distributions for the Hermite-Gaussian Beam. Secondly, distribution orders zero through three are shown.

The Complex amplitude of the Hermite-Gaussian beam labeled by indexes l,m (orders):

Laguerre-Gaussian Beams

The Laguerre-Gaussian Beam is a solution to the Helmholtz equation in cylindrical coordinates.

The shape of the Laguerre-Gaussian Beam intensity distribution is of a toroid which increases in radius for orders where m = 0 and for orders m > 0, it takes the form of multiple rings.

The Bessel Beam

The Bessel Beam, by comparison to the Gaussian Beam differs in that it has a ripple effect by oscillation in addition to a similar gaussian curve. The complex amplitude of the Bessel Beam is an exact solution to the Helmholtz equation, while the complex amplitude of the Gaussian beam is an approximate solution (paraxial solution).

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