Mirrors in Geometrical Optics, Paraxial Approximation

The main types of mirrors used as simple optical components are planar mirrors, paraboloidal mirrors, spherical mirrors and elliptical mirrors.

Planar Mirrors reflect rays in a manner that the apparent object location reflects rays from a position that forms a reflected angle (Snell’s law) with the angle between the point of reference and the mirror.

mirror1

Paraboloidal Mirrors focus all incident rays to a single point, the focus or focal point. The distance from the end of the paraboloidal mirror to the focal point is the focal length. Paraboloidal mirrors are used in telescopes to collect light. Paraboloidal mirrors are also used in flashlight bulbs and light-emitting diodes to direct rays in one direction from a source of light.

Elliptical Mirrors reflect all rays from one source point to another point. Hero’s principle concludes that any path traveled from either point to another will be equal in distance, no matter the direction.

ellipticalmirror

Spherical Mirrors will direct all rays in varying directions. Spherical mirrors may be concave and convex. A spherical mirror acts like a paraboloidal mirror of focal length f = radius/2.

spmirror

Rays that make small angles with the mirrors axis are called paraxial rays. For paraxial rays, a spherical mirror exhibits a focusing property similar to an elliptical mirror and an imaging property as present in elliptical mirrors. The paraxial approximation considers only paraxial rays and therefore allows spherical mirrors to be considered for the above tendencies. Paraxial Optics is an approach to optics which operates under a set of rules derived from the paraxial approximation. Paraxial Optics is also referred to as first-order optics or Gaussian optics.

In spherical mirrors, considering the paraxial approximation, a focal point is assigned for each source point. All rays that are emitted from a a very far distance (approaching infinite distance) are focused to a point at distance f = (-R)/2.

spmir2

The following is an example of a use of a paraxial approximation for an image formation using a spherical mirror:

sp

Images are credit of Fundamentals of Photonics, Wiley Series in Pure and Applied Optics

Electroacoustic Transducers

An electroacoustic transducer converts energy from electrical to mechanical. Transducers in general convert energy from various forms. For example a “piezoelectric” (stress electric) transducer converts a mechanical force into a voltage. These transducers can be used to convert speech or music signals into electrical signals for processing or to serve as measuring instruments for acoustic quantities. A transducer can be modeled as a two port network relating electrical and mechanical properties. All of these values are RMS (effective) values.

Various electrical quantities can be transformed into mechanical quantities:

Voltage<—>Force

Current<—>Velocity

Inductance<—>Mass

Capacitance<—>Inverse of stiffness

Resistance<—>Mechanical Resistance or Damping

Transducers can be reciprocal or nonreciprocal. Crystal or ceramic electroacoustic transducers are considered reciprocal (Transduction coeffecients for electrical and mechanical are equivalent).

Two major types of electroacoustic transducers are the electromagnetic variety (which use the principle of Faraday’s Law of Induction to stimulate charge flow) or electrostatic transducers which store charge on capacitive plates which then vibrate to create changes in pressure. They can be further classified as “active” (not requiring external power) or “passive”. There are resistive, inductive, capacitive or light dependent transducers. Transducers can be characterized by their frequency response or directivity pattern. For example, a microphone with a circular directive pattern would be seen as “omnidirectional” or equal amplification in all directions. The following is the frequency response of a microphone that appears very flat. This means the microphone is very neutral and does not amplify certain frequencies more than others.

freqresponse.jpg

Fermi Level in Semiconductor Materials

The Fermi level in a semiconductor is the probability that energy levels in a valence band and conduction band in the atoms are occupied. At absolute zero temperature, a semiconductor acts as a perfect insulator. As the temperature increases, free electrons are made available.  An intrinsic semiconductor is a pure crystal with no impurities or defect atoms. In an intrinsic semiconductor, the probability of occupation of energy levels in either the conduction band or the valence band are equal. The Fermi level of an intrinsic semiconductor lies between the valence band and the conduction band. This area between both bands is known as the forbidden band.

fermi

Where KB is the Boltzmann constant (1.3806503 × 10-23 m2 kg s-2 K-1), T is the absolute temperature of the intrinsic semiconductor, Nv is the density of states in the valence band, the hole concentration in the valence band is:

hole

Where Nc is the density of states in the conduction band, the electron concentration in the conduction band is calculated:

electron

The Fermi level for an intrinsic semiconductor is given as the average of the conduction band level and the valence band level.

fermilev

 

Electron Doping

A intrinsic semiconductor may be altered by adding controlled amounts of specific atoms, called dopants to the crystal. This alters the number of electrons in the conduction bands or electron holes in the valence bands.

Displacement Current

One of Maxwell’s equations, Ampere’s circuit law, tells us that there are two sources of magnetic fields: conduction currents and displacement currents. Conduction current is very familiar to most people: it is flow of electrons through a conductor due to an applied electric field. The electrons hop from atom to atom within the conductor and rate at which this happens is termed displacement current.

The differential form of the equation also shows another source of magnetic field: displacement current. For a static field (not time varying/DC), there is no displacement current and the Ampere equation is

ampere.PNG

However for time varying fields, the right hand side contains an extra term, which is displacement current density.

dcurrent.PNG

“J” is the conduction current density which is equivalent to conductivity multiplied to E field (also known as Ohm’s Law in point form). Taking the surface integral of the second term on the right hand side yields displacement current.

The important distinction here is that displacement current is not due to the flow of electrons directly, but rather a time varying electric field. A common example is that of a capacitor with an AC voltage source applied to the device. While there is no conduction current flowing through the dielectric which separates the plates, there is still a current through the capacitor (displacement current) .

capacitor

The following image shows two surfaces about a capacitor with an AC voltage applied. If Ampere’s law is applied to surface one, the right hand side is equal to the conduction current flowing in the wire. However, if the law is applied to surface two it demonstrates that no conduction flows through the capacitor. The same closed path is used in the integration (L) and therefore the right hand side cannot be zero. This means a new term for displacement current must be inserted to satisfy the equation. This is demonstrated in the equation below the figure.

displacement.PNG

provingd.PNG

Crystal Structures

Crystal Structures

Crystalline structures are noted by their regular, predictable and periodic arrangement of atoms or molecules. The  arrangement of atoms and molecules for crystal structures is called a lattice. Crystalline materials include many metals, chemical salts and semiconductors.

copper

Solid crystals are classified by the cohesive forces that hold the lattice together and the shape or arrangement of the atoms in the material. Different arrangements include a simple cubic crystal, a face-centered cubic structure and a body-centered cubic structure.

cubic

In metals, each atom contributes at least one loosely bound electron to build an electron gas of nearly free electrons that move throughout the lattice structure. When an electric field E is applied to a metal, a current flows in the direction of the field. The flow of charges is described in terms of a current density J, or current per unit cross-sectional area. The current density is proportional to the applied electric field by a factor of the electrical conductivity σ of the material.

J = σ*E

The electrons in the lattice material experience a force F = -e*E due to the field and become accelerated. The velocity of electrons in the lattice is known as the drift velocity.

 

Bonding and the formation of Semiconductors

In atomic structures, different types of molecules have a varying number of electrons in the outer atomic rings or shells (valence electrons). Ionic bonding is performed by electrons present in the outermost shell, easily forming a positive ion by releasing the outer electron (net positive charge) or enter the outermost shell of another atom to make it a negative ion (net negative charge). Metallic bonding uses a loosely bound electron in an outermost shell to contribute to the crystal as a whole, creating a metallic crystal.

periodic table

The method of bonding for Ge, C and Si can be quite different however, since they have four valence electrons in the outermost shell. These four electrons can be shared with four neighboring molecules. The bonding force that results from this phenomenon is covalent bonding. In this formation however, electrons belonging to the same bond do not have a definite position in any one atom, meaning they may move between atoms that are bonded. Compound semiconductors such as GaAs (Gallium Arsenide), AlAs (Aluminum Arsenide) and InP (Indium Phosphide) have mixed bonding including both covalent and ionic bonding. These bonding characteristics and the ability for electrons to both move throughout atoms in the structure and to form ionic bonds are the basis for the use of semiconductor materials.

Reflection and Transmission of Sound

Similar to electromagnetics, sound waves that are incident upon a medium with different properties will experience reflection, transmission/refraction or absorption depending on multiple factors. The analysis of the transmission and reflection of sound is greatly simplified when the boundary between media and the incident wave are planar. The amount of transmission and reflection depends on each material’s acoustic or characteristic impedance (r = p*c) and the angle that the incident wave makes with the boundary.

Much like in electromagnetics, transmission and reflection and reflection coefficients can be defined as shown.

transreflect.PNG

The figure shows that the transmission coefficient is the ratio of the transmitted pressure wave to the incident pressure wave. The reflection coefficient is the ratio of the reflected pressure wave to the incident pressure wave. This is similar to the voltage reflection coefficient from transmission line theory.

 

The angle of the sound wave with the boundary can either be “normal” (at a 90 degree angle) or “oblique”. For normal incidence, the problems are greatly simplified, as there is no refracted wave, only a reflected component and a transmitted component. These are described by plane wave equations:

pi

pr

pt

The propagation vector (k) is dependent on the material. The transmitted wave has a different propagation vector value because it has surpassed the boundary into the second medium. This is due to the fact that a material’s characteristic impedance is dependent on the speed of sound.

normalincidence.PNG

In order for boundary conditions to be satisfied, both the normal component of velocity and pressure must be continuous. This means the acoustic pressure on both sides of the boundary must be equal, leaving no net force on the planar boundary separating the fluids. The fluids must also remain in contact, meaning the normal component of the velocity must be continuous.

boundary.PNG

These equations can be used to derive the reflection and transmission coefficients.

It is important to note that the reflection coefficient is always real. When the second medium (the medium the transmitted wave propagates into) has a greater acoustic impedance than the first, the reflection coefficient is positive. This makes sense because when a sound pressure wave comes in contact with a rigid boundary, sound echoing occurs. When r2 is much greater than r1, this defines the rigid boundary condition. When r1 is much greater than r2, the boundary is termed “pressure release” and there is an 180 degree phase shift between incident and reflected wave.

 

Having an incident angle other than 90 degrees complicates the solution process a bit. This is termed “oblique incidence”. The pressure equations become a bit more complex.

oblique

The angles made by an obliquely incident wave are shown below.

angles.PNG

Applying the same continuity of pressure boundary condition from before leads to Snell’s law.

snell.PNG

The critical angle can be defined as

critical.PNG

This angle, as well as the comparison of speeds of the materials determines the bending of the refracted wave.

rules.PNG

Postulates of Ray Optics

The following principles of ray optics may be used to describe many optical systems. The numbering system is of no significance.

1. Light travels in the form of a ray. This means that light will travel from a source and is observed when reaching a detector.

2. Optical rays are vector which point in the direction of energy flow.

3. An optical medium is characterized by a refractive index, n = c0 / c, where c0 is the speed of light in free space and c is the speed of light in the medium. The time taken by light to travel a distance d is d/c = nd/c0. The optical pathlength is n*d.

4. In an inhomogeneous medium, the refractive index n(r) is a function of the position r(x,y,z). The optical pathlength along a path between A and B is the integral of A to B of n(r)*ds.

5. Fermat’s Principle states that optical rays travel from A to B following the path that requires the least amount of travel time.

6. Hero’s Principle states that light travels in straight lines in a homogeneous medium. A homogeneous medium means that the refractive index is consistent throughout.

7. Light reflects from mirrors in accordance with the law of reflection: The angle of reflection equals the angle of incidence and the reflected ray lies in the plane of incidence. This may be proven using Hero’s principle.

planeofincidence

8. At a boundary between two mediums of different refracting indexes, a ray is split in two. One resulted ray is a reflected ray and the other is a refracted or transmitted ray. The reflected ray is shown in figure (b) above as vector C, while the refracted ray is C’.

9. The refracted ray lies in the place of incidence. The angle of refraction is related to the angle of incidence by Snell’s Law:

snell

10. The proportion of reflected light to refracted light is not dealt with in ray optics.

ray1

Introduction to Acoustic Waveguides/Cavities

Waveguides are of great importance to both electromagnetics (for example, guiding microwaves into the cooking chamber of a microwave oven to cook food) and the world of acoustics. Waveguides are hollow tubes that guide waves by reflecting them. Without waveguides, waves propagate spherically and decay with range. The waveguides restricts the propagation of the wave to one dimension. These devices can be rectangular or circular in shape. A major area of importance when studying waveguides is boundary conditions. For acoustic waveguides, the boundary conditions are governed by the linearized force equation:

force

This equation shows that the spacial derivative (gradient) of the pressure (a scalar field) is proportional to density and acceleration. The equation can be used to show that at a rigid boundary where the pressure is maximum, the velocity must be minimum. Calculus tells us that taking the derivative of a function and setting it to zero (and hence the velocity in this case) will yield maximum values. The opposite is also true for a “pressure release” boundary.

We will first consider the case of a rectangular boundaried cavity. A cavity is similar to a waveguide, however the dimensions of a cavity are comparable to each other whereas a waveguide will have one direction that is much longer than the others (to propagate the waves). Applying boundary conditions, it is apparent that if all boundaries are rigid, only standing waves can be contained within the cavity. A pressure equation can be derived from the boundary conditions through substitution into the wave equation. The wave equation will always be satisfied for any kind of wave, including pressure waves. The following equation is substituted into the wave equation:

pressure1

This results in an equation of three sinusoids in each direction. In this case, the boundary conditions result in cosine (which is maximal at zero (x=0,y=0,z=0)).

pressure2.PNG

Then, from the boundary conditions it must be true that these cosine functions be equal to zero at the boundaries x =Lx, y = Ly, z = Lz.

pressure3

This leads to solving for the cutoff frequency of the cavity. The wave number (k) is defined as w/c. Solving for w leads to

cutoff

For waveguides, it is best for the frequency of propagation to be much higher than the cutoff for a decrease in waveguide dispersion. Frequencies below cutoff produce evanescent waves, or waves that die off without propagating.

Review of Fourier Series

The French mathematician Fourier discovered that any periodic waveform can be expressed as a series of harmonically related sinusoids.

Any periodic waveform can be expressed as the following:

series

The first term a0/2 is the constant DC or average component of f(t). The terms with coefficients a1 and b1 represent the fundamental frequency components of f(t). Coefficients a2 and b2 are the second harmonic components at frequency 2w. The frequency doubling on the second order harmonic is computed as a result of the multiplication of sinusoids.

harmonics

In order to determine the coefficients of a harmonic series ai and bi, multiply both sides of the above formula by 2sin(2wt). In this case for simplicity, let w = 1.

calcaibi

Next, integrate from zero to 2*pi.

sereis2

The following relations are then found:

seriesrelations

The Fourier series are often expressed in exponential form:

series3

The MATLAB function int(f,t,a,b) is often a useful tool, where f is the function, t is the symbolic variable, and a and b are the bounds of integration.

matlabint

 

 

Simple Harmonic Oscillator

Any spring that obeys Hooke’s Law is appropriately named a “simple harmonic oscillator”. This law of physics describes the behavior of a mass-spring system disturbed from its equilibrium by pulling or pushing on the mass in such a way that the mass experiences a “restoring force” described by Hooke’s Law,  F = -sx where “F” is the restoring force, “s” is the spring constant (units – Newtons/meter) and “x” is the displacement of the mass. If x is positive, this refers to stretching the spring, and if x is negative this refers to compression. The spring constant determines how easily the spring is deformed (also known as stiffness). The negative sign refers to the fact that the force opposes the force applied (it is a restoring force), which aligns with Newton’s Third Law of Motion.

Another important Newtonian equation is F = ma which states that force is proportional to both an object’s mass and acceleration. If these two equations are brought together, using the relation that acceleration is the second derivative of position, a second order linear differential equation can be formed.

Mechanical-Oscillator

It is seen that the solution to this undamped mechanical system is sinusoidal in nature. This makes intuitive sense, because sinusoids are proportional to their second derivative, meaning if a sine or cosine is plugged into the differential equation, it can be shown to be a solution. The system’s “natural frequency” is obtained by equating the potential and kinetic energy of the system. This is intuitively satisfying because the kinetic energy dominates the system at frequencies below the resonance/undamped natural frequency and above this frequency, potential energy dominates.

For damped system, the equation becomes slightly more complex as a new force must be considered (the product of mechanical resistance and acceleration). This is a much more realistic approach because in a real mass spring system, air creates a frictional force effect on the spring as it oscillates, causing the oscillations to die out as mechanical/motional energy is lost to heat. The solution to this new differential equation contains a decaying exponential term.

damped

Three cases of damping are shown above: critical damping, overdamping and underdamping. These cases depend upon whether the resonance/undamped natural frequency of the system is equal to the real part of gamma (called temporal absorption coefficient) in which case the system would be critically damped. An underdamped system would occur when the absorption coefficient is lower than the resonance frequency and the opposite would be considered an overdamped system. As shown above, critically damped systems have no oscillations because the imaginary part of gamma is zero (no reactance). Underdamped systems experiences oscillations that decay to zero over time. Overdamped systems decay to equilibrium without oscillating, but not as quickly as critically damped systems.

Ray Optics & Geometrical Optics (Introduction)

Ray Optics

In describing the nature of light, numerous theories have been described. One of the oldest and most simplest of explanations of the nature of light is Ray Optics. In variable contrast to Wave Optics, Electromagnetic Optics or Quantum Optics, the theory of Ray Optics describes light as obeying a set of geometrical rules. Ray Optics assumes that the wavelength of light is infinitesimally smaller than the objects that light “rays” interact with. Ray Optics is also referred to as Geometrical Optics due to the geometrical nature of the understanding of the theory and the manner of calculations involved.

2

Ray Optics has limitations and does not describe many phenomenon. However, Ray Optics or Geometrical Optics is is useful in determining the conditions in which light travels and is guided within various mediums, such as in relation to a lens, mirror or glass fiber. Optical rays may also be described as vectors which point in the direction of travel of a light ray.

optics1

The above diagram describes the relationship between Ray Optics to other important theories regarding the nature of light. Electromagnetic Optics describes light as an electromagnetic wave phenomenon and therefore assesses light using concepts applied to electromagnetic radiation, such as the form of electric field waves and magnetic field waves coupled. Wave Optics approximates this wave phenomenon as a scalar wave. Electromagnetic Optics, Wave Optics and Ray Optics encompass what is known as Classical Optics. To describe the nature of light in a manner consistent with quantum mechanics, the theory of Quantum Optics meets these purposes.

 

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.

crystal1

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.

photoelectriceffect

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.

u444444444444444

The following is an example problem for photoelectric effect calculations:

photoexample

 

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.

waveparticleduality

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.

optics2

photonics1

 

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.

laser1

 

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.

led

 

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.