# Lecture notes

### Intro

• Office hours: Monday, 4-5pm, Shapiro CEPSR 810
• TA office hours: Fri 2-3pm
• 40% ps’s
• 25% midterm: Thu 10-21-10
• 35% final: 12-16-10
• Exams are closed book, open notes
• Assignments due 1 week after assigned…11-12 assignments in total

### Lecture 1

• Ray optics
• Principles
• Rectilinear propagation
• Law of reflection
• Snell’s law
• Unity in Fermat’s principle
• Optical path length
• Photonic components
• Mirrors

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#### Photonics overview

• Lasers are essential to photonics in their control of the properties of light
• Crucial that light can be described by photons
• Electro-optics: light goes into electric field, which modifies the light’s polarization
• Nano-optics: devices that are smaller in size than wavelength of light, used to manipulate light
• Quantum optics (QED, quantum electrodynamics): what does it cover that’s not in classical optics?
• Quantum nature of light
• particle and wave nature of probability distributions
• entanglement - fundamental aspects of quantum theory: principle of non-locality

#### Optics overview

• Scope of optics (in chronological order), and increasing in scope
• Ray (geometric) optics
• Wave optics (polarization not included)…in this light properties is described by the scalar wave equation
• Electromagnetic optics: Maxwell’s equations (polarization important)
• Quantum optics: quantized EM fields
• This describes everything previously too (highest scope)

#### Ray optics

• Valid when wavelength is small compared to dimensions of optical devices, and also when photon energy is small relative to energy sensitivity of measurements
• Postulates (hierarchy?):
• Light is described by lines and density of lines
• Speed of light is v =
• Important: optical pathlength: λ = nd…formally:

• Fermat’s principle: path taken by light ray is that which minimizes the time it takes to travel that path
• Physics: want constructive interference along only one path
• Reflection and refraction
• There’s always some reflection and refraction
• Ray optics doesn’t tell us the proportions
• Refraction law: Snell’s law: ni sinθi = nr sinθr
• Proved by minimizing time with repsect to distance and substituting trig (sine) definitions
• Alternative to Fermat’s principle: Huygen’s principle:
• Light considered as a a spreading, circular wave front from current position
• Can also derive Snell’s law
• Evanescent waves carry no energy
• Optical components:
• Mirrors (flat, spherical, elliptical)
• Paraxial approximation: light rays are close to axis of spherical mirror

### Intro

• Will add hint for #5 on homework
• Subscribe to Google course calendar for up-to-date information on the class

### Lecture 2

• Optical components:
• Spherical mirror: imaging
• Prisms
• Lenses
• Matrix optics

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#### Review

• Wave eqn:

• Superlensing: refractive index is less than zero…try working through this
• Allows you to image well below the diffraction limit!
• Paraxial approximation: 1
• Paraxial approximation is what we’re working in!
• sinθ ~ θ; tan θ ~ θ
• θ1 - θ2 = 2θ0 (no paraxial approximation here)
• Now applying approx, we obtain the imaging equation:

• Where f =
• So rays parallel to axis image to the focus
• Magnification in off-axis image formation and magnification:

• Beamsplitter/combiner: a common component
• Light has a different refractive index for different colors
• Total internal reflection: confinement of light in an optical fiber…the (sine of the) critical angle is the “numerical aperture of the fiber”
• Spherical boundary between two media: use two image equations
• Lense is made from intersection of two spherical surfaces
• θ3 = θ1 -
• Lensmaker equation:

#### Matrix optics

• Prescription to trace optical rays through complicated optical systems while always in paraxial approximation
• Ray transfer matrix if the principal object in this
• General prescription to see if optical system is bounded:

### Review

• GRIN fiber…a fiber with a parabolically decaying refractive index

### Wave equations

• Note: v = …understand this “coordinate differentiation”
• Dispersion relation of a wave is the relation between the spatial frequency and temporal frequency of the wave

• There are many different wave equations…u is the solution to all polynomial wave equations…the dispersion relation is probably different though
• Or, we can start from the dispersion relation and obtain the wave equation from that, without worrying about the solution u
• For photons you can derive the Schrodinger equation, starting from the dispersion relation…interesting definitions between wave and heat equations
• Wave optics ignores vector nature and just uses scalar version
• Wave equation is a linear equation so principal of superposition for the solutions applies
• Monochromatic wave oscillates at just one frequency
• Take real part to represent physical quantity
• In complex representation of solution to wave equation, separate spatial and temporal parts; plugging back in yields the Hemholtz equation
• It assumes exponential time dependence but general spatial dependence
• This simple exponential time dependence describes a monochromatic wave
• Then, for plane waves, if you plug in that form for the spatial dependence, you end up with the expected dispersion relation
• Plane waves:
• Wavefronts are surfaces of constant phase (in general)
• λ =
• In media: λ =
• Spherical waves:
• U(r) = exp[-ikr]
• Wavefronts are surfaces of constant phase (in general): kr = 2πq
• Fresnel approximation occurs when z2 x2 + y2

### 9-16-10

#### Review

• If waveform stays same at different times we have a non-dispersive wave v in the wave equation is constant
• To solve any PDE you can perhaps plug in plane wave form: u = u0ei(ωt-kt)…gets you an algebraic equation after plugging in trial form
• For wave equation; ω = vk; for Schrodinger equation: ω = + V
• Group velocity for Schrod does depend on k (dispersive then I believe), but not so for wave equation
• Optical waves described by u(r,t)
• Spherical waves (also) solve the Hemholtz equation
• All phases (in a wavefront) are equal to multiples of 2π: -ik r = 2πq

#### New material

• He’s plotting the real part (of a Gaussian?) in the slide
• Wavelength is much smaller than decrease of envelope in the slowly-varying envelope approximation
• Phasefront = wavefront
• Now plug general solution into Hemholtz equation and then apply slowly-varying approx
• Paraxial = slowly varying envelope…this is a typical situation
• 2 = T2 +
• Now we have diffraction-limited spot size

##### Interference
• Interference only works if two sources are phase-locked?…I think this probably means they have the same frequency…it’s the difference in phase that causes interference

### 9-21-10

• Average on HW1 was 88%
• In for instance focusing onto a lense, the image is independent of the incident angle, so you can solve the matrix multiplication more quickly…shortcuts

### Review

• Gaussian wave comes directly from paraxial approx to Hemholtz eqn
• visibility = …maximum visiblity you can get is one
• Formula for constructive interference in m slits: = sinθd

### New material

#### Waves, ETALON

• Finesse basically tells you how many times the field bounces
• The “sharper” the peaks, the high the finesse

#### Polychromatic waves

• Wave packets I think
• Constructive interference with all different frequencies

#### Beam optics

• Gaussian waveform maximizes energy transfer between two points
• Parabolic solution blows up (cuz of ) at origin, which is unphysical and can’t be our solution
• Argues that shifting z by complex constant ξ, and as long as it has complex component denominator does not go to zero

### 9-23-10

Came in a couple minutes late…things written on the board…see his slides…I think this is a calculation of Etalon properties.

This is something like a filter. (Etalons and cavities). It increases intensity inside the cavity by a lot…constructive interference I think

• These are “resonators” I think.
• If you expand intensity about small θ, you see the line width is proportional to , just as before it was .
• Cavity amplifies electric field by factor F.
• Effectively in the cavity we’re slowing light down, so it must be getting amplified?

#### Now polychromatic waves looking at same cavity phenomena

• Effectively we have same effect on intensity…ϕ-→Δωt
• But now we’re describing a different phenomenon…a pulsed laser system.
• Cavity supports certain modes…every round trip must equal some integral number of wavelengths:

(m ~ 107)

• Spacing between consecutive modes inside the resonator is ΔνFSR = (he derived this on the board)…called Free Spectral Range of cavity
• So there’s an optimal pulse rate for a cavity to maximize the intensity inside I think
• We can also determine the pulse duration.
• τ =

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• Gaussian modes are used everywhere (applications)
• Gaussian modes remove divergence issue
• Depth of focus: what distance in z is my minimum width (approximately) maintained?
• Minimum radius ends at z0
• Two measurements are needed to fully characterize the beam (origin and z0 or W0)
• Standard measure of beam quality: M2 factor
• For a Gaussian the product is always

#### Transmission of Gaussian beams (optical components)

• Thin lense adds an extra spherical phase dependence
• Divergence and lense? have similar effects…Gaussian beam in and Gaussian beam out!
• Convenient to define magnification factor
• Spherical mirror very similar to effect of thin lense
• Problem with Etalons…light not really confined laterally, so cavities are actually concave mirrors not plane ones…R1 = -R?? (p91 of text, bottom right image)
• Stopped at Hermite Gaussian modes, higher order modes, the last solutions to Helmholtz equation

### 9-28-10

#### Homework 3 problem 4 notes

• See notebook notes
• Finesse is just like a grating but with more bounces.
• Can solve for the 3 db point, but no need to be too exact.

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• You really need 3 parameters to characterize a Gaussian beam…one of his slides should say 3 not 2, in the transmission through a lense review slide
• Gaussian waves can be focused to a minimum spot size, and have a minimum divergence for a given something (width or radius of curvature?)?…so perhaps at the moon a Gaussian beam area is the smallest

### 9-30-10

Came in ~30 min late…

• r is reflectivity I think…look into the appropriate equation for finesse in terms of r or R
• Finesse refers directly to number of round trips in a resonator…Q refers to the time, the response time I think, time light remains in the cavity perhaps…they are both measures of resonator quality
• ΔνFWHM Δτ = 1

#### Spherical mirror resonators

• What does “stability” really refer to?…see my outline on chapter 10
• Symmetrical confocal resonator is actually the most stable resonator I think
• To find resonance modes: after one round trip, phase must equal integer multiple of 2π
• Resonance in Gaussian beams has an extra phase (“extra” over that of plane wave)
• He did first 2/3 of ch 10

### 10-5-10

• Note B is not the magnetic field but rather the combination:

• H is the magnetic field
• All six fields are interchangeable
• At a boundary:
• Normal component of electric fields are discontinuous by ~ σ
• Normal component of magnetic fields are continuous
• Tangential components of electric fields are continuous
• Tangential components of magnetic fields are discontinuous by ~ K
• I should try to derive these boundary conditions by myself from first principles…done
• Look into possible error in Poynting theorem

### 10-7-10

• Perhaps the negative sign in the possible error in homework 5 is due to a cross product switching order perhaps
• P = ε0χeE
• D = εE
• Nondispersive: P changes instantaneously to a change in E
• Isotropy: In general: Pi = ε0χijEj

### Reflection

• There are two possible configurations: B parallel to surface or E parallel to surface
• Because the magnitudes of k are equal, i = i
• Jackson ch 7 explains the details of the reflection at an interface slide
• The transmission and reflection coeffients are ratios of electric fields
• “Bother Jake”

### 10-12-10

• Midterm on 10/21
• Review session next week (instead of class I think)
• Photonic crystals is first topic not on exam…so we’re tested on everything up to that

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• Linear relationship between P and E only if electromagnetic field is weak…I think so
• I should probably derive the reflection and transmission coefficients myself…?? (for both TE and TM modes)…done
• We should just always make the coefficients positive and just add the phase change ourselves manually I think…correct
• The wave equation is in fact valid if the spatial derivative of ϵ is small (from when we assumed this in chapter 2)
• We’re talking about electron frequencies in the optics of metals section??…no
• E&M wave incident on metal is what we’re talking about…here the “penetration depth” is introduced
• Using the complex Maxwell’s equations for the rest of this lecture I think
• Use real quantities of E, D, B, H for the EM plane wave energy derivation (not resulting formula, which only includes E0)
• Energy density in an EM field can be calculated from only E or only H
• Through today’s class is what we’re responsible for on the midterm

### 10-14-10

• Problem at hand: describe E&M phenomena in periodic structure
• There are different band gaps for electrons, photons, phonons, etc…perhaps study these overall
• Band gap means light is reflected…so otherwise light is trapped!…depends on where it starts…can it just go through? (of course…transparence)

### 10-19-10

• Understand, given the ABCD problem, what is the image and what is the object, and understand the order of things, why, for the sphere example we can’t just set A = 0.
• FP interferometer extends effective path length by the finesse
• Coherence?
• Understand how LIGO works
• Understand interferometers
• Have definitions listed perhaps!
• A mode-locked system is essentially an interference effect
• For a given divergence angle, the Gaussian beam has the smallest width (or minimum width?)
• For a given beam width, the divergence angle is minimized in a Gaussian wave
• For example problem we want to determine = 0
• Check his equations if I use them…there are typos!
• Race through the beginning (true-false-like) questions on the exam

### 10-26-10

• Two ways to confine light: TIR and DBR (distributed Bragg reflection)
• Periodic dielectric constant…look into this and how it occurs, etc.…got it I think
• Bloch theorem applies to electric field?…yes I believe
• Photonic band gap emits only evanescent waves
• Index contrast is what matters, e.g., silicon/air…if it’s large enough there is a band gap
• “Dielectric index of beads [materials]”??
• CMOS?
• At some point the wavevector is such that the light is no longer confined in the crystal…at this point I think in the vertical direction the wavevectors are imaginary in order for ω equation to hold…I think (yes I think) this is due to TIR no longer applying
• To normalize coordinates multiply by a∕π, so you get a normalized frequency a∕λ
• 3 confinement mechanisms:
• Everything below light line is confined by TIR
• Band gap shows us confinement in in-plane direction
• To do this we can either lower sizes of (one of the) holes in the band gap, which pushes down band structure…this causes air band to be pushed into band gap
• Or we can enlarge holes locally, pushing dielectric band up
• For a waveguide we do this for a whole line
• Third mechanism?
• FDTD just keep iterating through Maxwell’s equations solving for the fields successively in time
• Idea is to excite all modes at once and all but resonant ones will decay in time…or just take FT and you won’t have to wait in time!…then you have the eigenmodes of the structure
• Advantage of FDTD is that one simulation covers a very broad spectrum
• TIR is always associated with evanescent field in transmitted side
• For m large enough, BC is that evanescent field goes immediately to zero…this is the whispering gallery mode approximation

### 10-28-10

• TIR in the vertical direction, DBR otherwise in planar photonic crystals
• Q = = 106…solve for Δω then the lifetime (ω 1015)
• The higher the index contrast, the broader the band gap
• Spacing inside a cube or something: kn = n, so energy spacing between modes is ΔE = c
• Zero-point energy is responsible for spontaneous emission
• Think of zero-point energy as oscillating electric field

### 11-4-10

• Whenever we have binomial distrib with no memory, we must derive Poisson statistics
• If there’s memory, I think the observations are no longer independent, which results in non-coherence (to some degree), from which I believe the Boltzmann distribution results??
• Coherent vs. thermal fit? field? f-something (types of randomness?)

### 11-9-10

• Amplitude noise vs. phase noise, good for quantum communication I think
• Most interaction between light and matter occurs with electrons
• GaAs has a direct band gap, that’s why it’s used to much
• By making the quantum well smaller (via quantum confinement methods), we change the energy levels and can tune where the transition will occur
• Understand this dipole interaction with light…accelerating dipole emits photons
• Couple ways to look at absorption
• z = z(τ,t), mess with partial derivatives to do transformation into photon’s frame of reference (where there’s a retarded time), and we end up with a simplifying equation to specify the process
• The slowly-varying approx has to do with a radiating dipole slowly decaying in its radiation (but staying at the same frequency)
• If incident light is not exactly at resonance but rather off by Δ, multiply polarization (I think) by

• Out of phase radiation because of the i times i…there are two π∕2 out-of-phases occurring (I belive this is for a dipole emitting radiation…the dipole and photon are out of phase)

### 11-16-10

• Interesting note: particle in a box can be applied to band gaps and stuff…electrons stuck in lower band or something
• ω( + n)…the 1/2 is spontaneous, the n is stimulated emission
• Δω = ω01…sometimes he write ω10
• Classical spectrum blackbody radiation gives us the uv energy catastrophe I think.

### 11-23-10

• In the 3-level system it should say “level 1 [not 2] is pumped to level 3”
• Stimulated emission is caused
• Decay occurs quickly so N3 0
• You need to pump it to get transparency (N0 = 0) and then even more to get a gain (N0 > 0)
• τS is the stimulated emission lifetime??

• In our case τ21,NR is typically large, so τ21 τSP
• Lower transition is done by stimulated emission I believe…yes, of course
• For 4-level system N0 is stricly positive–this is what differentiates the systems
• Negative N0 means absorption
• 4-level system seems easier…yes, it’s better I think
• Gain = I think the gain of light passing through some atomic medium = I think

…no, that’s not the definition of gain, it’s just ratio of fluxes between final and initial points

• Lineshape function is g(ν)
• ϕ actually depends on ν I believe (ϕ is not the phase, but the flux…totally different quantities…however, BOTH depend on ν!)

### 11-30-10

• Gain equation is only valid in the non-saturation regime
• No population inversion possible in two-level system…can only get N = 0 from negative start
• Gain will saturate when input power is too high
• Gain drops to zero with increasing distance (along z)
• The absorption of saturable abosrbers saturates and absorbs no more as input is increased
• Average speed comes from 32kT = 12mv2

### 12-7-10

• Gain saturates with distance
• For inhomogeneous broadening the gain is different for each of the populations
• Q = τω
• loss, gain, reflectivity given, you can determine transmission for the laser
• NT
• Q-switching makes super strong pulses
• Now rate equations are observing photon number inside cavity (per unit volume): n
• ns = τp, where τp is the length of time they’re in the cavity, τs is stimulated emission
• τp is related to Q value

### 12-9-10

• Know Gaussian beams
• Divergence of Poynting vector gives us the flow of energy into/out of a volume
• Isotropic media means polarizability tensor is diagonal only?
• Spherical waves won’t be on exam
• Dipole radiation is like a donut
• Don’t worry about derivation of periodic coupling of Bloch equations…
• A particular k point describes the real space direction of the wave
• Boundary in the 2D photonic crystals slide is that between TIR and DBR
• Complex k vector I believe describes decaying wave (evanescent?)
• TIR leads to an evanescent field
• Understand WGM mode of 16,2, conceptually
• Effective index: neff =
• Vibrational transitions are of much lower energy than electronic transitions
• S
• Coherence has to do with phase
• Be familiar with the rate equations!!!…study orders of magnitude times τ??
• Know population inversion.…it starts to drop if there’s saturation.

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# Ray and matrix optics (chapter 1)

### Intro

• Light = electromagnetic optics
• “Wave optics” is a scalar treatment describing only the wavevector, not the E or B fields
• Ray optics = wave optics in which wavelength of light is infinitesimally small…like photons
• Light follows geometrical rules

### Postulates of ray optics

• Refractive index: n(r) = 1
• Optical pathlength: nd, where d is some distance considered through which a ray travels. More precisely:

The time it takes light to travel along a path between two points is proportional to the optical pathlength

• Fermat’s principle (the main postulate of ray optics): Light rays travel along the path of least time, implying by the property of optical pathlength above that:

(It’s usually a minimum)

• Cool: sometimes the minimum time is shared by more than one path, which are then all followed simultaneously by the rays
• This says to minimize the time, i.e., minimize the optical pathlength

#### Propagation in a homogeneous medium

In this the path of minimum time is also the path of minimum distance (the latter of which is Hero’s principle), since c is constant.

#### Reflection from a mirror

Law of reflection:

• The reflected ray lies in the plane of incidence
• Angle of reflection equals the angle of incidence

#### Reflection and refraction at the boundary between two media

• At the boundary between two media of refractive indexes n1 and n2, an incident ray is split into a reflected ray and a refracted (or transmitted) ray.
• The reflected ray obeys the law of reflection.
• The refracted ray obeys the law of refraction:
• The refracted ray lies in the plane of incidence
• Snell’s law:

• Ray optics does not describe the proportions reflected and refracted

### Simple optical components

#### Mirrors

• Planar mirror: causes rays to appear to come from an image, what light rays appear to come from upon reflection
• Paraboloidal mirror:
• All incident light rays parallel to its axis are reflected to its focus
• Focal length: distance between “minimum” point and the focus
• Elliptical mirror: all rays emitted from one focus are “imaged” onto the other focus
• From Hero’s principle, all these total distances along any of the paths are equal
• Spherical mirror: no cool properties, but parallel rays close to the axis approximately reflect to the axis at the same point
• The focal length F = (the radius of curvature R is correspondingly positive for convex mirrors)
• Paraxial approximation: Consider only paraxial rays, which are rays that make small angles with the axis, i.e., sinθ θ, i.e., θmax 1
• In this approximation, the spherical mirror has approximately both the properties of the paraboloidal and elliptical mirrors (I don’t see the elliptical mirror similarities)
• This approximation defines paraxial optics (also: first-order optics; Gaussian optics)
• Imaging equation for paraxial rays (both incident and reflected rays are paraxial):

• In a spherical mirror, rays from the same point (y1,z1) meet at the same point (y2,z2), so that essentially we have an object plane z = z1 and an image plane z = z2; the mirror serves to magnify the object by -. (Negative magnification means that the image is inverted.)

#### Planar boundaries

• Prism deflection angle:

• When α and θ are very small, this becomes

#### Spherical boundaries and lenses

• In paraxial approximation at a spherical boundary:

• For lenses, if we further assume a thin lense:

#### Light guides

• For an optical fiber:

• θa is called the acceptance angle

• Since here n = n(r), we use Fermat’s principle δ ABn(r)ds = 0, which results in the ray equation:

• If we’re in the paraxial approximation so that ds dz, the ray equation becomes:

• A graded-index fiber is a glass cylinder with a refractive index that varies as a function of the radial distance from its axis.
• Numerical aperture of it: NA sinθa n0, where a is the fiber radius and θa is the maximum acceptance angle for which the ray trajectory is confined within the fiber
• The scalar function S(r), called the eikonal, is like electric potential V …its gradient S describes the ray directions, just like the gradient V describes the electric field directions.
• Thus, its constant-value surfaces (S =constant) are everywhere normal to the rays. Ray trajectories are normal to the surfaces of constant S(r).
• To satisfy Fermat’s principle, S must satisfy the PDE eikonal equation:

• This equation is an alernative principal postulate of ray optics.
• Its integral between two points gives the optical pathlength,

so that the difference S(rB) - S(rA) is analagous to the potential difference.

• To determine the ray trajectories in an inhomogeneous medium, we can either solve the ray equation, or we can now solve the eikonal equation for S, from which was calculate the gradient S.

### Matrix optics

• Matrix optics is a technique for tracing paraxial rays.
• Consider a system of a circularly symmetric fiber along the z-axis…thus consider a section along the fiber in the y - z plane, where y represents the distance from the z-axis. So a ray is completely generally described by its coordinates y and θ, where the latter is the angle with respect to the optical axis (the z-axis).
• An optical system is something…we don’t know what exactly (can contain multiple optical components)…that between two planes z1 and z2 changes the ray’s coordinates. In the paraxial approximation, in which sinθ θ, the transformation is linear in the coordinates:

where the matrix is called the ray-transfer matrix.

#### Matrices of simple optical components

• Free space propagation:

where d is the distance between the input and output planes

• Refraction at a planar boundary:

• Refraction at a spherical boundary:

• Transmission through a thin lens:

• Reflection from a planar mirror (note convention of the z-axis changing direction going from incident to reflected rays):

• Reflection from a spherical mirror:

• Recurrence relation for ray position in a periodic system:

where

• This is obtained by simply eliminating angular dependence from the standard iterative equation for the coordinates of the light ray between steps.
• Assuming the first and last stages are in air, the solution to the recurrence relation is:

where

and ymax and ϕ0 are determined from the initial conditions y0 and y1.

• Further:

• Condition for harmonic trajectory (stability condition):

• Condition for periodic trajectory: must be a rational number , where s is the period

### Very cool fact from homework solutions (and that he mentioned in class)

• The height y2 of the image through an optical system must be independent of the angle θ1 at the object, since all rays converge at the image, as this is what is meant by an “image.” So we know that there can be no dependence of y2 on θ1…in other words, if the matrix describing the composite optical system is

we know B = 0. (Since the system can be described from

we can get a relation between variables once we’ve done the matrix multiplication to obtain B and set the result to zero.)

# Wave optics (chapter 2)

### Introduction

• Optical wavelengths:
• Infrared: 760 to 300,000 nm
• Visible: 390 to 760 nm
• UV: 10 to 390 nm
• In our daily lives, light to us is like a particle (a ray…the limit of zero wavelength), since the wavelength of light is much smaller than the scale of objects we normally encounter. It’s when we look closely using more precise (smaller) instruments that the ray limit breaks down and we thus observe the wave nature of light.
• New phenomena: diffraction (optical-wave transmission through apertures), interference
• Light is described by a scalar function, called the wavefunction, which obeys the wave equation.

### Postulates of wave optics

• Wave equation:

• Physical significance of the wavefunction: an electromagnetic-field component
• Optical intensity (optical power per unit area):

where the time average is over a time interval much longer than an optical cycle but much shorter than any other time of interest (such as the duration of a pulse of light).

• Optical power flowing into an area A normal to the propagation direction:

• Optical energy collected in a given time interval:

### Monochromatic waves

• Monochromatic = harmonic time dependence
• Condition for optical phenomena: harmonic time dependence, in which frequency ν is independent of time and position. So boundary conditions are conditions w.r.t. position only.
• General form of solution for optical phenomena:
• Complex amplitude: U(r) = a(r)exp
• For optical phenomena, the general form reduces to the Hemholtz equation for the complex amplitude U:

where

is the wavenumber.

• Optical intensity:

• Wavefronts are surfaces of equal phase: ϕ(r) = 2πq, where q is an integer
• The wavefront normal at position r is parallel to the gradient vector ϕ(r)
• Phase of a wave: arg

#### Solutions to Hemholtz equation

• Plane wave:

where A is a complex constant called the complex envelope and k is called the wavevector, whose magnitude is the wavenumber k

• I(r) = 2
• arg X = -ilog X
• As a monochromatic plane wave propagates through media of different refractice indexes its frequency ν remains the same, but its velocity, wavelength, and wavenumber are altered:

• Spherical wave:

• I(r) = I(r) =
• Fresnel approximation for a spherical wave:
• z: The paraxial approximation of ray optics would be applicable if the points described by this condition were the endpoints of rays beginning at the origin.
• For phase: r z +
• For magnitude (less sensitive to errors): r z
• U(r) = expexp
• Simplifies theory of diffraction
• A wave is paraxial if its wavefront normals are paraxial rays:

• Within a distance of a wavelength λ = 2π∕k, the variation of the envelope A(r) and its derivative with position z must be slow so that the wave approximately maintains its underlying plane-wave nature.
• The complex envelope satisfies the slowly varying envelope approximation (simply called the paraxial Hemholtz equation)

under the conditions

• Solutions to this equation include the paraboloidal wave (the paraxial approximation of a spherical wave) and the more-important Gaussian beam
• The eikonal equation 2 n2 is the limit of the Hemholtz equation when λ0 0.
• Given n(r) we may use the eikonal equation to determine S(r), and thus a, and thus the wavefunction. (p50)

### Simple optical components

• Reflection of wavevectors of plane waves behaves like that of optical rays.
• Reflection from a planar mirror:
• BC’s: phases of the waves are equal at interface
• k1 r = k2 r
• The total wavefunction must satisfy the Hemholtz equation, not just the w.f. of each part
• Transmittances:
• of a transparent plate:

• of a variable-thickness plate:

• of a thin lens:

• of a graded-index thin plate:

• Transmittances multiply.
• A diffraction grating periodically modulates the phase or amplitude of an incident wave.
• Grating equation (exact):

• Grating equation (approximate):

where q is an integer and is called the diffraction order

• To have an effect on incident waves, you can vary either the refractive index or the thickness (or both)

### Interference

• Interference equation:

• Interference of two plane waves propagating at an angle θ to each other:

• Coherent light: light with constant phases (no random fluctuations in phase). Here we limit ourselves to the study of coherent light.
• An interferometer is an optical instrument that splits a wave into two waves using a beamsplitter, delays them by unequal distances, redirects them using mirrors, recombines them using another (or the same) beamsplitter, and detects the intensity of their superposition.
• Energy conservation in an interferometer requires that the phases of the waves reflected and transmitted at a beamsplitter differ by .
• Interference of M waves of the same frequency and amplitude each separated by a phase difference of ϕ:

• For the superposition of waves of constant phase differences and decreasing amplitudes, we have

where F is the finesse.

• The point of all this seems to be that we can filter out different frequencies of light primarily by manipulating the distance traveled between the plantes.

### Polychromatic and pulsed light

• Waves whose time dependence is arbitrary are polychromatic.
• Nothing special, just expand them using monochromatic waves and use superposition and things we already know.
• A quasi-monochromatic wave has a range Δν of frequencies about ν0, where Δν ν0.
• The optical intensity of a quasi-monochromatic wave:

#### Pulsed plane wave

• Before, the complex envelope varied only in space; the wave’s time-dependence was purely harmonic. For example, a spherical wave had the form A = , and the plane wave had no spatial dependence at all: A = A0.
• For a pulsed plane wave, the envelope varies in time in addition to position, and has the form A = A.
• Interference equation of two plane waves of different frequencies:

• In this whole book they seem to suppress space-dependence but express time-dependence.

_______________________________________________________________________________________________________

• If you have M waves of equal intensity and phase and constant frequency difference, in their superposition their frequency differences are similar to constant phase differences, so we now have pulses (in intensity) of light in time, whereas before the pulses were in phase difference.
• The Fabry-Perot interferometer is also called an “etalon,”…this is the device that consists of two parallel plane mirrors between which light bounces, resulting in constant phase differences and decreasing amplitudes. The intensity (interference) pattern is described by the finesse F.

# Beam optics (chapter 3)

### The Gaussian beam

• A paraxial wave is a plane wave traveling along the z direction, modulated by a complex envelope that is a slowly varying function of position…here is the complex amplitude:

• The amplitude must satisfy the Helmholtz equation, so the complex envelope must satisfy the paraxial Helmholtz equation.
• A solution to the paraxial Helmholtz equation is a parabaloidal wave, which via a transformation can be expressed as a Gaussian wave (of a single wavelength), whose complex amplitude is

where

The two parameters, in addition to the wavelength λ, are A0 and z0 (Rayleigh range), which are determined from the boundary conditions.

• Intensity:
where P is the total optical power carried by the beam.
• The Gaussian beam decays in all spatial directions: radially like a Gaussian curve, axially similar to an inverse square drop-off.
• The beam width is defined as the radius at which the beam intensity decreases by , which might be twice a standard deviation σ.
• The beam waist is where the smallest beam width occurs, i.e., at z = 0, where W = W0.
• The spot size is defined as the waist diameter, 2W0.
• Divergence angle of cone of propagating Gaussian beam: 2θ0 = . Note the tradeoff between wavelength and waist size.
• In any cross-section of this cone, there is a constant amount of beam power.
• Depth-of-focus (or confocal parameter), total length of “cone” whose cross-sectional area is within twice that at the z = 0 plane:

• A measure of the quality of an optical beam is its deviation of its profile from Gaussian form.

which is always 1.

### Transmission through optical components

• As long as the component is cirularly symmetric and aligned with the beam axis, the result is itself a Gaussian beam, though the waist and curvature are altered.
• Transmission through a thin lense:

• Beam radius is positive if the beam is diverging, and negative if converging
• Beam parameters through components depending on the magnification factor M:

• Reflection from a spherical mirror:

#### Transmission through an arbitrary optical system

• ABCD law:

• This is a general rule for paraxial Gaussian beams through an optical system
• Use previous M matrices of chapter 1’s ray optics
• In free space, q = z + jz0
• The ABCD law applies even to inhomogeneous media, since they can be thought of as series of optical components

### Hermite-Gaussian beams

• HG beams are other solutions of the paraxial Helmholtz equation
• They have non-Gaussian intensity distributions, but share the wavefronts (and angular divergence) of the Gaussian beam, whose wavefronts are nice because they can match the curvatures of spherical mirrors of large radii (think optical resonators)
• HG beams have the same phase as long as the excess phase Z(z) varies slowly with respect to z
• We obtain the solutions by plugging the above (not explicitly stated here) general form of the solutions (Gaussian times a couple other functions) into the paraxial Helmholtz equation and then applying separation of variables. The solutions to the first two resulting ordinary differential equations are the Hermite polynomials; the solution to the last is
• This result shows that Z(z) indeed varies slowly for any z in
• Plugging the solutions back into the general form (that we initially put into the paraxial Helmholtz eqn), we obtain the complex amplitude of a Hermite-Gaussian beam of order (l,m):

where

is known as the Hermite-Gaussian fuction of order l, where Hl(u) (where l = 0,1,2,) is the Hermite polynomial of order l

• Gl(u), where l is even, is an even function, whereas if l is odd, Gl(u) is an odd function of u.
• The HG beam of order (0,0) is the simple Gaussian beam.
• Of course, the intensity distributions of the HG beams look like the probability distributions of a particle in a parabolic well.

# Resonator optics (chapter 10)

### Math notes

• If a differential equation is linear, the solution satisfying it and the initial conditions is a unique solution.

### Intro

• Optical resonator parameters:
• Degree of temporal light confinement: quality factor Q, which is proportional to the storage time of the resonator in units of optical period
• Degree of spatial light confinement: modal volume V , which is the volume occupied by the confined optical mode
• Resonator uses:
• Confinement and storage of light at resonance frequencies–laser light is generated and built up
• Optical transmission system–filters, or spectrum analyzers
• Pulsed lasers, via their confinement properties

### From lecture slides

• Whispering gallery mode approximation: no penetration outside the boundary; zero field outside the resonator for large m (large θi)
• Effective index approximation: assume disk is an infinite cylinder and change the index inside the cylinder accordingly, to neff
• So there’s symmetry perpendicular to the propagation direction, just like for a plane wave.
• If the wavevector in the propagation direction is β, we calculate neff from the formula:

• I believe the effective index is typically smaller.
• For a slab waveguide, we have:

where b is the “field confinement factor”:

and V is the normalized frequency:

• So solving for the modes of a microdisk, we assume whispering gallery mode approx in which there is no field outside the cylinder (this gives us a boundary condition of zero at the boundary), and the effective index approx in which the disk becomes an infinite cylinder by modifying the index inside and in which we approximate the waves as plane waves. Since we have plane waves, the time dependence cancels out of the wave equation and we have the Helmholtz equation, which we need to solve in cylindrical coordinates (polar coords since there’s symmetry in the z-direction), which we solve via separation of variables. Another boundary condition is that the field must be periodic in ϕ. We’re choosing to solve for a magnetic field H rather than an electric field component. Use first BC above to solve for the mode frequency. The solutions are degenerate (clockwise and counterclockwise modes).
• The intensity has 2m nodes (along the perimeter), where m is the azimuthal mode number.
• R(r) has n - 1 zeros between r = 0 and r = radius, where n is the radial mode number
• Frequency:

where smn is the nth zero of the mth-order Bessel function

• Magnetic field:

### Planar-mirror resonators

• Monochromatic wave of frequency ν wavefunction:

which represents a transverse component of the electric field

• The resonator modes are of course the solutions to the Helmholtz equation under the appropriate BCs, which, for planar planar-mirrors are that the transverse components of the E fields vanish at the mirror surfaces
• Using U(r) = Asinkz as an Ansatz, we obtain the limits on the wavevector k:

where q is known as the mode number.

• Also, we have

• The difference between adjacent mode frequencies is the free spectral range:

• The restriction on the wavelengths is:
so we see that the round-trip distance the light travels must be an integer multiple of the light’s wavelength
• That was resonance in thought of in terms of standing waves; we can also think of resonance in terms of traveling waves: the phase difference for constructive interference, i.e. for a mode to exist and not die out, must be the case…this condition is that in a round-trip the change in phase must be a multiple of 2π:
• Of course, constructive interference results in buildup of intensity and thus of power, so the traveling wave way of thinking about this appropriately relates resonance with a constant buildup.
• Further, we can think of the traveling wave picture as showing feedback, in which the output of one round-trip is fed back into the input, only if the input and output are in phase.
• Density of modes: the number of modes per unit frequency (per unit length)…note this is for each orthogonal polarization (for 1D resonator):
• An arbitrary wave that doesn’t die out in the resonator is a superposition of modes:

#### How things change when there are losses

• The precise frequencies allowed now have a finite width about those frequencies.
• The plot of I vs. ν goes from equally-separated delta functions to equally-separated peaks.
• The general case is when we’re not in a mode (qinteger) and there are losses, in which the amplitude steadily decreases upon the reflections. The analysis is then the same as that in chapter 2; here are the results:

Note: |r| is the magnitude of the round-trip attenuation factor.

• Sources of loss:
• Imperfect reflection
• Light escaping the mirrors I believe in a vertical direction…this can be expressed as an effective imperfect reflection though
• So these first two sources can be expressed by reflectances R1 = |r1|2 and R2 = |r2|2
• Losses in the medium between the mirrors…this source of loss can be expressed by exp(-2αsd), where αs is the loss coefficient of the medium associated with absorption and scattering
• So the total loss can be expressed as a round-trip intensity attenuation factor:
where

where αr is called the loss coefficient.

• The finesse becomes when R1 R2 1:
• The loss per unit length is αr, so r is the loss per unit time, so we have a characteristic decay time:

• We thus have an uncertainty relation between time interval and frequency interval:

• Defining the quality factor Q as 2π times the stored energy E over the energy loss per cycle rE∕ν, we have:
which we see is understood to be the storage time of the resonator in units of the optical period T = 1∕ν. Note the dependence on frequency…the higher the frequency normally the better the quality.

### Spherical-mirror resonators

• Better because they are more stable, geometrically…light doesn’t as easily wander out, as the geometry in planar-mirror resonators needs to be perfect
• Optical axis, my definiton: the axis along which energy is carried by light
• “Stable” means if the ray’s positions ym in the optical system are bounded, like in ym = ymax sin(+ ϕ0)
• “Conditionally stable” means the condition which is the “most” while still being stable
• Confinement condition:

where

• Types of stable resonators:
• Planar (symmetric)
• Symmetric confocal (R1 = R2 = -d) (symmetric)
• Symmetric concentric (R1 = R2 = -) (symmetric)
• Confocal/planar (R1 = -d, R2 = )
• Concave/convex (R1 < 0, R2 > 0)
• Rays in the symmetric confocal resonator retrace themselves after two round-trips
• All paraxial rays are confined, as opposed to the planar resonator, in which the rays must have an angle of exactly 0
• Here the depth of focus 2z0 = d
• The waist radius is proportional to the square root of the mirror spacing: W0
• The width of the beam at each of the mirrors is greater than it is at the waist by a factor of
• A resonator is symmetric if R1 = R2

#### Gaussian modes

• A Gaussian beam is a mode if its radius of curvature equals the radius of curvature of each mirror, and if the phase also retraces itself
• Solutions of the above condition (ignoring the phase condition for now):

Everything about the beam is now known, since we’ve determined the beam center at the depth of focus 2z0.

• z0 must be real for this all to represent a Gaussian beam; otherwise, it represents a paraboloidal wave, which is an unconfined solution of the paraxial Helmholtz equation.
• It can be shown that the above satisfies the confinement condition 0 g1g2 1.
• For symmetric spherical resonators, if the quantity d∕|R| is zero, we have plane waves of course as the type of resonant wave, and of course the resonator is planar. If the quantity is 1, the resonator is confocal, and if the quantity is 2, the resonator is concentric and we have a spherical wave as the resonant wave.
• Since the wavefronts match the curvature of the mirrors, the phase is the same everywhere at the mirror.
• The resonance frequencies of the Gaussian modes are

where νF is still c∕2d, and Δζ = ζ(z2) - ζ(z1)

• Since the wavefronts of Hermite-Gaussian waves are the same as those of Gaussian waves, they too represent modes of a spherical-mirror resonator. For them, the resonance frequencies are

• Longitudinal (or axial) HG modes have the same (l,m) but different q; transverse HG modes have the same q (perhaps) but different (l,m). In other words, the mode number for longitudinal HG waves is q, whereas the mode numbers for the transverse HG waves are l and m.

### Two- and three-dimensional resonators

• Resonance frequencies for a 2D square resonator of side length d:

• Whispering gallery modes (WGM) are when a light ray bounces off the perimeter of a circular mirror at grazing incidence:

where a is the radius of the mirror.

• Resonance frequencies for a 2D rectangular resonator:

• Density of modes only comes into play in the “classical” limit, in which the spacing of the resonator is much larger than the wavelength of the light. In this case, the density of modes (the number of modes per unit volume of the resonator, per unit bandwidth surrounding the frequency ν) is

• The frequency spacing between adjacent modes decreases as the frequency increases.

### Microresonators

• These have one or more spatial dimensions of size on of a few wavelengths of light or smaller.
• Microcavity resonator, or microcavity, is one in all spatial directions.
• Examples of microresonators:
• Micropillar - Bragg-grating reflectors - DBR axially and TIR perpendicularly (around)
• Microdisk and microsphere - ligh reflects near the surface in whispering-gallery modes - TIR
• Microspheres:
• Light travels in great circles around the inside
• Formed by surface tension in a molten state, so very high reflectivities and thus very high quality factors
• Microtoroid - resembles small fiber rings formed by surface tension in a molten state - TIR
• 2D photonic crystals containing light-trapping defects that function as microcavities
• The goal of microresonators: (1) small modal volume V , (2) high quality factor Q
• To solve for the modes of dielectric microresonators: use the Helmholtz equation in the appropriate coordinate system, fit appropriate BCs, for all components of the EM fields
• A Fabry-Perot resonator confines light via distributed Bragg-grating reflectors (DBRs). I believe these just represent materials whose reflectivity is high.
• Think of DBR as periodic lattices that have bandgaps!

# Electromagnetic optics (chapter 5)

### Electromagnetic theory of light

• H is the magnetic field
• Maxwell’s equations:
• So, compared to what I’m used to: B μ0H
• All six components of E and H, which are externally applied, must satisfy the wave equation:

• The speed of light in a medium:

• Derivation of the wave equation (for the electric field) from Maxwell’s equations, in free space:
• Since the wave equation is linear, we can apply superposition to the fields.
• If μ or ϵ are not equal to their free space values, we must be in a medium, so we must also describe the induced fields in the medium; hence we have something of effective electric and magnetic fields:
• D = ϵ0E + P is the electric flux density, or displacement.
• B = μ0H + μ0M is the magnetic flux density.
• The pattern here is, for fields/(charges and currents), <applied> + <induced> = <effective/free>.
• Basically, “free” current and charge refers to the amount of current and charge effectively being applied, since “incuded” inherently subtracts from the initial “applied.”
• P is the polarization density, while M is the magnetization density.
• Thus, Maxwell’s equations become, doing what we need to do to get rid of the constants:
• So there’s a difference between “free space” and “source-free.”

#### _______________________________________________________________________________________________________ Boundary conditions between two dielectrics

##### In general
• Parallel-to-surface E is continuous
• Parallel-to-surface H is not continuous, by Kf ×
• Normal-to-surface D is not continuous, by σf
• Normal-to-surface B is continuous
• Extra rule to remember: for parallel components consider the applied fields E and H; for normal components consider the effective fields D and B. Observing Maxwell’s equations, we relate “parallel” to the curl, and “normal” to the divergence…it’s “backwards.” The RHSs of Maxwell’s equations are all “effective” and “free.”

##### No free charges/currents
• Parallel-to-surface E is continuous
• Parallel-to-surface H is continuous
• Normal-to-surface D is continuous
• Normal-to-surface B is continuous

_______________________________________________________________________________________________________

• A perfect mirror is a perfect conductor.
• The parallel-to-surface component of the electric field of a perfect conductor must be zero, so by the boundary conditions above the total E&M wave normal to the surface on the incident side must have an electric field equal to zero. Since <incident> + <reflected> = <transmitted>. Since all of the incident light is reflected, the magnitudes of the incident and reflected light must be equal. But since the waves must now add to zero, there must be a phase shift of π.
• To go from orig ME’s to ME’s in a dielectric:
• Need equations for D and B
• Need ρ = ρf + ρb = ρf -∇⋅ P
• Need J = Jf + Jb + Jp = Jf + Jb + = Jf + ∇× M +
• To get S&T’s form of curl of electric field equation, just memorize it; can’t find derivation, though haven’t looked in Jackson: ∇× E = - (for no sources)

#### _______________________________________________________________________________________________________ Fundamental theorems of vector calculus

• 2D divergence theorem:

• 3D divergence theorem (Gauss’s theorem):

• Stokes’ theorem:

_______________________________________________________________________________________________________

• Poynting vector describes the flow of electromagnetic energy flux: S = E × H
• Intensity (power [Watts] flow across a unit area normal to S) is the magnitude of the time-averaged Poynting vector: I(r,t) = = , where T is larger than the optical period but small otherwise
• Note the “extra terms” when calculating the divergence of the Poynting vector describe the energy stored in forming the dipoles in the dielectric material.
• Overall, including the extra terms (the terms describing energy in the electric and magnetic dipoles in the dielectric), the Poynting theorem describes fully the conservation of energy, seen when applying the 3D divergence theorem to the divergence of S.
• Linear momentum density:

### Electromagnetic waves in dielectric media

• Nondispersive: instantaneous response of the system to applied field (no time lag)
• Spatially nondispersive: same but with space…no spatial “lag”

#### Linear, nondispersive, homogeneous, isotropic media

• P = ϵ0χE
• Electric susceptibility: χ
• D = ϵE (true for homogeneous and isotropic media, period [not nec linear])
• Electric permittivity: ϵ = ϵ0(1 + χ)
• Relative permittivity (aka the dielectric constant): = 1 + χ
• B = μH (true for homogeneous and isotropic media, period [not nec linear])
• Magnetic permeability: μ = μ0(1 + χm)
• μ0 =
• Maxwell’s equations:
• Identical to Maxwell’s equations in free space, except with the substitutions ϵ0 ϵ and μ0 μ
• For nonmagnetic material, μ = μ0 and n = =

#### Relaxing these conditions (only listed are the changes from the previous state)…

##### Inhomogeneous media (e.g., GRIN materials)
• χ χ(r)
• ϵ ϵ(r)
• μ stays the same

##### Anisotropic media (but homogeneous!)
• χ χij
• ϵ ϵij
• Ditto for χm and μ

##### Dispersive media
• Now there’s a linear differential equation relating P and E with respect to time
• Point I think is that χ χ(t)
• Can describe system using a linear-systems approach, using impulse response functions, convolutions, transfer functions (FT of impulse response function)

##### Nonlinear media
• In these the relation between P and E is nonlinear.
• Wave equation for all homogeneous and isotropic dielectric media, regardless of linearity or dispersivity:

• For nonlinear and nondispersive, this becomes (with P being some nonlinear function Ψ(E)):

• Most dielectric media are approximately linear unless the optical intensity is substantial.

_______________________________________________________________________________________________________

#### Derivation of particular equations from Maxwell’s equations

Maxwell’s equations in matter (general):

Assuming no currents or charges (source-free):

This becomes (source-free):
Starting with the second equation, we have (source-free all the way through):
Our grand starting point for the electric field (assuming all is similar for the magnetic field) is (source-free):

Dirk said that the permeability is typically constant throughout the material, so I’m pretty sure that implies ∇× M = 0:

• It’s okay to always assume the magnetic properties throughout a material do not change (unless we’re talking about rather cutting-edge research).

_______________________________________________________________________________________________________

• Useful vector identities:

_______________________________________________________________________________________________________

### Monochromatic electromagnetic waves

• Assume waves of the form
• Assuming Maxwell’s equations are satisfied by the full, complex expressions
Maxwell’s equations become
• From now on, we will write the complex amplitudes as regular letters. i.e., (r) D(r) D. Until this section when we’re written D we’ve referred to the real, r- and t-dependent field. So now, it is complex, and only dependent on r. So now, the true field (r,t) = .
• D = ϵ0E + P
• B = μ0H + μ0M
• = , where

is the complex Poynting vector

• Optical intensity:
• We know the full complex field vectors must obey the wave equation

For waves that have time-dependence X(r,t) = X(r)eiωt, this becomes the Helmholtz equation:

where c = , the dispersion relation for light in a common (linear, nondispersive, homogeneous, isotropic) medium.

• This Helmholtz equation holds in inhomogeneous media if ϵ = ϵ(r) varies slowly compared to the wavelength (just with the appropriate substitution k k(r).
• In dispersive media the constants χ, ϵ, k all pick up ν-dependence (and are generally complex quantities in dispersive media):

### Elementary electromagnetic waves

• Since we’re going to talk about monochromatic waves, the complex amplitudes of the fields satisfy the Helmholtz equation

• We’re also dealing with common (linear, nondispersive, homogeneous, isotropic) media

#### The transverse electromagnetic (TEM) plane wave

• Here we have
• Overall, it’s Maxwell’s equations -→ wave equation-→ Helmholtz equation
• Plugging into Maxwell’s equations for monochromatic waves in common media
we have
which becomes
This is why the TEM plane waves are transverse!
• Note:

• The previous equations imply that the ratio of the magnitudes are

where η is called the impedance of the medium.

• Intensity:

• Time-averaged energy density:

The electric and magnetic contributions to the energy are equal.

• Thus

• Linear momentum density per unit volume:

#### The spherical wave

• My definitions:
• Derivation from the vector potential:
• Then,
and from Maxwell’s equation, the electric field is
where the approximations are for where r λ. Further,

• Note that though of course the wavefronts are spherical, the magnitudes vary as sinθ, due to the representation of the spherical wave as radiation far away from a dipole.

##### Paraxial rays (points near the z-axis)
• We have

• These imply that
• If z x, this becomes
Since U(r) e-ikz, we approach a TEM plane wave.

#### The Gaussian beam

• As before, we obtain this by making the substitution z z + iz0 in the paraboloidal wave equations:

### Optics of conductive media

• All we have is one of Maxwell’s equations that changes…the Jf term is finally added:

• Now if this material has both linear dielectric properties (D = ϵE) and linear conducting properties (J Jf = σE), we have
where

is called the effective permittivity.

• All previous equations with a permittivity ϵ apply here for ϵe

# Polarization optics (chapter 6)

### Intro

• Polarization of an E&M wave is determined by the time course of its electric field
• TEM means transverse electromagnetic, in which the electric-field vectors lie approximately in planes transverse to propagation

### Reflection and refraction

• Again, at a boundary the tangential components of E and H are continuous, as are the normal components of D and B.
• Impedance η is a characteristic of the medium.
• If the electric field is perpendicular to the plane of incidence, the mode is called the transverse electric (TE) polarization, the orthogonal polarization, or the s polarization.
• If the magnetic field is perpendicular to the plane of incidence, the mode is called transverse magnetic (TM) polarization, the parallel polarization, or the p polarization.
• Record more equations from my notebook!!!

##### TM Fresnel equations

• Of course, a phase shift of π corresponds to a multiplication by -1 via Euler’s formula: e = -1.
• The phase shift of π upon reflection by a higher-index material is simply a result of the argument (the phase) of the reflection coefficient being π, and since E= rE, this phase shift shifts to the new electric field E.
• My confusion with the step function change of the phase could possibly be resolved as: Consider a smooth function that crosses the independent-variable axis. Instead of it doing that, plot the absolute value of it, so we still have a smooth function, except the phase changes suddenly by π instead of the absolute value going negative. In other words, we have z = |z|e, where perhaps what we really care about is |z|…??
• In general r and t are ratios of complex amplitudes and are thus themselves complex.
• Brewster angle:

• Power reflectance: R = |r|2
• Power transmittance: T = 1 - R|t|2 in general
• Power reflectance at normal incidence:

# Photonic-crystal optics (chapter 7)

### Intro

• Study various interferometers for the exam
• For monochromatic waves in inhomogeneous materials the Helmholtz equations become generalized:

where η(r) ϵ0∕ϵ(r).

• While waves in a homogeneous medium are plane waves, modes in a periodic medium are Bloch modes.
• Bloch modes are traveling waves modulated by standing waves.

### From lecture slides

• First use “generalized Helmholtz equations” (i.e., “master equations”), which are the Maxwell curl equations, to solve for E or H, and then use the divergence equations to obtain the other.
• Reciprocal and direct lattice vector relations: bi aj = 2πδij
• Mode (Bloch mode or Bloch state) of field E: EK(r) = eiKruK(r), where uK(r) has the periodicity of the lattice: uK(r + a) = uK(r):

• Plug this into the master equations to obtain the A (amplitudes) and the ω(K) (frequencies = dispersion relations)
• Thus, a general field E can be expressed as an LC of these modes.
• Only a 3D photonic crystal can exhibit a full band gap.
• Index contrast is what matters, e.g., silicon/air…if it’s large enough there is a band gap
• Note that in a region a traveling wave can written as a sum of forward- and backward-traveling waves.
• The “n” terms in periodic lattices usually refer to the offset (i.e., multiply n by the spacing), whereas the actual variable is the total distance, so that something like z - (n - 1)a simply represents position at the point z…it’s such because this is part of the argument of a periodic function, e.g., sin, and it’s the distance from the beginning of the period location that matters. You can always evaluate sin37π but you may as well evaluate sin = sinπ = 0.

### One-dimensional photonic crystals

• Unique properties occur especially when the period is the same order as the wavelength.
• Just as a wave in a homogeneous medium, because of translational invariance, must have the form U(z) = ce-iKz, a wave in an inhomogeneous medium must have the form

which is called a Bloch mode, and K is called the Bloch wavenumber. It is thus essentially a (traveling) plane wave (harmonic wave) modulated by a standing wave.

• Since the amplitude pK(z) is periodic, it can be expanded in a Fourier series of the form exp(-imgz), where g = 2π∕Λ. Thus a Bloch wave is a superposition of plane waves of multiple spatial frequencies K + mg.
• Now we just plug the general solution UK(z) into the generalized Helmholtz equations, expanding η(z) and pK(z) as Fourier series and solving numerically.

### Two- and three-dimensional photonic crystals

• 1D means a 3D crystal with 1D symmetry. It’s not actually simply one dimension. Ditto for 2D and 3D of course.

# Photon optics (chapter 12)

### Intro

• The optical part of quantum electrodynamics (QED) is called quantum optics.
• Photon optics is a set of rules from QED that permits us to deal with optical phenomena that lie beyond the reach of classical theory, but cannot provide an explanation for all the effects that can be explained by quantum optics.

### From lecture slides

• Uncertainty:

• As a wave travels forward (whether it’s a left-traveling wave or a right-traveling wave), its phase decreases by the amplitude of its wavevector times the distance traveled.

### The photon

• The spin of a photon is associated with its polarization. This angular momentum of the photon is intrinsic (no reference point needed).
• Photons in a resonator are like phonons in a crystal, in which properties of the mode are assigned to the quanta.
• For photons, these properties are for example frequency, spatial distribution, direction of propagation, and polarization.
• A photon of wavelength 1 μm has energy 1.24 eV, so we have

• Energy of a photon is proportional to its frequency and thus inversely proportional to its wavelength:

• Associated with each photon of frequency ν is a wave described by the complex wavefunction U(r)exp of the mode.
• Photon position; The probability of observing a photon at r within an incremental area dA, at any time, is proportional to the local optical intensity I(r) 2, so that:

• So an optical photon behaves as part wave and part particle.
• Through a beamsplitter a photon must choose the path and cannot go through both. It’s statistical.
• Energy formula to remember:

• So momentum of a photon in a plane wave mode is

• Radiation pressure is just the force (per area) emitted by a photon…think of it in terms of the fact that a change in momentum must be caused by an impulse (units: force * time = momentum)
• Photon position and time:

• The Fourier transform indeed is what basically implies the quantum uncertainty principles.
• Time-energy uncertainty, in which σt is the duration of the photon function I(t):

• So a monochromatic photon has an eternal duration. This still satisfies the uncertainty principle.
• A wavepacket photon (polychromatic…LC of monochromatic waves) passes a point in a finite time σt and thus has a finite frequency spread.
• All radiation can be described as a sum of modes (a basis), and if our modes are monochromatic uniform plane waves, we have various occupations of the different modes.

### Photon streams

• The number of photons occupying any mode is generally random.
• Photon streams often contain numerous propagating modes, each carrying a random number of photons.
• Mean photon-flux density (analog to optical intensity) is the average number of photons incident per unit area per unit time:

• Mean photon-flux (analog to optical power) is the average number of photons incident on area A per unit time:

where hν is the average photon energy and the optical power P = AI(r)dA

• Mean photon number (analog to optical energy) is the average number of photons incident on area A in time T:

where the optical energy E = PT

• In general, i.e., for polychromatic light, we subscript any quantity above with frequency so we have spectral density. E.g., Pνrepresents the optical power in the frequency range ν to ν +
• The properties of the light source determine the flucuations in ϕ(r,t).
• Coherent light results from a deterministic power source, as opposed to partially coherent light, which is generated by a time-varying random power source. If the power source is not completely coherent, fluctuations in photon detection are not independent, altering the statistics of the process. I believe for our purposes we’re considering coherent light as resulting from constant optical power P.
• In a time T, the expected number of photons is thus n = , and for coherent light the probability distribution is the Poisson distribution:

• PT above in general is:

so that in general the Poisson expectation number (of photons) is

• This is the result of the binomial distribution when dividing our interval T into N subintervals such that the probability of observing a photon in time T∕N is pretty much 0 or 1…then we take the limit as N →∞.
• Mean and variance of a random number:

• The variance of a Poisson distribution is equal to its mean:

• Signal to noise:

• For Poisson distrib, SNR = n
• Laser light follows Poisson
• For an optical resonator whose walls are held at a temperature T in thermal equilibrium, so that the photons are emitted into the modes (each of energy En) of the resonator, this is thermal light and it follows the Boltzmann probability distribution:

• So in thermal equilibrium, the energy associated with each mode is random and follows the Boltzmann distribution above.
• For photons, in which En = (n + ), we have
Calculating the mean of this distribution we find

Eliminating exp we end up with the Bose-Einstein distribution:

• So the emission of photons from atoms in thermal equilibrium follows the Bose-Einstein distribution!
• This “thermal” case was not “coherent.”
• SNR for the Bose-Einstein distribution is

which is poor due to the randomness of the amplitude and phase of thermal light.

# Photons and atoms (chapter 13)

### Intro

• If a photon is absorbed by an atom, it’s said to be annihilated; stimulated emission, a photon is said to be created

### Energy levels

• The Schrodinger equation,

is mathematically similar to the paraxial Helmholtz equation of wave optics

• The eigenfunctions represent stationary states.
• A system can move between stationary states only if there is thermal excitation or an external field. I believe this implies a time-dependent thermal excitation or external field.
• For hydrogen-like atom, in which there’s only one electron, the energy levels are:

where Mr is the reduced mass of the atom.

• Born postulate:

• Dyes have dense levels each of which are spaced far apart.

### Occupation of energy levels

• Temperature is the principal determinant of both the average and fluctuations of energy-level occupancy
• Distinguishable particles follow the Boltzmann distribution:

• Indistinguishable fermions (half-integer spin) follow the Fermi-Dirac distribution:

• Indistiguishable bosons (integer spin), like photons emitted from a blackbody, follow the Bose-Einstein distribution:

• Calculating occupancy:

1. Normalize probability: iP(Ei) = 1
2. (Occupancy of state) = (probability of state being occupied) × (degeneracy g(Ei) of state)
• Check: If there are N particles in the system we must have:

• Note: In the degeneracy (density of states) g(Ei), don’t forget to include the contribution from the Pauli exclusion principle (e.g., for electrons there’s a factor of 2 in it)
• Pauli exclusion principle: For two identical fermions, the total wave function is anti-symmetric
• Note that if E μ, the quantum distributions become the Boltzmann distribution…this is usually the case when optical transitions occur and thus for such transitions we apply the Boltzmann distribution.

### Interactions of photons with atoms

• Photons can interact with:
• Electrons in atoms
• Electrons or vibrations or rotations in molecules
• Electrons and holes in semiconductors
• Phonons in a solid
• More?
• These results come from quantum electrodynamics…these are laws that govern photon-atom interactions

#### Spontaneous emission

• This transition is independent of the number of photons that are already in the electromagnetic mode to which the energy of the created photon adds.
• Spontaneous emission into a prescribed mode ν:

where

• p is the probability density (probability per second), or rate, for the transition
• V is the volume of the cavity in which the transition occurs
• σ(ν) is called the transition cross section, which is centered about the atomic resonance frequency ν0 =
• σ can be calculated from the Schrodinger equation but this is hard, so σ is usually determined experimentally
• σ = σmax cos2θ, where θ is the angle between the dipole moment of the atom and the field direction of the mode
• c is the speed of light
• τ =
• pspΔt is the probability of a transition within Δt, and if there are N atoms in the higher-energy state, then within Δt we’d expect the number of atoms that undergo a transition to be pspNΔt. Thus the change in the number of atoms in this higher state is
• Doesn’t depend on number of photons n already in the mode…this number doesn’t matter
• Can be regarded as stimulated emission by fluctuations in the zero-point energy

#### Absorption

• Electric field of light induces dipole in atom so that there is both a P and an E present.
• The P oscillates out of phase with the E, leading to destructive interference and hence an absorption line at the atom’s resonance frequency.
• Out of phase radiation because of the i times i…there are two π∕2 out-of-phases occurring
• Oscillating dipole emits photons
• Single atom polarization:

(this γ is not the gain)

• This absorption occurs with a probability proportional to the transition cross section σ(ν)…I think this Lorentzian lineshape comes from the finite decay time and thus Fourier uncertainty (finite frequency uncertainty)
• The Lorentzian lineshape indeed comes from the Fourier transform of the exponential decay function!
• Also, photon doesn’t have to be exactly at resonance frequency ν0 for absorption to occur, that’s why lineshape function is not a delta function but rather a Lorentzian…
• Probability of absorption by an atom from an unoccupied mode of the quantized field:

(same as for spontaneous emission)

• Probability of absorption by an atom from a mode of the quantized field already containing n photons:

• Unlike spontaneous emission, this process is induced, and in particular induced by the photon or zero-point energy.

#### Stimulated emission

• Like absorption, only occurs when the mode already contains a photon
• Is inverse of absorption
• Produces a photon identical in every way to the incident photon
• Law:

_______________________________________________________________________________________________________

• So total probability densities are

• Transition strength or oscillator strength…describes strength of the interaction:

• Normalized transition cross section is the lineshape function:

• Linewidth is Δν, which is the FWHM of g(ν)
• Also, Δν 1∕g(ν0)
• σ0 = σ(ν0)

#### Spontaneous emission into all modes

• psp = σ(ν) is for spontaneous emission into one mode described by ν. For all modes, we need to integrate this. Where i corresponds to one of three emission directions, we have, for a volume V :
where, questionably, λ is the wavelength of the light in the medium.
• Thus,

which allows us to calculate S by measuring tsp, which is hard to do theoretically.

• tsp is known as the spontaneous lifetime, which I believe is the same as any τsp above
• σ(ν) = Sg(ν) = g(ν)
• If we use the non-averaged value of σ(ν), we called the corresponding tsp the effective spontaneous lifetime.

#### Induced (stimulated) transitions: absorption and stimulated emission

• Now switch from considering an atom in a resonator to a stream of photons (monochromatic of frequency ν and intensity I) incident on an atom, where this photon flux is described by

• If we construct a cylinder of volume V , length c (the distance light travels in a second), and base area A, parallel to the stream, at any given time in the cylinder is n = ϕA = ϕ, so

• We thus have
• So ϕ is like the total photon flux (which is basically probability density per unit area), σ is like the effective area that captures the photons for a transition, and Wi is the resulting probability density for the induced transition.
• Note that whereas spontaneous emission is independent of n(ν) of the mode, absorption and stimulated emission can be greatly enhanced by the presence of photons n(ν) already in the mode.
• Spectral energy density ρ(ν) is the energy per unit bandwidth per unit volume
• Overall probability of absorption or stimulated emission is
where

represents the mean number of photons per mode and “atomic linewidth” refers to the width of σ(ν).

• I believe ϕν = , which is the mean photon-flux spectral density (photons per second per unit area per unit frequency)
• λ = c∕ν0 is the wavelength in the medium at the central frequency ν0.
• Einstein coefficients:

• Ratio of basically rates of spontaneous to stimulated transitions:

This is when all of the atoms of a medium are taken to be identical and to have identical lineshape functions.

• Radiative transitions (in which atoms undergo transitions between energy levels) result in photon absorption and emission
• Nonradiative transitions permit energy transfer by mechanisms such as lattice vibrations (phonons), inelastic collisions among the constituent atoms, and inelastic collisions with the walls of the vessel.
• Each atomic energy level has a lifetime τ, which is the inverse of the rate at which its population decays, radiatively or nonradiatively, to all lower levels.
• Remember: Erad(t) e10te-γt (damped oscillator)…this represents oscillation enveloped by a decaying function
• τ2 tsptsp corresponds to the decay from E2 to only E1 and excludes possible lower levels.
• If E1 is the ground state, τ2 = tsp and τ1 = .
• As always, uncertainty in the time τ corresponds to uncertainty in the frequency and thus the energy (Fourier-type uncertainty):

• The lineshape function g(ν) has a Lorentzian profile:

so the transition cross section σ(ν) does too.

• The peak cross section (occurring when the decay is from the first excited state) is of the order of one square wavelength:

where fcol is the collision rate (mean number of collisions per second)

• Think of collision broadening as resulting from shifts in phase due to elastic collisions…there are random phase shifts of the wavefunction associated with the energy level.
• This results in a random phase shift of the radiated field at each collision time.
• Inelastic collisions are obviously a bit harder to calculate.

• Since this time not all atoms in the medium have the same lineshape function or center frequencies, we define an average lineshape function:

where β refers to the different lineshape functions existing in the medium.

• g(ν) is obtained by weighting gβ(ν) by the fraction of the atomic population endowed with the property β.
• If the atom is moving, there is a Doppler effect which results in a shift of its lineshape function. This is an example of a mechanism that would result in different lineshape functions for the atoms, requiring an average lineshape function g(ν) to be calculated.
• This is called Doppler broadening.
• The shift of an atom with velocity v is ±(v∕c)ν0
• So now v plays the role of the parameter β.
• If p(v)dv is the probability of an atom having a velocity between v and v + dv, the average lineshape function is

### Thermal light

• Blackbodies absorb all light incident on them, and they emit what is called thermal light.
• Blackbody radiation only occurs when there are no energy sources except for thermal equilibrium at temperature T.
• Photons interacting (via the three transition methods) with atoms in thermal equilibrium at temperature T are themselves in thermal equilibrium at the same temperature T. This collection of such photons is called a photon gas.
• Mean number of photons per mode near frequency ν:

This is a pretty general formula (not for just two possible energy levels).

• Thus, the average energy of a mode in thermal equilibrium is
• Density of modes for a 3D resonator (number of modes per unit volume of the resonator, per unit bandwidth surrounding the frequency):

• So for a 3D blackbody the energy density (power? per unit volume per unit bandwidth), called the spectral energy density for blackbody radiation, or the blackbody radiation spectrum:
• Integrating this we get the total power per unit volume of a blackbody…this is the Stefan-Boltzmann law:
This is the total power emitted by a blackbody at a given temperature.
• Planck’s law was evidence of quantum mechanics because it’s derived using the Bose-Einstein statistics for photons and results in accurate radiation intensity for even small wavelength particles

# Laser amplifiers (chapter 14)

### Tip

• In a laser amplifier system never forget that iNi = Na, where Ni is the population density of level i and Na is the total population density of the system.
• Assume Na is a given, a parameter, of the system.

### Intro

• A coherent optical amplifier is a device that increases the amplitude of an optical field while maintaining its phase.
• An incoherent optical amplifier increases the intensity of an optical wave without preserving its phase.
• A laser is basically an optical oscillator; LASER refers to the underlying principle for achieving coherent amplification of light and stands for Light Amplification by Stimulated Emission of Radiation.
• Electronic amplifiers use cavities to select the principal frequency; lasers use such resonators only for auxiliary frequency tuning and simply rely on the energy levels of the material to select the principal frequency.
• Whereas light transmitted through matter in (thermal) equilibrium is attenuated due to more absorption than stimulated emission occurring, in a laser instead the higher energy levels are already excited (via pumping) and thus more stimulated emission occurs…there is instead amplification rather than attenuation.
• The amplifier gain is the increase of amplitude of an input signal.

### Theory of laser amplification

• Spontaneous emission is responsible for amplifier noise and is otherwise not too important it seems.
• The probability density (s-1) that an unexcited atom absorbes a single photon (absorption) and that stimulated emission occurs is

where

• Wi is a rate, whose inverse is a time constant.
• Ni refers to the number of photons per unit volume in energy level i.
• The average density of absorbed photons is N1Wi and the average density of clone photons generated as a result of stimulated emission is N2Wi, so the net number of photons gained per second per unit volume is NWi, where N = N2 - N1.
• N is the population density difference, or simply the population difference.
• If N > 0, we have a population inversion.
• If N = 0, the medium is transparent.
• As incident photons travel through an inverted medium, the photon-flux density ϕ(z) increases exponentially.
• A gain is described by the differential equation

where

is called the gain coefficient. This refers to the increase of photon-flux density as photons go through a medium, and it depends on the current photon-flux density too.

• γ also represents the gain in intensity per unit length of the medium.
• α(ν) = -γ(ν) is the attenuation coefficient.
• The gain in a finite region of length d is the ratio of the photon density or intensity between the ends:
This is called the amplifier gain.
• Linewidth conversion:
• Skipping phase shift it seems…
• If something is frequency-dependent, frequency dependence of the phase shift must be examined as well. E.g., since the gain of the resonant medium is frequency-dependent, the medium is dispersive and a frequency-dependent phase shift must be associated with its gain.
• I think overall gain refers to photons exponentially increasing in number as they move through the material.

### Amplifier pumping

• Pumping can often most easily be done by using auxiliary energy levels, not just a straight pump of the transition of interest.
• The rate equations describe both the pumping and the absorption/emission rates.
• The steady state population difference N in the absence of amplifier radiation is found from two differential equations and is

where

• R1 is the pumping rate out of level 1
• R2 is the pumping rate into level 2
• τ1 and τ2 are the overall lifetimes of each level
• τ21 is the lifetime of the decay from level 2 to level 1
• Not shown: τ20 is the lifetime of the decay from level 2 to all lower levels besides 1
• Not shown: tsp is the radiative spontaneous emission component time constant
• Ideally we want τ21 tsp τ20 so that τ2 tsp and τ1 tsp; under these conditions our rate equation becomes

• If there’s no depumping (R1 = 0), or when R1 (tsp∕τ1)R2, this becomes

• I think R2 can include “pumping” coming from higher energy levels.
• In the presence of amplifier radiation the steady-state population difference is now

where

is the saturation time constant, which is always positive.

• As radiative processes get stronger, N 0, as the radiative processes balance each other and push the system this way…independent of the initial sign of N.
• When Wi = 1∕τs, N is reduced by a factor of 2 from is value when Wi = 0.

#### Pumping schemes

• Again, the purpose is to increase the population in the higher-energy level and to decrease that in the lower-energy level.
• W is the pumping transition probability
• N0 and τs saturate as W increases in both schemes below.
• Four-level laser amplifiers typically have larger gain than three-level amplifiers

##### Four-level system
• Note that there is a total atomic population density Na available, so that the pumping rate from level 0 (ground state) to level 3 (some state above level 2) is not independent of N1 and N2.
• Rate equations?

• Levels 1 and 3 are short-lived
• If E1 kT, this is especially true
• The pump is from level 0 to level 3, and since level 3 is short-lived, and since level 1 quickly decays to level 0, this is essentially a pump from level 1 to level 2
• Note that nominally:

so that any finite pumping R will yield a population inversion.

##### Three-level system
• This time the lower-energy state of interest is the ground state: E1 = E0
• τs,3 = τs,4
• Unlike for the four-level system, E1 is more easily populated since it’s the ground state
• Rate equations?

• Note that nominally:

so that significant pumping R is needed just to create a population inversion, let alone N0 > Nt > 0.

### Amplifier nonlinearity

• The gain coefficient γ(ν) depends on the population difference N:

so γ = γ(ν,N)

• N depends on the pumping rate R
• e.g., in a four-level system, using approximations, we have

so N = N(R)

• N also depends on the transition rate Wi:

so N = N(R,Wi)

• Wi depends on the radiation photon-flux density ϕ:

so Wi = Wi(ϕ)

• Thus, γ = γ(ν,R,Wi) = γ(ν,R,ϕ). However, R can be written in terms of other variables, so I believe we have γ = γ(ν,ϕ): The gain coefficient of a laser medium is dependent on the photon-flux density that is to be amplified. This is the origin of gain saturation and laser amplifier nonlinearity:
where
where again, λ = c∕ν0 is the wavelength of the light at the transition frequency ν0. (It would be more clear if λ was written as λ0.) ϕs(ν) is known as the saturation photon-flux density.
• The point is that ϕ depends on N (photon flux depends on population difference) and N depends on ϕ, so there’s a nonlinearity.
• Finally,
where
γ(ν) is the most general gain coefficient, known as the saturated gain coefficient (for homogeneously broadened media). γ0(ν) is the small-signal gain coefficient.
• Again, the point is that the gain coefficient, which is responsible for amplifying ϕ, in turn depends on ϕ.
• ϕs(ν) represents the photon-flux density at which the gain coefficient decreases to half its maximum value γ0(ν).
• Unsaturated gain coefficient γ0(ν) has linewidth Δν, whereas saturated linewidth has linewidth:

• So gain coefficient, as saturation nears (ϕ ~ ϕs) gets short and fat w.r.t. ν. The fattening corresponds to reduced frequency selectivity, of course.
• Remember, gain coefficient γ is gain per unit length.
• Gain:

• Small signal approximation is when γ γ0 (not near saturation).
• Solving

when γ depends on z (implying saturation) this time yields

where X ϕ(0)∕ϕs and Y ϕ(d)∕ϕs are the input and output photon-flux densities normalized to the saturation photon-flux density and the gain G = Y∕X.

• Examining limiting cases:
• X and Y both much smaller than 1:

This is the solution we obtained way above for γ γ0.

• X and Y both much greater than 1:

which implies

This is the situation of heavy saturation in which the atoms of the medium are “busy,” emitting a constant photon-flux density , which is independent of the input ϕ(0).

• If the gain coefficient is negative (N0 < 0), the absorption coefficient α = -γ is positive, and instead of amplification we have attenuation, and instead of gain we have transmittance. The equation

still applies, except now γ0d is a negative quantity. As the input transmits through the medium, N 0 (but might not reach zero since the medium has a finite size). As the input is increased, eventually N will actually approach 0 in the medium, so that the transmittance eventually equals 1.

• This relation holds:

• Such a medium is called a saturable absorber.
• So now transmittance = Y∕X.

#### Saturated gain in inhomogeneously broadened media

• Average small-signal gain coefficient:

• Saturated gain coefficient (be careful when doing this one!):

where

where

• Here gβ(ν) = g(ν - νβ), where

and we have

where

• I believe the subset of atoms β is homogeneously broadened.
• NOTE THAT νβ IS CENTERED AROUND ZERO, SO BE MINDFUL OF THIS WHEN CALCULATING THE INTEGRAL (THE AVERAGE γ)!!!!
• For normally distributed (σD) velocities of the atoms and for this distribution being much broader than the gains γβ(ν):

where

Since the average gain coefficient γ at ν = ν0 is and that for a homogeneously broadened (due to saturation) medium is

we see that with an increase in the photon flux at ν = ν0 the gain coefficient increases more slowly in inhomogeneously broadened media.

• Note that *only here* ϕ is for ν = ν0. ν0 is the transition frequency corresponding to the subset of atoms that is not moving at all.
• I believe homogeneous broadening can and does result from saturation.
• If there is a normally distributed inhomogeneously broadened medium, the average gain curve at an arbitrary frequency γ(ν) will also be normal, I believe with a standard deviation that’s the same as that of the broadened medium. We just showed that the gain coefficient for photons of frequency ν0 is

so if a monochromatic beam is incident on the medium we can have saturation at only the frequency ν1 of the beam, which means there will be a drop in the average gain curve at ν1. This is called spectral hole burning.

• Note the actual ν dependence of ϕ:

• So if the photon flux were the same for all populations (with difference speeds and different νs), Δνs would be the same for all populations, but if the flux at ν1 is larger than at other frequencies, there will be a hole at ν1 of width Δνs(ν1), so that by increasing the flux ϕ(ν1) the hole will become both deeper and wider.
• Understand when these holes are symmetric in the curve.

# Lasers (chapter 15)

### Intro

• The output of a laser is fed back to the input with matching phase (multiple of 2π) to do more stimulated emission.
• The laser initially bursts I believe and then the steady-state is the actual observed laser state.
• Steady state occurs because the gain saturates.
• Both the gain and phase shift are functions of frequency.

### Theory of laser oscillation

• Coherent = at the same phase
• Phases of a laser:
• Increase the pumping rate R (increasing N0 [not N]) until the pumping threshold is exceeded (gain condition) and thus so is the minimum population difference needed for laser oscillation (or “lasing”). ϕ and I will then go from not increasing to increasing exponentially.
• No amplifier radiation (no stimulated emission):

• As radiative processes get stronger, N 0, as the radiative processes balance each other and push the system this way…independent of the initial sign of N.
• In the presence of amplifier radiation the steady-state population difference becomes
where

• Gain coefficient (gain per unit length): γ(ν)
• Governs the rate at which the photon-flux density ϕ or the optical intensity I = hνϕ increases via

• For small ϕ the (small-signal) gain coefficient is:

where:

• N0 is the equilibrium density difference (the subscript stands for small-signal, not equilibrium [but equilibrium is implied])…equilibrium refers to, I believe, steady-state + no stimulated emission
• σ(ν) is the transition cross section
• tsp is the effective spontaneous lifetime (for stimulated emission?)
• g(ν) is the transition lineshape
• λ = λ0∕n is the wavelength in the medium
• ϕ will increase and at some point the amplifier enters a region of nonlinear operation in which it saturates and its gain decreases.
• Then its population difference becomes, for a homogeneously broadened medium,

where

• ϕs(ν) = is the saturation photon-flux density
• τsis the saturation time constant, which depends on the decay times of the energy levels involved (for a four-level pump, τs tsp; for a three-level pump, τs = 2tsp)
• Thus, a saturated amplifier has a gain coefficient (for homogeneous broadening):
• The amplifier phase shift per unit length that is due to the laser amplification, when the lineshape is Lorentzian with linewidth Δν and

is

• This phase shift is in addition to that introduced by the medium hosting the laser atoms.
• Wavenumber (wavevector) is like phase shift per unit length
• αr is overall loss per unit length
• Photon lifetime: τp = 1∕αrc
• When resonator losses are small and the finesse is thus large
• Gain condition is that the small-signal gain coefficient be greater than the loss coefficient:

• Thus,

where

is the threshold population difference. This last equation implies that the threshold the lowest at the frequency of the peak of the lineshape function (ν0).
• Phase condition for coherence is

• When the contribution arising from the active laser atoms (second term) is not negligible, “frequency pulling” results.
• This pulling results in all the resonant frequencies being pulled toward the central frequency of the resonant medium ν0.
• Under the steady state operation of the laser in which the loss equals the gain, an approximate solution of this equation is

where νq are the “cold-resonator” solutions to the equation, δν is the spectral width of the cold resonator modes, and Δν is the linewidth of the atomic transition of interest.

• Perhaps typically the atomic transition linewidth is much wider than the resonator mode linewidths.

### Characteristics of the laser output

#### Power

• Steady state occurs when N = Nt or when

• The middle of the inverted “S” curve occurs when N0 = 2Nt (seen from the above equation).
• The facts that the steady state N value can never increase above Nt and that the steady state gain γ(ν) can never increase above αr is called gain clamping.
• Note that the photon-flux density (steady state) for photons traveling in a single direction is ϕ∕2.
• Thus, the output photon flux density is
where T is the intensity transmittance of the transmitting mirror.
• The optimization of the transmittance is based off the principle that to increase the output flux you can increase the transmittance, but this will add to the loss inside the resonator and thus decrease the steady state flux inside the oscillator. That’s where the tradeoff comes in.
• We can relate ϕO and T using equations from this subsection to obtain:

• Assuming T 1 (which is fine as can be seen by plotting this for common parameters and seeing that the maximum occurs at T 1) we have ln(1 - T) ≈-T and minimizing ϕO with respect to T we find:

#### Spectral distribution

• B is the width (units of frequency) of the allowable ragion in which the γ0(ν) curve is greater than the loss αr. If νF is the spacing between resonator modes, the number of possible laser oscillation modes is

This rule is based on the two conditions for oscillation, the gain and phase conditions.

• νF = c∕2d
• The actual number of modes depends on the nature of the atomic line broadening mechanism.

• Only the mode or two modes (in the symmetric case) closest to ν0 will remain in the end, since the entire gain curve will reduce in size because the most dominant modes will deplete (to a point) the population difference:

• UNLESS there’s spatial hole burning, because the different modes occupy different spatial portions of the active medium. The bottom line is that spatial hole burning allows another mode, whose peak fields are located near the energy nulls of the central mode, the opportunity to lase as well.

• In an inhomogeneously broadened medium, the gain γ0(ν) represents the composite envelope of gains of different species of atoms.
• Now there are different populations of atoms for resonator modes to draw from, so some populations may not be drawn from at all while others are drawn from, causing holes in the overall profile where modes overlap with atomic populations (whose gain curves are greater than the loss of course). The result of this is that the final spectrum has peaks separated by the resonator mode spacing, νF = c∕2d.
• This phenomenon is called spectral hole burning.
• The more central the frequency the depper and wider the hole (if it overlaps with a population whose gain is above the loss).
• Width of the holes was calculated earlier:

where ϕ is governed as found above in this section (in the cases brackets).

• Thus, there ends up being many more modes than for homogeneously broadened media.

Spectral hole burning in a Doppler-broadened medium

• The interaction radiation of a group of an atom moving with velocity v toward the direction of propagation interacts with radiation of frequency ν0(1 + v∕c).
• Since radiation of frequency νq interacts with populations of atoms νq = ν0(1 ± v∕c)v = ±(νq - ν0), symmetric hole burning occurs for the frequencies νq - ν0 = ±.
• Note that this symmetricity is due to the fact that the photons are in a resonator and thus are bouncing in two different directions.
• Of course, holes have widths Δνs.

#### Spatial distribution and polarization

• A laser with two planar mirrors outputs a plane wave propagating along the axis of the resonator.
• A spherical mirror resonator supports Hermite-Gaussian beams.
• Because of their different spatial distributions, different transverse modes undergo different gains and losses (less so for longitudinal modes unless there’s spatial hole burning).
• Higher-order modes can generate larger optical power, while the Gaussian mode has the smallest beam diameter and can be focused to the smallest spot size.
• Of course, each mode (l,m,q) has two polarizations (two independent modes). If everything else is equal, the laser will oscillate on each of the two polarizations simultaneously, independently, and with the same intensity. The laser output is then unpolarized.
• A greater portion of the gain medium contributing to the laser output power as a result of the availability of a larger model volume
• Higher output powers attained from operation on the lowest-order mode, rather than on higher-order transverse modes as in the case of stable resonators
• High output power with minimal optical damage to the resonator mirrors, as a result of the use of purely reflective optics that permits the laser light to spill out around the mirror edges (this configuration also permits the optics to be water-cooled and thereby to tolerate high optical powers without damage).

#### Mode selection

• The point is to use an element within the resonator to provide loss sufficient to prevent oscillation of the undesired modes.
• Selection of a laser line (certain frequency): You can use a prism to deflect lines of undesired modes (which won’t be reflected from the other mirror and hence not participate in feedback).
• Selection of a transverse mode: Since they have different spatial distributions, you can physically use an aperture to prevent some modes, or even design the mirrors to favor a particular transverse mode.
• Selection of a polarization: Use a polarizer, optimally inside the resonator. Use of Brewster windows aids in this.
• Selection of a longitudinal mode:
• Increase loss so that only the largest remains. Bad because the surviving mode would itself be weak.
• Increase the spacing νF = c∕2d by making the resonator length shorter. Not too good either, since this reduces the volume of the active medium. Better ways to increase the spacing:
• Put an etalon inside the resonator that filters out the undesired modes. Usually tilted for fine-tuning.
• Using multiple-mirror resonators, i.e., coupled resonators (both active or just one active) of different lengths or coupled resonator/interferometers

### Common lasers (skipping details)

#### Solid-state lasers

Examples are ruby, alexandrite, Nd3+:YAG, Nd3+:glass, Er3+:silica fiber, Nd3+:YVO4, Yb3+:YAG, Ti3+:sapphire, fiber lasers, Raman fiber lasers, random lasers (scattering itself provides feedback)

#### Gas lasers

Examples are He-Ne, Ar+, Kr+ (atomic and ionic lasers), CO2, methanol, water (molecular gas lasers), excimer (“excited dimer”) lasers, chemical lasers such as HF

#### Other lasers

Organic dye lasers, solid-state dye lasers, extreme UV and x-ray lasers, free-electron lasers

### Pulsed lasers

Why pulse lasers? Doing it greatly increases optical power (total energy is the same of course, it’s just stored in the “off time”).

#### Methods of pulsing lasers

Obviously we should do this internally, not externally, to not waste energy.

##### Gain switching (periodically increasing gain, constant loss)

Turn the laser pump on and off. These is easy to do using electrical pulses.

##### Q-switching (constant gain, periodically decreasing loss)

Reduce the loss via a modulated absorber inside the resonator. Energy is stored in the atoms.

##### Cavity dumping (constant gain, periodically increasing loss)

Increase the loss via altering the mirror transmittance. Energy is stored in the photons. Like periodically dumping a bucket of water that’s constantly being filled.

##### Mode locking

Couple together the modes of a laser and lock their phases to each other. Like Fourier components.