General Flashcards
What factors influence light-solid interaction
Light matter interactions are fundamentally a function of the material and the light source where the optical properties are described through the refractive index
Light (light-matter interactions)
This is a transverse electromagnetic wave with an electric and magnetic field which oscilate perpindicular to the direction of propogation at an optical frequency of 1013-1017 hz which includes UV, visible, and IR.
Fundamentally light is EMR governed by Maxwell’s equations and produced by the acceleration of electric charges.
Solids (light-matter interactions)
These are considered to be materials of positive and negative charged particles with inducible dipoles
Interaction (light-matter)
The fundamental interaction of light and matter is that light polarizes the atoms by nature of its transverse electromagnetism to produce oscillating dipole moments. The interaction produces either reflection, transmission, absorption, or scattering.
Dipole moments
This is defined as qx where q is the partial charge and x is the distance of the partial charge from equilibrium
Electronic oscillations
This is the concept of electrons surrounding the nucleas behaving like springs where they oscillate about the positively charged protons. In this model there are several electrons attached to springs surrounding the nucleas that oscillates like a hula hoop.
Because mass of electrons is so much less than that of the nucleas the shifting electronic field of EMR makes the electrons shift with respect to the nucleas and not vice versa
Resonant frequency
This describes the frequency of light that coincides with the frequency of oscillations in order to create constructive interference.
Resonant frequency of electronic oscillations
ω is proportional to (1/me + 1/mn).5 = (1/μ).5 where m=mass and μ = reduced mass. Because me<
Resonant frequency of electronic oscillations speed
These are the easiest to induce and therefore and the fastest and have frequencies in the UV and visible spectra. They are a function of collisions, electronegativity, and orbital arrangements
Vibrational oscillators
This is a type of dipole based on charged atoms that are bonded to one another vibrating from their equilibrium position. It is dependent on the rigidity of the bond and the degree of covalent nature.
Because the speed of this type of action is slower the resonant frequency is in the IR range. The oscillating atoms create phonons.
Quantum Theory version of vibrational oscillators
Quantum theory says that electrons do not vibrate in a bond but are elevated from the inner orbitals into the conductance bands at the resonant frequency where they emit photons when returning to the ground state
Free electron oscillator’s
These are electrons that are within the outer orbits of large atoms (like metals) that can oscillate from light interactions without any true restoring force. These oscillations have a resonant frequency of 0.
In metals these are known as the conductance electrons and are responsible for the reflective nature of metals.
Polarization of a Medium Equation
P = εo χ E where
P=Polarizability of the medium (dependent variable)
εo = permittivity of a vacuum =
χ = Dialectric susceptibility of the medium
E = electric field that the material is within
This is the basic way of describing how a material behaves but is best applied to net effects
Diaeletric
A dielectric is an insulator that can be polarized in an electric field not because of electron flow but electron drift. The negative charges in atoms align with the negative end of the electric field and the positive charges to the positive side of the field.
These are effectively capacitors
Electric Susceptibility
χ represents the electric susceptibility of a dielectric material. It is a property of the material that represents the ease at which is polarizes in response to an electric field.
It directly influences the electric permittivity of the material.
Vaccum permittivity
This represents the ability for an electric field to exist in a vacuum. It is given in terms of F/m which represents the capacitance per meter of space.
εo = 8.854 * 10-12
Polarizability of a medium meaning
This represents the density of moment dipoles in the medium. Thus, it is given in C/m2 It represents how easily a material accumulates charge at a point that is not equilibrium.
Moment dipole = qx
3 levels of polarizability with EMR
At low frequency (radio/microwave) χ (magnetic susceptibility) is highest because the wavelength is long enough for materials to reorient to constructively interfere with the EMR. It is a rotational shift.
In the IR/Vis there is vibrational and electronic dipoles. It slowly decreases with frequency in this space. In the lower frequencies (IR) the vibrational dipoles create heat through distortion of the bonds. This is phonons
In the visible and higher frequencies only, electronic vibrations occur because these are the fastest so they can occur rapidly. They resonate with electronic vibrations to create photons.
polarizability with consideration of induced dipoles
P = εo ( χ *E + χ2*EE + χ3*EEE+…)
This equation fundamentally says that the various oscillations of the atoms and charged particles induce impacts onto the polarizability of a material
In most optics (except lasers nothing above the second order term matters.
This represents that the polarizability of the material in electronic oscillations is influenced by the induced electric fields produced within oscillations.
The Dipole Oscillator Model
This says that within electronic vibrations we consider the electron to fluctuate with their natural resonant frequency (ωo) with dampening. Their position is given by
me ( d2x/dt2 + γ (dx/dt) + (ωo)2x) =qE
where γ = dampening constant and x is given by the motion of the electron from the equilibrium point along the vector of E
This fundamentally says that F = ma where a = d2x/dt2
Dipole moment of a Lorentz oscillator (e with springs)
For N = #of charges/volume
P = Σ qx = E*N [q2/m /(ωo2 - ω2 + iγω)] where ω = the frequency of light and ωo =natural frequency of the oscillator
This is found by using E = Eo eiωt as an induced oscillation created by the EMR with the dipole oscillator equation. It represents how polarizable a material is within an EMR interaction
Real Refractive index of a material with 1+ oscillator
n2 = 1 + (q2/εom) (Σin Ni/(ωi2 -ω + iγi ω)
where n = refractive index
εo = permittivity of an electric field in a vacuum
Ni = number of i atoms
ωi = resonant frequency of atom i
γi = dampening of the atom i
ω = frequency of light
The Complex Refractive index
n = n + ik
n = real refractive index = c/v
k = extinction coefficient which is proportional to the absorption coefficient (α) which is related to the dissapation/attenuation of light
How does n vary with ω
IF ω i then n is ~constant and low
IF ω = ωi then n is increases rapidly because iγi ω is ~0. This produces a resonance line.
IF ω > ωi then n is ~ 1
How does n vary with frequency in visible light?
n is proportional to c* ω so at higher ω n increases which changes the angle of refraction and creates dispersion
When are materials transparent?
This occurs between resonant frequencies where n ~n and thus almost no dampening occurs because dampening is given by the term iγi ω
Electric field for a light wave
E = Eo ei(kz - ωt) where
k = the wave vector = nω/c if transparent = (n + iK)(ω/c) with absorption
Plugging this in
E = Eo e-(ωKz)/c ei(ωnz/c - ωt)
Where Eo e-(ωKz)/c is the amplitude and ei(ωnz/c - ωt) represents the real refractive index of a wave
The important part about this function is that it exponentially decays with z
Absorption coefficient
α = 2Kω/c where K = extinction coefficient ω = angular frequency of light
Claussis Masetti equation for isotropic media
n2 - 1 / n2 + 2 = χ/3 This says that for sufficiently dense media that the dipoles of nearby atoms influence the polarization of each dipole. χ is the electric susceptibility which is directly related to P
Four Measurable Optical Parameters
You can measure reflection, absorption, scattering, and transmission
Reflection
R = IR/Io = [(n-1)2 + k2]/ [(n+1)2 + k2]
where: I = beam intensity (lumens)
* n*=real refractive index of the material
k = extinction coefficient ~0 for transparent materials
Absorption
This occurs when the angular frequency of light = resonant frequency and vibrational oscillations lead to changes in kinetic energy due to attenuation of the wave (friction)
α = attenuation efficiency
Absorption Equation
This is given by Beer’s law which says
I(z) = Io e-αz so it decays exponentially with α being the absorption efficiency coefficient.
Scattering
This is the idea that certain light is not attenuated in a solid but the waves are split and change direction which lowers E
Scattering Equation
I(z) = Io e-Sz Where: S = scattering coefficient
S is proportional to 1/λ4
Scattering and attenuation
Because we cannot measure the difference between S and α we say αT = S + α but S is ~0
Conservation of Energy when measuring light
For light with an intensity of Io upon entering a material
1 = (IR/Io) + (I<span>T</span>/Io) + (IA/Io) = R + T + A
Where R = reflection, T = transmission, and A = absorption
Transmission equation
T = (1-R)2 e-αl
Where: R = reflection coefficient
α = attenuation efficienvy
l = thickness of the material
T = proportion of light transmitted
Tool to measure light-matter interactions
A spectrometer is used to measure light-matter interactions by measuring Io , IR and IT as a function of wavelength
Fourier Transform Infrared (FTIR)
This uses a glow bar to create a light source over IR and NIR for spectrometry
UV-VIS double beam spectrometer
This uses a tungsten and deuterium lamp for analyzing UV to NIR
For dialectrics and semiconductors how does A change with wavelength?
At the lower bound of transparency the short wavelengths are absorbed by electron oscillators which are excited to the conductance band.
At the upper bound is when atomic vibrations begin to occur which create heat in the IR range. Thus a higher atomic mass leads to a greater maximum wavelength
Transparency
When materials are transparent (when resonant frequency matches the angular frequency of light) k (extinction coefficient) is very small which means that n ~ n
n is the real refractive index and is a function of N, the number of electronic oscillators within a volume of material where an increase in N leads to greater n and greater R. Thus transparent materials with a higher electron density (like semiconductors) are more reflective
Ways to find n
n is principally found via snell’s law that says n1 sin (θ1) = n2 sin (θ2) where monochromatic light needs to be used to find n because n = f(λ)
There is also ellipsometry which uses the properties of a thin film and changing angles of incidence to derive n
Refractometers: Use and types
These are used to find the angle of refraction for an incident ray to find the real refractive index.
Abbe refractometers use a glass hemisphere
V-block refractometers use a V-shaped prism
Prism gonimeter uses a prism specimin and a prism of glass
When is ellipsometry preferred for finding n
n is better found using ellipsometry when investigating light that is at a frequency above the transparency frequency (where electrons are excited across the conductance band and absorption dominates)
This would be most useful for semiconductors that have smaller eg gaps so they become opaque at lower frequencies
Plasma frequency
ωp2 = plasma frequency = Nq2/εome and n2 = 1-ωp2/ω2
This is the frequency that metals stop reflecting light. It is usually near UV.
When ω< ωp n2 is imaginary so it is opaque
When ω> ωp n2 is real so it transmits light
n2 formula for metals
n2 = 1-q2N/εo m ω
Assumptions for metals in the lorentz model of e- oscillators
This assumes that there is not a dampener because electrons flow and ωo = 0
It also assumes that all electrons are equal
Plasma frequency formula
1-(q2N/ε m)= ωp2
Classes of Diffraction Gratings
Diffraction Grating are either reflective or transmissive
Diffraction Gratings
These are optical elements with repetitive parallel patterns embedded into a grating that create dispersion which is used to separate wavelengths of light.
Transmissive Diffraction Gratings
These function by having many tiny slits which influence the output waves to create interference
Diffraction Grating Formula
a[sin(θm -+sin θi] = +-mλ
Where λ = wavelength of interest
θm = angle from the surface normal to the transferred ray
θi = angle to normal of the incident ray
Sign convention for θ
If both rays exit the transmission grating on the same side of the surface normal then they are both positive
Sign convention for grating formula
Both signs are negative for transmission and positive for reflection
Order of Principle Maxima
Diffraction grating create diffraction patterns where m = the order of principle maxima which refers to which “hump” it is on the diffraction pattern. At every m not equal to 0 the light wavelength is a function of θm
Reflective gratings
This usually describes a metal deposited onto an optic with parallel groves ruled onto the surface. This has a specular reflection given by θr
Blazed Gratings
These are gratings that use a step-like pattern to, in theory, maximize efficiency at one value of m for the blaze wavelength (design wavelength). Instead of the light energy being dispersed over many humps, it is concentrated on one m. They do this by having each reflective surface be individually tapered.
Blaze grating anatomy
a = length of one step
blaze arrow = the “up” direction where the non-lee side faces
γ = the angle from the surface normal to the step normal.
Blazed Grating zero order geometry
For m=0 -θr = θi
Blazed grating general relation between θr and θi
If theta is on the same side of the normal then θr <0 and θi-θr = 2*γ
Blazed Grating θi =0
When the angle of incidence = 0 then a * sin (-2*γ ) = mλ
Littrow Configuration Equation
2a sin(θL) = mλD
θL = γ
where θL = Littrow configuration angle for the design wavelength λD
Angular separation formula
ΔλFSR = λ/m
ΔλFSR = the free spectral range which is the distance of spread for the specta which decreases as m increases because spectra begin to overlap. At higher m the efficiency decreases too.
Littrow Configuration
This is a configuration for maximizing the intensity of one order (m=1) of light. It is used in monochromoters and spectrometers
Echelle Grating
This is a ruled diffraction grating with extremely high blaze angles and large a to concentrate higher modes of principle diffraction. It is mainly used for producing very high resolution monochromatic light
Ruled Gratings
These are mechanically cut and polished gratings like blaze gratings
Holographic gratings
These are etched by lasers and have a sinusoidal pattern. They have less error at a reduced efficiency where dispersion = f(grooves/mm)
Gratings being used in monochromators
The idea is that you input polychromatic light which colimmeates (becomes parallel rays) in a focusing mirror which interacts with a grating and the spectrum is then mechanically selected
Diffraction Efficiency
This is defined as the incident optical power (W/m2)/entrant power
Pattern density influence on pattern
With increased density of pattern there are more reflectors and there are more “small humps”
Diffraction gratings: missing orders
With transmissive gratings if the distance between slits is given by w=a/m where a is the size of the opening then the m’th order will not appear in the dispersion
Spectrogrpahs
These are things that use gratings to disperse light which is then measured using an electric analyzer. They take a broadband source and disperse it so that way each wavelength is in a different location to be read by the signal detector.
Gratings use with laser pulses
Gratings are used to disperse the laser pulse to make the energy/area less by effectively lengthening the pulse time. This is helpful for protecting sensitive instruments/objects. The lasers can be reconstructed to the original amplitude too.
What is light fundamentally?
Light is generated as the result of the acceleration of electric charges
Maxwell’s equation for light propagation
E(z,t) = x Eo cos( ωt-kz+ ϕ) for monochromatic light propagating in the z direction
ω = 2pi v (v=frequency)
k = 2pi/λ = wavenumber
ϕ = arbitrary phase correction
E = the electric field vector (perpendicular to the z direction)
Types of Light Polarization
Light is polarized when it is constrained to oscillate within specific planes
Linearly polarized light is light that is polarized within a single plane
Circular polarization is when the oscilations are in a perfect circle
Elliptical polarization: This describes when light’s E and h fields are not equal creating an elliptical cross section
How does polarization influence reflection
s-polarized light reflects better than p-polarized light
Types of reflectance polarization
s-polarization refers to when light oscillates parallel to the surface it is reflecting on.
p-polarization is when the light is oscillating normal to the propagation direction and thus intersects the surface.
Brewster angle
this is the incident angle where p-polarization has zero reflectance. It is the minimum reflectance in the reflectance vs incident angle curve
angular dispersion
This is a a measure of how wide the wavelength is displayed for a diffraction grating
dθ/dλ = L/Acosθi
where A = spacing =a and L = order = m
This says that with lower spacing, higher order, and higher incidence angles that the radians/wavelength increases
Monochromator
This takes polychromatic light and reflects it on a colimeating light then a diffraction grating. This seperates the wavelengths by distance. The specified wavelength of interest is the only one that is allowed through a slit based on where it appears in the diffraction.
Thermal polychromatic light sources
These work by heating a material above some critical temperature or by passing an electric current through a high MP metal. These are good sources because they are characteristically broad and continuous.
Blackbodies
These are materials that absorb all incident radiation independent of wavelength and conversely emits all wavelength as a function of temperature
Radiant Emittance Formula for blackbodies
M(T) = C1 λ-5(exp (C2/λT) -1)-1
Where M = radiant emmittence (W/cm2*unit wavelength)
λ = emitted wavelength
This says that as wavelength increases the power decreases
Weins Displacement Law
This says that the maximum power of a blackbody as a function of temperature plots along .289878 cmK
Real Refractive index of a material derivation
Using maxwell’s displacement vector which describes how an electric field permeates through a material (D = εo E + P = εE)
and n2 = εr = ε/εo and P = Eεoχ where χ = f(oscilations within E) to get that
n2 = 1 +χ
n2 = 1 + (q2/εom) (Σin Ni/(ωi2 -ω + iγi ω)
Transparency of glass with wavelength curve
Glass become transparent in the near vis range until the UV range because the conductance band gap for SiO2 is relatively large compared to a semiconductor which set a low upper limit for wavelength (high upper limit for omega). Additionally because the atoms in glass are smaller than those of a semidconductor which usually have metals, phonons perterbate at lower IR wavelengths.
Transparancy of a semiconductor with wavelength
Semiconductors are typically doped glasses with heavier metals (EX: Ge)
This means that on the low wavelength (high frequency) bound of transparency it is lower than dielectrics because the conductance band is smaller so less energy is needed to excite electrons.
On the high wavelength (low frequency) bound of transparency, the heavier metallic atoms oscillate slower and therefore require longer wavelengths to resonate. This effectively shifts the transparency curve to higher wavelengths.
Plasma frequency meaning
ωp2 = plasma frequency = Nq2/εome and n2 = 1-ωp2/ω2
n>=1 hence,
When ω< ωp n2 is imaginary because n is less than one. This means that kappa (extinction coefficient) is large so R approaches unity so the metal is opaque. This is at high wavelengths.
When ω> ωp n2 is real so R becomes a function of n and not kappa. This means that it decreases significantly
Reflection Equation meaning
R = IR/Io = [(n-1)2 + k2]/ [(n+1)2 + k2]
IF the material is transparent then k ~0 and n dominate the equation. This reduces to R = [(n-1)2 ]/ [(n+1)2] where R ~ 0 when n = 1 and increases quasi linearly as n increases to approach 1.
When n is complex k>>n and R approaches 1 (R = k2/k2) This is why metals are reflective at low frequencies where n is complex
Reflectance of a metal as a function of wavelength
At low wavelengths (high frequencies) usually in the UV region, metal is non-reflective (n is real and R~0). At the plasma frequency n becomes imaginary and metals become highly reflective. This is because the band gap for metals is very small and can be excited at a variety of wavelengths.
Plasma frequency derivation
ωp2 = plasma frequency = Nq2/εome and n2 = 1-ωp2/ω2
Basically, because we assume metals are homogenous with little variation in their electrons, which are in the conductance band and have a natural frequency of zero we assume that ωi = 0; γ=0;
This means that the big long equation for the complex refractive index reduces and we can take out a constant and call it the plasma frequency.
Why is the reflectance term in the transparency equation squared?
This is because reflectance (R) occurs twice when light transmits through a material (front and back).
