Equations Flashcards
Wave particle duality
λ = c/v
E = hv
De broglies equation
λ = h/p = h/(mv/sqrt(1-v^2/c^2)))
Stefan’s law
I = P/A = σT^4
Wiens law
λ𝑚𝑎𝑥 = b/T
Radiance
L = dP/(dA cosθ dΩ)
Radiant exitance
I = M = ( ∫ Ω) L cosθ dΩ
Source strength
S = ( ∫ S) M dA = 2 pi R^2 L
Raleigh’s criterion
sin θ = 1.22 λ/D
Nyquist theorem
λN = 2px
λ > λN
Thin lens
1/s0 + 1/si = 1/f
Magnification
MT = yi/yo = -si/s0
pixel size on detector
d = M/2d
where fN = 1/λ = 1/2d
detector limited resolution
Δx = 1/fN
Lens maker formula
1/f = (n-1) (1/R1 - 1/R2)
Convex
biconvex - R1 > 0, R2 < 0
planar convex - R1 = inf, R2 < 0
meniscus convex - R1 > 0, R2 > 0
Concave
Biconcave - R1 < 0, R2 > 0
Planar concave - R1 = inf, R2 > 0
Meniscus concave - R1 > 0, R2 > 0
Dioptric power
D = 1/f
Magnifying power = angular magnification
MP = 𝛼a / 𝛼u
MP = d0 x D
Numerical aperture
NA = ni sin 𝜃max
Depth of field
DOF ∝ λ/NA^2
Contrast
Contrast = (Imax - Imin)/(Imax + Imin)
Resolution
Δ𝑥 = 1.22 𝜆/2NA
Dry objectives
NA max = 1
Immersive objectives
NA max > 1
Contrast to noise ratio
CNR = (I1 - I2) / σ
Modulation transfer function
MTF = image modulation / object modulation
Mobj = (Fmax - Fmin)/(Fmax + Fmin)
Mimg = b1(k)/b0 < Mobj
PSF in relation to MTF
MTF = ℱ(PSF)
image irradiance
= object * PSF
here * is the convolution
Smells law
n1 sin θ1 = n2 sin θ2
Critical angle
sin θ1c = n2/n1 sin (90)
Immersion objective
n1 < n2
θ1 > θ2
photoelectric effect
Ek = hv - q𝜑m
Where 𝜑m is the work function
Conductivity
Δ𝜎 = Δnq𝜇
Bolometer
Δ𝑇 = P/G
G is thermal conductance
Width affects junction capacitance
Cj = (εr ε0)/w
Shockley equation
J = Js [ exp(qV/kT - 1) ]
Total current density
J = Js [ exp(qV/kT - 1) ] + Jp
Where Jp is the total photo current density
and Js the dark current density
Sampling rate
Fs = 1/Δt
Frame size
T = N Δt
where N is the block size
Bandwidth
Fmax = Fs/2
Frequency resolution
Δf = Fmax/SL
where SL is the spectral lines
Spectral lines
SL = N/2
Noise in detectors
𝜎TOT = sqrt(𝜎1^2 + 𝜎2^2 + 𝜎3^2 + …)
Photon noise
𝜎N = sqrt(<N>) = sqrt(ΦΔt)</N>
Shot noise
is = sqrt(2q I ∆f) = sqrt(2q(Ip + ID)∆f)
Johnson noise
vJ = sqrt(4kTR ∆𝑓)
Bandwidth of a photodetector with a stray capacitance
∆𝑓 = 1/(2𝜋𝑅L C)
Photo current shot dominates when
is RL > 2kT/e
Background shot dominates when
ib RL > 2kT/e
Johnson noise dominate when
(is + ib) RL < 2kT/e
Quantum noise limit
Ip»_space; I0
Small signal regime
Ip «_space;I0
Quantum efficiency
𝜂 = Ip/q / Φp/hv
Responsivity
ℜ(𝜆,f) = Ip/Φp(𝜆)
Quantum efficiency and responsivity are related via
𝜂 = hv/q ℜ
and if gain is considered
ℜ(𝜆,f) = G𝜂q/hv
Noise equivalent power
NEP = in / ℜ
Detectivity
D = 1/NEP
Dynamic range
DR = 20log10(Well Size/Read Noise)
Bandwidth response of a photo detector
∆𝑓 = 1/2πτ
Electron and hole ionisation rates
k = 𝛼p / 𝛼n
What is the total noise in an APD
in^2 = is^2 + ij^2
= 2q(Ip+Id)Δ𝑓𝐺^2𝐹 + 4kTΔ𝑓/RL
F(G) = G(bar)^2/G^2
F(G) = kiG + (2 - 1/G)(1-ki)
ki = 𝛼p / 𝛼n
Total current from a PMT
I(PMT) = G I(pc)
The PMT gain
G = Pd V^(km)
Pd - dynode collection efficiency
k - voltage power constant
m - number of dynodes
Dark current in PMTs
Richardson Equation
I(D) = αAT^2 exp(-eψ/kT)
Brewster angle
θ(B) = tan^(-1) (n2/n1)
Birefringence
Δn = ne - no
retardation
Γ = t|no - ne|
convert retardation to phase difference
δ = 2πΓ/λ
The introduction of multiple elements is given by the multiplication of all matrices
Et = 𝒜n …. 𝒜2𝒜1 E1
Optical path
OP = nt
where number of waves = nt/λ
Dissipative absorption
E photon = E atomic transition
Non resonant scattering
E photon < E atomic transitions
Refractive index for x-rays
n(𝜔) = 1 - δ + iβ
δ, β < 1
δ dictates the critical angle for total external reflection.
β imaginary term dictates absorption by a material.
Propagating wave
E(r,t) = E0 exp (-i(ωt - k.r))
Phase velocity
ω/k = c/n = c/1-δ+iβ
can rearrange for k and substitute into propagating wave
Snells law
n1cos θ1 = n2 cos θ2
Fraunhofer
R > b^2/λ
Obliquity factor
K(θ) = 1/2(1+cos θ)
Fresnel zones contribute
E ~ |E1|/2
Even and odd zones
Even E ~ 0
Odd E ~ |E1|
Radiation absorption
dI/dx = -𝜇I => I = I0 exp(-𝜇x)
Mass absorption coefficient
𝜇/ρ = NA/A Σi 𝜎i
generation of visible light
L_R(E) = NTr(Al)b(sc) x 65900 x E
Variance of the produced visible light
𝜎^2_LR(E) = NTr(Al)Ab(sc) x (65900E)^2
where N is the number of photons produced per incident x-ray
Tral is the transmission of the x-ray through the aluminium material
Absc is the absorption of the scintillator material
65900 is the average number of photons that a scintillator yields.
Swank noise
SNR(E) = L_R(E)/sqrt(𝜎^2LR(E)) = sqrt(NTr(Al)Ab(sc))
Relativistic wavelength
𝜆 = h(sqrt(2𝑚0𝑒𝑉 (1 + 𝑒𝑉
/2𝑚0𝑐^2)))
Brightness
𝛽 = Δ𝐼/Δ𝑆ΔΩ = j/𝜋𝛼^2 A/m^2 sr
j = I/A where A = 𝜋d^2/4
where d can be found from the spherical aberrations and diffraction limited spot size.
Reduced brightness
𝛽r = 𝛽/V0
Lorentz force
me dv/dt = -ev x B
Resolution of TEM
d(min) = 1.3 𝜆^3/4 Cs^1/4
Resolution = d(min)/2
Malus law
I = I0 cos^2 theta
Fresnel zones
1/f = mλ/Rm^2
Scherzer defocus is
Δf = -1.2sqrt(Cs λ)
Refractive index of x rays
n ~ 1
Angular acceptance of the zone plate
sin θ = R/f
Jones matrix for a horizontally linear polariser
[ 1 0, 0 0]
Jones matrix Vertically linear polarisation
[ 0 0, 0 1]
Jones matrix linear polariser at 45 degrees
1/2 [ 1 1, 1 1]
Jones matrix linear polariser at -45 degrees
1/2[ 1 -1, -1 1]
Jones matrix quarter wave plate vertical
exp(ipi/4) [ 1 0, 0 -i]
Jones matrix quarter wave plate horizontal
exp(ipi/4) [ 1 0, 0 i]
x-rays: refractive optics
Makes use of the lensmaker equation
1/f = (n(lm) - 1) (1/R1 - 1/R2)
Bragg condition
n λ = 2dsin θ
Electrostatic phase change
Δ Φ = π/λE ∫ V(r) dz
phase change through a material a result of mean inner potential
Phase contrast can be described by the contrast transfer function
χ(k) on formula sheet
exposure time
t = Ne/I
N = number of photons
I =
current
I = (ηNe)
