Equations Flashcards
Einsteins relations
A21/B21 = ℏω21^3/π^2c^3
g1B12 = g2B21
Condition for optical gain
N2B21p(ω21) > N1B12 p(ω21)
g1B12 = g2B21
=> N2/g2 > N1/g1
Condition for steady state population inversion
(R2 τ2 g1)/(R1 τ1 g2) (1- g2/g1 A21 τ1) > 1
=>
A21 < g1/g2 1/τ1
Gain coefficient
Is described by a differential equation
∂I(ω,z)/∂z = α(ω-ω0) I(ω,z)
Where α(ω-ω0) is the gain coefficient
Detuning
(ω-ω0) = Δf = Δω = δ
Line width of the transition
Δω = 1/τ1 + 1/τ2
Doppler shift
f’ = f(1 ± vz/c)
Maxwell Boltzmann distribution
P(f)df
Wave vector
|k| = 2π/λ
Optical cavity modes
v(lmp) = c/2Lc p [1+ (l^2 + m^2)/2p^2 (Lc/a) + … ] ~ c/2Lc p
Lx = Ly = a , Lz = Lc, and v(lmp) = ω(lmp)/2π
Length of the cavity
Lc = λ(lmp)/2 p
Free spectral range
Δv = c/2nd
Reflection through the etalon on
I^(r)/I^(i) = (Fsin^2 δ/2)/(1+Fsin^2 δ/2)
Reflected intensity/Incident intensity
Transmission through the etalon
I^(t)/I^(i) = 1/(1+ Fsin^2 δ/2)
Finesse of cavity
F = 4ℜ/(1-ℜ)^2
Where ℜ is the mirror reflectivity
Phase difference between adjacent rays
δ = 2π/λ 2nd cosα
g-parameters
g1 = 1 - L/R1
g2 = 1 - L/R2
Plano cavity
R1 = R2 = inf
Confocal cavity
R1 = R2 = L
Concentric cavity
R1 = R2 = L/2
A cavity is stable if
0 < g1g2 < 1
Laser threshold condition
R1R2 exp[2α(0th)(ω)lg] exp[-2κ(ω)Lc] = 1
R1R2 - product of mirror reflection coefficients
α(0th) - threshold value of optical gain coefficient (small signal)
lg - length of gain medium
κ(ω) - attenuation of beam throughout cavity by scattering/other losses.
free space propagation of a ray
r2 = r1 + (z2 -z1)r1’
r2’ = r1’
describing a ray into a vector
r = (r r’)^T
ray transfer matrices
r2 = ( r2 r2’)^T = (A B , C D)^T (r1 r1’) = Mr1
ray transfer matrix - for propagation through free space
M = (1 z2 - z1, 0 1)^T
ray transfer matrix - for transmission through a lens
M = ( 1 0, -1/f 1 )^T
What is the determinant of both ray transfer matrices?
one
We can represent the effect of the combination of optical components on any (paraxial) ray with a single matrix M
M = M(N) M(N-1) ,,, M2M1
A ray vector r is an eigenray of the cavity if it satisfies the equation
Mr = γr
γ is the corresponding eigenvalue
How do we find the eigenvalue?
|M - γI| = 0
where I is the identity matrix
superposition of eigenrays
r = c(a) r(a) + c(b) r(b)
eigenray desciprtion is attractive
M^(N) r(a) = γ(a)^Nr(a)
stability requirement
m^2 <= 1
stability condition
m^2 ≤ 1 i.e. - 1 ≤ m ≤ 1
where m = (A+D)/2
Number of photons left in the cavity after time t
Np = Np,0 exp(-t/τp)
where
τp = (2nL/c)/(1-R1R2)
Cavity linewidth
Δω ~ 1/τp
Quality factor
Q = 2π energy stored in cavity/energy lost per cycle
Q = 2π/T τp
Q = ω0τp = ω0/Δω
Laser beam equation
u = i Uo/zR on formula sheet
spot size
ω(z) = ω0 sqrt(1+(z/zR)^2)
Rayleigh range
zR = π ω0^2/λ
Radius of curvature of the phase front
R(z) = z + zR^2/z
Gouy phase shift
α(z) = arctan (z/zR)
beam divergence
θ = λ0/πw0
confocal parameter
twice the rayleigh range
=> b = 2zR
For a gaussian beam the effect of an optical element is given by
q2 = (Aq1 + B)/(Cq1+D)
Polarisation
P = χ ε0 E
Displacement field
D = ε0 E + P
D = (1+ χ) ε0 E
relative permatibiltiy?
εr = 1 + χ
Dispersion equation
n = 1 + Nq^2/(2 ε0m(ω0^2-ω^2))
Complete dispersion equation
n = 1 + q^2/(2 ε0m) (Σ k) Nk/(ωk^2-ω^2+ iγkω))
Birefringence
b = Δ𝑛 = ne - no
speed of light for an o-ray
c/v⟂ = no
Speed of light for an e-ray
ne = c/v∥
Refractive index of a birefringence material
n = 1/sqrt(cos^2 θ/no^2 + sin^2 θ/ne^2)
Polarisation varies non linearly with field
P = ε0{χ^(1)E + χ^(2)E^2 + χ^(3)E^3 + …}
Power density
S = n ε0 c/2 E^2
Conservation of energy implies
χ ω3; ω1, ω2 = 0
Unless ω3 = ω1 + ω2
χij ω3; ω1 = 0
Unless ω3 = ω1
Phase mismatch
Δk = k ω3 - k ω1 - k ω2
= k 2 ω - 2k ω
Perfect phase matching
The phase matched condition is
Δk = k 2 ω - 2k ω = 0
Phase mismatch parameter
ΔkL/2
Coherence length
L < |2 π / Δk|
Exploit birefringence to match the index at the two frequencies
no^n ω = ne^m ω(θ)
Maximise efficiency
η(SHG) = I^2ω/I^ω = C^2L^2I^ω = C^2L^2 P/A = C^2 P L^2/A
η(SHG) ∝ L^2/A
Confinement of a beam
L^2/A = 2L/λ
difference frequency
ωi = ωp - ωs
energy conservation
1/λp = 1/λs + 1/λi
kerr effect
b = n∥ - n⊥ = λ0KE^2
where K is the Kerr constant
Kerr effect arises from the third order nonlinear susceptibility?
(εr)i = 1 + 3χ^(3)iikk (Ek(kerr))^2
pockels effect
(no - ne) = no^3 r63 E = no^3 r63 V/z
(no - ne)z = no^3 r63 V
Δφ = 2π no^3 r63 V/λo
