Equations Flashcards by grace dobson

Einsteins relations

A21/B21 = ℏω21^3/π^2c^3

g1B12 = g2B21

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Condition for optical gain

N2B21p(ω21) > N1B12 p(ω21)

g1B12 = g2B21

=> N2/g2 > N1/g1

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Condition for steady state population inversion

(R2 τ2 g1)/(R1 τ1 g2) (1- g2/g1 A21 τ1) > 1

A21 < g1/g2 1/τ1

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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

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Detuning

(ω-ω0) = Δf = Δω = δ

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Line width of the transition

Δω = 1/τ1 + 1/τ2

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Finite lifetimes

Δω = 1/2πτ

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Doppler shift

f’ = f(1 ± vz/c)

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Maxwell Boltzmann distribution

P(f)df

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Wave vector

|k| = 2π/λ

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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

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Length of the cavity

Lc = (λ(lmp)/2) p

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Longitudinal modes are separated by the free spectral range

Δv = c/2nd

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longitudinal mode

k2Lcosθ = 2πn

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Reflection through the etalon on

I^(r)/I^(i) = (Fsin^2 δ/2)/(1+Fsin^2 δ/2)

Reflected intensity/Incident intensity

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Transmission through the etalon

I^(t)/I^(i) = 1/(1+ Fsin^2 δ/2)

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Finesse of cavity

F = 4ℜ/(1-ℜ)^2

Where ℜ is the mirror reflectivity

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Phase difference between adjacent rays

δ = 2π/λ 2nd cosα

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g-parameters

g1 = 1 - L/R1

g2 = 1 - L/R2

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Plano cavity

R1 = R2 = inf

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Confocal cavity

R1 = R2 = L

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Concentric cavity

R1 = R2 = L/2

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A cavity is stable if

0 < g1g2 < 1

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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.

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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

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

FWHM

FWHM = 2sqrt(2δ) get δ when 1/2 = exp(-mc^2δ^2/2kTf^2)

Equations Flashcards

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