If we have two laser beams with central wavelengths lambda_1 and lambda_2 (one with negligible spectral width), interacting in a non-linear crystal, how would we calculate the spectral width of the difference-frequency-generated (DFG) output? I know it's a Lorentzian given by Delta lambda=(4 lambda_1^2 Delta lambda_2)/(4(lambda_2-lambda_1)^2-Delta lambda_2^2). But I don't know how to prove this.

Kade Rosales 2022-08-06 Answered
If we have two laser beams with central wavelengths λ 1 and λ 2 , (one with a spectral width of Δ λ 2 and one with negligible spectral width), interacting in a non-linear crystal, how would we calculate the spectral width of the difference-frequency-generated (DFG) output? I know it's a Lorentzian given by Δ λ = 4 λ 1 2 Δ λ 2 4 ( λ 2 λ 1 ) 2 Δ λ 2 2 . But I don't know how to prove this.
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Answers (1)

Dereon Parker
Answered 2022-08-07 Author has 11 answers
Difference frequency generation (DFG) is a stimulated parametric down-conversion process, which implies conservation of energy. It manifests in the relationship among the frequencies of the three beams (the pump, the seed and the idler or DFG beam):
ω p ω s = ω i .
In terms of wavelengths it becomes
λ i = λ 2 λ 1 λ 2 λ 1 ,
where λ 1 is the pump wavelength and λ 2 is the seed wavelength. Assuming the pump has a negligible bandwidth, the bandwidth of the idler is determined by the bandwidth of the seed. To determine the idler bandwidth, we use
Δ λ i = | λ i ( λ 2 ) λ 2 | Δ λ 2 = | λ 1 λ 2 λ 1 λ 2 λ 1 ( λ 2 λ 1 ) 2 | Δ λ 2 = λ 1 2 Δ λ 2 ( λ 2 λ 1 ) 2 .
The resulting expression follows from that provided by the OP by setting Δ λ 2 2 = 0 since it would be small compared to the other term in the denominator. If this is not the case, the bandwidth would also depend on the shape of the spectrum.
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