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.

Question

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&#039;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&#039;t know how to prove this.

Dereon Parker · Accepted Answer

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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