ηSHGP=Ps(,t1)Pp(0,t1)=20r1dr1er12tanh2(K1er12)

(4.226)

using Eqn. 4.197 for the case of no phase mismatch.

For pulsed incident beams with Gaussian temporal and spatial shapes

Es()=2πdt0rdrIs()=2πr0p2t0pIp0dt10r1dr1er12et12×tanh2(K1er12et12)

(4.227)

so that the energy conversion efficiency for SHG can be written as

ηSHGE=Es()Ep(0)=2πdt10r1dr1er12et12 ×tanh2(K1er12et12).

(4.228)

When K1 ≪ 1, i.e., for the case of no pump depletion, using the relation tanh xx for small x, we have

ηSHGP=2K10r1dr1e2r12=K12=Ip02Ia

(4.229)

and

ηSHGE=2K1πdt10r1dr1e2r12e2t12=

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