SFG () with undepletion at

Boulanger, B.; Zyss, J.

doi:10.1107/97809553602060000634

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 208

Section 1.7.3.3.4.2. SFG ( $[\omega_s+\omega_p=\omega_i]$ ) with undepletion at $[\omega_p]$

B. Boulanger^a ^* and J. Zyss^b

^a Laboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and ^bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail: benoitb@satie-bourgogne.fr

1.7.3.3.4.2. SFG ( $[\omega_s+\omega_p=\omega_i]$ ) with undepletion at $[\omega_p]$

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$[(\omega_s,\omega_p,\omega_i) = (\omega_1,\omega_2,\omega_3)]$ or $[(\omega_2,\omega_1,\omega_3)]$ .

The undepleted wave at ω_p, the pump, is mixed in the nonlinear crystal with the depleted wave at ω_s, the signal, in order to generate the idler wave at $[\omega_i=\omega_s+\omega_p]$ . The integrations of the coupled amplitude equations over ( [X,Y,Z] ) with $[E_s(X,Y,0)\ne 0]$ , $[E_p(X,Y,0)\ne 0]$ , $[\partial E_p(X,Y,Z)/\partial Z=0]$ and [E_i(X,Y,0)= 0] give $[\eqalignno{P_p(L)&=T_p^2P_p(0)&(1.7.3.84)\cr P_i(L)&={\omega_i\over \omega_s}P_s(0)\Gamma^2L^2{\sin^2\{\Gamma^2L^2+[(\Delta k\cdot L)/2]^2\}^{1/2}\over \Gamma^2L^2+[(\Delta k\cdot L)/2]^2}&\cr&&(1.7.3.85)\cr P_s(L)&=P_s(0)\left[1-{\omega_s\over\omega_i}{P_i(L)\over P_s(0)}\right],&(1.7.3.86)}%fd1.7.3.86]$ with $[\Delta k=k_i-(k_s+k_p)]$ and $[\Gamma^2=[B_sP_p(0)]/w_o^2]$ , where $[B_s={8\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over \lambda_s\lambda_i}{T_sT_pT_i\over n_sn_pn_i}.]$ Thus, even if the up-conversion process is phase-matched ( $[\Delta k=0]$ ), the power transfers are periodic: the photon transfer efficiency is then 100% for $[\Gamma L=(2m+1)(\pi/2)]$ , where m is an integer, which allows a maximum power gain $[\omega_i/\omega_s]$ for the idler. A nonlinear crystal with length $[L = (\pi/2\Gamma)]$ is sufficient for an optimized device.

For a small conversion efficiency, i.e. ΓL weak, (1.7.3.85) and (1.7.3.86) become $[P_i(L)\simeq P_s(0){\omega_i\over \omega_s}\Gamma^2L^2\sin c^2{\Delta k\cdot L\over2}\eqno(1.7.3.87)]$ and $[P_s(L)\simeq P_s(0).\eqno(1.7.3.88)]$ The expression for P_i(L) with $[\Delta k=0]$ is then equivalent to (1.7.3.83) with $[\omega_p = \omega_1]$ or $[\omega_2]$ , $[\omega_i=\omega_3]$ and $[\omega_s = \omega_2]$ or $[\omega_1]$ .

For example, the frequency up-conversion interaction can be of great interest for the detection of a signal, ω_s, comprising IR radiation with a strong divergence and a wide spectral bandwidth. In this case, the achievement of a good conversion efficiency, P_i(L)/P_s(0), requires both wide spectral and angular acceptance bandwidths with respect to the signal. The double non-criticality in frequency and angle (DNPM) can then be used with one-beam non-critical non-collinear phase matching (OBNC) associated with vectorial group phase matching (VGPM) (Dolinchuk et al., 1994): this corresponds to the equality of the absolute magnitudes and directions of the signal and idler group velocity vectors i.e. $[{\rm d}\omega_i/{\rm d}{\bf k}_i={\rm d}\omega_s/{\rm d}{\bf k}_s]$ .

References

Dolinchuk, S. G., Kornienko, N. E. & Zadorozhnii, V. I. (1994). Noncritical vectorial phase matchings in nonlinear optics of crystals and infrared up-conversion. Infrared Phys. Technol. 35(7), 881–895.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 208

Section 1.7.3.3.4.2. SFG () with undepletion at

1.7.3.3.4.2. SFG () with undepletion at

References

Section 1.7.3.3.4.2. SFG ( $[\omega_s+\omega_p=\omega_i]$ ) with undepletion at $[\omega_p]$

1.7.3.3.4.2. SFG ( $[\omega_s+\omega_p=\omega_i]$ ) with undepletion at $[\omega_p]$