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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 211

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

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:  benoit.boulanger@grenoble.cnrs.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 ωp, the pump, is mixed in the nonlinear crystal with the depleted wave at ωs, 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)[link] and (1.7.3.86)[link] 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 Pi(L) with [\Delta k=0] is then equivalent to (1.7.3.83)[link] 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, ωs, comprising IR radiation with a strong divergence and a wide spectral bandwidth. In this case, the achievement of a good conversion efficiency, Pi(L)/Ps(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[link]): 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.








































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