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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 209-210

Section SHG ([\omega+\omega=2\omega]) and SFG ([\omega+2\omega=3\omega]) in different crystals

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: SHG ([\omega+\omega=2\omega]) and SFG ([\omega+2\omega=3\omega]) in different crystals

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We consider the case of the situation in which the SHG is phase-matched with or without pump depletion and in which the sum-frequency generation (SFG) process ([\omega+2\omega=3\omega]), phase-matched or not, is without pump depletion at [\omega] and [2\omega]. All the waves are assumed to have a flat distribution given by ([link] and the walk-off angles are nil, in order to simplify the calculations.

This configuration is the most frequently occurring case because it is unusual to get simultaneous phase matching of the two processes in a single crystal. The integration of equations ([link] over Z for the SFG in the undepleted pump approximation with [E_1^\omega(Z_{\rm SFG}=0)=] [E_1^\omega(L_{\rm SHG})], [E_2^{2\omega}(Z_{\rm SFG}=0)=] [E_2^{2\omega}(L_{\rm SHG})] and [E_3^{3\omega}(Z_{\rm SFG}=0)=] [], followed by the integration over the cross section leads to[\displaylines{P^{3\omega}(L_{\rm SFG})\hfill\cr\quad=B_{\rm SFG}[aP^\omega(L_{\rm SHG})]P^{2\omega}(L_{\rm SHG}){L^2_{\rm SFG}\over w_o^2}\sin c^2{\Delta k_{\rm SFG}L_{\rm SFG}\over 2}\quad({\rm W})\hfill}]with[\displaylines{B_{\rm SFG}={72\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over\lambda_\omega^2}{T^{3\omega}_3T_1^\omega T_2^{2\omega}\over n^{3\omega}_3n_1^\omega n_2^{2\omega}}\quad({\rm W}^{-1})\cr a=1\hbox{ for type-I SHG,}\quad a={\textstyle{1 \over 2}}\hbox{ for type-II SHG}.\cr\hfill(}]Pω(LSHG) and P2ω(LSHG) are the fundamental and harmonic powers, respectively, at the exit of the first crystal. LSHG and LSFG are the lengths of the first and the second crystal, respectively. [\Delta k_{\rm SFG} = k^{3\omega} - (k^\omega + k^{2\omega})] is the SFG phase mismatch. λω is the fundamental wavelength. The units and other parameters are as defined in ([link].

For type-II SHG, the fundamental waves are polarized in two orthogonal vibration planes, so only half of the fundamental power can be used for type-I, -II or -III SFG ([a=1/2]), in contrast to type-I SHG ([a=1]). In the latter case, and for type-I SFG, it is necessary to set the fundamental and second harmonic polarizations parallel.

The cascading conversion efficiency is calculated according to ([link] and ([link]; the case of type-I SHG gives, for example,[\eqalignno{\eta_{\rm THG}(L_{\rm SHG},L_{\rm SFG})&={P^{3\omega}(L_{\rm SFG})\over P_{\rm tot}^\omega(0)}&\cr &=B_{\rm SFG}(T^\omega)^4P_{\rm tot}^\omega(0)\tanh^2(\Gamma L_{\rm SHG})&\cr&\quad\times{\rm sech}^2(\Gamma L_{\rm SHG}){L^2_{\rm SFG}\over w_o^2}\sin c^2\left({\Delta k_{\rm SFG}L_{\rm SFG}\over 2}\right),&\cr&&(}]where Γ is as in ([link].

(nω, Tω) are relative to the phase-matched SHG crystal and ([n_1^\omega,n_2^{2\omega},n_3^{3\omega}, T_1^\omega,T_2^{2\omega},T_3^{3\omega}]) correspond to the SFG crystal.

In the undepleted pump approximation for SHG, ([link] becomes (Qiu & Penzkofer, 1988[link])[\displaylines{\eta_{\rm THG}(L_{\rm SHG},L_{\rm SFG})\hfill\cr\quad=BT^\omega\left[{P^\omega(0)\over w_o^2}\right]^2L^2_{\rm SHG}L^2_{\rm SFG}\sin c^2\left({\Delta k_{\rm SFG}L_{\rm SFG}\over 2}\right)\hfill\cr\hfill(}]with[\eqalign{B&=B_{\rm SHG}\cdot B_{\rm SFG}\cr&={576\pi^2\over\varepsilon_o^2c^2}\left({2N-1\over N}\right)^2{d_{{\rm eff}_{\rm SHG}}^2d_{{\rm eff}_{\rm SFG}}^2\over \lambda_\omega^4}\left({T_{\rm SHG}^3\over n_{\rm SHG}^3}\right)\left({T_{\rm SFG}^3\over n_{\rm SFG}^3}\right)}]in W−2, where[{T_{\rm SHG}^3\over n_{\rm SHG}^3}={(T^\omega)^3\over(n^\omega)^3}\;\hbox{ and }\;{T_{\rm SFG}^3\over n_{\rm SFG}^3}={T_3^{3\omega}T_1^\omega T_2^{2\omega}\over n_3^{3\omega}n_1^\omega n_2^{2\omega}}.]The units are the same as in ([link].

A more general case of SFG, where one of the two pump beams is depleted, is given in Section[link].


Qiu, P. & Penzkofer, A. (1988). Picosecond third-harmonic light generation in β-BaB2O4. Appl. Phys. B, 45, 225–236.

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