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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 210

Section SHG ([\omega+\omega=2\omega]) and SFG ([\omega+2\omega=3\omega]) in the same crystal

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 the same crystal

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When the SFG conversion efficiency is sufficiently low in comparison with that of the SHG, it is possible to integrate the equations relative to SHG and those relative to SFG separately (Boulanger, Fejer et al., 1994[link]). In order to compare this situation with the example taken for the previous case, we consider a type-I configuration of polarization for SHG. By assuming a perfect phase matching for SHG, the amplitude of the third harmonic field inside the crystal is (Boulanger, 1994[link])[\eqalignno{E^{3\omega}(X,Y,Z)&=jK^{3\omega}(\varepsilon_o\chi_{{\rm eff}_{\rm SFG}})&\cr&\quad\times\textstyle \int \limits_{0}^{L}E_{\rm tot}^\omega(X,Y,Z)E^{2\omega}(X,Y,Z)\exp(j\Delta k_{\rm SFG}Z)\,\,{\rm d}Z&\cr&&(}]with[\eqalignno{E^{2\omega}(X,Y,Z)&=(T^\omega)^{1/2}|E_{\rm tot}^\omega(0)|\tanh(\Gamma Z)&\cr\hbox{and }\,\,E_{\rm tot}^\omega(X,Y,Z)&=(T^\omega)^{1/2}|E_{\rm tot}^\omega(0)|\,{\rm sech}(\Gamma Z).&\cr&&(}]Γ is as in ([link].

([link]) can be analytically integrated for undepleted pump SHG; [{\rm sech}(m)\rightarrow 1], [\tanh(m)\rightarrow m], and so we have[\eta_{\rm THG}(L)=P^{3\omega}(L)/P_{\rm tot}^{\omega}(0)\eqno(]with[\displaylines{P^{3\omega}(L)\hfill\cr\quad={576\pi^2\over\varepsilon_o^2c^2}\left({2N-1\over N}\right)^2T^{3\omega}{d_{{\rm eff}_{\rm SHG}}^2d_{{\rm eff}_{\rm SFG}}^2\over n^{3\omega}(n^\omega)^3(n^{2\omega})^2}{[T^\omega P^\omega_{\rm tot}(0)]^3\over w_o^4\lambda_\omega^4}J(L),\hfill}]where the integral J(L) is[J(L)=\left|\textstyle \int \limits_{0}^{L}Z\exp(i\Delta k_{\rm SFG}Z)\;{\rm d}Z\right|^2.\eqno(]

For a nonzero SFG phase mismatch, [\Delta k_{\rm SFG}\ne 0],[J(L)\simeq L^2/(\Delta k_{\rm SFG})^2.\eqno(]

For phase-matched SFG, [\Delta k_{\rm SFG}=0],[J(L)=L^4/4.\eqno(]

Therefore ([link] according to ([link] is equal to ([link] with [L_{\rm SHG}=L_{\rm SFG}=L/2], [\Delta k_{\rm SFG}=0] and 100% transmission coefficients at ω and 2ω between the two crystals.


Boulanger, B. (1994). CNRS–NSF Report, Stanford University.
Boulanger, B., Fejer, M. M., Blachman, R. & Bordui, P. F. (1994). Study of KTiOPO4 gray-tracking at 1064, 532 and 355 nm. Appl. Phys. Lett. 65(19), 2401–2403.

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