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 Third harmonic generation (THG)

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: Third harmonic generation (THG)

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Fig.[link] shows the three possible ways of achieving THG: a cascading interaction involving two χ(2) processes, i.e. [\omega+\omega=2\omega] and [\omega+2\omega=3\omega], in two crystals or in the same crystal, and direct THG, which involves χ(3), i.e. [\omega+\omega+\omega=3\omega].


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Configurations for third harmonic generation. (a) Cascading process SHG ([\omega+\omega=2\omega]): SFG ([\omega+2\omega=3\omega]) in two crystals NLC1 and NLC2 and (b) in a single nonlinear crystal NLC; (c) direct process THG ([\omega+\omega+\omega=3\omega]) in a single nonlinear crystal NLC. 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]. 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. Direct THG ([\omega+\omega+\omega=3\omega])

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As for the cascading process, we consider a flat plane wave which propagates in a direction without walk-off. The integration of equations ([link] over the crystal length L, with [E_4^{3\omega}(X,Y,0)=0] and in the undepleted pump approximation, leads to[\eqalignno{E_4^{3\omega}(X,Y,L)&=jK^{3\omega}_4[\varepsilon_o\chi^{(3)}_{\rm eff}]E_1^{\omega}(X,Y,0)E_2^{\omega}(X,Y,0)E_3^{\omega}(X,Y,0)&\cr&\quad\times L\sin c[(\Delta k\cdot L)/2]\exp(-j\Delta kL/2).&\cr&&(}]

According to ([link] and ([link], the integration of ([link] over the cross section, which is the same for the four beams, leads to[\eta_{\rm THG}(L)={P^{3\omega}(L)\over P^\omega(0)}=B_{\rm THG}[P^\omega(0)]^2{L^2\over w_o^4}\sin c^2[(\Delta k\cdot L)/2]]with[B_{\rm THG}={576\over \varepsilon_o^2c^2}{d_{\rm eff}^2\over\lambda_\omega^2}{T_4^{3\omega}(T_1^\omega)^2T_2^\omega\over n_4^{3\omega}(n_1^\omega)^2n_2^\omega}\quad({\rm m}^{2}\;{\rm W}^{-2}),\eqno(]where [d_{\rm eff}=(1/4)\chi_{\rm eff}^{(3)}] is in m2 V−2 and λω is in m. The statistical factor is assumed to be equal to 1, which corresponds to a longitudinal single-mode laser.

The different types of phase matching and the associated relations and configurations of polarization are given in Table[link] by considering the SFG case with [\omega_1=\omega_2=\omega_3=\omega_4/3].


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

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