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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 206-208

Section Non-resonant SHG with depleted pump in the parallel-beam limit

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: Non-resonant SHG with depleted pump in the parallel-beam limit

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The analytical integration of the three coupled equations ([link] with depletion of the pump and phase mismatch has only been done in the parallel-beam limit and by neglecting the walk-off effect (Armstrong et al., 1962[link]; Eckardt & Reintjes, 1984[link]; Eimerl, 1987[link]; Milton, 1992[link]). In this case, the three coordinate systems of equations ([link] are identical, ([X,Y,Z]), and the general solution may be written in terms of the Jacobian elliptic function [{\rm sn}(m,\alpha)].

For the simple case of type I, i.e. [E_1^\omega(X,Y,Z)=] [ E_2^\omega(X,Y,Z) =] [ E^\omega(X,Y,Z)= ] [ E_{\rm tot}^\omega(X,Y,Z)/(2^{1/2})], the exit second harmonic intensity generated over a length L is given by (Eckardt & Reintjes, 1984[link])[I^{2\omega}(X,Y,L)=I_{\rm tot}^\omega(X,Y,0)T^{2\omega}T^{\omega}v_b^2{\rm sn}^2\left[{\Gamma(X,Y)L\over v_b}, v_b^4\right].\eqno(][I_{\rm tot}^\omega(X,Y,0) = 2 I^\omega(X,Y,0)] is the total initial fundamental intensity, [T^{2\omega}] and [T^\omega] are the transmission coefficients, [{1 \over v_b}={\Delta s\over 4}+\left[1+\left({\Delta s\over 4}\right)^2\right]^{1/2}]with[\Delta s=(k^{2\omega}-k^{\omega})/\Gamma]and[\Gamma(X,Y)={\omega d_{\rm eff} \over cn^{2\omega}}(T^\omega)^{1/2}|E_{\rm tot}^\omega(X,Y,0)|.\eqno(]For the case of phase matching ([k^\omega = k^{2\omega}], [T^\omega = T^{2\omega}]), we have [ \Delta s=0] and [v_b=1], and the Jacobian elliptic function [{\rm sn}(m,1)] is equal to [\tanh(m)]. Then formula ([link] becomes[I^{2\omega}(X,Y,L)=I_{\rm tot}^\omega(X,Y,0)(T^\omega)^2\tanh^2[\Gamma(X,Y)L], \eqno(]where [\Gamma(X,Y)] is given by ([link].

The exit fundamental intensity [I^\omega(X,Y,L)] can be established easily from the harmonic intensity ([link] according to the Manley–Rowe relations ([link], i.e.[I^{\omega}(X,Y,L)=I_{\rm tot}^\omega(X,Y,0)(T^\omega)^2{\rm sech}^2[\Gamma(X,Y)L].\eqno(]For small [\Gamma L], the functions [\tanh^2(\Gamma L) \simeq \Gamma^2L^2] and [{\rm sn}^2[(\Gamma L/v_b),v_b^4]\simeq\sin^2(\Gamma L/v_b)] with [v_b\simeq 2/\Delta s].

The first consequence of formulae ([link]–([link] is that the various acceptance bandwidths decrease with increasing ΓL. This fact is important in relation to all the acceptances but in particular for the thermal and angular ones. Indeed, high efficiencies are often reached with high power, which can lead to an important heating due to absorption. Furthermore, the divergence of the beams, even small, creates a significant dephasing: in this case, and even for a propagation along a phase-matching direction, formula ([link] is not valid and may be replaced by ([link] where [k(2\omega) - k(\omega)] is considered as the `average' mismatch of a parallel beam.

In fact, there always exists a residual mismatch due to the divergence of real beams, even if not focused, which forbids asymptotically reaching a 100% conversion efficiency: [I^{2\omega}(L)] increases as a function of ΓL until a maximum value has been reached and then decreases; [I^{2\omega}(L)] will continue to rise and fall as ΓL is increased because of the periodic nature of the Jacobian elliptic sine function. Thus the maximum of the conversion efficiency is reached for a particular value (ΓL)opt. The determination of (ΓL)opt by numerical computation allows us to define the optimum incident fundamental intensity [I_{\rm opt}^\omega] for a given phase-matching direction, characterized by K, and a given crystal length L.

The crystal length must be optimized in order to work with an incident intensity [I_{\rm opt}^\omega] smaller than the damage threshold intensity [I_{\rm dam}^\omega] of the nonlinear crystal, given in Section 1.7.5[link] for the main materials.

Formula ([link] is established for type I. For type II, the second harmonic intensity is also an sn2 function where the intensities of the two fundamental beams [I_1^\omega(X,Y,0)] and [I_2^\omega(X,Y,0)], which are not necessarily equal, are taken into account (Eimerl, 1987[link]): the tanh2 function is valid only if perfect phase matching is achieved and if [I_1^\omega(X,Y,0)=I_2^\omega(X,Y,0)], these conditions being never satisfied in real cases.

The situations described above are summarized in Fig.[link].


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Schematic SHG conversion efficiency for different situations of pump depletion and dephasing. (a) No depletion, no dephasing, [\eta = \Gamma^2L^2]; (b) no depletion with constant dephasing δ, [\eta = \Gamma^2L^2\sin c^2\delta]; (c) depletion without dephasing, [\eta = \tanh^2(\Gamma L)]; (d) depletion and dephasing, [\eta] [=] [\eta_m{\rm sn}^2(\Gamma L/v_b,v_b^4)].

We give the example of type-II SHG experiments performed with a 10 Hz injection-seeded single-longitudinal-mode ([N=1]) 1064 nm Nd:YAG (Spectra-Physics DCR-2A-10) laser equipped with super Gaussian mirrors; the pulse is 10 ns in duration and is near a Gaussian single-transverse mode, the beam radius is 4 mm, non-focused and polarized at π/4 to the principal axes of a 10 mm long KTP crystal (Lδθ = 15 mrad cm, Lδϕ = 100 mrad cm). The fundamental energy increases from 78 mJ (62 MW cm−2) to 590 mJ (470 MW cm−2), which correponds to the damage of the exit surface of the crystal; for each experiment, the crystal was rotated in order to obtain the maximum conversion efficiency. The peak power SHG conversion efficiency is estimated from the measured energy conversion efficiency multiplied by the ratio between the fundamental and harmonic pulse duration ([\tau_\omega/\tau_{2\omega}=2^{1/2}]). It increases from 50% at 63 MW cm−2 to a maximum value of 85% at 200 MW cm−2 and decreases for higher intensities, reaching 50% at 470 MW cm−2 (Boulanger, Fejer et al., 1994[link]).

The integration of the intensity profiles ([link] and ([link] is obvious in the case of incident fundamental beams with a flat energy distribution ([link]. In this case, the fundamental and harmonic beams inside the crystal have the same profile and radius as the incident beam. Thus the powers are obtained from formulae ([link] and ([link] by expressing the intensity and electric field modulus as a function of the power, which is given by ([link] with [m=1].

For a Gaussian incident fundamental beam, ([link], the fundamental and harmonic beams are not Gaussian (Eckardt & Reintjes, 1984[link]; Pliszka & Banerjee, 1993[link]).

All the previous intensities are the peak values in the case of pulsed beams. The relation between average and peak powers, and then SHG efficiencies, is much more complicated than the ratio [\tau^{2\omega}/\tau^\omega] of the undepleted case.


Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. (1962). Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939.
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.
Eckardt, R. C. & Reintjes, J. (1984). Phase matching limitations of high efficiency second harmonic generation. IEEE J. Quantum Electron. 20(10), 1178–1187.
Eimerl, D. (1987). High average power harmonic generation. IEEE J. Quantum Electron. 23, 575–592.
Milton, J. T. (1992). General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves. IEEE J. Quantum Electron. 28(3), 739–749.
Pliszka, P. & Banerjee, P. P. (1993). Nonlinear transverse effects in second-harmonic generation. J. Opt. Soc. Am. B, 10(10), 1810–1819.

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