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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 199-200

Section Spatial and temporal profiles

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: Spatial and temporal profiles

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The resolution of the coupled equations ([link] or ([link] over the crystal length L leads to the electric field amplitude [E_i(X,Y,L)] of each interacting wave. The general solutions are Jacobian elliptic functions (Armstrong et al., 1962[link]; Fève, Boulanger & Douady, 2002[link]). The integration of the systems is simplified for cases where one or several beams are held constant, which is called the undepleted pump approximation. We consider mainly this kind of situation here. The power of each interacting wave is calculated by integrating the intensity over the cross section of each beam according to ([link]. For our main purpose, we consider the simple case of plane-wave beams with two kinds of transverse profile:[\eqalignno{{\bf E}(X,Y,Z)&={\bf e}E_o(Z)\quad\hbox{for }(X,Y)\in[-w_o,+w_o]&\cr{\bf E}(X,Y,Z)&=0\phantom{E_o(Z)\quad}\hbox{elsewhere}&(}]for a flat distribution over a radius wo;[{\bf E}(X,Y,Z)={\bf e}E_o(Z)\exp[-(X^2+Y^2)/w_o^2]\eqno(]for a Gaussian distribution, where wo is the radius at ([1/e]) of the electric field and so at ([1/e^2]) of the intensity.

The associated powers are calculated according to ([link], which leads to[P(L)=m(n/2)(\varepsilon_o/\mu_o)^{1/2}|E_o|^2\pi w_o^2\eqno(]where [m=1] for a flat distribution and [m = 1/2] for a Gaussian profile.

The nonlinear interaction is characterized by the conversion efficiency, which is defined as the ratio of the generated power to the power of one or several incident beams, according to the different kinds of interactions.

For pulsed beams, it is necessary to consider the temporal shape, usually Gaussian:[P(t)=P_c\exp(-2t^2/\tau^2)\eqno(]where Pc is the peak power and τ the half ([1/e^2]) width.

For a repetition rate f (s−1), the average power [\tilde P] is then given by[{\tilde P}=P_c\tau f(\pi/2)^{1/2}={\tilde E}f\eqno(]where [\tilde E] is the energy per Gaussian pulse.

When the pulse shape is not well defined, it is suitable to consider the energies per pulse of the incident and generated waves for the definition of the conversion efficiency.

The interactions studied here are sum-frequency generation (SFG), including second harmonic generation (SHG: [\omega+\omega=2\omega]), cascading third harmonic generation (THG: [\omega+2\omega=3\omega]) and direct third harmonic generation (THG: [\omega+\omega+\omega=3\omega]). The difference-frequency generation (DFG) is also considered, including optical parametric amplification (OPA) and oscillation (OPO).

We choose to analyse in detail the different parameters relative to conversion efficiency (figure of merit, acceptance bandwidths, walk-off effect etc.) for SHG, which is the prototypical second-order nonlinear interaction. This discussion will be valid for the other nonlinear processes of frequency generation which will be considered later.


Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. (1962). Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939.
Fève, J. P., Boulanger, B. & Douady, J. (2002). Specific properties of cubic optical parametric interactions compared with quadratic interactions. Phys. Rev. A, 66, 063817–1–11.

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