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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 190-191

Section Coupled electric fields amplitudes equations

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: Coupled electric fields amplitudes equations

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The nonlinear crystals considered here are homogeneous, lossless, non-conducting, without optical activity, non-magnetic and are optically anisotropic. The nonlinear regime allows interactions between γ waves with different circular frequencies [\omega_i,i=1,\ldots,\gamma]. The Fourier component of the polarization vector at ωi is [{\bf P}(\omega_i)=\varepsilon_0\chi^{(1)}(\omega_i){\bf E}(\omega_i)+{\bf P}^{NL}(\omega_i)], where [{\bf P}^{NL}(\omega_i)] is the nonlinear polarization corresponding to the orders of the power series greater than 1 defined in Section 1.7.2[link].

Thus the propagation equation of each interacting wave ωi is (Bloembergen, 1965[link])[\nabla x\nabla x{\bf E}(\omega_i) = (\omega_i^2/c^2)\varepsilon (\omega _i){\bf E}(\omega_i) + \omega_i ^2\mu_0 {\bf P}^{NL}(\omega_i).\eqno(]The γ propagation equations are coupled by [{\bf P}^{NL}(\omega_i)]:

  • (1) for a three-wave interaction, γ = 3,[\eqalign{{\bf P}^{NL}(\omega_1)&={\bf P}^{(2)}(\omega_1)=\varepsilon_0\chi^{(2)}(\omega_1=\omega_3-\omega_2)\cdot{\bf E}(\omega_3)\otimes{\bf E}^*(\omega_2),\cr {\bf P}^{NL}(\omega_2)&={\bf P}^{(2)}(\omega_2)=\varepsilon_0\chi^{(2)}(\omega_2=\omega_3-\omega_1)\cdot{\bf E}(\omega_3)\otimes{\bf E}^*(\omega_1),\cr {\bf P}^{NL}(\omega_3)&={\bf P}^{(2)}(\omega_3)=\varepsilon_0\chi^{(2)}(\omega_3=\omega_1+\omega_2)\cdot{\bf E}(\omega_1)\otimes{\bf E}^*(\omega_2)\semi}]

  • (2) for a four-wave interaction[\eqalign{{\bf P}^{NL}(\omega_1)={\bf P}^{(3)}(\omega_1)&=\varepsilon_0\chi^{(3)}(\omega_1=\omega_4-\omega_2-\omega_3)\cr&\quad\cdot{\bf E}(\omega_4)\otimes{\bf E}^*(\omega_2)\otimes{\bf E}^*(\omega_3),\cr {\bf P}^{NL}(\omega_2)={\bf P}^{(3)}(\omega_2)&=\varepsilon_0\chi^{(3)}(\omega_2=\omega_4-\omega_1-\omega_3)\cr&\quad\cdot{\bf E}(\omega_4)\otimes{\bf E}^*(\omega_1)\otimes{\bf E}^*(\omega_3),\cr}][\eqalign{{\bf P}^{NL}(\omega_3)={\bf P}^{(3)}(\omega_3)&=\varepsilon_0\chi^{(3)}(\omega_3=\omega_4-\omega_1-\omega_2)\cr&\quad\cdot{\bf E}(\omega_4)\otimes{\bf E}^*(\omega_1)\otimes{\bf E}^*(\omega_2)\cr {\bf P}^{NL}(\omega_4)={\bf P}^{(3)}(\omega_4)&=\varepsilon_0\chi^{(3)}(\omega_4=\omega_1+\omega_2+\omega_3)\cr&\quad\cdot{\bf E}(\omega_1)\otimes{\bf E}(\omega_2)\otimes{\bf E}(\omega_3).}]

The complex conjugates [{\bf E}^*(\omega_1)] come from the relation [{\bf E}^*(\omega_i)={\bf E}(-\omega_i)].

We consider the plane wave, ([link], as a solution of ([link], and we assume that all the interacting waves propagate in the same direction Z. Each linearly polarized plane wave corresponds to an eigen mode E+ or E defined above. For the usual case of beams with a finite transversal profile and when Z is along a direction where the double-refraction angles can be nonzero, i.e. out of the principal axes of the index surface, it is necessary to specify a frame for each interacting wave in order to calculate the corresponding powers as a function of Z: the coordinates linked to the wave at ωi are written ([X_i,Y_i,Z]), which can be relative to the mode (+) or (−). The systems are then linked by the double-refraction angles ρ: according to Fig.[link], we have [X_j^+=X_i^+ +Z\tan[\rho^+(\omega_j)-\rho^+(\omega_i)], Y_j^+=Y_i^+] for two waves (+) with [\rho^+(\omega_j)>\rho^+(\omega_i)], and [X_j^- =] [X_i^-, Y_j^- =] [Y_i^-] [ +] [Z\tan[\rho^-(\omega_j)-\rho^-(\omega_i)]] for two waves (−) with [\rho^-(\omega_j)>\rho^-(\omega_i)].

The presence of [{\bf P}^{NL}(\omega_i)] in equations ([link] leads to a variation of the γ amplitudes Ei) with Z. In order to establish the equations of evolution of the wave amplitudes, we assume that their variations are small over one wavelength λi, which is usually true. Thus we can state[\eqalignno{{1 \over k(\omega_i)}\left| {\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}\right| &\ll \left| E(\omega _i, X_i, Y_i, Z)\right|\hbox{ or}&\cr \left| {\partial ^2 E(\omega _i, X_i, Y_i, Z)\over \partial Z^2 }\right| &\ll k(\omega_i)\left| {\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}\right|.&\cr&&(}]This is called the slowly varying envelope approximation.

Stating ([link], the wave equation ([link] for a forward propagation of a plane wave leads to[\eqalignno{{\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}&= j\mu_0 {\omega _i^2 \over 2k(\omega_i)\cos^2 \rho (\omega_i)}{\bf e}(\omega_i)\cdot{\bf P}^{NL}(\omega _i, X_i, Y_i, Z)&\cr&\quad\times\exp[- jk(\omega_i)Z].&(}]We choose the optical frame ([x,y,z]) for the calculation of all the scalar products [{\bf e}(\omega_i)\cdot{\bf P}^{NL}(\omega _i)], the electric susceptibility tensors being known in this frame.

For a three-wave interaction, ([link] leads to[\eqalignno{{\partial E_1 (X_1, Y_1, Z)\over \partial Z} &= j\kappa_1 \left[{\bf e}_1 \cdot\varepsilon_0\chi^{(2)}(\omega_1 = \omega_3 -\omega_2)\cdot{\bf e}_3 \otimes {\bf e}_2 \right] &\cr&\quad\times E_3 (X_3, Y_3, Z)E_2^* (X_2, Y_2, Z)\exp(j\Delta kZ)&\cr{\partial E_2 (X_2, Y_2, Z)\over \partial Z}&= j\kappa_2 \left [{\bf e}_2 \cdot \varepsilon_0\chi ^{(2)}(\omega_2 = \omega _3 - \omega_1)\cdot{\bf e}_3 \otimes {\bf e}_1 \right]&\cr&\quad\times E_3 (X_3, Y_3, Z)E_1^* (X_1, Y_1, Z)\exp(j\Delta kZ)&\cr {\partial E _3 (X_3, Y_3, Z)\over \partial Z}&= j\kappa_3 \left [{\bf e}_3 \cdot \varepsilon _0 \chi ^{(2)}(\omega _3 = \omega _1 + \omega _2)\cdot{\bf e}_1 \otimes{\bf e}_2 \right] &\cr&\quad\times E _1 (X_1, Y_1, Z) E _2 (X_2, Y_2, Z)\exp(- j\Delta kZ), &\cr&&(}]with [{\bf e}_i =] [{\bf e}(\omega_i)], [E_i(X_i,Y_i,Z_i) = ] [E(\omega_i,X_i,Y_i,Z)], [\kappa_i =] [ (\mu_o\omega_i^2)/[2k(\omega _i)\cos^2 \rho (\omega_i)]] and [\Delta k=k(\omega_3)-[k(\omega_1)+k(\omega_2)]], called the phase mismatch. We take by convention [\omega_1\,\lt\,\omega_2\,\,(\lt\,\omega_3)].

If ABDP relations, defined in Section[link], are verified, then the three tensorial contractions in equations ([link] are equal to the same quantity, which we write [\varepsilon_0\chi^{(2)}_{\rm eff}], where [\chi^{(2)}_{\rm eff}] is called the effective coefficient:[\eqalignno{\chi _{\rm eff}^{(2)} &= {\bf e}_1 \cdot\chi^{(2)}(\omega_1 = \omega _3 - \omega _2)\cdot{\bf e}_3 \otimes {\bf e}_2 &\cr& = {\bf e}_2 \cdot\chi ^{(2)}(\omega_2 = \omega _3 - \omega _1)\cdot{\bf e}_3 \otimes {\bf e}_1 &\cr& = {\bf e}_3\cdot\chi^{(2)}(\omega _3 = \omega _1 + \omega _2)\cdot{\bf e}_1\otimes {\bf e}_2. &(}]The same considerations lead to the same kind of equations for a four-wave interaction:[\eqalignno{{\partial E_1 (X_1, Y_1, Z)\over \partial Z}&= j\kappa _1 \varepsilon _0 \chi _{\rm eff}^{(3)}E_4 (X_4, Y_4, Z)E_2^* (X_2, Y_2, Z)&\cr&\quad\times E_3^* (X_3, Y_3, Z)\exp(j\Delta kZ)&\cr{\partial E_2 (X_2, Y_2, Z)\over \partial Z}&= j\kappa _2 \varepsilon _0 \chi _{\rm eff}^{(3)}E_4 (X_4, Y_4, Z)E_1^* (X_1, Y_1, Z)&\cr &\quad\times E_3^* (X_3, Y_3, Z)\exp(j\Delta kZ)&\cr{\partial E_3 (X_3, Y_3, Z)\over \partial Z}&= j\kappa _3 \varepsilon _0 \chi _{eff}^{(3)}E_4 (X_4, Y_4, Z)E_1^* (X_1, Y_1, Z)&\cr&\quad\times E_2^* (X_2, Y_2, Z)\exp(j\Delta kZ)&\cr{{\partial E_4 (X_4, Y_4, Z)}\over {\partial Z}}&= j\kappa _4 \varepsilon _0 \chi _{eff}^{(3)}E_1 (X_1, Y_1, Z)E_2 (X_2, Y_2, Z)&\cr&\quad\times E_3 (X_3, Y_3, Z)\exp(- j\Delta kZ).&\cr&&(}]The conventions of notation are the same as previously and the phase mismatch is [\Delta k=k(\omega_4)-[k(\omega_1)+k(\omega_2)+k(\omega_3)]]. The effective coefficient is[\eqalignno{ \chi _{\rm eff}^{(3)} &= {\bf e}_1 \cdot\chi ^{(3)}(\omega _1 = \omega _4 - \omega _2 - \omega _3)\cdot{\bf e}_4 \otimes{\bf e}_2 \otimes {\bf e}_3 &\cr&= {\bf e}_2 \cdot\chi ^{(3)}(\omega _2 = \omega _4 - \omega _1 - \omega _3)\cdot{\bf e}_4 \otimes {\bf e}_1 \otimes {\bf e}_3 &\cr &= {\bf e}_3 \cdot\chi ^{(3)}(\omega _3 = \omega _4 - \omega _1 - \omega _2)\cdot{\bf e}_4 \otimes {\bf e}_1 \otimes {\bf e}_2 &\cr&= {\bf e}_4 \cdot\chi ^{(3)}(\omega _4 = \omega _1 + \omega _2 + \omega _3)\cdot{\bf e}_1 \otimes {\bf e}_2 \otimes {\bf e}_3. & \cr&&(}]Expressions ( for [\chi_{\rm eff}^{(2)}] and ( for [\chi_{\rm eff}^{(3)}] can be condensed by introducing adequate third- and fourth-rank tensors to be contracted, respectively, with [\chi^{(2)}] and [\chi^{(3)}]. For example, [\chi_{\rm eff}^{(2)}=\chi^{(2)}\cdot e_3\otimes e_1\otimes e_2] or [\chi_{\rm eff}^{(3)} =] [\chi^{(3)}\cdot e_4\otimes e_1\otimes e_2\otimes e_3], and similar expressions. By substituting ([link] in ([link], we obtain the derivatives of Manley–Rowe relations ([link] [\partial N(\omega_3,Z)/\partial Z =] [-\partial N(\omega_k,Z)/\partial Z] [(k =] [1,2)] for a three-wave mixing, where [N(\omega_i,Z)] is the Z photon flow. Identically with ([link], we have [\partial N(\omega_4,Z)/\partial Z =] [-\partial N(\omega_k,Z)/\partial Z] [(k = 1,2,3)] for a four-wave mixing.

In the general case, the nonlinear polarization wave and the generated wave travel at different phase velocities, [(\omega_1+\omega_2)/[k(\omega_1)+k(\omega_2)]] and [\omega_3/[k(\omega_3)]], respectively, because of the frequency dispersion of the refractive indices in the crystal. Then the work per unit time Wi), given in ([link], which is done on the generated wave Ei, Z) by the nonlinear polarization PNLi, Z), alternates in sign for each phase shift of π during the Z-propagation, which leads to a reversal of the energy flow (Bloembergen, 1965[link]). The length leading to the phase shift of π is called the coherence length, [L_c=\pi/\Delta k], where Δk is the phase mismatch given by ([link] or ([link].


Bloembergen, N. (1965). Nonlinear optics. New York: Benjamin.

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