Coupled electric fields amplitudes equations

Boulanger, B.; Zyss, J.

doi:10.1107/97809553602060000634

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.7, pp. 187-188

Section 1.7.3.2.1. Coupled electric fields amplitudes equations

B. Boulanger^a ^* and J. Zyss^b

^a Laboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and ^bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail: benoitb@satie-bourgogne.fr

1.7.3.2.1. 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 ω_i 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.

Thus the propagation equation of each interacting wave ω_i is (Bloembergen, 1965) $[\nabla x\nabla x{\bf E}(\omega_i) = (\omega_i^2/c^2)\varepsilon (\omega _i){\bf E}(\omega_i) + \omega_i ^2\mu_0 {\bf P}^{NL}(\omega_i).\eqno(1.7.3.19)]$ 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, (1.7.3.3), as a solution of (1.7.3.19), 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 ω_i 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. 1.7.3.1, 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 (1.7.3.19) leads to a variation of the γ amplitudes E(ω_i) with Z. In order to establish the equations of evolution of the wave amplitudes, we assume that their variations are small over one wavelength λ_i, which is usually true. Thus we can state $[\eqalignno{{1 \over k(\omega_i)}\left| {\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}\right| &\ll \left| E(\omega _i, X_i, Y_i, Z)\right|\hbox{ or}&\cr \left| {\partial ^2 E(\omega _i, X_i, Y_i, Z)\over \partial Z^2 }\right| &\ll k(\omega_i)\left| {\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}\right|.&\cr&&(1.7.3.20)}]$ This is called the slowly varying envelope approximation.

Stating (1.7.3.20), the wave equation (1.7.3.19) for a forward propagation of a plane wave leads to $[\eqalignno{{\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}&= j\mu_0 {\omega _i^2 \over 2k(\omega_i)\cos^2 \rho (\omega_i)}{\bf e}(\omega_i)\cdot{\bf P}^{NL}(\omega _i, X_i, Y_i, Z)&\cr&\quad\times\exp[- jk(\omega_i)Z].&(1.7.3.21)}]$ 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, (1.7.3.21) leads to $[\eqalignno{{\partial E_1 (X_1, Y_1, Z)\over \partial Z} &= j\kappa_1 \left[{\bf e}_1 \cdot\varepsilon_0\chi^{(2)}(\omega_1 = \omega_3 -\omega_2)\cdot{\bf e}_3 \otimes {\bf e}_2 \right] &\cr&\quad\times E_3 (X_3, Y_3, Z)E_2^* (X_2, Y_2, Z)\exp(j\Delta kZ)&\cr{\partial E_2 (X_2, Y_2, Z)\over \partial Z}&= j\kappa_2 \left [{\bf e}_2 \cdot \varepsilon_0\chi ^{(2)}(\omega_2 = \omega _3 - \omega_1)\cdot{\bf e}_3 \otimes {\bf e}_1 \right]&\cr&\quad\times E_3 (X_3, Y_3, Z)E_1^* (X_1, Y_1, Z)\exp(j\Delta kZ)&\cr {\partial E _3 (X_3, Y_3, Z)\over \partial Z}&= j\kappa_3 \left [{\bf e}_3 \cdot \varepsilon _0 \chi ^{(2)}(\omega _3 = \omega _1 + \omega _2)\cdot{\bf e}_1 \otimes{\bf e}_2 \right] &\cr&\quad\times E _1 (X_1, Y_1, Z) E _2 (X_2, Y_2, Z)\exp(- j\Delta kZ), &\cr&&(1.7.3.22)}]$ 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 1.7.2.2.1, are verified, then the three tensorial contractions in equations (1.7.3.22) 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: $[\eqalignno{\chi _{\rm eff}^{(2)} &= {\bf e}_1 \cdot\chi^{(2)}(\omega_1 = \omega _3 - \omega _2)\cdot{\bf e}_3 \otimes {\bf e}_2 &\cr& = {\bf e}_2 \cdot\chi ^{(2)}(\omega_2 = \omega _3 - \omega _1)\cdot{\bf e}_3 \otimes {\bf e}_1 &\cr& = {\bf e}_3\cdot\chi^{(2)}(\omega _3 = \omega _1 + \omega _2)\cdot{\bf e}_1\otimes {\bf e}_2. &(1.7.3.23)}]$ The same considerations lead to the same kind of equations for a four-wave interaction: $[\eqalignno{{\partial E_1 (X_1, Y_1, Z)\over \partial Z}&= j\kappa _1 \varepsilon _0 \chi _{\rm eff}^{(3)}E_4 (X_4, Y_4, Z)E_2^* (X_2, Y_2, Z)&\cr&\quad\times E_3^* (X_3, Y_3, Z)\exp(j\Delta kZ)&\cr{\partial E_2 (X_2, Y_2, Z)\over \partial Z}&= j\kappa _2 \varepsilon _0 \chi _{\rm eff}^{(3)}E_4 (X_4, Y_4, Z)E_1^* (X_1, Y_1, Z)&\cr &\quad\times E_3^* (X_3, Y_3, Z)\exp(j\Delta kZ)&\cr{\partial E_3 (X_3, Y_3, Z)\over \partial Z}&= j\kappa _3 \varepsilon _0 \chi _{eff}^{(3)}E_4 (X_4, Y_4, Z)E_1^* (X_1, Y_1, Z)&\cr&\quad\times E_2^* (X_2, Y_2, Z)\exp(j\Delta kZ)&\cr{{\partial E_4 (X_4, Y_4, Z)}\over {\partial Z}}&= j\kappa _4 \varepsilon _0 \chi _{eff}^{(3)}E_1 (X_1, Y_1, Z)E_2 (X_2, Y_2, Z)&\cr&\quad\times E_3 (X_3, Y_3, Z)\exp(- j\Delta kZ).&\cr&&(1.7.3.24)}]$ 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 $[\eqalignno{ \chi _{\rm eff}^{(3)} &= {\bf e}_1 \cdot\chi ^{(3)}(\omega _1 = \omega _4 - \omega _2 - \omega _3)\cdot{\bf e}_4 \otimes{\bf e}_2 \otimes {\bf e}_3 &\cr&= {\bf e}_2 \cdot\chi ^{(3)}(\omega _2 = \omega _4 - \omega _1 - \omega _3)\cdot{\bf e}_4 \otimes {\bf e}_1 \otimes {\bf e}_3 &\cr &= {\bf e}_3 \cdot\chi ^{(3)}(\omega _3 = \omega _4 - \omega _1 - \omega _2)\cdot{\bf e}_4 \otimes {\bf e}_1 \otimes {\bf e}_2 &\cr&= {\bf e}_4 \cdot\chi ^{(3)}(\omega _4 = \omega _1 + \omega _2 + \omega _3)\cdot{\bf e}_1 \otimes {\bf e}_2 \otimes {\bf e}_3. & \cr&&(1.7.3.25)}]$ Expressions (1.7.3.23) for $[\chi_{\rm eff}^{(2)}]$ and (1.7.3.25) 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 (1.7.3.8) in (1.7.3.22), we obtain the derivatives of Manley–Rowe relations (1.7.2.40) $[\partial N(\omega_3,Z)/\partial Z =]$ $[-\partial N(\omega_k,Z)/\partial Z]$ [(k =] [1,2)] for a three-wave mixing, where $[N(\omega_i,Z)]$ is the Z photon flow. Identically with (1.7.3.24), 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 W(ω_i), given in (1.7.2.39), which is done on the generated wave E(ω_i, Z) by the nonlinear polarization P^NL(ω_i, 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). 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 (1.7.3.22) or (1.7.3.24).

References

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

International Tables for Crystallography (2006). Vol. D. ch. 1.7, pp. 187-188