Coupling-length phase matching for nonlinear optical frequency conversion in parallel waveguides

We describe and analyze a quasi-phase-matching scheme for nonlinear optical frequency conversion where the spatial modulation of mode intensity in coupled parallel waveguides provides the required modulation in the generation of the frequency conversion signal, instead of a variation of any material parameter or propagation constant. We analyze this coupling-length phase-matching (CLPM) scheme both for second-order frequency conversion, such as second harmonic generation or difference-frequency generation, as well as for third-order four-wave mixing processes, for which we consider the example of generating a longer wavelength by third-order nonlinear mixing of two shorter wavelength waves. Numerous phase-matching conditions are identified in each case. We show that the maximum photon conversion efficiencies reached after an optimum propagation length are always higher than half those obtained for perfect phase matching in a single waveguide, with nearly 100% photon conversion possible for several of the CLPM conditions we studied.

Figure 1

Schematic diagram of the principle of coupling-length phase-matching (CLPM). The two waveguides $a$ and $b$ are assumed to be identical. The pump beams (at frequency ${\omega }_1$ and ${\omega }_2$ in this example) are injected in waveguide $a$. The evanescent coupling between the two waveguides causes the power of the pump waves to oscillate between the two waveguides (schematically represented in this sketch by the two lines meandering between the two waveguides). Under the appropriate CLPM condition it is possible to arrange for the signal wave at frequency ${\omega }_3$ that is created by nonlinear optical interaction of the two pump waves to grow constructively with propagation length, in both waveguides. The qualitative depiction in this figure corresponds to the case of sum-frequency generation $\left({\omega }_3={\omega }_1+{\omega }_2\right)$ that will be presented later in Fig. 2.

Figure 2

Spatial evolution of the absolute values of the wave amplitudes in the case of SFG under the CLPM condition $K_2=0$ in parallel waveguides of length $L$. All solid curves are obtained in the pump depletion regime by a numerical solution of Eqs. (5, 6, 7) and the corresponding equations for waveguide $b$. Panels (a) and (b) give the amplitudes, in waveguide $a$, of the pump waves at frequency ${\omega }_1$ and ${\omega }_2$, respectively. The generated sum-frequency wave amplitude is shown for both waveguides in $\left(\text{c}\right)$ and (d), respectively. The amplitudes are normalized to $\tilde{A}={|A_{1,0}^{\left(a\right)}{A}_{2,0}^{\left(a\right)}|}^{1/2}$ and the nonlinearity constant used for the plots is given by $Ld=2/\tilde{A}$, with $L$ the length of the waveguide. The other parameters are ${\kappa }_{1}L=160,{\kappa }_{2}L=80,{\kappa }_{3}L=40$, and $\Delta kL=200$. The nondepleted analytical solution (12) corresponds to the initial part at short distances and is given explicitly in $\left(\text{c}\right)$ as a dotted black curve.

Figure 3

Spatial evolution of the absolute values of the SFG wave amplitudes under the CLPM condition $K_1=K_2=0$ (row 5 in Table 1) in parallel waveguides of length $L$. The curves are obtained as in Fig. 2 and for the same nonlinearity. Panels (a) and (b) give the amplitudes, in waveguide $a$, of the pump waves at frequency ${\omega }_1$ and ${\omega }_2$, respectively. The generated sum-frequency wave amplitude is shown for both waveguides in $\left(\text{c}\right)$ and (d), respectively. The solid curves are all in the pump depletion regime. The black dotted curve in $\left(\text{c}\right)$ corresponds to the analytic undepleted regime according to (12) and (14). For these plots $\tilde{A}={|A_{1,0}^{\left(a\right)}{A}_{2,0}^{\left(a\right)}|}^{1/2},\Delta kL={\kappa }_{1}L=200,{\kappa }_{2}L={\kappa }_{3}L=60$, and $Ld\tilde{A}=2$.

Figure 4

Growth of the second-harmonic signal amplitude for the optimum CLPM condition (first line of Table 2) and for the same effective nonlinearity $(Ld=2/|A_{1,0}^{\left(a\right)}\left|\right)$ as in Figs. 2 and 3. Both curves are for $\Delta kL={\kappa }_{3}L=40$; the solid red curve is for ${\kappa }_{1}L(={\kappa }_{2}L)=120$, and the dotted blue curve is for ${\kappa }_{1}L=70$. The general growth behavior is independent of ${\kappa }_1$, which only affects the small oscillations shown enlarged in the inset.

Figure 5

Spatial evolution of the absolute values of the DFG wave amplitudes under the CLPM condition $K_1=K_2=0$ (row 5 in Table 1) in parallel waveguides of length $L$. The curves are obtained as in Fig. 2 and for the same nonlinearity. Panels (a) and (b) give the amplitudes, in waveguide $a$, of the pump waves at frequency ${\omega }_3$ and ${\omega }_2$, respectively. The amplitude of the generated difference-frequency wave is shown for both waveguides in $\left(\text{c}\right)$ and (d), respectively. The solid curves are all in the pump depletion regime. The black dotted curve in $\left(\text{c}\right)$ corresponds to the analytical solution in the undepleted regime according to Eqs. (19) and (20). For these plots $\tilde{A}\equiv {\left|A_{3,0}^{\left(a\right)}{A}_{2,0}^{\left(a\right)}\right|}^{1/2},\Delta kL={\kappa }_{1}L=200,{\kappa }_{2}L={\kappa }_{3}L=60$, and $Ld\tilde{A}=2$.

Figure 6

Spatial evolution of the amplitudes of the interacting waves for third-order frequency down-conversion, ${\omega }_1=2{\omega }_2-{\omega }_3$ under CLPM condition $K_2=0$ (row 1 in Table 3) in parallel waveguides of length $L$. Panels (a) and (b) give the amplitudes, in waveguide $a$, of the pump waves at frequency ${\omega }_3$ and ${\omega }_2$, respectively. The generated wave amplitude at the frequency ${\omega }_1$ is shown for both waveguides in $\left(\text{c}\right)$ and (d), respectively. The solid curves are all in the pump depletion regime. The black dotted curve in $\left(\text{c}\right)$ corresponds to the analytical solution from (28) and (29) in the undepleted regime. For these plots $\tilde{A}\equiv \left(\right|A_{3,0}^{\left(a\right)}\left|\right|A_{2,0}^{\left(a\right)}{{|}^2)}^{1/3},\Delta kL=200,{\kappa }_{1}L=240,{\kappa }_{2}L=60,{\kappa }_{3}L=40$, and $L\chi {\tilde{A}}^2=2$. The curves are normalized to the initial amplitude of the wave ${\omega }_3$ injected in waveguide $a$.

Figure 7

Periodic behavior of second-harmonic generation in the pump depletion regime. (a) Dependence of the photon-conversion efficiency ${\eta }^{\mathrm{SH}}$ on propagation distance under the CLPM condition in the first line of Table 2 $\left(\Delta k={\kappa }_3\right)$. (b) Evolution of the corresponding total normalized pump wave photon intensity, where $I_{\omega }\left(z\right)\propto |A_1^{\left(a\right)}{\left(z\right)|}^2+{\left|A_1^{\left(b\right)}\left(z\right)\right|}^2$ and $I_{\omega }(z=0)\propto {\left|A_1^{\left(a\right)}\left(0\right)\right|}^2$. The inset in (b) shows the photon intensity of the fundamental wave in each individual waveguide around the point of maximum pump depletion. The parameters are as in Fig. 4, $\Delta k{L}_0={\kappa }_3{L}_0=40,{\kappa }_1{L}_0=120$, and $L_{0}d=2/\left|A_{1,0}^{\left(a\right)}\right|$.