Pulsed-squeezed-light generation in a waveguide with second-subharmonic generation and periodic corrugation

Quantum pulsed second-subharmonic generation in a planar waveguide with a small periodic corrugation at the surface is studied. Backscattering of the interacting fields on the corrugation enhances the nonlinear interaction, giving larger values of squeezing. The problem of backscattering is treated by perturbation theory, using the Fourier transform for nondispersion propagation, and by numerical approach in the general case. Optimum spectral modes for squeezed-light generation are found using the Bloch-Messiah reduction. An improvement in squeezing and increase of the numbers of generated photons are quantified for the corrugation resonating with the fundamental and second-subharmonic field. Splitting of the generated pulse by the corrugation is predicted.

Figure 1

Scheme of a periodically poled nonlinear waveguide with a rectangular transverse profile (thickness $t$ and width $\Delta y$) and length $L$ having ${\chi }^{\left(2\right)}$ susceptibility. A linear corrugation with period ${\Lambda }_l$ and depth $t_l$ occurs on the waveguide upper surface; ${\Lambda }_{nl}$ denotes the period of nonlinear poling. The waveguide is made of LiNbO${}_3$ in which the optical axis coincides with the $x$ axis. Four fields interact inside the waveguide: forward-propagating fundamental (electric-field amplitude $A_{p_F}$), backward-propagating fundamental ($A_{p_B}$), forward-propagating second-subharmonic ($A_{s_F}$), and backward-propagating second-subharmonic ($A_{s_B}$) fields.

Figure 2

(a) Absolute value of linear coupling constant $K_p$ and (b) linear phase mismatch ${\delta }_p-{\delta }_p^0$ for the fundamental field as they depend on relative frequency ${\omega }_p/{\omega }_p^0$; $t=5\times {10}^{-7}$ m, $t_l=5\times {10}^{-8}$ m.

Figure 3

(a) Absolute value of linear coupling constant $K_s$ and (b) linear phase mismatch ${\delta }_s-{\delta }_s^0$ for the SSF field as functions of relative frequency ${\omega }_s/{\omega }_s^0$; values of parameters are the same as in Fig. 2.

Figure 4

Contour plots of (a) absolute value $|K_{nl,1}|$ of nonlinear coupling constant and (b) nonlinear phase mismatch ${\delta }_{nl,1}-{\delta }_{nl,1}^0$ as they depend on relative frequencies ${\omega }_s/{\omega }_s^0$ and ${\omega }_s^{\prime }/{\omega }_s^0$. Values of parameters are the same as in Fig. 2.

Figure 5

Intensity spectral profiles $|{\mathbf{X}}_{s,n}{|}^2$ for the scattered (a) fundamental and (b) SSF field for the first (solid curve without symbols), second (solid curve with $*$), third (solid curve with $\circ$), and fourth (solid curve with $▵$) eigenmode obtained in the model with nondispersion propagation. The profiles are normalized such that $\int d\omega |{\mathbf{X}}_{s,n}{\left(\omega \right)|}^2/{\omega }_s^0=1$. In (a), ${\Lambda }_l=1.151\times {10}^{-7}$ m and ${\Lambda }_{nl}=3.5510\times {10}^{-6}$ m. In (b), ${\Lambda }_l=2.459\times {10}^{-7}$ m and ${\Lambda }_{nl}=3.5547\times {10}^{-6}$ m. $P_{p_F}=1\times {10}^{-6}$ W, $L=1\times {10}^{-3}$ m.

Figure 6

Principal squeeze variances ${\lambda }_{s,n}$ of the maximum-squeezed 15 eigenmodes for the scattered fundamental (solid curve with $*$) and SSF (solid curve with $▵$) field determined assuming nondispersion propagation. Also, the principal squeeze variances ${\lambda }_{s_F,n}$ of the forward-propagating SSF field are shown (solid curve with $\circ$). For comparison, principal squeeze variances ${\lambda }_{s,n}$ of the real waveguide without the scattered fields (solid curve with $\diamond$) are drawn; $P_{p_F}=1\times {10}^{-6}$ W, $L=1\times {10}^{-3}$ m.

Figure 7

Intensity spectral profiles $|{\mathbf{X}}_{s,n}{|}^2$ for the first (solid curve without symbols), second (solid curve with $*$), third (solid curve with $\circ$), and fourth (solid curve with $▵$) SSF-field eigenmode for (a) no scattered field and (b) scattered fundamental field. In (a), ${\Lambda }_{nl}=3.5516\times {10}^{-6}$ m, and in (b), ${\Lambda }_l=1.151\times {10}^{-7}$ m, ${\Lambda }_{nl}=3.5510\times {10}^{-6}$ m; $P_{p_F}=1\times {10}^{-6}$ W, $L=1\times {10}^{-3}$ m.

Figure 8

Principal squeeze variances ${\lambda }_{s,n}$ depending on the mode number $n$ for $P_{p_F}=1\times {10}^{-7}$ W (solid curve with $\circ$), $P_{p_F}=1\times {10}^{-6}$ W (solid curve with $▵$), and $P_{p_F}=1\times {10}^{-5}$ W (solid curve with $*$) for (a) no scattered field and (b) scattered fundamental field. Values of the parameters are the same as in the caption of Fig. 7.

Figure 9

(a) Intensity spectrum $N_{ss,\omega }^d\left(\omega \right)\equiv N_{ss,\omega }(\omega ,\omega )$ and (b) cut $N_{ss,\omega }^c\left(\omega \right)\equiv \left|N_{ss,\omega }({\omega }_s^0,\omega )\right|$ across the amplitude correlation function of the overall SSF field for the waveguide with the fundamental-field scattering (solid curve without symbols) and without scattering (solid curve with $*$). The spectra are normalized such that $\int d\omega N_{ss,\omega }^d\left(\omega \right)/{\omega }_s^0=1$. Values of the parameters are the same as in the caption to Fig. 7.

Figure 10

Flux $N_{ss,\tau }^d\left(\tau \right)\equiv N_{ss,\tau }(\tau ,\tau )$ of photon numbers in the SSF field for the waveguide with the fundamental-field scattering (solid curve without symbols) and without scattering (solid curve with $*$). It holds that $\int d\tau N_{ss,\tau }^d\left(\tau \right)=1$ and values of the parameters are the same as in the caption to Fig. 7.

Figure 11

(a) Principal squeeze variances ${\lambda }_{s,n}$ and (b) numbers $N_{s,n}$ of generated photons as they depend on incident fundamental-field power $P_{p_F}$. The quantities are shown for the waveguide with fundamental-field scattering (${\Lambda }_l=1.151\times {10}^{-7}$ m, ${\Lambda }_{nl}=3.5510\times {10}^{-6}$ m, solid curves) as well as without scattering (${\Lambda }_{nl}=3.5516\times {10}^{-6}$ m, dashed curves). Curves without symbols are for $n=1$, whereas the curves with $*$ are for $n=23$ (with scattering) and $n=14$ (without scattering) equal to the number $K$ of effectively populated modes; $L=1\times {10}^{-2}$ m. In (b), the logarithmic $y$ axis is used.

Figure 12

Numbers $N_s$ of generated photons depending on incident fundamental-field power $P_{p_F}$ for the waveguide with fundamental-field scattering (solid curve) and without scattering (dashed curve); $L=1\times {10}^{-2}$ m. The logarithmic $y$ axis is used.