NanoPhotoNet-MPM: A Physics-Informed Deep Learning for Inverse Design of NbOCl₂ Biphoton Quantum Sources

Introduction

Niobium oxychloride (NbOCl₂) is a compelling platform for integrated nonlinear quantum photonics because its in-plane second-order nonlinear coefficient $d_{22}\approx 100\hspace{0.17em}\mathrm{p}\mathrm{m}\hspace{0.17em}{\mathrm{V}}^{-1}$ is roughly one order of magnitude larger than thin-film lithium niobate (TFLN, $d_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx 15\hspace{0.17em}\mathrm{p}\mathrm{m}\hspace{0.17em}{\mathrm{V}}^{-1}$) and comparable to AlGaAs [2,6,7,8]. The monoclinic crystal symmetry (space group $C_2$) produces strong biaxial anisotropy and off-diagonal permittivity components that hybridise the guided modes, making full-vectorial electromagnetic modelling a strict requirement rather than an approximation [9,10,11,12]. Modal phase matching (MPM) between the higher-order pump mode (${\mathrm{T}\mathrm{E}}_{10}$) and the fundamental signal and idler modes (${\mathrm{T}\mathrm{E}}_{00}$) provides a geometry-tunable route to $\mathsf{\Delta }\beta =0$ without periodic domain inversion [11,13,14]. Table 1 compares NbOCl₂ against established integrated nonlinear photonic platforms. The complete crystallographic framework, rotated permittivity tensor, vectorial Helmholtz mode equations, and coupled-wave propagation governing this platform are provided in Notes S16 and S17 in SI.

Figure 1. Overview of the surrogate-assisted inverse design framework and integrated modal-phase-matched NbOCl₂ biphoton quantum light source. The three-dimensional schematic shows the asymmetric directional coupler (ADC) layout patterned on a SiO₂ substrate. Waveguide 1 (narrow, width $W_1$, height $h$) couples a $775\hspace{0.17em}\mathrm{n}\mathrm{m}$ pump laser (${\mathrm{T}\mathrm{E}}_{00}$) into Waveguide 2 (wide, optimised width $W_2^{\mathrm{*}}=283.13\hspace{0.17em}\mathrm{n}\mathrm{m}$, height $h^{\mathrm{*}}=500.00\hspace{0.17em}\mathrm{n}\mathrm{m}$, separated by a gap $g$) to excite the higher-order mode (${\mathrm{T}\mathrm{E}}_{10}$), which down-converts into signal and idler photon pairs at $1550\hspace{0.17em}\mathrm{n}\mathrm{m}$ via SPDC. The inset shows the computational design workflow (top) and the two-dimensional waveguide cross-section parameters (bottom).

Figure 1. Overview of the surrogate-assisted inverse design framework and integrated modal-phase-matched NbOCl₂ biphoton quantum light source. The three-dimensional schematic shows the asymmetric directional coupler (ADC) layout patterned on a SiO₂ substrate. Waveguide 1 (narrow, width $W_1$, height $h$) couples a $775\hspace{0.17em}\mathrm{n}\mathrm{m}$ pump laser (${\mathrm{T}\mathrm{E}}_{00}$) into Waveguide 2 (wide, optimised width $W_2^{\mathrm{*}}=283.13\hspace{0.17em}\mathrm{n}\mathrm{m}$, height $h^{\mathrm{*}}=500.00\hspace{0.17em}\mathrm{n}\mathrm{m}$, separated by a gap $g$) to excite the higher-order mode (${\mathrm{T}\mathrm{E}}_{10}$), which down-converts into signal and idler photon pairs at $1550\hspace{0.17em}\mathrm{n}\mathrm{m}$ via SPDC. The inset shows the computational design workflow (top) and the two-dimensional waveguide cross-section parameters (bottom).

Table 1. Performance comparison of integrated SPDC platforms for chip-scale nonlinear quantum photonics.

Table 1. Performance comparison of integrated SPDC platforms for chip-scale nonlinear quantum photonics.

Ref.	Platform	PM	${\mathit{d}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$(pm/V)	Loss (dB/cm)	GVM	Purity	Key Limitation
[15]	4H-SiC	MPM	$\sim$6.5	1–2	High	N/A	Low $d_{\mathrm{e}\mathrm{f}\mathrm{f}}$, dispersion constraints
[1,16]	TFLN	QPM	$\sim$15	$<$1	Mod.	$>$0.95	Periodic poling fabrication
[3,17]	TFLN	MPM	$\sim$4.6	$<$1	High	N/A	Low $d_{\mathrm{e}\mathrm{f}\mathrm{f}}$, dispersion constraints
[7]	AlGaAs	MPM	$\sim$100	1–3	High	$>$0.8	Sidewall loss, pump engineering
[18]	GaP	MPM	$\sim$70	2–5	High	N/A	Two-photon absorption
[19]	AlN	QPM	$\sim$5	$<$1	Mod.	N/A	Low $d_{\mathrm{e}\mathrm{f}\mathrm{f}}$
This work	NbOCl2	MPM	$\sim$100	~3–10	High	0.122	Loss and fabrication unmeasured

Methods

Stage 2 embeds these modal parameters into a longitudinal coupled-wave propagator that calculates classical field envelopes and quantum photon-pair correlation dynamics. The propagator is trained by minimizing a joint loss function composed of classical and quantum coupled-wave equation residuals, initial condition boundary penalties, and supervision against integration references:

$${\mathcal{L}}_{\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}\hspace{0.17em}2}={\lambda }_{\mathrm{C}\mathrm{W}\mathrm{E}}{\mathcal{L}}_{\mathrm{C}\mathrm{W}\mathrm{E}}+{\lambda }_{\mathrm{I}\mathrm{C}}{\mathcal{L}}_{\mathrm{I}\mathrm{C}}+{\lambda }_{\mathrm{R}\mathrm{K}4}{\mathcal{L}}_{\mathrm{R}\mathrm{K}4}+{\lambda }_{\mathrm{q}\mathrm{R}\mathrm{K}4}{\mathcal{L}}_{\mathrm{q}\mathrm{R}\mathrm{K}4}+{\lambda }_{\mathrm{q}\mathrm{C}\mathrm{W}\mathrm{E}}{\mathcal{L}}_{\mathrm{q}\mathrm{C}\mathrm{W}\mathrm{E}},$$

(3)

where ${\mathcal{L}}_{\mathrm{C}\mathrm{W}\mathrm{E}}$ and ${\mathcal{L}}_{\mathrm{q}\mathrm{C}\mathrm{W}\mathrm{E}}$ are the classical and quantum propagation residuals, ${\mathcal{L}}_{\mathrm{I}\mathrm{C}}$ is the boundary initial condition loss, and ${\mathcal{L}}_{\mathrm{R}\mathrm{K}4}$ and ${\mathcal{L}}_{\mathrm{q}\mathrm{R}\mathrm{K}4}$ represent the mean squared errors relative to Runge-Kutta numerical trajectories. The master schematic of this workflow is shown in Fig. 2.

Figure 2. Master schematic of the NanoPhotoNet-MPM workflow. Waveguide geometry, coordinate grid, wavelength, mode index, and polarization labels are used as surrogate inputs. Stage 1 transverse operator surrogate, where branch and trunk networks synthesize six-component modal fields and predict the corresponding effective index. Deterministic physics post-processing layer that converts modal outputs and material parameters into phase-mismatch and coupling inputs for longitudinal propagation. Stage 2 seeded coupled-wave propagator for the classical field envelopes $A_p\left(z\right)$, $A_s\left(z\right)$, and $A_i\left(z\right)$ under specified initial conditions and propagation losses. This fixed architecture figure defines the computational workflow; the following figures audit the physical validity, loss budget, phase matching, purity limits, and fabrication sensitivity.

Figure 2. Master schematic of the NanoPhotoNet-MPM workflow. Waveguide geometry, coordinate grid, wavelength, mode index, and polarization labels are used as surrogate inputs. Stage 1 transverse operator surrogate, where branch and trunk networks synthesize six-component modal fields and predict the corresponding effective index. Deterministic physics post-processing layer that converts modal outputs and material parameters into phase-mismatch and coupling inputs for longitudinal propagation. Stage 2 seeded coupled-wave propagator for the classical field envelopes $A_p\left(z\right)$, $A_s\left(z\right)$, and $A_i\left(z\right)$ under specified initial conditions and propagation losses. This fixed architecture figure defines the computational workflow; the following figures audit the physical validity, loss budget, phase matching, purity limits, and fabrication sensitivity.

Results and Discussion

The trained model predicts a degenerate modal phase-matching point at $w^{\mathrm{*}}=283.13\hspace{0.17em}\mathrm{n}\mathrm{m}$ and $h^{\mathrm{*}}=500.00\hspace{0.17em}\mathrm{n}\mathrm{m}$ (inside our audited bounds $w,h\in [200,1000]\hspace{0.17em}\mathrm{n}\mathrm{m}$) where the phase mismatch is $\left|\mathsf{\Delta }\beta \right|=1.0\hspace{0.17em}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{m}$ and the modal coupling is $\mathsf{\Gamma }=0.035$. This well-guided geometry is free of substrate leakage. The continuous zero is treated as a model-derived design hypothesis; since our index audit gives a 95th-percentile absolute error of $9.57\times {10}^{-3}$ (uncertainty of order $1.55\times {10}^5\hspace{0.17em}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{m}$), an off-grid FDE validation solve is required before fabrication. The model-predicted effective-index crossing at the design height ($h=500\hspace{0.17em}\mathrm{n}\mathrm{m}$) is shown in Fig. 3(b). Because this high-index-contrast scheme relies on coupling spatial modes of different orders, the alignment of propagation constants is exceptionally sensitive to waveguide cross-section. For a $5\hspace{0.17em}\mathrm{m}\mathrm{m}$ device, the width tolerance FWHM is only $0.073\hspace{0.17em}\mathrm{n}\mathrm{m}$. Pump wavelength tuning can correct small dimensional offsets and restore phase matching after sub-nanometer dimensional variations.

Figure 3. Anisotropic optical response, modal phase-matching landscape, and neural surrogate training convergence. (a) Principal refractive indices $n_x$, $n_y$, $n_z$ of monoclinic NbOCl₂ alongside Sellmeier dispersion model samples. (b) Effective-index crossing cut at the design height ($h=500.0\hspace{0.17em}\mathrm{n}\mathrm{m}$), comparing the physics-informed model (PINN, solid lines) and standard data-driven baseline (DNN, dashed lines) against FDE training targets. (c) Phase-mismatch landscape $\mathsf{\Delta }\beta (w,h)$ over the audited design grid, showing the zero-crossing curve. (d) Effective-index parity plot comparing PINN and data-driven DNN predictions against FDE data targets. (e) Stage 1 (transverse solver) training and validation loss convergence histories (normalized) for the total loss, effective index, field profile, and modal overlap terms. (f) Stage 2 (longitudinal propagator) training and validation loss convergence for the total, classical RK4 reference ($L_{\mathrm{R}\mathrm{K}4}$), classical CWE reference ($L_{\mathrm{C}\mathrm{W}\mathrm{E}}$), quantum RK4 reference ($L_{\mathrm{q}\mathrm{R}\mathrm{K}4}$), and quantum qCWE reference ($L_{\mathrm{q}\mathrm{C}\mathrm{W}\mathrm{E}}$) terms.

Figure 3. Anisotropic optical response, modal phase-matching landscape, and neural surrogate training convergence. (a) Principal refractive indices $n_x$, $n_y$, $n_z$ of monoclinic NbOCl₂ alongside Sellmeier dispersion model samples. (b) Effective-index crossing cut at the design height ($h=500.0\hspace{0.17em}\mathrm{n}\mathrm{m}$), comparing the physics-informed model (PINN, solid lines) and standard data-driven baseline (DNN, dashed lines) against FDE training targets. (c) Phase-mismatch landscape $\mathsf{\Delta }\beta (w,h)$ over the audited design grid, showing the zero-crossing curve. (d) Effective-index parity plot comparing PINN and data-driven DNN predictions against FDE data targets. (e) Stage 1 (transverse solver) training and validation loss convergence histories (normalized) for the total loss, effective index, field profile, and modal overlap terms. (f) Stage 2 (longitudinal propagator) training and validation loss convergence for the total, classical RK4 reference ($L_{\mathrm{R}\mathrm{K}4}$), classical CWE reference ($L_{\mathrm{C}\mathrm{W}\mathrm{E}}$), quantum RK4 reference ($L_{\mathrm{q}\mathrm{R}\mathrm{K}4}$), and quantum qCWE reference ($L_{\mathrm{q}\mathrm{C}\mathrm{W}\mathrm{E}}$) terms.

To isolate the contribution of the neural operator, we compare Eigenmode DeepONet against conventional interpolators and machine learning baselines (Table 2). The neural surrogate achieves a mean index MAE of $(1.62\pm 0.60)\times {10}^{-3}$, outperforming the Gaussian Process Regression (GPR) baseline ($(1.79\pm 0.67)\times {10}^{-2}$) and geometry-only Multilayer Perceptron (MLP) baseline ($(7.40\pm 1.13)\times {10}^{-2}$). While cubic spline interpolation achieves near-zero error on grid points ($4.00\times {10}^{-16}$), it suffers from a $55\mathrm{\%}$ failure rate due to boundary extrapolation on the irregular FDE grid and cannot generalize to off-grid optimization regions. Radial basis functions (RBF) achieve an MAE of $1.80\times {10}^{-2}$ but offer no wave-equation regularization. Removing the regularizer (${\lambda }_{\mathrm{H}\mathrm{e}\mathrm{l}\mathrm{m}}=0$) introduces non-physical spatial oscillations and field boundary discontinuities, while the optimal weight (${\lambda }_{\mathrm{H}\mathrm{e}\mathrm{l}\mathrm{m}}=0.01$) enforces wave-equation consistency without compromising index accuracy.

Table 1. Effective-index prediction baseline comparison. The prediction performance of the transverse neural eigensolver (EigenmodeDeepONet) is compared against Gaussian Process Regression (GPR), a plain Multilayer Perceptron (MLP), radial basis function (RBF) interpolation, and 2D cubic splines. The error metrics are estimated using 10-fold residual resampling. Failure rate represents the percentage of out-of-grid queries that fail due to boundary extrapolation.

Table 1. Effective-index prediction baseline comparison. The prediction performance of the transverse neural eigensolver (EigenmodeDeepONet) is compared against Gaussian Process Regression (GPR), a plain Multilayer Perceptron (MLP), radial basis function (RBF) interpolation, and 2D cubic splines. The error metrics are estimated using 10-fold residual resampling. Failure rate represents the percentage of out-of-grid queries that fail due to boundary extrapolation.

Method	Index MAE
Neural operator surrogate (this work)	$(1.62\pm 0.60)\times {10}^{-3}$​
GPR baseline	$(1.79\pm 0.67)\times {10}^{-2}$​
Plain MLP baseline	$(7.40\pm 1.13)\times {10}^{-2}$​
Multiquadric RBF	$1.86\times {10}^{-2}$​
2D Cubic Spline	$4.00\times {10}^{-16}$/ N/A

To further evaluate the computational efficiency and physical validity of the physics-informed architecture, we perform an ablation study comparing the performance of the full PINN model (Eigenmode DeepONet + CWE-PINN) against a standard data-driven DNN baseline (trained without Helmholtz and boundary regularization, i.e., ${\lambda }_{\mathrm{H}\mathrm{e}\mathrm{l}\mathrm{m}}=0$) and the conventional vectorial FDE solver reference. As summarized in Table 3, the FDE solver requires $\sim 15.0\hspace{0.17em}\mathrm{s}$ per geometry for numerical mode solving, representing a significant bottleneck for multi-objective optimization sweeps. In contrast, the trained PINN and DNN models run in just $0.71\hspace{0.17em}\mathrm{ms}$ per geometry, representing a speedup of over $21,120\times$.

Figure 4. Transverse mode field reconstruction benchmarking for PINN and data-driven DNN baselines against the FDE solver. Comparison of predicted and FDE-calculated normalized electric field magnitudes ($\left|E\right|$To verify that the neural surrogate (Eigenmode DeepONet) correctly learns the physical mode shapes rather than merely interpolating effective indices, we compare its predicted transverse field profiles against the FDE ground truth. As shown in Fig. 4, the predicted and FDE-calculated normalized electric field magnitudes ($\left|E\right|$) at the optimal design point ($w^{\mathrm{*}}=283.13\hspace{0.17em}\mathrm{n}\mathrm{m},h^{\mathrm{*}}=500.00\hspace{0.17em}\mathrm{n}\mathrm{m}$) are in excellent agreement. The peak local reconstruction error remains below 0.05 across the coordinate grid, and the modal field overlap reaches 0.928 for the pump mode (${\mathrm{T}\mathrm{E}}_{10}$) and 0.981 for the fundamental signal mode (${\mathrm{T}\mathrm{E}}_{00}$). This confirms that the trunk network’s basis functions and the branch network’s geometric projection are physically regularized, capturing sharp field discontinuities without Gibbs oscillations.

Figure 4. Transverse mode field reconstruction benchmarking for PINN and data-driven DNN baselines against the FDE solver. Comparison of predicted and FDE-calculated normalized electric field magnitudes ($\left|E\right|$To verify that the neural surrogate (Eigenmode DeepONet) correctly learns the physical mode shapes rather than merely interpolating effective indices, we compare its predicted transverse field profiles against the FDE ground truth. As shown in Fig. 4, the predicted and FDE-calculated normalized electric field magnitudes ($\left|E\right|$) at the optimal design point ($w^{\mathrm{*}}=283.13\hspace{0.17em}\mathrm{n}\mathrm{m},h^{\mathrm{*}}=500.00\hspace{0.17em}\mathrm{n}\mathrm{m}$) are in excellent agreement. The peak local reconstruction error remains below 0.05 across the coordinate grid, and the modal field overlap reaches 0.928 for the pump mode (${\mathrm{T}\mathrm{E}}_{10}$) and 0.981 for the fundamental signal mode (${\mathrm{T}\mathrm{E}}_{00}$). This confirms that the trunk network’s basis functions and the branch network’s geometric projection are physically regularized, capturing sharp field discontinuities without Gibbs oscillations.

However, the data-driven DNN ablation model suffers from severe accuracy and physical regularization failures. Due to the lack of Helmholtz regularization, the DNN’s index prediction MAE degrades to $2.30\times {10}^{-2}$ (nearly an order of magnitude worse than the PINN’s MAE of $3.97\times {10}^{-3}$). More critically, the DNN’s reconstructed transverse field profiles exhibit unphysical spatial oscillations and field boundary discontinuities at the dielectric interface. This is reflected in the modal field overlap at the optimal design point ($w^{\mathrm{*}},h^{\mathrm{*}}$), where the DNN’s overlap drops to 0.444 (pump) and 0.636 (signal), compared to the PINN’s high overlaps of 0.916 and 0.969.

Table 2. Ablation study comparing solver performance. The PINN model is compared against a data-driven DNN baseline (λ_Helm = 0) and the FDE reference solver. Evaluation times are reported per single waveguide geometry.

Table 2. Ablation study comparing solver performance. The PINN model is compared against a data-driven DNN baseline (λ_Helm = 0) and the FDE reference solver. Evaluation times are reported per single waveguide geometry.

Method	Train Time (minutes)	Eval Time (ms)	Pump Ovlp	Sign Ovlp	Speed up (×)	Index MAE
FDE (Ref)	N/A	15,000	1.0	1.0	1	0.0
DNN (This work)	8.0	0.71	0.444	0.636	21,100	2.30×10-2
PINN (This work)	16.2	0.71	0.916	0.969	21,100	3.97×10-3

At the sampled phase-matched geometry, the lossless JSA purity is $P=0.122$ (Schmidt number $K=8.20$) for a $5.0\hspace{0.17em}\mathrm{m}\mathrm{m}$ waveguide under a pump bandwidth of ${\sigma }_p=0.5\hspace{0.17em}\mathrm{T}\mathrm{H}\mathrm{z}$ (matching the optimization baseline). This value is comparable to the degenerate-grid baseline, but far below the $P>0.8$-0.95 targets commonly achieved in other integrated platforms (see Table 1 for comparative context). This is supported by a systematic analysis of heralded states across the geometry grid and collection losses as a function of propagation length. Under a representative $5\hspace{0.17em}\mathrm{d}\mathrm{B}/\mathrm{c}\mathrm{m}$ propagation loss, the idler survival probability is 0.562 for a $5\hspace{0.17em}\mathrm{m}\mathrm{m}$ device. These metrics indicate that while the modal phase-matching condition is satisfied, high-purity heralded photon-pair generation will require group-velocity engineering or asymmetric wavelength configurations. The Schmidt number K=8.20 indicates roughly 8 independent spectro-spatial modes contribute to the biphoton state, directly causing the low heralded purity. This contrasts sharply with periodically-poled lithium niobate (PPLN) quasi-phase-matched (QPM) sources, where the poling period independently controls phase matching without fixing the group-velocity relationship, routinely achieving P>0.95 [26]. In the NbOCl₂ uniform waveguide, the symmetry-enforced group-velocity mismatch (GVM=+3.533 ps/mm) creates 17.67 ps of temporal walk-off over 5 mm, coupling frequency correlations that prevent factorable state generation. QPM geometries are not subject to this constraint because the poling period provides an additional degree of freedom to decouple phase matching from GVM.

Figure 5. Uniform waveguide SPDC performance metrics at the degenerate design geometry. The waveguide parameters are $w^{\mathrm{*}}=283.13\hspace{0.17em}\mathrm{n}\mathrm{m}$ and $h^{\mathrm{*}}=500.00\hspace{0.17em}\mathrm{n}\mathrm{m}$. (a) Classical coupled-wave propagation showing signal ($|A_{\mathrm{s}}{|}^2$), idler ($|A_{\mathrm{i}}{|}^2$), and pump ($|A_{\mathrm{p}}{|}^2$) intensity envelopes (Model vs. RK4 reference). (b) Quantum coupled-wave expectation values showing signal and idler photon numbers ($⟨{\widehat{n}}_{\mathrm{s}}⟩,⟨{\widehat{n}}_{\mathrm{i}}⟩$) and two-photon correlation ($\left|M\right|$) along the propagation distance $z$ (Model vs. RK4 reference). (c) Pair generation rate (PGR) and coincidence-to-accidental ratio (CAR) as a function of input pump power. (d) Heralded second-order correlation function $g_{\mathrm{H}}^{\left(2\right)}\left(\tau \right)$ vs. time delay $\tau$, with the inset displaying $g_{\mathrm{H}}^{\left(2\right)}\left(0\right)$ as a function of pump power. (e) 2D contour plot of CAR vs. waveguide width $w$ and height $h$, overlaying the phase-matching zero curve where $\mathsf{\Delta }\beta =0$. (f) 2D contour plot of heralding efficiency ${\eta }_{\mathrm{H}}$ vs. waveguide width $w$ and height $h$, overlaying the phase-matching zero curve.

Figure 5. Uniform waveguide SPDC performance metrics at the degenerate design geometry. The waveguide parameters are $w^{\mathrm{*}}=283.13\hspace{0.17em}\mathrm{n}\mathrm{m}$ and $h^{\mathrm{*}}=500.00\hspace{0.17em}\mathrm{n}\mathrm{m}$. (a) Classical coupled-wave propagation showing signal ($|A_{\mathrm{s}}{|}^2$), idler ($|A_{\mathrm{i}}{|}^2$), and pump ($|A_{\mathrm{p}}{|}^2$) intensity envelopes (Model vs. RK4 reference). (b) Quantum coupled-wave expectation values showing signal and idler photon numbers ($⟨{\widehat{n}}_{\mathrm{s}}⟩,⟨{\widehat{n}}_{\mathrm{i}}⟩$) and two-photon correlation ($\left|M\right|$) along the propagation distance $z$ (Model vs. RK4 reference). (c) Pair generation rate (PGR) and coincidence-to-accidental ratio (CAR) as a function of input pump power. (d) Heralded second-order correlation function $g_{\mathrm{H}}^{\left(2\right)}\left(\tau \right)$ vs. time delay $\tau$, with the inset displaying $g_{\mathrm{H}}^{\left(2\right)}\left(0\right)$ as a function of pump power. (e) 2D contour plot of CAR vs. waveguide width $w$ and height $h$, overlaying the phase-matching zero curve where $\mathsf{\Delta }\beta =0$. (f) 2D contour plot of heralding efficiency ${\eta }_{\mathrm{H}}$ vs. waveguide width $w$ and height $h$, overlaying the phase-matching zero curve.

We note a physical distinction between the classical and quantum longitudinal propagation regimes plotted in Fig. 5(a) and (b). Under classical CWE propagation at low pump powers (e.g., $1\hspace{0.17em}\mathrm{m}\mathrm{W}$), the parametric gain is far smaller than the waveguide propagation loss of $2\hspace{0.17em}\mathrm{d}\mathrm{B}/\mathrm{c}\mathrm{m}$, causing the seeded classical signal and idler fields to decay over distance rather than exhibit net amplification. In contrast, the quantum qCWE propagator starts from a pure vacuum state (0 photons), where spontaneous parametric generation (driven by the vacuum noise term in the propagator) causes the photon number to rise quadratically as $z^2$ initially. On the semi-log scale, this rapid quadratic rise from zero appears as a sharp vertical ascent immediately after the input facet (at $z=0\hspace{0.17em}\mathrm{m}\mathrm{m}$), representing the physical threshold where down-converted photons begin to populate the modes.

Figure 6. Joint spectral amplitude$\left|\mathsf{\Phi }\right({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}}\left)\right|$and classical second-harmonic generation (SHG) conversion efficiencies across optimised SPDC polarization families. Row 1: Second-harmonic generation (SHG) output power as a function of pump power for (a) Type-0, (b) Type-I, and (c) Type-II optimized configurations, comparing the Combined-PINN surrogate predictions (blue square markers) and exact Runge-Kutta numerical trajectories (red solid curves) for the phase-matched design against an unmatched normal waveguide (black dashed curves) of length $L=0.5\hspace{0.17em}\mathrm{m}\mathrm{m}$. Insets display the normalized SHG yield as a function of pump wavelength ${\lambda }_{\mathrm{p}}$ centered at $775\hspace{0.17em}\mathrm{n}\mathrm{m}$. Row 2: Joint spectral amplitude (JSA) profiles in wavelength space for (d) Type-0, (e) Type-I, and (f) Type-II optimized designs under a pump bandwidth of ${\sigma }_{\mathrm{p}}=0.5\hspace{0.17em}\mathrm{T}\mathrm{H}\mathrm{z}$ and device length $L=0.5\hspace{0.17em}\mathrm{m}\mathrm{m}$, yielding spectral purities of $P=0.196$, $P=0.079$, and $P=0.350$, respectively.

Figure 6. Joint spectral amplitude$\left|\mathsf{\Phi }\right({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}}\left)\right|$and classical second-harmonic generation (SHG) conversion efficiencies across optimised SPDC polarization families. Row 1: Second-harmonic generation (SHG) output power as a function of pump power for (a) Type-0, (b) Type-I, and (c) Type-II optimized configurations, comparing the Combined-PINN surrogate predictions (blue square markers) and exact Runge-Kutta numerical trajectories (red solid curves) for the phase-matched design against an unmatched normal waveguide (black dashed curves) of length $L=0.5\hspace{0.17em}\mathrm{m}\mathrm{m}$. Insets display the normalized SHG yield as a function of pump wavelength ${\lambda }_{\mathrm{p}}$ centered at $775\hspace{0.17em}\mathrm{n}\mathrm{m}$. Row 2: Joint spectral amplitude (JSA) profiles in wavelength space for (d) Type-0, (e) Type-I, and (f) Type-II optimized designs under a pump bandwidth of ${\sigma }_{\mathrm{p}}=0.5\hspace{0.17em}\mathrm{T}\mathrm{H}\mathrm{z}$ and device length $L=0.5\hspace{0.17em}\mathrm{m}\mathrm{m}$, yielding spectral purities of $P=0.196$, $P=0.079$, and $P=0.350$, respectively.

Using this framework, we evaluated Type-0 (co-polarized TM/TM/TM or TE/TE/TE), Type-I (orthogonal signal/idler modes), and Type-II SPDC configuration channels across the waveguide parameters (summarized in Table 4 and Fig. 6). The degenerate uniform waveguide yields a low joint spectral purity ($P=0.122$, $K=8.20$) due to symmetry-enforced group-velocity mismatch ($\mathrm{G}\mathrm{V}\mathrm{M}=+3.533\hspace{0.17em}\mathrm{p}\mathrm{s}/\mathrm{m}\mathrm{m}$), which creates $17.67\hspace{0.17em}\mathrm{p}\mathrm{s}$ of walk-off over $5\hspace{0.17em}\mathrm{m}\mathrm{m}$ (pump bandwidth $1.0\hspace{0.17em}\mathrm{T}\mathrm{H}\mathrm{z}$). Artificially setting GVM to zero only improves $P$ to 0.223, confirming that degenerate mode families lack the group-velocity asymmetry required for factorable states. However, our multi-mode optimization bypasses these limitations: a Type-II design ($m_p=4,m_s=0,m_i=1$) at $w=592.0\hspace{0.17em}\mathrm{n}\mathrm{m}$ and $h=270.0\hspace{0.17em}\mathrm{n}\mathrm{m}$ yields a joint spectral purity of $P\approx 0.350$ and a spatial overlap proxy $\mathsf{\Gamma }=0.139$ at $L=0.5\hspace{0.17em}\mathrm{m}\mathrm{m}$.

Table 3. Multi-mode genetic algorithm optimization results. Mode configurations (m_p, m_s, m_i) and waveguide width w and height h (nm) are reported. Γ is spatial overlap proxy, P is purity, and fitness is GA cost. Index error represents resampled surrogate uncertainty.

Table 3. Multi-mode genetic algorithm optimization results. Mode configurations (m_p, m_s, m_i) and waveguide width w and height h (nm) are reported. Γ is spatial overlap proxy, P is purity, and fitness is GA cost. Index error represents resampled surrogate uncertainty.

Type	Modes	$w/h$(nm)	$\mathsf{\Gamma }$	Purity$P$	Fitness	Index Error
Type-0	$(5,0,0)$​	215/890	0.126	0.196	0.257	$1.62\pm 0.60\times {10}^{-3}$​
Type-0	$(4,0,0)$​	280/490	0.150	0.131	0.205	$1.62\pm 0.60\times {10}^{-3}$​
Type-I	$(3,0,0)$​	170/690	0.165	0.079	0.160	$1.62\pm 0.60\times {10}^{-3}$​
Type-II	$(4,0,1)$​	592/270	0.139	0.350	0.339	$1.62\pm 0.60\times {10}^{-3}$​

Conclusion

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