Dual Quasi-Phase Matching (D-QPM)

• Dual QPM is a domain-engineering approach that enables simultaneous phase matching of two nonlinear processes via tailored periodic poling or geometric patterning.
• It employs dual-period or dual-order architectures that use independent grating vectors or Fourier components to meet multiple phase mismatch conditions.
• This technique supports advanced applications such as quantum photonics and frequency conversion, offering high efficiency and precise entanglement control.

Dual quasi-phase matching (D-QPM) is a domain-engineering approach in nonlinear optics that enables the simultaneous phase-matching of two distinct parametric processes—such as spontaneous parametric down-conversion (SPDC) or second-harmonic generation (SHG)—within a single nonlinear medium. This is achieved either by multiplexing separate quasi-phase-matching (QPM) grating periods, harnessing multiple Fourier components of a single periodic poling, or by geometric patterning. D-QPM architectures facilitate the generation of dual-wavelength, dual-channel, or dual-type photon sources with coherent amplitude control and entanglement, supporting advanced quantum photonic and frequency conversion functionalities (Warke et al., 2021, Kaneda et al., 2018, Meetei et al., 2020, Zhou et al., 22 Jul 2024, Hendra et al., 20 May 2025, Phillips et al., 2015, Brambilla et al., 2019, Liu et al., 2022).

1. Fundamental Principles of Quasi-Phase Matching and Dual QPM

Quasi-phase matching in a nonlinear crystal compensates the inherent phase mismatch, $\Delta k_0 = k_p - k_s - k_i$, between pump, signal, and idler fields by introducing periodic sign reversal of the nonlinear coefficient with a poling period $\Lambda$ (Warke et al., 2021). The resulting grating wavevector $K = 2\pi/\Lambda$ supplies the necessary momentum so that $\Delta k_0 + K \approx 0$, enabling efficient energy transfer through the three-wave mixing process.

D-QPM extends this principle by engineering either:

• Two spatially multiplexed poling periods ($\Lambda_1$, $\Lambda_2$), each providing independent $K_j = 2\pi/\Lambda_j$,
• Multiple QPM orders (harmonics) within the same period ($m_1$, $m_2$ in Fourier decomposition),
• Or, more generally, by geometric, temporal, or spatial patterns that compensate multiple phase-mismatch conditions simultaneously (Zhou et al., 22 Jul 2024, Phillips et al., 2015, Liu et al., 2022).

This results in simultaneous phase matching of two parametric processes with different wavelengths, polarization states, or nonlinear tensor elements, allowing coherent superposition and entanglement of multiple photon pairs.

2. Mathematical Formulation and Design Criteria

The phase-matching condition for each process $j=1,2$ is governed by:

$\Delta k_j = k_p - k_{s,j} - k_{i,j}$

and is satisfied by selecting appropriate grating vectors:

$$\Delta k_j + K_j = 0,\quad K_j = 2\pi m_j/\Lambda_j,\quad m_j\in\mathbb{Z}^+$$

For dual-period poling ($\Lambda_1$, $\Lambda_2$), design formulas under first-order QPM ($m_1=m_2=1$) yield:

$$\Lambda_j = \frac{1}{n_p/\lambda_p - n_{s,j}/\lambda_{s,j} - n_{i,j}/\lambda_{i,j}}$$

where $n_m$ is the refractive index and $\lambda_m$ the wavelength of each field (Warke et al., 2021).

Alternatively, when relying on higher QPM orders within a single period $\Lambda$, both phase-matching conditions are simultaneously satisfied if:

$$\frac{\Delta k^{(1)}}{m_1} = \frac{\Delta k^{(2)}}{m_2}$$

The effective nonlinearities $d_{\text{eff}}^{(j)}$ for each process are scaled by Fourier coefficients $G_{m}$, with $d_{\text{eff}}^{(m)} = d_j G_m = d_j (2/(m\pi))$ for $m$ odd and $50:50$ duty cycle (Liu et al., 2022, Hendra et al., 20 May 2025).

In 2D-patterned QPM, the phase-matching vector is generalized to $\mathbf{K}_g = K_x \hat{x} + K_z \hat{z}$, enabling control over both longitudinal and transverse mismatch components (Phillips et al., 2015, Brambilla et al., 2019).

3. D-QPM Architectures and Implementation

Dual-Period and Sequential Poling

Physical realization of D-QPM can be achieved by inscribing two poling periods ($\Lambda_1$, $\Lambda_2$) either interleaved or sequentially along the propagation direction of the crystal (Warke et al., 2021, Kaneda et al., 2018). For example, in LiNbO$_3$ waveguides, two periods ($6.797\,\mu$m and $6.832\,\mu$m) simultaneously yield two distinct signal-idler pairs ($780$/$1551$ nm and $775$/$1571$ nm), with near-maximal entanglement $\gamma\approx0.98$ (Warke et al., 2021).

In frequency-bin entanglement schemes, sequential poling generates a coherent superposition state in two discrete frequency bins:

$$|\psi_f\rangle = \frac{1}{\sqrt{2}} \left( |\omega_1\rangle_A |\omega_2\rangle_B + e^{i\phi} |\omega_2\rangle_A |\omega_1\rangle_B \right)$$

where $\phi$ is a controllable relative phase (Kaneda et al., 2018).

Dual-Order Periodic QPM

A single period $\Lambda$ can support multiple QPM orders via its Fourier spectrum. For example, in Rb-doped KTP waveguides, third-order QPM ($m=3$) with $d_{33}$ supports type-0 SPDC, while first-order QPM ($m=1$) with $d_{24}$ supports type-II SPDC, both with comparable effective nonlinearities and conversion efficiencies (Hendra et al., 20 May 2025).

Phase-Reversal Optical Superlattice (PROS)

PROS structures insert $\pi$ phase-reversal domains at specified positions along a crystal of length $L$. For equal-interval PR placements ($z_m = mL/(N+1)$), exactly two equal-intensity SHG peaks are obtained. For unequal intervals, multi-peak SHG spectra with tailored intensity distributions are achievable (Meetei et al., 2020).

Table: Dual-QPM Grating Architectures

Scheme	Grating Physical Realization	Dual Processes Phase-Matched
Dual-period poling	Two spatially sequenced/interleaved periods	$$(\omega_{s_1}, \omega_{i_1}); (\omega_{s_2}, \omega_{i_2})$$
Dual-order QPM	Single period, multiple QPM orders $m$	Type-0 & type-II SPDC, type-I & type-II SPDC, ...
PROS	Regular $\Lambda$ with phase-reversal domains	Dual or multi-peak SHG

4. Interaction Hamiltonian and Quantum Output States

The interaction Hamiltonian in a D-QPM waveguide is expressed as:

$$\hat{H}_{\text{int}} = -\varepsilon_0 \int d^3r\,d(x)\,E_p(x,t)\,\hat{E}_s(x,t)\,\hat{E}_i(x,t) + \text{h.c.}$$

where $d(x)$ accommodates all grating components. Substitution yields output states of the form:

$$|\Psi\rangle \propto \int d\omega_{s,1} C_1|\omega_{s,1},\omega_{i,1}\rangle + \int d\omega_{s,2} C_2|\omega_{s,2},\omega_{i,2}\rangle$$

with $C_j$ amplitudes determined by the grating coefficients, overlap integrals, and phase-matching (Warke et al., 2021). Quantum interference between the two amplitudes produces maximally entangled frequency-bin states. In time-resolved HOM interference, the beat period $T_{\text{beating}}$ is set by the frequency detuning, with observed visibilities up to $93\%$ and deterministic conversion between frequency-bin and polarization entanglement (Kaneda et al., 2018).

In 2D QPM crystals pumped by dual beams, coherent amplitudes sum, leading to parametric gain enhancement by $\sqrt{2}$ for the central phase-matched mode, and tunable multi-mode coupling via pump amplitude and phase control (Brambilla et al., 2019).

5. Spatiotemporal Dual QPM and Non-Conventional Regimes

Recent work generalizes D-QPM to spatiotemporal domains. In Si$_3$N$_4$ microresonators, a self-organized traveling $\chi^{(2)}$ grating arises via all-optical poling and the photogalvanic effect, forming a concurrent spatial and temporal modulation. This enables quasi-phase matching where both the momentum ($\Delta k + K = 0$) and energy ($\Delta\omega + \Omega = 0$) mismatches are compensated, implementing a Doppler-shifted second harmonic and reconfigurable, broadband SHG (Zhou et al., 22 Jul 2024). This mechanism requires no lithographic poling and extends D-QPM to new physical platforms and parametric processes.

6. Applications, Performance, and Advantages

D-QPM supports a range of functionalities:

• On-chip simultaneous generation of two frequency-entangled photon pair states for quantum information channels at different wavelengths (e.g., 780 nm for atomic interfaces, 1550 nm for telecom) (Warke et al., 2021, Liu et al., 2022).
• Frequency-bin entanglement sources, enabling high-dimensional quantum key distribution and frequency-multiplexed linear optics quantum computing (Kaneda et al., 2018).
• Dual-order QPM fabrication for dual-type SPDC in KTP waveguides, with brightness figures $>250$ MHz/mW (type-0) and $56$ MHz/mW (type-II) (Hendra et al., 20 May 2025).
• Multi-peak SHG converters based on phase-reversal domain engineering (Meetei et al., 2020).
• Integrated nonlinear photonics: broadband, agile SHG/SFG parametric sources in microresonators without lithographic poling (Zhou et al., 22 Jul 2024).
• Systems with common-mode thermal and mechanical stability, reduced footprint, and cost via single-chip multi-photon sources and Sagnac interferometers (Liu et al., 2022).

The technique enables tunable amplitude control through domain-engineering (duty cycle, length), scaling to multi-grating/multi-order QPM for higher-dimensional entanglement, and synchronous multi-type photon-pair generation with minimized crosstalk and maximized source stability.

7. Notable Research Directions and Future Implications

Recent trends focus on:

• Scaling the D-QPM principle to higher dimensions by n-periodic or n-order QPM, supporting arbitrary spectral and polarization entanglement (Kaneda et al., 2018).
• Generalization to 2D patterned and spatiotemporal QPM for simultaneous longitudinal and transverse phase-matching, geometric pulse shaping, and frequency conversion (Phillips et al., 2015).
• Applications in energy-time entanglement quantum metrology, ultrafast mid-IR pulse engineering, and quantum communications (Warke et al., 2021, Phillips et al., 2015).
• Systematic analysis of dual QPM design constraints—temperature and period precision, waveguide geometry, and mode overlap—for scalable, stable, and brightness-optimized sources (Liu et al., 2022).
• Exploration of new photonic materials (e.g., Rb-doped KTP, Si$_3$N$_4$) and nonlinear architectures (self-organized QPM) for silicon photonics and CMOS compatibility (Zhou et al., 22 Jul 2024, Hendra et al., 20 May 2025).

Dual quasi-phase matching thus constitutes a unified and adaptable framework for multi-channel quantum states, multidimensional parametric conversion, and integrated nonlinear photonics across a broad application spectrum.