Non-Phase-Matched SPDC in Thin Nonlinear Media

• Non-phase-matched SPDC is a nonlinear optical process in ultrathin media where relaxed longitudinal phase matching enables a broad acceptance of signal–idler frequencies and angles.
• Experiments with micrometer-scale and subwavelength LN and GaP films demonstrate ultra-broad spectral spans and tight spatial-angular correlations, offering rich high-dimensional entanglement.
• The trade-off in non-phase-matched SPDC is reduced per-mode efficiency for enhanced integration and flexibility, opening new avenues in quantum spectroscopy, imaging, and metrology.

Searching arXiv for the specified papers to ground the article in current records. Non-phase-matched spontaneous parametric down-conversion (SPDC) denotes SPDC in a nonlinear medium whose longitudinal extent is sufficiently small that strict longitudinal momentum conservation is relaxed, while energy conservation remains enforced as $\omega_p=\omega_s+\omega_i$. In this regime, the usual longitudinal phase-matching condition $\Delta k = k_p-k_s-k_i=0$ no longer sharply selects a narrow set of signal–idler frequencies and angles; instead, the finite interaction length produces a broad acceptance function $L\,\mathrm{sinc}(\Delta k L/2)$ that tolerates large phase mismatch. Experiments in micrometer-scale and subwavelength lithium niobate and gallium phosphide films have shown that such sources can generate entangled photon pairs with ultra-broad spectral and angular distributions, tight spatio-temporal correlations, and large frequency or transverse-wavevector entanglement, while retaining the efficiency penalty associated with amplitudes scaling as $L$ and pair probabilities as $L^2$ (Okoth et al., 2019, Santiago-Cruz et al., 2020, Okoth et al., 2019).

1. Definition and thin-medium limit

In standard SPDC, a pump photon spontaneously decays into lower-frequency signal and idler photons in a $\chi^{(2)}$ medium. The joint two-photon state may be written as

$$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$

with joint amplitude $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$, where $\alpha$ encodes the pump envelope and

$$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$

For a uniform $\Delta k = k_p-k_s-k_i=0$0 layer of thickness $\Delta k = k_p-k_s-k_i=0$1, the longitudinal probability envelope is

$\Delta k = k_p-k_s-k_i=0$2

and for a Gaussian pump of waist $\Delta k = k_p-k_s-k_i=0$3 the transverse acceptance is

$\Delta k = k_p-k_s-k_i=0$4

These dependencies make explicit that the pair probability scales as $\Delta k = k_p-k_s-k_i=0$5, where $\Delta k = k_p-k_s-k_i=0$6 is the relevant nonlinear tensor element (Okoth et al., 2019).

The defining limit is the ultrathin regime $\Delta k = k_p-k_s-k_i=0$7, for which $\Delta k = k_p-k_s-k_i=0$8 over a very broad range of $\Delta k = k_p-k_s-k_i=0$9. In subwavelength films this can be expressed through the coherence length $L\,\mathrm{sinc}(\Delta k L/2)$0: when $L\,\mathrm{sinc}(\Delta k L/2)$1, the phase-mismatch penalty is negligible and the phase-matching function becomes effectively flat. For degenerate SPDC at $L\,\mathrm{sinc}(\Delta k L/2)$2, the reported estimates are $L\,\mathrm{sinc}(\Delta k L/2)$3 in LN and $L\,\mathrm{sinc}(\Delta k L/2)$4 in GaP, while the experimental film thicknesses were $L\,\mathrm{sinc}(\Delta k L/2)$5 and $L\,\mathrm{sinc}(\Delta k L/2)$6, respectively, giving $L\,\mathrm{sinc}(\Delta k L/2)$7 in both cases (Santiago-Cruz et al., 2020).

This regime differs from both birefringent phase matching and quasi-phase-matching. In QPM one writes $L\,\mathrm{sinc}(\Delta k L/2)$8 or, more generally, $L\,\mathrm{sinc}(\Delta k L/2)$9, but the micrometer-scale LN experiments discussed here used unpoled material, so the thickness, rather than periodic poling, dominated the phase-matching behavior (Okoth et al., 2019, Okoth et al., 2019).

2. Relaxed longitudinal phase matching and the meaning of “without momentum conservation”

The phrase “without momentum conservation” refers to the loss of strict longitudinal translational invariance in a finite nonlinear layer, not to a violation of fundamental conservation laws. In a medium of thickness $L$0, translational symmetry along the pump axis is broken, so strict longitudinal momentum conservation $L$1 is not enforced. The allowed longitudinal mismatch is instead determined by the Fourier transform of the interaction profile, yielding the familiar $L$2 dependence. Large $L$3 values are therefore admitted when $L$4 is small, and many combinations of $L$5 remain compatible with energy conservation (Okoth et al., 2019).

In the thin-crystal formulation used for transverse-wavevector entanglement, the biphoton amplitude is $L$6, and in the limit $L$7 the state is governed predominantly by pump energy conservation and transverse momentum conservation. For a broad pump waist, the pump factor is sharply peaked at $L$8, producing near-ideal transverse momentum anti-correlations over a very large angular bandwidth. In the angular representation employed for the LN measurements, the state is written as

$L$9

with $L^2$0. In the idealized limit this approaches

$L^2$1

which the authors describe as an almost ideal EPR-like state (Okoth et al., 2019).

A related misconception is that relaxed longitudinal phase matching implies unconstrained transverse emission. The reported theory and experiments instead separate the roles of the two acceptance factors: $L^2$2 determines the overall spectral and angular support and broadens as $L^2$3, whereas $L^2$4 fixes the transverse correlations with width of order $L^2$5. Consequently, the source can simultaneously exhibit a broad far-field angular spread and tight pairwise spatial correlations set by the pump mode (Okoth et al., 2019).

3. Experimental platforms and material choices

The first reported entangled-photon generation in this regime used an ultra-thin magnesium-oxide-doped lithium niobate wafer on a $L^2$6 fused silica substrate, with an LN layer thickness varying from $L^2$7 to $L^2$8 across the wafer. The interaction was type-0, with pump, signal, and idler all polarized along $L^2$9, exploiting the largest LN tensor element $\chi^{(2)}$0, described as about $\chi^{(2)}$1 larger than the effective $\chi^{(2)}$2 in typical BBO phase-matched experiments. For $\chi^{(2)}$3 pumping with degenerate SPDC at $\chi^{(2)}$4, the coherence length was reported as $\chi^{(2)}$5, so scanning the beam across the sample accessed effective thicknesses of approximately $\chi^{(2)}$6 to $\chi^{(2)}$7; emission maxima occurred at odd multiples of $\chi^{(2)}$8 and vanished at even multiples (Okoth et al., 2019).

The pump configurations included a CW $\chi^{(2)}$9 laser up to $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$0, a CW $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$1 pump focused to a $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$2 spot for optimized coupling to superconducting nanowire detectors, and a stimulated-emission-tomography configuration based on a frequency-doubled Nd:YAG at $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$3 providing $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$4 pulses at $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$5 with an OPA seed tunable from $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$6 to $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$7 (Okoth et al., 2019).

Subwavelength-film experiments extended the regime to flat nonlinear films. The reported devices were a $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$8 thick X-cut LN film on fused silica and a $$|\Psi\rangle = \int d\omega_s\, d\omega_i\, d^2q_s\, d^2q_i\; \Phi(\omega_s,\omega_i,q_s,q_i)\,\hat a_s^\dagger(\omega_s,q_s)\hat a_i^\dagger(\omega_i,q_i)|0\rangle,$$9 GaP film on fused silica on sapphire. In both cases the optimal observed SPDC was type-0 and vertically polarized. Correlation measurements used a CW diode at $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$0, up to approximately $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$1, with degenerate detection at $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$2 through a $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$3 FWHM bandpass; fibre spectroscopy used a CW $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$4 pump at $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$5 for LN and $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$6 for GaP. Collection employed an aspheric lens and a $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$7 off-axis gold parabolic mirror, with longpass filters at $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$8 and $$\Phi \propto \alpha(\omega_s+\omega_i,q_s+q_i)\,\phi(\Delta k)$$9, a broadband polarizer, a $\alpha$0 broadband single-mode fibre splitter, a $\alpha$1 DCF4 fibre arm, and one visible plus one infrared SNSPD (Santiago-Cruz et al., 2020).

The angular-entanglement experiments in micrometer-scale LN used a $\alpha$2 CW pump, spatially mode-cleaned through a single-mode fibre and focused with an $\alpha$3 lens; a high-NA aspheric lens with $\alpha$4 and $\alpha$5 collected the broad emission. Stimulated emission tomography used a $\alpha$6 pump with $\alpha$7 pulses at $\alpha$8 and a seed centered at $\alpha$9 with $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$0 pulse energy, while the angularly resolved signal near $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$1 was recorded in the Fourier plane with a SPIRICON camera (Okoth et al., 2019).

4. Spectral breadth, temporal correlations, and cavity effects

A primary characteristic of non-phase-matched SPDC is ultra-broad spectral acceptance. For the ultra-thin LN source pumped at $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$2, the calculated non-phase-matched frequency–angle spectrum at degeneracy near $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$3 had an expected full spectral span of approximately $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$4. Single-photon spectroscopy based on $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$5 of SMF-28 fibre produced a normalized correlation peak corresponding to an observed bandwidth of approximately $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$6; the reported interpretation was that this measured width was limited by detector sensitivity and frequency-dependent coupling, whereas the predicted width was approximately $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$7. Using the inverse-bandwidth estimate $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$8, the authors inferred a biphoton correlation time of approximately $$\phi(\Delta k)=e^{i\Delta k L/2}L\,\mathrm{sinc}(\Delta k L/2).$$9 for the measured $\Delta k = k_p-k_s-k_i=0$00 bandwidth and approximately $\Delta k = k_p-k_s-k_i=0$01 for the full $\Delta k = k_p-k_s-k_i=0$02 span (Okoth et al., 2019).

In the subwavelength-film work, fibre spectroscopy with a $\Delta k = k_p-k_s-k_i=0$03 DCF4 fibre yielded SPDC spectra with approximately $\Delta k = k_p-k_s-k_i=0$04 bandwidth in both LN and GaP when pumped at $\Delta k = k_p-k_s-k_i=0$05. The reported limitations were instrumental: on the red side, low group-velocity dispersion near the fibre zero-dispersion wavelength at approximately $\Delta k = k_p-k_s-k_i=0$06; on the blue side, fibre attenuation and the fibre-splitter ratio. The intrinsic bandwidth of subwavelength-film SPDC was stated to be capable of spanning more than an octave (Santiago-Cruz et al., 2020).

Subwavelength films also introduce interface and cavity effects absent from the simplest uniform-layer model. A high-index film between lower-index media forms a Fabry–Perot etalon, and the signal and idler vacuum modes can experience resonant field buildup inside the film. The reported intracavity enhancement factor was written as

$\Delta k = k_p-k_s-k_i=0$07

or equivalently in Airy-like form, and the SPDC spectral density acquires a multiplicative factor $\Delta k = k_p-k_s-k_i=0$08. In the GaP film the measured SPDC spectrum exhibited clear Fabry–Perot fringes and a resonant enhancement of up to approximately $\Delta k = k_p-k_s-k_i=0$09 at peaks, whereas the LN film showed no pronounced FP modulation and its measured spectral shape was mainly set by detector efficiency (Santiago-Cruz et al., 2020).

These observations establish that non-phase-matched SPDC is not a single spectral regime but a family of regimes in which thickness-induced acceptance broadening can coexist with interface filtering, vacuum-field enhancement, and detection-window truncation. This suggests that in integrated implementations the intrinsic flat phase-matching response and the extrinsic cavity response become comparable design variables.

5. Spatial structure, EPR-like correlations, and entanglement measures

The thin-crystal regime supports unusually strong transverse-wavevector entanglement because the longitudinal filter becomes broad while the pump still imposes a narrow constraint on the sum transverse momentum. The LN analysis emphasizes that the source exhibits tight angular correlations governed by $\Delta k = k_p-k_s-k_i=0$10 together with an ultra-broad far-field angular spread permitted by $\Delta k = k_p-k_s-k_i=0$11. With a pump spot size of approximately $\Delta k = k_p-k_s-k_i=0$12, the transverse acceptance in $\Delta k = k_p-k_s-k_i=0$13-space is of order $\Delta k = k_p-k_s-k_i=0$14, yielding near-field signal–idler position correlations as tight as the pump mode, while the far-field angular distribution remains broad and is truncated mainly by total internal reflection and the collection optics (Okoth et al., 2019).

The dedicated EPR-state study quantified this structure through the Fedorov ratio $\Delta k = k_p-k_s-k_i=0$15, where $\Delta k = k_p-k_s-k_i=0$16 is the unconditional angular width and $\Delta k = k_p-k_s-k_i=0$17 the conditional coincidence width. Stimulated-emission tomography of the angular two-photon intensity reported an unconditional width $\Delta k = k_p-k_s-k_i=0$18, a conditional width $\Delta k = k_p-k_s-k_i=0$19, and therefore $\Delta k = k_p-k_s-k_i=0$20, with theory predicting $\Delta k = k_p-k_s-k_i=0$21. Accounting for the full transverse field, the estimated two-dimensional Fedorov ratio was $\Delta k = k_p-k_s-k_i=0$22, and the theoretical Schmidt number for the corresponding one-dimensional distribution was $\Delta k = k_p-k_s-k_i=0$23 (Okoth et al., 2019).

The same work formulated the near-field scaling in terms of $\Delta k = k_p-k_s-k_i=0$24 for the coincidence width and $\Delta k = k_p-k_s-k_i=0$25 for the single-photon positional width set by the pump waist. This directly links thinner nonlinear layers to tighter position correlations. The authors explicitly connect the broad angular bandwidth to increased spatial-frequency support in imaging protocols, and note that moving to still thinner platforms could drive the position-correlation scale deeply below the wavelength (Okoth et al., 2019).

Frequency entanglement was characterized independently in the ultra-thin LN wafer by stimulated-emission tomography of the joint spectral intensity (JSI). For $\Delta k = k_p-k_s-k_i=0$26 pumping and an LN thickness of $\Delta k = k_p-k_s-k_i=0$27, the reconstructed JSI fragment probed a $\Delta k = k_p-k_s-k_i=0$28 unconditional window around collinear emission, within which the conditional width was $\Delta k = k_p-k_s-k_i=0$29. Using the Fedorov ratio $\Delta k = k_p-k_s-k_i=0$30, this yielded at least $\Delta k = k_p-k_s-k_i=0$31, with the explicit caveat that the true entanglement was likely an order of magnitude larger when the full spectral width was taken into account. The reconstructed JSI matched the predicted anti-diagonal structure characteristic of energy-conserving, frequency-anticorrelated biphotons (Okoth et al., 2019).

6. Rates, backgrounds, modeling, and prospective directions

Non-phase-matched SPDC trades broad modal acceptance for reduced per-mode efficiency. In the thin-layer LN formulation, the amplitude scales as $\Delta k = k_p-k_s-k_i=0$32 and the pair probability as $\Delta k = k_p-k_s-k_i=0$33, so the short interaction length suppresses brightness even while admitting many more signal–idler mode pairs than a phase-matched crystal. The micrometer-scale LN source nevertheless produced over $\Delta k = k_p-k_s-k_i=0$34 pairs/s under moderate CW pumping, and with a $\Delta k = k_p-k_s-k_i=0$35 pump the coincidence-to-accidental ratio, defined as $\Delta k = k_p-k_s-k_i=0$36, reached $\Delta k = k_p-k_s-k_i=0$37 at approximately $\Delta k = k_p-k_s-k_i=0$38. The measured $\Delta k = k_p-k_s-k_i=0$39 showed a pronounced peak at $\Delta k = k_p-k_s-k_i=0$40, and the dependence $\Delta k = k_p-k_s-k_i=0$41 was verified, consistent with $\Delta k = k_p-k_s-k_i=0$42 and accidentals proportional to $\Delta k = k_p-k_s-k_i=0$43 (Okoth et al., 2019).

The subwavelength-film experiments reported much lower bare-film rates in narrowband coincidence measurements: $\Delta k = k_p-k_s-k_i=0$44 for LN with a $\Delta k = k_p-k_s-k_i=0$45 pump at $\Delta k = k_p-k_s-k_i=0$46 and a $\Delta k = k_p-k_s-k_i=0$47 minute acquisition, and $\Delta k = k_p-k_s-k_i=0$48 for GaP over $\Delta k = k_p-k_s-k_i=0$49 minutes under the same pump. The reduced GaP rate was attributed to non-optimal orientation and strong fluorescence near the absorption edge, which elevated accidental coincidences. The same study notes that incoherent fluorescence scales as $\Delta k = k_p-k_s-k_i=0$50 and can dominate accidental backgrounds when $\Delta k = k_p-k_s-k_i=0$51 is very small, whereas the pair-generation rate in a narrow spectral band scales as $\Delta k = k_p-k_s-k_i=0$52 (Santiago-Cruz et al., 2020).

Background discrimination is therefore a central experimental issue. In the LN wafer pumped at $\Delta k = k_p-k_s-k_i=0$53, fluorescence in the visible exceeded the SPDC singles by more than an order of magnitude, so identification of the source relied on correlation-based methods, including HBT measurements and single-photon spectroscopy. Polarization analysis further showed that real coincidences appeared only for $\Delta k = k_p-k_s-k_i=0$54-polarized emission, with count rate proportional to $\Delta k = k_p-k_s-k_i=0$55 as a function of pump polarization angle, while no coincidences were detected for $\Delta k = k_p-k_s-k_i=0$56-polarized emission; this confirmed mediation by $\Delta k = k_p-k_s-k_i=0$57 and the type-0 process (Okoth et al., 2019).

A general first-principles simulation framework has also been formulated for SPDC and parametric up-conversion under nonzero vector phase mismatch. In that treatment, the signal-mode amplitude for a Gaussian pump and a crystal of length $\Delta k = k_p-k_s-k_i=0$58 contains the same structural factors that recur across the thin-film experiments,

$\Delta k = k_p-k_s-k_i=0$59

and absolute counts are obtained by integrating $\Delta k = k_p-k_s-k_i=0$60 over idler modes, propagating the resulting rays through the optical system, and applying wavelength- and angle-dependent efficiency factors. The reported implementation reproduced spectrally and spatially resolved absolute counts, including non-phase-matched “tails” in frequency–angle maps, and explicitly separated SPDC from thermally seeded down-conversion and up-conversion contributions (Riexinger et al., 2021).

Reported implications include quantum spectroscopy, optical coherence tomography, ultrafast quantum metrology, clock synchronization, ghost imaging, imaging with undetected photons, and high-dimensional encoding in frequency and transverse momentum. The subwavelength-film work frames these directions in terms of flat SPDC sources and quantum optical metasurfaces, with Fabry–Perot, Mie, and BIC resonances identified as possible control mechanisms; the micrometer-scale LN studies emphasize the availability of more than $\Delta k = k_p-k_s-k_i=0$61 transverse modes and the use of the largest tensor components without birefringent phase matching or poling (Santiago-Cruz et al., 2020, Okoth et al., 2019). A plausible implication is that non-phase-matched SPDC is best understood not as a replacement for phase matching in bulk nonlinear optics, but as a distinct thin-medium operating regime in which extreme bandwidth, high-dimensional entanglement, and microscale integration are obtained at the cost of short-interaction-length efficiency.