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arXiv:2608.16524

Spectral filtering and crystal length as control parameters for conditional correlations in quantum imaging

Hashir Kuniyil, Asad Ali, Mohammadreza Rezaee, Saif Al-Kuwari

quant-phSubmitted Aug 17, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

We establish spectral filtering as a control parameter for the conditional momentum and position correlations of SPDC biphotons used in quantum imaging. The conditional momentum uncertainty in spontaneous parametric down-conversion (SPDC) is strongly crystal-length and spectral-filter dependent along the walk-off axis; therefore, the effect is observed only in critically phase-matched (CPM) crystals such as $β$-barium borate (BBO) crystals, while quasi-phase-matched (QPM) crystals and the non-walk-off axis of BBO remain scaled strictly according to the standard pump waist size ($w_0$) dependent scaling law $1/w_0$-independent of the filter. In position space, the spectral-filter effect is universal and produces a flat-dip-rise (FDR) profile in every crystal class examined. Although this FDR profile was previously demonstrated in BBO only in the nondegenerate regime, our results establish its generality: the FDR dip is also present at exact degeneracy in QPM crystals, an unexpected feature that was previously thought to deliver a resolution advantage in the nondegenerate regime alone. Our treatment applies to all SPDC-based quantum-imaging regimes (including CPM and QPM crystals, walk-off and non-walk-off axes, degenerate and nondegenerate emission, and signal and idler filtering) and offers enhanced quantum-imaging resolution via the design rules presented-most prominently along the walk-off axis of CPM crystals (in far field) and across all transverse axes in the near field.

Comments
13 pages including references, 3 figures

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We establish spectral filtering as a control parameter for the conditional momentum and position correlations of SPDC biphotons used in quantum imaging. The conditional momentum uncertainty in spontaneous parametric down-conversion (SPDC) is strongly crystal-length and spectral-filter dependent along the walk-off axis; therefore, the effect is observed only in critically phase-matched (CPM) crystals such as βββ-barium borate (BBO) crystals, while quasi-phase-matched (QPM) crystals and the non-walk-off axis of BBO remain scaled strictly according to the standard pump waist size (w0w_0w0​) dependent scaling law 1/w01/w_01/w0​-independent of the filter. In position space, the spectral-filter effect is universal and produces a flat-dip-rise (FDR) profile in every crystal class examined. Although this FDR profile was previously demonstrated in BBO only in the nondegenerate regime, our results establish its generality: the FDR dip is also present at exact degeneracy in QPM crystals, an unexpected feature that was previously thought to deliver a resolution advantage in the nondegenerate regime alone. Our treatment applies to all SPDC-based quantum-imaging regimes (including CPM and QPM crystals, walk-off and non-walk-off axes, degenerate and nondegenerate emission, and signal and idler filtering) and offers enhanced quantum-imaging resolution via the design rules presented-most prominently along the walk-off axis of CPM crystals (in far field) and across all transverse axes in the near field.
Spatial correlations of photon pairs from spontaneous parametric down-conversion (SPDC) underpin quantum imaging and entanglement certification. We present the first systematic study of spectral filter bandwidth effects on transverse spatial correlations in bulk Type-I BBO for degenerate and non-degenerate configurations. In the far field, the degenerate conditional momentum width is pump-limited and filter-invariant, while non-degenerate configurations exhibit monotonic growth in both marginal and conditional momentum widths -- with the walk-off axis ≈100\approx 100≈100 times more sensitive than the non-walk-off axis. In the near field, we identify a previously unreported flat-dip-rise profile: the conditional position width narrows by ≈10%\approx 10\%≈10% at an optimal bandwidth Δdip≈1.35 ΔλSPDCΔ_\mathrm{dip} \approx 1.35\,Δλ_\mathrm{SPDC}Δdip​≈1.35ΔλSPDC​ before rising due to geometric displacement. When the filter is placed on the idler arm, the dip shifts by the exact factor (λi/λs)2(λ_i/λ_s)^2(λi​/λs​)2. Both results are universal for any non-degenerate SPDC source, requiring only a finite crystal length, dθ/dλ≠0dθ/dλ\neq 0dθ/dλ=0, and incoherent spectral averaging. The Reid EPR uncertainty product is consistently smaller on the walk-off axis -- a structural advantage of bulk birefringent geometry absent in quasi-phase-matched sources. The optimal filter bandwidth ΔF=ΔdipΔ_F = Δ_\mathrm{dip}ΔF​=Δdip​ is determined entirely by the intrinsic phase-matching bandwidth of the crystal and is directly readable from the X-entanglement spectral width of the source.
Abstract Spontaneous parametric down-conversion (SPDC) is the most widely used process for generating photon pairs entangled in various degrees of freedom such as polarization, time-energy, position-transverse momentum, and angle-orbital angular momentum (OAM). In SPDC, a pump photon interacts with a non-linear optical crystal and splits into two entangled photons called the signal and the idler photons. The SPDC process has been studied extensively in the last few decades for various pump and crystal configurations, and the entangled photon pairs produced by SPDC have been used in numerous experimental studies on quantum entanglement and entanglement-based real-world quantum-information applications. In this tutorial article, we present a thorough study of phase matching in BBO crystals for spontaneous parametric down-conversion and thereby also investigate the generation of entangled photons in such crystals. First, we present a theoretical derivation of two-photon wavefunction produced by SPDC in the frequency and transverse momentum bases. We then discuss in detail the effects due to various crystal and pump parameters including the length of the crystal, the angle between the optic axis and the pump propagation direction, the pump incidence angle on the crystal surface, the refraction at the crystal surfaces, and the pump propagation direction inside the crystal. These effects are extremely relevant in experimental situations. We then present our numerical and experimental results in order to illustrate how various experimental parameters affect the phase matching and thus the generation of entangled photons. Finally, using the two-photon wavefunction in the transverse wave-vector basis, we show how to derive the two-photon wavefunction in the OAM basis and thereby calculate the two-photon angular Schmidt spectrum. We expect this article to be useful for researchers working in various capacities with entangled photons generated by SPDC in BBO crystals.
Spontaneous Parametric Down-Conversion (SPDC) is key to generating broadband spectrally entangled photons for quantum optical spectroscopy. Here we use a common-path birefringent interferometer to experimentally map the spatial-spectral distribution of SPDC in Type I and II β-barium borate crystals using hyperspectral k-space imaging. Under broadband pulsed pumping, we observe strong spatial variations in the spectra, with distinct features for each phase-matching type. Simulations based on biphoton amplitudes and phase-matching conditions qualitatively match our results. This spatial-spectral mapping enables control over biphoton entanglement entropy, paving the way for optimized quantum light sources for material spectroscopy.
Optimizing the brightness of a spontaneous parametric down-conversion (SPDC) source is an important task for many quantum information applications. We investigate the optimal focusing conditions to maximize the number of photons produced in an SPDC process and coupled with single-mode fibers. We provide a general expression for the two-photon wave function, generalizing previous known results, by considering collinear and noncollinear emission. We present analytical expressions for our results in the thin-crystal limit and clarify the relation between different focusing conditions already existing in the literature. Differently from what was previously reported, we show that the optimal ratio between the pump waist and the generated photons waist depends on the emission angle: It is 1/21/\sqrt{2}1/2​ for collinear degenerate emission and approaches 1/21/21/2 for larger collection angles. The role of spectral filters is also analyzed. We support and enrich our discussion with numerical simulations, performed for type-I SPDC in a $\ensuremath{\beta}$ barium borate crystal. For this type of emission, we also investigate the role of the transverse walk-off outside the thin-crystal regime.
Image resolution of quantum imaging with undetected photons is governed by the spatial correlations existing between the photons of a photon pair that has been generated in a nonlinear process. These correlations allow for obtaining an image of an object with light that never interacted with that object. Depending on the imaging configuration, either position or momentum correlations are exploited. We hereby experimentally analyse how the crystal length and pump waist affect the image resolution when using position correlations of photons that have been generated via spontaneous parametric down conversion in a nonlinear interferometer. Our results support existing theoretical models for the dependency of the resolution on the crystal length. In addition, we probe the resolution of our quantum imaging scheme for varying pump waists over one order of magnitude. This analysis reveals the intricate dependency of the resolution on the strength of the correlations within the biphoton states for parameter combinations in which the crystal lengths are much larger than the involved photon wavelengths. We extend the existing models in this parameter regime to properly take nontrivial effects of finite pump waists into account and demonstrate that they match the experimental results.
Spontaneous Parametric Down-Conversion (SPDC) is key to generating broadband spectrally entangled photons for quantum optical spectroscopy. Here we use a a common-path birefringent interferometer to experimentally map the spatial-spectral mode structure of SPDC in Type I and II β\betaβ-barium borate crystals using hyperspectral k-space imaging. Under broadband pulsed pumping, we observe strong spatial variations in the spectra, with distinct features for each phase-matching type. Simulations based on biphoton amplitudes and phase-matching conditions qualitatively match our results. This spatial-spectral mapping enables control over biphoton entanglement entropy, paving the way for optimized quantum light sources for material spectroscopy.