Annular SPDC Emission Ring

• Annular SPDC emission ring is a spatially structured distribution of entangled photon pairs, emerging from energy and momentum conservation in non-collinear down-conversion.
• Its spatial profile and dual-ring formation critically depend on pump beam properties, such as beam mode (Gaussian or vortex) and beam radius, influencing FWHM and ring separation.
• Experimental setups use tailored phase matching, precise temperature control, and optical elements like axicons to achieve invariant ring diameters for optimized quantum random number generation and entanglement.

An annular SPDC emission ring is a spatially structured distribution of photon pairs generated by spontaneous parametric down-conversion (SPDC) in a nonlinear optical crystal, where phase-matching geometry and pump beam properties produce a ring-like (annular) emission pattern. This structure arises due to the conservation of energy and momentum in the non-collinear down-conversion process, and exhibits strong spatial correlations, especially in setups using optical vortices or quasi-phase-matched crystals. The annular geometry has significant implications for entangled photon generation, quantum random number generation, and the engineering of spatial mode entanglements.

1. Physical Principles and Generation Mechanism

SPDC is a nonlinear process whereby a high-frequency pump photon ($\omega_p$) interacts with a second-order ($\chi^{(2)}$) nonlinear crystal, leading to the probabilistic splitting into lower-frequency signal ($\omega_s$) and idler ($\omega_i$) photons such that $\omega_p = \omega_s + \omega_i$. For annular SPDC emission, the process is realized in a non-collinear phase-matching regime, whereby the conservation of transverse momentum leads to the emission of photon pairs at diametrically opposite points on a ring in the far field.

The phase-matching condition may be expressed as:

$$\vec{k}_p - \vec{k}_s - \vec{k}_i - \frac{2\pi}{\Lambda}\hat{z} = 0$$

for periodically poled crystals (quasi-phase-matching, QPM), where $\Lambda$ is the poling period and $\vec{k}$ are the wave vectors. In type-0 phase matching, all generated photons have the same polarization. Adjusting the crystal parameters (temperature, poling period) and pump wavelength controls the emission angle and consequently the diameter of the annular ring. When pumped below the collinear degenerate temperature or with vortex beams, the emission pattern transitions from Gaussian to annular (Singh et al., 2022, Nai et al., 1 Oct 2024).

2. Spatial Distribution and Dependence on Pump Properties

The spatial profile of the SPDC emission is highly sensitive to both the pump beam's mode and its radius:

• Gaussian Pump: The emission forms a single annular ring centered around the pump direction. Empirical data show that the ring's thickness (full width at half maximum, FWHM) scales linearly with the pump's radius, $\sigma_{\text{ring}} \propto \sigma_{\text{pump}}$ (Prabhakar et al., 2013).
• Optical Vortex Pump: Using beams with orbital angular momentum (OAM, topological charge $l$), described by $$I_l(x, y) = I_0 (x^2 + y^2)^{|l|} e^{-(x^2 + y^2)/\sigma^2}$$, leads to the formation of two concentric annular rings if the beam's radius exceeds a critical value. Both the FWHM and separation of these rings increase with vortex order $l$, reflecting the spatial width and phase singularity of the vortex (Prabhakar et al., 2013).

The presence of a critical beam radius below which no change in ring structure is observed, and above which pronounced broadening and dual-ring formation occur, signifies a threshold in spatial mode involvement in the down-conversion process.

3. Experimental Techniques and Optical Configuration

Typical experimental realization is based on the following components:

• Laser Source: A continuous-wave diode laser (405 nm) passes through spatial light modulators (SLM) for mode-shaping into Gaussian or vortex beams, and through polarization optics (HWP, PBS) to match the crystal's accepted polarization.
• Nonlinear Crystal: Beta-barium borate (BBO) or periodically poled KTP (PPKTP) are common choices, with precise orientation and temperature control to set phase-matching conditions.
• Detection: Down-converted photons are filtered (e.g., 810 nm, ±5 nm) and imaged using EMCCD cameras or coupled into fiber arrays. Mirrors and lenses section and collimate the annular emission for further analysis or collection (Singh et al., 2022, Nai et al., 1 Oct 2024).

In advanced schemes, annular emission is spatially transformed into a "perfect" ring (fixed diameter regardless of SPDC ring variations) using axicons and Fourier lenses. For example, a Bessel-like beam transformation followed by imaging at a Fourier plane enables stable collection efficiency even with significant temperature or wavelength fluctuations (Singh et al., 2022).

Table: Influence of Pump Mode on SPDC Ring Structure

Pump Type	SPDC Ring Structure	Dependence on Pump Parameters
Gaussian Beam	Single annular ring	FWHM $\uparrow$ with radius
Optical Vortex	Dual concentric rings (for $>r_c$)	FWHM/separation $\uparrow$ with $l$
Temperature Drift	Annular to "perfect ring" (w/ axicon/lens)	Diameter invariant after transformation

4. Correlations and Their Exploitation

Photon pairs generated in annular SPDC exhibit strong spatial and temporal correlations:

• Spatial Correlation: Due to momentum conservation, photons are emitted in pairs at diametrically opposed positions on the ring. Sectioning the ring into quadrants or smaller segments, one can assign binary events to coincidence detections between specific sections (e.g., $U_1$ vs $D_2$) (Nai et al., 1 Oct 2024).
• Temporal Correlation: The generation process is inherently simultaneous. Precision time-tagging within a nanosecond window further enhances signal-to-noise ratio for correlated detection.
• Quantum Random Number Generation (QRNG): By exploiting correlated detection between ring sections, QRNGs can assign random bits without requiring a beamsplitter (thus avoiding its bias and loss), enabling higher bit rates. For example, sectioning the ring into four quadrants yields multiple bits per coincidence, with bit rates confirmed at $\sim$3 Mbps for 17 mW pump, and extraction efficiency $>95\%$ (Nai et al., 1 Oct 2024).

$$H_{\infty}(X) = -\log_2 \left( \max_{x \in \{0, 1\}^N} P[X = x] \right)$$

5. Numerical Modeling and Validation

Quantitative modeling employs phase-matching equations and numerical integration across the transverse pump profile:

• Energy and Momentum Conservation: Enforces relations $\hbar\omega_p = \hbar\omega_s + \hbar\omega_i$, $\vec{k}_p = \vec{k}_s + \vec{k}_i$.
• Simulation Methods: Runge–Kutta and related algorithms integrate the contributions from each spatial element of the pump beam, using measured refractive indices, crystal orientation, and pump geometry. These models accurately reproduce observed annular structures, critical radius effects, and FWHM scaling with beam parameters (Prabhakar et al., 2013).
• Empirical Agreement: Experimental images of SPDC rings, and their dependence on pump mode and size, are consistent with numerical predictions.

6. Impact, Applications, and Prospects

The annular SPDC emission ring offers substantive advantages and new capabilities:

• Entangled Photon Source Optimization: Control over ring thickness and spatial correlations allows tailoring of photon collection efficiency and mode selectivity, crucial for entanglement-based protocols and high-dimensional quantum systems (Prabhakar et al., 2013).
• Quantum Communication and QKD: Stable, bright sources with robust entanglement and reduced sensitivity to temperature or wavelength enable practical deployment in field or satellite-based quantum key distribution (Singh et al., 2022).
• Quantum Random Number Generation: Exploitation of both spatial and temporal correlations yields compact, device-independent QRNG devices with greatly enhanced bit rates, validated by rigorous statistical test suites (NIST, TestU01) (Nai et al., 1 Oct 2024).
• Quantum Imaging and Multiplexed Entanglement: Structured ring emission and concurrent spatial-mode entanglement suggest new avenues for high-resolution quantum imaging and increases in quantum channel capacity.
• A plausible implication is that further sectioning of the ring and addressing more spatial modes may enable even higher extraction efficiency and faster bit rates.

7. Generalization and Future Directions

The techniques are generic across materials, wavelengths, and collection methods, extensible to other nonlinear crystals and spectral regimes. Transformation of variable annular rings into invariant "perfect" rings by simple optical means supports robust operation outside laboratory conditions (Singh et al., 2022). Further research could focus on optimizing coupling efficiency, extending robustness to wider parameter drifts, and integrating with advanced single-photon detectors. Multi-channel quantum networks and on-chip architectures may benefit from these enhanced spatial correlation capabilities.

In summary, the annular SPDC emission ring represents a critical structure for the spatial engineering of entangled photon pairs, manipulation of quantum correlations, and advancement of quantum communication, computation, and randomness applications. Its physical basis in non-collinear phase-matching and pump-mode control enables a versatile platform for future quantum photonic technologies.