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Propagation of Two-Photon Zernike States in Atmospheric Turbulence

Hakob Avetisyan, Vahagn Abgaryan

Abstract

We analyze propagation and detection of two-photon states expanded in Zernike modes through atmospheric turbulence using the extended Huygens-Fresnel formalism. For SPDC states prepared with a single Zernike pump mode, we analytically reduce the 8-dimensional continuous propagation integrals to an exact, discrete modal expansion. In the absence of turbulence, Zernike addition enforces conservation of azimuthal index and a strict radial-order bound. Turbulence relaxes these constraints, driving structured azimuthal and radial crosstalk dominated by low-order aberration modes. By explicitly removing the lowest-order terms from the discrete turbulence sum, we demonstrate that partial adaptive optics correcting only up to the sixth radial order is sufficient to heavily suppress this crosstalk and restore near-ideal spatial correlations.

Propagation of Two-Photon Zernike States in Atmospheric Turbulence

Abstract

We analyze propagation and detection of two-photon states expanded in Zernike modes through atmospheric turbulence using the extended Huygens-Fresnel formalism. For SPDC states prepared with a single Zernike pump mode, we analytically reduce the 8-dimensional continuous propagation integrals to an exact, discrete modal expansion. In the absence of turbulence, Zernike addition enforces conservation of azimuthal index and a strict radial-order bound. Turbulence relaxes these constraints, driving structured azimuthal and radial crosstalk dominated by low-order aberration modes. By explicitly removing the lowest-order terms from the discrete turbulence sum, we demonstrate that partial adaptive optics correcting only up to the sixth radial order is sufficient to heavily suppress this crosstalk and restore near-ideal spatial correlations.
Paper Structure (21 sections, 72 equations, 1 figure)

This paper contains 21 sections, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Zernike-mode formalism and field propagation through turbulence.
  3. Zernike modes and mode-coupling identities.
  4. Zernike expansion of the pupil function and field representation.
  5. Field operators and extended Huygens–Fresnel principle.
  6. Propagation and detection of two-photon Zernike states
  7. No-turbulence limit and recovery of exact selection rules.
  8. Ensemble-averaged two-photon detection probability in turbulence.
  9. Discussion
  10. Conclusion
  11. Derivation and summary of Zernike and Fourier–Zernike identities
  12. Representations of $A$ and $\Gamma$
  13. New identities
  14. No turbulence limit.
  15. Thin crystal approximation.
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Top Row: Radial-mode correlations in Zernike-entangled photon pairs under varying turbulence strengths. Shown is the base-10 logarithm of the normalized joint detection probability $P_{N}^{M}(\tilde{Z}_{N_1}^{M_1}, \tilde{Z}_{N_2}^{M_2})$. The color scale is lower-bounded at $10^{-4.5}$ to isolate physically meaningful crosstalk from negligible numerical background. Rows correspond to fixed detector azimuthal indices $(M_1, M_2) = (1, -1)$. Columns compare increasing turbulence strengths for a pump mode $(N,M) = (2,0)$. Left ($\sigma_R = 0.0$): In a perfect vacuum, exact momentum-matching and Zernike selection rules perfectly restrict the transition to a single state $(N_1=1, N_2=1)$. Center and Right ($\sigma_R = 0.01$ and $0.1$): Increasing turbulence relaxes the radial-order constraints and breaks the macroscopic azimuthal selection rule, forming an extended, exponentially decaying distribution across higher radial orders, reflecting the relaxation of the strict free-space Zernike constraints. Bottom Row: Ideal partial AO is mathematically modeled by truncating the lowest-order macroscopic modes from the turbulence tensor ($n_5 > 6$ in \ref{['eq:P-final']}, correcting up to primary spherical aberration). This truncation perfectly restores the free-space limit at $\sigma_R = 0.01$ and heavily suppresses crosstalk at weak-to-moderate turbulence ($\sigma_R = 0.1$). Even under moderate-to-strong turbulence ($\sigma_R = 0.5$), the target correlation peak remains highly dominant. The residual leakage is confined to adjacent modes, demonstrating that high-frequency turbulent eddies cannot efficiently drive simultaneous, multi-mode radial scattering. Parameters: $k=10^7 \text{ m}^{-1}$, $z=5\times 10^3 \text{ m}$, $R=5\times 10^{-3} \text{ m}$.