Beam correlation¶

bolojax computes photon noise correlations between detector pixels in densely packed focal plane arrays. The original BoloCalc included pre-computed amplitude coherence curves based on an undocumented beam model. This implementation follows the formalism of Hill & Kusaka (2024), “Photon noise correlations in millimeter-wave telescopes,” Applied Optics 63(6), 1654. arXiv:2309.01153.

Physical picture¶

When detector pixels are packed with pitch $p_\text{pix} \lesssim 1.2 F\lambda$, neighboring detectors sample overlapping spatial modes of the incoming radiation field. The Bose (wave noise) term of the photon noise then correlates between pixels through the Hanbury Brown & Twiss (HBT) effect. As a result, array-averaged sensitivity scales less favorably than the uncorrelated $N_\text{det}^{-1/2}$ limit.

This matters for mm-wave CMB experiments where the mean photon occupation number $\bar{n} \sim 1$ places the detector in the crossover regime between shot-noise-dominated (optical, $\bar{n} \ll 1$) and wave-noise-dominated (radio, $\bar{n} \gg 1$).

Theory¶

Mutual intensity and coherence¶

The mutual intensity between detectors $i$ and $j$ is (Hill Eq 6):

\[B_{ij}(\nu) = \sum_k S_{ik}^*(\nu) S_{jk}(\nu) n(T_k, \nu) \label{eq:mutual-intensity}\]

where $S_{ik}$ is the scattering matrix coupling source mode $k$ to detector $i$, and $n(T,\nu) = [\exp(h\nu/k_BT) - 1]^{-1}$ is the Bose-Einstein occupation number.

The amplitude coherence (“van Cittert-Zernike Thoerem” (VCZT) coefficient) is the normalized mutual intensity (Hill Eq 10):

\[\gamma_{ij}(\nu) = \frac{B_{ij}(\nu)}{\sqrt{B_{ii}(\nu) B_{jj}(\nu)}} \label{eq:amplitude-coherence}\]

The intensity coherence (HBT coefficient) that enters the noise covariance is its squared magnitude (Hill Eq 17):

\[\gamma_{ij}^{(2)}(\nu) = |\gamma_{ij}(\nu)|^2 \label{eq:intensity-coherence}\]

Aperture and stop decomposition¶

For a simplified telescope model (Hill Sec 3A), the discrete sum over source modes in $\eqref{eq:mutual-intensity}$ becomes a continuous integral over the aperture plane. Two steps are involved:

First, the scattering matrix $S_{ik}$ reduces to the pupil function $G_i(u, v)$: detector $i$’s beam back-propagated to coordinates $(u, v)$ on the aperture plane. This is the van Cittert-Zernike theorem: for spatially incoherent thermal sources (coherence length $\sim \lambda \ll D_\text{ap}$), the coupling between a source at position $(u, v)$ and detector $i$ is simply $G_i$ evaluated there.

Second, the occupation number $n(T_k)$ factors out of the integral. In $\eqref{eq:mutual-intensity}$, each source mode $k$ has its own temperature $T_k$. But within the aperture region, all sources are treated as having a single effective occupation number $n(T_\text{ap})$ (defined by the weighted sum over optical elements in Hill Eq 46). Since $n$ is constant across the integration domain, it becomes a prefactor. The integral becomes just the overlap of the beam patterns between dector $i$ and $j$:

\[B_{\text{ap},ij}(\nu) \propto n(T_\text{ap}) \iint G_i^*(u,v) G_j(u,v) e^{2\pi i (u x_{ji} + v y_{ji}) / D_\text{ap} F\lambda} \mathrm{d}u \mathrm{d}v \label{eq:general-mutual}\]

where $(x_{ji}, y_{ji})$ is the displacement between detectors $j$ and $i$ in the focal plane. Since a phase tilt in the pupil plane corresponds to a pointing offset in the focal plane, the phase factor encodes the detector separation $p_{ij} = \sqrt{x_{ji}^2 + y_{ji}^2}$. The factored-out $n(T_\text{ap})$ reappears later as the occupation number weight in $\eqref{eq:combined-hbt}$.

Under the assumption that all detectors share the same beam pattern ($G_i \simeq G_j \equiv G$; Hill Eq 52 $\to$ 53), the cross-correlation $G_i^* G_j$ simplifies to the power spectrum $|G|^2$, and the coherence becomes a function of $p_{ij}$ alone.

The aperture stop divides the pupil plane into two regions: radiation passing through the aperture (sky signal, as seen through the optical chain before the stop, at effective temperature $T_\text{ap}$) and radiation from the stop itself (thermal emission at physical temperature $T_\text{stop}$). Since each region has a uniform occupation number, the mutual intensity separates into two independent integrals (Hill Eq 51), with $n(T_\text{ap})$ factoring out of the aperture term and $n(T_\text{stop})$ out of the stop term.

Working in normalized coordinates where $\rho = 2\sqrt{u^2 + v^2}/D_\text{ap}$ (so $\rho = 1$ at the aperture edge):

Aperture radiation: sky-side sources, $\rho \in [0, 1]$ (Hill Eq 53):

\[\gamma_{\text{ap},ij}(\nu) = \frac{1}{\eta_\text{ap}} \iint_{\rho \in [0, 1]} |G(u, v)|^2 e^{2\pi i p_{ij} u / F\lambda} \mathrm{d}u \mathrm{d}v \label{eq:aperture-coherence}\]

Stop radiation: thermal emission from beyond the aperture edge, $\rho \in [1, \infty)$ (Hill Eq 56):

\[\gamma_{\text{stop},ij}(\nu) = \frac{1}{1-\eta_\text{ap}} \iint_{\rho \in [1, \infty)} |G(u, v)|^2 e^{2\pi i p_{ij} u / F\lambda} \mathrm{d}u \mathrm{d}v \label{eq:stop-coherence}\]

where $\eta_\text{ap} = \iint_{\rho \leq 1} |G(u,v)|^2 \mathrm{d}u \mathrm{d}v$ is the aperture spillover efficiency. The frequency dependence enters through $F\lambda = Fc/\nu$ in the phase factor. For identical beams, $\gamma_{ij}$ depends on the detector pair only through their separation $p_{ij}$.

For a circularly symmetric pupil $G(u, v) = G(\rho)$, the 2D Fourier transforms reduce to 1D Hankel transforms. Writing $p = p_{ij}/F\lambda$ as the separation in diffraction units:

\[\gamma(p) = \frac{1}{\eta} \int_{\rho_{\mathrm{min}}}^{\rho_{\mathrm{max}}} |G(\rho)|^2 J_0(2\pi p \rho) 2\pi \rho \mathrm{d}\rho \label{eq:hankel}\]

with limits $[0, 1]$ for the aperture and $[1, \infty)$ for the stop. This is the form implemented in by the preset models in the beam_coherence module.

Combined HBT coefficient¶

Because the aperture and stop radiate at different temperatures, their contributions to the wave noise must be weighted by their respective photon occupation numbers $n(T, \nu) = [\exp(h\nu/k_BT) - 1]^{-1}$. The combined intensity coherence is (Hill Eq 59):

\[|\gamma_{ij}^{\text{np}}|^2 = \frac{n(T_{\text{ap},i})}{n(T_{(i)})} \eta_{\text{ap},i} |\gamma_\text{ap}|^2 + \frac{n(T_\text{stop})}{n(T_{(i)})} (1-\eta_{\text{ap},i}) |\gamma_\text{stop}|^2 \label{eq:combined-hbt}\]

Each weight is the fraction of photons that sources deliver to detector $i$:

$\eta_{\text{ap},i}$ is the aperture spillover efficiency of detector $i$, i.e. the fraction of $G_i$’s beam power passing through the aperture.
$n(T_{\text{ap},i})$ is the effective photon occupation number of all radiation entering through the aperture as seen by detector $i$ (Hill Eq 46): $n(T_{\text{ap},i}, \nu) = \sum_\rho \epsilon_\rho H_\rho(\nu) n(T_\rho, \nu)$, where the sum runs over every optical element on the sky side of the aperture stop (CMB, atmosphere, mirrors, windows, filters, etc), weighted by their emissivities $\epsilon_\rho$ and cumulative throughput $H_\rho$.
$T_\text{stop}$ is the physical temperature of the cold stop.
Thus, $n(T_{(i)}) = \eta_{\text{ap},i} n(T_{\text{ap},i}) + (1-\eta_{\text{ap},i}) n(T_\text{stop})$ is the total photon occupation number at detector $i$ from both sources (Hill Eq 60).

Under the identical-beam assumption ($G_i = G$), all per-detector quantities collapse: $\eta_{\text{ap},i} \to \eta_\text{ap}$, $T_{\text{ap},i} \to T_\text{ap}$, and $T_{(i)} \to T$.

Impact on array sensitivity¶

The array noise variance including correlations is (Hill Eq 68):

\[\sigma_\text{arr}^2 = \frac{\sigma_\text{shot}^2 + (1 + \gamma^{(2)}) \sigma_\text{wave}^2 + \sigma_\text{int}^2}{N_\text{det}} \label{eq:array-noise}\]

where $\gamma^{(2)} = N_\text{det}^{-1} \sum_{ij} (1 - \delta_{ij}) \gamma_{ij}^{(2)}$ is the array-averaged HBT coefficient (Hill Eq 67). In compute/noise.py, this enters as a per-element correlation factor that modifies the wave-noise contribution to the photon NEP.

Implementation¶

Beam models¶

bolojax implements three beam illumination models in the compute.beam_correlation module:

Polynomial taper (poly_taper): $[\max(1 - a_1 r - a_2 r^2, 0)]^n$

Truncated Gaussian (trunc_gauss): $\exp(-2\sigma^2 r^2) \Pi(r / R)$, a Gaussian illumination truncated at radius $R$. Generalization of Hill Eq 54.

HE11 mode (he11): $J_0(2.405 r/R)^2 \Pi(r / R_\text{taper})$, the fundamental corrugated horn mode truncated at $R_\text{taper}$.

$\Pi(x)$ is a rect/top-hat function, implemented as a smooth sigmoid for numerical stability/differentiability.

The Hankel transform $\eqref{eq:hankel}$ is evaluated numerically by beam_coherence, with integration limits selecting aperture or stop.

Presets¶

Three presets bundle default beam parameters:

Preset	Model	Parameters
`bolocalc`	poly_taper	a1=1.0825, a2=-0.0413, n=1.300, R_zero=0.961
`trunc_gauss`	trunc_gauss	sigma=1.33, R=1.0
`he11`	he11	R=1.05, R_taper=1.05

The bolocalc preset is the default and reproduces original BoloCalc behavior. For the aperture coherence, it uses the polynomial taper beam model (fit using the original coherent aperture correlation curves). For the stop coherence, it loads the original BoloCalc curve directly from a stored array. For the physically motivated presets (trunc_gauss, he11), both aperture and stop coherence derive from the same beam model with different integration limits.

The beam model is configured with the beam_model field on the camera. To use a preset by name:

camera:
  beam_model: trunc_gauss

To set custom beam parameters, pass a dict:

camera:
  beam_model:
    model: trunc_gauss
    sigma: 1.5
    R: 0.9

Any parameter not specified falls back to the preset default. The available parameters for each model are listed in the table above.

Correlation factor computation¶

Noise.corr_facts computes per-element correlation factors by:

Computing aperture and stop coherence curves $\gamma(p)$ on a pitch grid from 0 to 5 $F\lambda$.
Evaluating $\gamma$ at each shell separation allowed by a hex grid: $p_s \in {p, \sqrt{3}p, 2p, 2\sqrt{3}p, \ldots}$ out to a cutoff (default 3 $F\lambda$).
Summing $\sum_s n_s \gamma(p_s)$ with $n_s = 6$ at every shell.
Returning $\sqrt{1 + \sum_s n_s \gamma(p_s)}$ as the per-element factor.

Each optical element uses either the aperture or stop coherence curve: elements on the sky side of the aperture stop use $\gamma_\text{ap}$, the aperture stop itself uses $\gamma_\text{stop}$, and elements on the detector side have a factor of 1 (no correlation).

Assumptions baked in:

Identical beams: a single coherence curve is used for all detector pairs at a given separation.
Hex-packed focal plane: the array-averaged HBT coefficient $\gamma^{(2)} = N_\text{det}^{-1} \sum_{j \neq i} \gamma^{(2)}(p_{ij})$ requires summing over all detector pairs. Since $\gamma^{(2)}$ depends only on separation, this reduces to $\sum_s n_s \gamma^{(2)}(p_s)$, where $n_s$ is the number of detectors at distance $p_s$ from a given detector. The code assumes $n_s = 6$ at every shell (exact for nearest neighbors in a hex lattice, approximate for farther shells).
Temperature-independent weighting: the implementation currently assumes temperature independence. In other words, setting $T_\text{ap} = T_\text{stop} \equiv T$ causes the occupation number ratios to drop out of $\eqref{eq:combined-hbt}$,

\[|\gamma_{ij}|^2 = \eta_\text{ap} |\gamma_\text{ap}|^2 + (1-\eta_\text{ap}) |\gamma_\text{stop}|^2\]

In practice $T_\text{ap} \gt T_\text{stop}$, so this approximation overstates the relative stop contribution.
Unpolarized: Hill Eq 62 gives the Stokes Q intensity coherence as $\gamma_{ij}^{Q,(2)} = \cos[2(\psi_i - \psi_j)] |\gamma_{ij}|^2$, where $\psi_i$ is the polarization angle of detector $i$. The current implementation ignores this modulation.
Truncated at 3 $F\lambda$: correlations beyond flamb_max are ignored.