1. INTRODUCTION

Modern millimeter and submillimeter (“mm-wave”) telescopes for astronomy are often limited by fluctuations in the background radiation. This is especially true for ground-based experiments where emission from the atmosphere and telescope is substantial. At high frequencies (e.g., optical wavelengths), the mode’s mean occupation number $\bar n \ll 1$ and photon fluctuations are dominated by uncorrelated shot noise such that $\Delta \bar n \approx \sqrt {\bar n}$. At low frequencies (e.g., radio wavelengths), $\bar n \gg 1$ and photon fluctuations are dominated by the Bose term of the photon noise (“wave noise”) which correlates such that $\Delta \bar n \approx \bar n$. Millimeter wavelengths lie in a cross-over regime where $\bar n \sim 1$, making the calculation of array-averaged sensitivity in general nontrivial.

In addition, many modern mm-wave telescopes, particularly those equipped with cryogenic bolometric detector arrays, are field-of-view-limited and therefore aim to increase the detector count by increasing the pixel density, which is typically cheaper than building more telescopes. In this high-pixel-density paradigm, it is possible to overfill the focal plane such that neighboring detectors sample the same spatial mode. As we will show, this mode sharing introduces photon noise correlations when the pixel spacing ${p_{{\rm pix}}} \lt 1.2F\lambda$, where $F$ is the effective focal ratio at the focal plane, and $\lambda$ is the operational wavelength of interest. In this oversampled regime, photon noise correlations can have substantial impacts on the array-averaged sensitivity.

The theory of intensity correlations of light from incoherent sources has been studied extensively [1–6], and the phenomenon was experimentally demonstrated by Hanbury Brown and Twiss (HBT) via measurements of the angular diameter of distant astronomical sources [7–9]. The impact of HBT correlations on mm-wave telescopes is discussed by Padin [10], where an empirical factor is introduced in an attempt to account for the corresponding sensitivity degradation.

In this paper, we present a prescription to estimate HBT correlations among detectors in millimeter- or submillimeter-wave telescopes based on a quantum optics formalism adopted from a circuit-based formalism for thermal photon correlations in quantum detectors developed by Zmuidzinas [11]. We then extend this formalism to polarization-sensitive detectors and use it to calculate the impact of HBT correlations on the sensitivity of a model mm-wave telescope.

This paper is organized as follows. In Section 2, we review the theoretical foundations of our formalism based on Ref. [11] and show how they relate to the HBT effect [7–9] and the van Cittert–Zernike theorem (VCZT) [12,13]. We also show the formalism’s relation to the standard single-detector sensitivity calculations for a bolometer (e.g., see Ref.  [14] and references therein). Section 3 defines a model optical system for estimating the array-averaged sensitivity impact of HBT correlations. In Section 4, we derive an expression for the intensity correlation using this model optical system. Section 5 discusses the impact of HBT correlations on the sensitivity of a telescope system with close-packed detectors on the focal plane. In Section 6, we discuss the implications of the presented sensitivity optimization for the design of mm-wave detector arrays. Finally, Section 7 presents our conclusions.

2. THEORETICAL FOUNDATIONS

In this section, we review the theory of photon-counting statistics and reformulate them to the context of astronomical telescope systems. We first adopt the treatment of thermal photon correlations derived by Zmuidzinas [11], which uses the machinery of transmission lines and scattering matrices to calculate the propagation of quantum modes in a linear circuit model. We then apply this treatment to optical systems, where the optical equivalence theorem [15] allows us to equate the scattering matrix for quantum modes with the mode-to-mode coupling of classical waves (e.g., those obtained via physical optics calculations). We then show a few simple examples that relate this formalism to the HBT effect, VCZT, and photon noise calculations.

A. Photon Correlations

We first consider a linear, lossy network of $k = 1,2, \cdots$ input ports detected at output port $i$ [16]. The input modes $k = 1,2, \cdots$ enter the network through their corresponding input ports along semi-infinite transmission lines via the photon creation operator $a_k^\dagger (\nu)$, and they are mapped onto the mode $i$ via a scattering matrix ${S_{{ik}}}$. The input mode $k$ originates from sources attached to the input ports, and the output modes $i$ are sensed by detectors attached to the output ports. Hereafter, we denote sources and detectors by their port indices, such as “source $k$” and “detector $i$.” Loss in the system is modeled by an orthogonal scattering matrix ${S_{{ik}}^\prime }$, which governs the noise added between the input mode $k$ and output mode $i$. Given this structure (Fig. 1), the creation operator $b_i^\dagger (\nu)$ at the output port $i$ and mode frequency $\nu$ is

 

Fig. 1. Schematic of the scattering matrix quantum circuit formalism. The creation operator for the incoming modes is ${a^\dagger}$, while that of the outgoing modes is ${b^\dagger}$. The scattering matrix $S$ maps the input modes onto the output modes, while the noise matrix $S^\prime $ calculates noise and loss within the system. $S$ and $S^\prime $ are assumed to be orthogonal.

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The two-photon expectation value at output ports $i$ and $j$ is given by

When the mixed states are in thermal equilibrium, which is a good approximation for the photon modes in the calculations that follow [17]:

Thermal detectors, which are commonly used in mm-wave applications, integrate photon power over time $\tau$ and sense mean intensity:

Finally, the covariance for quantum thermal detectors $\sigma _{{ij}}^2 \equiv \langle \Delta {d_i}\Delta {d_j}\rangle = \langle {d_i}{d_j}\rangle - \langle {d_i}\rangle \langle {d_j}\rangle$ can be written as

Given the thermal detector covariance $\sigma _{{ij}}^2$ in Eq. (11), the (quantum) second-order coherence $\Gamma _{{ij}}^{(2)}$ can be defined as

The normalized intensity coherence can be defined as

The normalized intensity and amplitude coherences can be related as

The intensity coherence $\gamma _{{ij}}^{(2)}$ is not affected by decoherence. Decoherence is present, or the complex phase of ${\gamma _{{ij}}}(\nu)$ rotates over the detection band such that $|{\gamma _{{ij}}}| \lt |{\gamma _{{ij}}}(\bar \nu)|$, when the path length difference between the light source and detectors $(i,j)$ is larger than the inverse of the detection bandwidth $|{R_i} - {R_j}| \mathbin{\lower.3ex\hbox{$\buildrel \gt \over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} c/\Delta \nu$. However, $\gamma _{{ij}}^{(2)}$ is a real-valued positive quantity; thus, such an effect is nonexistent [19]. The intensity signal would still de-correlate if $|{R_i} - {R_j}|$ were larger than $c\tau$ but, for typical mm-wave experiments, $\tau \sim {\cal O}{(10^{- 2}} {-} {10^{- 3}}\;{\rm s})$, and detectors are arranged such that $|{R_i} - {R_j}| \lesssim {10^{- 3}}\; {\rm m}$, leading to $|{R_i} - {R_j}| \ll c\tau$. Therefore, the de-correlation can be safely ignored.

B. Simple Example without Polarization

We now apply the formalism in Section 2.A to mm-wave optical systems. As noted in Appendix A of Ref.  [11], the quantum circuit treatment is readily applicable to free-space propagating waves, as the optical equivalence theorem [15] allows us to equate the scattering matrix to mode–mode coupling coefficients of classical electromagnetic wave amplitudes.

First, we consider the simplest case shown in Fig. 2 with two identical planar detectors at $z = {z_{{\rm pix}}}$ and a far-field planar source at $z = {z_s}$. We assume that $|{z_s} - {z_{{\rm pix}}}| \equiv L \gg 2D_{{\rm pix}}^2/\lambda$, where ${D_{{\rm pix}}}$ is each detector’s aperture diameter, and $\lambda \equiv c/\nu$ is the free-space electromagnetic wavelength, and we assume that the source is thermal with 100% emissivity. Given the classical-wave amplitude of the electric field ${E_i}$ detected by detector $i$, the partial field amplitude $\Delta {E_i}$ from an infinitesimal area of the planar source $\Delta {s_k}$ can be written as

 

Fig. 2. Simple example where two detectors ($i$ and $j$) on the $z = {z_{{\rm pix}}}$ plane observe light from a surface source on the $z = {z_{\rm s}}$ plane. The direction of the rays is defined by spherical coordinate $({\theta _{{\rm pix}}},{\phi _{{\rm pix}}})$.

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As noted previously, the optical equivalence theorem allows us to relate the right-hand side of Eq. (18) with the scattering matrix ${S_{{ik}}}(\nu)$. Thus, following Eq. (6), the mutual intensity can be calculated as

For a very distant circular source (such as a star) at $x = y = 0$ and with temperature $T$, the mutual intensity can be simplified to

C. Simple Example with Polarization

Building on Section 2.B, we now introduce the polarization degree of freedom. We adopt the same geometry as Fig. 2 and assume that each pixel is an ideal dual-polarization detector equipped with two orthogonal linear polarimeters. When decomposing the propagating field into two polarization degrees of freedom, it is convenient to adopt the Ludwig-3 basis set [20]:

Each detector pixel $i$ composes two detectors ${i_1}$ and ${i_2}$ with polarization angles ${\psi _i}$ and ${\psi _i} + \pi /2$, respectively. The angle ${\psi _i}$ is defined in the $x {\text -} y$ plane such that ${\psi _i} = 0$ when the polarization angle of detector ${i_1}$ is along the $x$-axis. Assuming ideal polarimetry [21–25], detector ${i_1}$ only responds to the propagating electric field with the polarization direction:

The surface source can also be decomposed into those emitting two polarization degrees of freedom, denoted by ${k_1}$ and ${k_2}$. This decomposition is arbitrary as long, as ${k_1}$ and ${k_2}$ are independent and orthogonal Gaussian-random emitters, which is the case for unpolarized thermal sources. Therefore, we assume for simplicity that sources ${k_1}$ and ${k_2}$ emit with polarization $\hat e_1^{L3}$ and $\hat e_2^{L3}$, respectively.

The coupling between detector ${i_1}$ and source ${k_1}$ is similar to Eq. (18), except for an additional polarization overlap factor $({\hat e_{i1}} \cdot \hat e_1^{L3})$:

It is convenient to calculate the covariance of the Stokes parameters, which is defined as the difference in power measured by two orthogonal polarimeters in a single detector pixel, ${Q_i} \equiv ({d_{i1}} - {d_{i2}})/2$. The Stokes-parameter covariance between two pixels can be calculated as

As previously mentioned and further discussed in Section 3, the form in Eq. (29) that only introduces a $\cos [2({\psi _i} - {\psi _j})]$ factor to the “no polarization” case is general, as long as the telescope and detector conform to the assumption of ideal polarimetry. It is worth noting that, according to Eq. (29), the focal plane can be designed to minimize the Stokes $Q$ correlation by assigning ${\pm}45^\circ$ polarization angles to the neighboring pixels.

D. Detector Photon Noise

We now relate the formalism in Section 2.A to the forms of photon noise for bolometric detectors often seen in the literature.

It is convenient to decompose the scattering matrix ${S_{{ik}}}$ between photon source $k$ and detector $i$ into that of the detector optics $S_{{ik}}^{{\rm d}i}$ and that of the telescope optics that couples to detectors $i$, $S_k^{(i)}$. The former includes the detector’s quantum efficiency (see below), and the latter includes coupling to the atmosphere, lossy optical elements, and any other detectable photon sources. The detector-only scattering matrix can be written as a simple three-port system, as shown in Fig. 3, where $i$, ${\rm L}i$, and ${\rm C}i$ represent the detection, loss, and optical-coupling ports, respectively. The three corresponding scattering matrix elements, $S_{i,i}^{{\rm d}i}$, $S_{i,{\rm L}i}^{{\rm d}i}$, and $S_{i,{\rm C}i}^{{\rm d}i}$, can be related to the reflection, loss, and transmission while satisfying the normalization

 

Fig. 3. Schematic representing the detector scattering matrix ${S^{{\rm d}i}}(\nu)$ and optics scattering matrix $S_k^{(i)}(\nu)$ for detector $i$. The detector scattering matrix consists of three ports: the detection port $i$, the lossy port ${\rm L}i$, and the optical coupling port ${\rm C}i$.

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For bolometric focal planes, the temperature of detection port $i$ and lossy port ${\rm L}i$ is typically ${\lt}0.5{\rm K}$. Therefore, when evaluating Eq. (6), the occupation numbers $n(\nu ,T)$ due to thermal emission from these detection ports are small at millimeter and submillimeter frequencies and can be neglected in further calculations. We can then write Eq. (6) as

We can define the occupation number at the detector as an average of the occupation numbers of all photon sources:

Given the detector’s quantum efficiency and the detected mode’s effective brightness temperature, the covariance between detectors $i$ and $j$ in Eq. (12) can now be written as

When $i = j$, Eq. (36) is consistent with the standard bolometer noise model (see, e.g., Ref. [14] and references therein). Each optical element can be simply expressed by its transmission, emission, and scattering, as shown in Fig. 4. The scattering matrix can then be written as

 

Fig. 4. Schematic showing how optical elements contribute to the impinging photons on a detector. Each optical element $\rho = 1,2,3, \cdots$ has a thermal emissivity ${\epsilon _\rho}(\nu)$ and temperature ${T_\rho}$. At the same time, each optical element $\rho = 1,2,3, \cdots$ scatters a fraction ${\delta _\rho}(\nu)$ of photons from sources $\rho = \delta ;\!1, \delta ;\!2, \delta ;\!3, \cdots$ into the line of sight. The transmission efficiency of the optical element $\rho$ is then given as ${\eta _\rho}(\nu) = 1 - {\epsilon _\rho}(\nu) - {\delta _\rho}(\nu)$.

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Equations (36) and (40) lead to a formalism consistent with that presented in literature, (e.g., Ref.  [14]). In our convention defined by Eq. (36), $\sqrt \tau {\sigma _{{ii}}}$ is the photon noise NEP with an S.I. unit of ${\rm W} \cdot \sqrt {\rm s}$ and may differ by a factor $\sqrt 2$ when compared to literature where the NEP is often presented in ${\rm W/}\sqrt {{\rm Hz}}$.

3. MODEL OPTICAL SYSTEM

Using the photon noise formulation in Eq. (36), we now move to quantify the impact of HBT correlations on the sensitivity of telescopes for mm-wave astronomy. The topic of intensity correlations from astronomical sources with various states of coherence is discussed extensively in the literature [5,6,27–31] and, in the sections that follow, we apply these findings to the mm-wave telescope design. Modern mm-wave telescopes employ a wide variety of lens and mirror systems, infrared filter stacks, anti-reflection coatings, and sensing architectures. Despite this variety in real experiments, we can distill a few key instrument characteristics to create a simple yet representative optical system in which to study the sensitivity impact of HBT correlations. More specifically, this simple system, characterized by its focal ratio and the effective brightness temperatures inside and outside its pupil stop, emulates the radiation environment at the focal plane of general, modern millimeter and submillimeter telescopes with sufficient accuracy.

Our goal is to calculate intensity correlations within a practical telescope system with ${\sim}10 \%$ accuracy. As shown in the following sections, the HBT correlation coefficient has a maximum sensitivity impact of ${\cal O}(10 \%)$; therefore, our HBT accuracy goal maps to a sensitivity accuracy of ${\sim}1 \%$, which is typically sufficient for the purposes of the telescope design. While our idealized optics and detector focal planes may not exactly reproduce those of real telescopes, they capture the characteristics needed to calculate the HBT coefficient of real systems with ${\sim}10 \%$ accuracy.

A. Telescope

The assumed example telescope model is depicted in Fig. 5. It consists of an objective lens with focal length ${f_{{\rm obj}}}$ within a blackened enclosure at ${T_{{\rm stop}}} = 4\;{\rm K}$. The cold box has a circular aperture of diameter ${D_{{\rm ap}}}$ at $z = {z_{{\rm ap}}}$ that truncates the incoming radiation from external field-filling sources. The objective lens, which is both cold and transparent, focuses the aperture-truncated radiation onto a circular focal plane at $z = {z_{{\rm fp}}}$ with a size determined by the telescope’s plate scale. The focal plane houses an array of close-packed detector pixels with diameter ${D_{{\rm pix}}}$ operating at ${T_{{\rm fp}}} = 0.1\;{\rm K}$. The detector + telescope system is assumed to be diffraction limited such that the optical throughput per mode is $A\Omega = {\lambda ^2}$.

 

Fig. 5. Assumed optical model for all calculations and simulations in this paper. The model includes an objective lens, aperture stop, and focal plane filled with an array of sensing antennas coupled to planar detectors. In the reverse-time sense, each pixel emits a collection of rays, defined by spherical coordinate $({\theta _{{\rm pix}}},{\phi _{{\rm pix}}})$, which the objective uniquely maps onto aperture–plane coordinate ${\vec r_{\rm a}}$ with idealized polarization fidelity and telecentricity. The system is enclosed in a black box of temperature ${T_{{\rm stop}}}$, and the focal plane is cooled to ${T_{{\rm fp}}}$.

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This model optical system does not include many common features of real telescopes—such as fore-optics, thermal filters, or additional lenses—which are needed to form high-fidelity images over a moderate field of view (FOV). Such details are experiment-dependent and are therefore beyond the scope of this paper, but we can capture their effects by imposing several assumptions onto our simple system. These assumptions are not strictly necessary to calculate photon noise correlations, as the scattering matrix formalism in Section 2.A is completely general, but they simplify the correlation calculation significantly while encapsulating the salient features of practical instruments.

First, we assume that all sources—both external and internal to the telescope—are isothermal blackbody emitters large enough to uniformly illuminate the aperture across the telescope’s FOV. The assumption of blackbodies allow us to readily evaluate each mode’s occupation number using the Bose–Einstein distribution $n(\nu ,{T_{\rm b}})$ in Eq. (4), given each source’s effective brightness temperature ${T_{\rm b}}$. The assumption that each source is FOV-filling [32] and isothermal generalizes the correlation integrals that follow and is a good approximation for experiments whose incoming photon power is mainly from extended sources. In practice, thermal gradients develop across optical elements and atmospheric brightness varies with elevation and cloud structure; however, these variations are typically small and experiment-dependent and are therefore beyond the scope of this paper.

Secondly, we assume a diffraction-limited, single-mode optical system that converts stop-truncated plane waves into spherical waves converging onto a telecentric focal plane. In other words, pixel rays with angle $({\theta _{{\rm pix}}},{\phi _{{\rm pix}}})$ are mapped via the objective lens onto parallel rays with aperture–plane location ${\vec r_{{\rm ap}}}$ (see Fig. 5). In this configuration, the optical path length between any given detector pixel and a spot on the aperture stop is identical regardless of $({\theta _{{\rm pix}}},{\phi _{{\rm pix}}})$ or ${\vec r_{{\rm ap}}}$, which simplifies the calculations that follow. We note that when re-imaging optics and a pupil stop are employed, the modes which map onto spherical waves at the focal plane do not in general correspond to plane waves passing through the pupil stop. However, such re-imaging optics can always be modeled as a simplified equivalent system with an aperture stop provided that each optic’s clear-aperture diameter is large enough to pass all pupil-permitted modes.

Thirdly, we assume an ideal aperture stop, such that all detector pixels have the same mapping between ray angle $({\theta _{{\rm pix}}},{\phi _{{\rm pix}}})$ and aperture plane location ${\vec r_{{\rm ap}}}$ (see Fig. 5). In other words, the aperture illumination is identical regardless the detector pixel location [33]. Strictly speaking, this condition is not generally satisfied for a system with a large FOV, as telecentricity and aperture truncation may differ significantly between the central and peripheral regions of the focal plane. However, as we discuss later, photon noise correlations arise predominantly between the neighboring pixels where such non-idealities are negligible.

Fourthly, we assume that the telescope optics achieve polarization fidelity across the focal plane. As shown in Fig. 6, incident linearly polarized plane waves with the propagation direction $({\theta _{{\rm in}}},{\phi _{{\rm in}}})$ and orthogonal polarization vectors $\hat e_1^{L3}({\theta _{{\rm in}}},{\phi _{{\rm in}}})$ and $\hat e_2^{L3}({\theta _{{\rm in}}},{\phi _{{\rm in}}})$ are focused onto detector pixels as spherical waves with Ludwig-3 polarization distributions (Eq. (22) [20]):

 

Fig. 6. Schematic of the ideal optics that focus incident plane waves onto detectors as spherical waves with Ludwig-3 polarization distributions. In this example, the incident wave has a propagation direction $({\theta _{{\rm in}}},{\phi _{{\rm in}}})$ and polarization direction $\hat \varepsilon = \hat e_1^{L3}({\theta _{{\rm in}}},{\phi _{{\rm in}}})$, and it is converted to a spherical wave with the polarization direction $\hat \varepsilon = \hat e_1^{L3}({\theta _{{\rm pix}}},{\phi _{{\rm pix}}})$ for all $({\theta _{{\rm pix}}},{\phi _{{\rm pix}}})$ mapped within the aperture stop.

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Finally, we assume that the objective lens is cold and transparent such that its emission and scattering terms are negligible compared to those of other internal and external thermal sources.

B. Focal Plane

The assumed focal plane model is shown in Fig. 7. We assume single-mode, dual-polarization detector pixels with circular apertures and diffraction-limited Gaussian beams. Each pixel’s angular response function is determined solely by its beam waist ${w_0}$ and takes the far-field form

 

Fig. 7. Assumed layout of detector pixels on the focal plane at ${D_{{\rm pix}}} = 1.2F\lambda$ spacing. Each pixel’s angular response is assumed to be a Gaussian whose width scales with the pixel diameter, as described in Eq. (43). Additionally, each pixel has two polarimeters (dotted lines) that sense orthogonal polarizations and whose noise outputs do not correlate, given the idealized optical system described in Section 3. The neighboring pixels are rotated by ${\pm}45\deg$ to minimize Stokes Q HBT correlations.

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Plugging Eq. (44) into Eq. (43) yields a simple relationship between ${D_{{\rm pix}}}$ and the aperture stop spillover efficiency:

While detector pixels can be arranged in a variety of ways, we must select a specific focal plane arrangement to find ${| {{\gamma _{{ij}}}(\nu)} |^2}$ explicitly. In the calculations that follow, we assume hex-packed circular pixels, and we assume that the pixel pitch is equal to pixel diameter ${p_{{\rm pix}}} = {D_{{\rm pix}}}$. This assumption allows us to relate the pixel packing density to the pixel size as ${n_{{\rm pix}}} \propto D_{{\rm pix}}^{- 2}$, where ${n_{{\rm pix}}}$ is the number of pixels per unit focal plane area. In practice, a small amount of dead space typically exists between pixels that does not scale with the pixel size and, in this case, a more complex relationship between ${n_{{\rm pix}}}$ and ${D_{{\rm pix}}}$ is needed [10]. However, these details are experiment specific and are therefore beyond the scope of the following discussions.

We note that the correlation calculation in Section 4 does not rely on most of the assumptions presented in this subsection. Specifically, only the detector beam’s assumed polarization properties are relevant to the HBT coefficient estimation. All the other assumptions related to the shape of the detector beam, the beam waist, and pixel packing are used only to calculate an explicit instrument sensitivity in Sections 5 and 6.

4. CORRELATION CALCULATION

Given the optical and detector models presented in Section 3, we now find the correlation patterns at the focal plane due to the thermal radiation within the aperture and from the stop. A schematic of the radiation model for the central detector pixel is shown in Fig. 8.

 

Fig. 8. Schematic of the aperture, stop, and pixel radiation models. The radiation within the aperture is decomposed into a basis set of plane waves, and the normally incident mode $({k_x},{k_y},{k_z}) = (0,0,2\pi /\lambda)$ is depicted here. The plane waves are assumed to uniformly illuminate a diameter ${D_{{\rm illum}}} \gt {D_{{\rm ap}}}$, and each mode’s Gaussian-random amplitude is defined by its effective brightness temperature ${T_{{\rm ap}}}$. The radiation from the stop is generated by a collection of blackbody point sources, which emit spherical Lambertian wavelets located at $(x,y,z) = (u,v,{z_{{ap}}})$ with $\sqrt {{u^2} + {v^2}} \ge {D_{{\rm ap}}}/2$. Both the aperture’s plane-wave modes and the stop’s wavelet modes couple to the objective lens, which focuses the thermal light onto the focal plane. A detector pixel then senses the incoming spherical wave via the Gaussian angular response in Eq. (43).

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A. Stop Radiation

The stop is located in the far field of the detectors and is effectively a black, annular source with temperature ${T_{{\rm stop}}}$. Therefore, we model the stop as a collection of infinitesimal, Gaussian-random, uncorrelated thermal emitters that generate Lambertian spherical wavelets, as shown in Fig. 8. These point sources represent atomic thermal motion within the stop’s absorbing material, and their wavelets superpose to form incoherent waves that the objective lens focuses onto the focal plane.

 

Fig. 9. Simulated correlation patterns at the focal plane due to the radiation within the aperture (left) and radiation from the stop (right). The points represent the Monte Carlo simulations described in Section 4.C, and the lines show the expectation of the van Cittert–Zernicke theorem for an aperture/stop with uniform illumination.

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The above stop radiation treatment relies on two assumptions, which we justify here. First, while we consider the stop as a collection of uncorrelated thermal sources, it is known that blackbody radiators have a non-zero correlation over the distance of a wavelength [5,27,38,39]. However, as we will show in Sections 4.B, 4.C, and Appendix B, this discrepancy leads to negligible errors for the calculations in this paper. Secondly, we assume for simplicity that all stop radiation reaches the detectors by propagating through all optics between the aperture plane and the focal plane, which impractically requires infinite optical throughput. That said, the cold box’s radiation environment has temperature ${T_{{\rm stop}}}$, and the detectors sense that radiation when ${\theta _{{\rm pix}}} \gt {\theta _{{\rm stop}}}$, regardless of the optical configuration. Therefore, our simplifying assumption of infinite optical throughput accurately accounts for stop-generated photons at each detector’s input.

B. Aperture Radiation

Radiation incident on the sky side of the aperture (right side of Fig. 8) can be regarded as blackbody emission [40,41] with an effective brightness temperature ${T_{{\rm (ap)}}}$. Here ${T_{{\rm (ap)}}}$ is defined such that $n({T_{{\rm (ap)}}},\nu)$ is the mean occupation number of the blackbody radiation within the aperture. This mean occupation number can be written by adopting the definitions in Section 2.D [Fig. 4 and Eqs. (37) –(39)] and using ${{\cal H}_\rho}(\nu)$, ${\epsilon _\rho}(\nu)$, ${\delta _\rho}(\nu)$ as

We then consider a virtual, infinitely large sheet at $z = {z_{{\rm ap}}}$ that consists of Gaussian-random, infinitesimal, uncorrelated thermal emitters of temperature ${T_{{\rm (ap)}}}$ that generate Lambertian spherical wavelets. The radiation from these virtual emitters has equivalent statistical properties to the sky-side radiation in Fig. 8 as long as the distances from the sheet are significantly larger than the wavelength $\lambda$. To calculate this virtual sheet’s aperture truncation, we can simply remove elements outside the aperture area $\sqrt {{x^2} + {y^2}} \ge {D_{{\rm ap}}}/2$. In Section 4.C and Fig. 9, we compare aperture-truncated plane waves with this sheet of emitters and demonstrate their equivalence.

There is a small error that arises from the presented aperture emitter treatment. As noted above, the virtual sheet’s statistics become that of Chajnantor plateau blackbody radiation at distances sufficiently larger than $\lambda$ and, as noted in Section 4.A, non-zero correlations arise between emitters separated by $\lesssim \lambda$. Thus, the emitter–sheet model deviates from that of aperture truncation in the region of $z \simeq {z_{{\rm ap}}}$ and ${D_{{\rm ap}}}/2 - \lambda \lesssim \sqrt {{x^2} + {y^2}} \lesssim {D_{{\rm ap}}}/2 + \lambda$. Therefore, when $\lambda \gtrsim {D_{{\rm ap}}}$, aperture truncation produces additional non-trivial correlations that include polarization [43] but, when $\lambda \ll {D_{{\rm ap}}}$, as is true for any reasonable telescope design, the correction can be safely neglected.

In summary, the VCZT coefficient ${\gamma _{{ij}}}(\nu)$ can be calculated using the radiation field from a sheet of infinitesimal thermal emitters at $z = {z_{{\rm ap}}}$ with temperature ${T_{{\rm stop}}}$ at $\sqrt {{x^2} + {y^2}} \ge {D_{{\rm ap}}}/2$ and ${T_{({\rm ap})}}$ at $\sqrt {{x^2} + {y^2}} \lt {D_{{\rm ap}}}/2$.

C. Intensity Correlation Patterns

Here we calculate spatial correlation patterns for the geometry in Fig. 5 while neglecting the polarization degree of freedom, similar to Section 2.B. We then consider polarization in the next subsection. Consider the classical-wave electric field amplitude ${E_i}$ detected by detector $i$ at location $(x,y,z) = ({x_i},{y_i},{z_{{\rm fp}}})$. The detected partial amplitude $\Delta {E_i}$ due to an infinitesimal source area at the aperture plane $(x,y,z) = (u,v,{z_{{\rm ap}}})$ can then be written as

The illumination function ${G_i}(u,v)$ is mapped from the detector’s angular-response function [Eq. (43)] via the telescope’s optics. We adopt the following normalization:

The non-polarized mutual intensity between detectors $i$ and $j$ can be written as a sum of contributions from the aperture radiation and stop radiation:

The stop radiation component $B_{{\rm stop},ij}^{{\rm np}}(\nu)$ and $\gamma _{{\rm stop},ij}^{{\rm np}}(\nu)$ can be calculated similarly to Eqs. (52) and (53), but using ${T_{{\rm stop}}}$ instead and integrating over $\sqrt {{u^2} + {v^2}} \ge {D_{{\rm ap}}}/2$:

Figure 9 shows spatial HBT correlation patterns for both the aperture and stop radiation calculated using their respective VCZT coefficients. As a demonstration, Fig. 9 also shows Monte-Carlo simulations where we directly calculate the (classical) intensity correlation coefficient by propagating fields to detectors $i$ and $j$ from complex Gaussian-random emitters on the stop and from complex Gaussian-random plane-wave modes incident on the aperture. We set the aperture diameter to ${D_{{\rm ap}}} = 100\lambda$, the stop’s outer diameter to $2{D_{{\rm ap}}}$, which simulates the case of ${\sigma _{{\rm ap}}} = {D_{{\rm ap}}}/2$, and the f-number to $F = 2.5$. We then simulate 3,000 realizations of the aperture and stop fields. For radiation from the stop, we integrate Gaussian-random emitters over a grid with the cell size ${\lambda ^2}$ [44]. For radiation within the aperture, we simulate 2,500 total plane-wave modes (50 for each of the $x$ and $y$ wave-number dimensions) distributed over the $2\pi {\rm sr}$ incident solid angle, which is enough to resolve the aperture’s edges and Nyquist sample its integration grid. As expected, the Monte-Carlo simulation yields consistent results with the semi-analytic calculation of HBT coefficients in Eqs. (53) and (56).

Combining the aperture- and stop-radiation components, we obtain the VCZT coefficient for the assumed optical system:

D. Polarized Correlation Patterns

Following the assumptions for telescope optics in Section 3.A and detector parameters in Section 3.B, we show here that the polarized correlation pattern becomes the same as those derived in Section 2.C.

We consider detector pixels $i$ and $j$ and assume that each pixel $i$ has two detectors $i1$ and $i2$ with orthogonal polarization angles ${\psi _i}$ and ${\psi _i} + \pi /2$ following the beam patterns defined in Eqs. (23) and (24), respectively. As described in Section 2.C, each infinitesimal thermal source area on the aperture plane emits radiation with two independent Ludwig-3 polarizations $\hat e_1^{L3}$ and $\hat e_2^{L3}$. This radiation couples to polarized plane-wave modes $\hat e_1^{L3}({\theta _i},{\phi _i})$ and $\hat e_2^{L3}({\theta _i},{\phi _i})$ which propagate along the direction $({\theta _i},{\phi _i})$, and the telescope transforms these plane waves into spherical waves with polarization patterns $\hat e_1^{L3}({\theta _{i,{\rm pix}}},{\phi _{i,{\rm pix}}})$ and $\hat e_2^{L3}({\theta _{i,{\rm pix}}},{\phi _{i,{\rm pix}}})$, as described in Eq. (41). In the reverse-time sense, the plane-wave modes radiating from detectors $i1$ and $i2$ correspond to plane-wave polarization vectors:

Given the above assumptions, the polarization degree of freedom introduces a factor of $\cos {\psi _i}$ and $\sin {\psi _i}$ for the coupling between detector $i1$ and the $\hat e_1^{L3}$ and $\hat e_2^{L3}$ emitters, respectively, with similar behavior for detector $i2$. The rest of the calculation proceeds exactly as in Section 2.C, following the results in Eqs. (26), (27), (28), and (29) that relate the unpolarized HBT coefficient $\gamma _{{ij}}^{{\rm np}}(\nu)$ to the Stokes $Q$ coherence as

It is worth noting that the results derived here indicate the zero correlation between two orthogonal detectors on different pixels $i1$ and $j2$ when ${\psi _i} = {\psi _j}$. This finding contrasts the situation for planar blackbody radiators (or absorbers), where a non-zero correlation structure arises between $\hat x$ radiation from one location and $\hat y$ radiation from another [38]. This difference comes from our assumption that the detector beam has an ideal Ludwig-3 polarization pattern with $E/H$-plane symmetry. For emitters on a surface, the modes project onto the $\hat x$ and the $\hat y$ directions with a factor of $\cos {\theta _i}$ and therefore are not $E/H$-symmetric, hence introducing coherence along the two orthogonal directions [45].

E. VCZT’s Assumptions and Applicability

The van Cittert–Zernike theorem (VCZT) relies on a few assumptions; thus, its application has limitations (e.g., see Ref. [46] and references therein). We now comment on these assumptions and justify the applicability of the VCZT formalism to the intensity coherence calculation for millimeter and submillimeter telescopes.

First, the VCZT in its simplest form assumes scalar waves without a polarization degree of freedom and that the detectors are in the far field of the source so that the Fraunhofer approximation can be used. That said, our $S$-matrix based formalism accommodates general mode-to-mode coupling and therefore does not rely on these assumptions. Secondly, the VCZT assumes completely incoherent sources, but our formalism assumes blackbody radiators, which exhibit a coherence length of ${\sim}\lambda$. Appendix B shows that this one-wave coherence length can be ignored and therefore that our application of the VCZT to blackbody sources is justified. Thirdly, when considering amplitude coherence, we require $|{R_i} - {R_j}| \ll c/\Delta \nu$ to avoid the problem of decoherence. In contrast, decoherence does not occur for intensity coherence, as noted in Section 2.A; therefore, no such bandwidth limit exists for calculations in this paper.

5. IMPACT OF CORRELATIONS ON SENSITIVITY

Using Eqs. (59), (17), and (62), we now investigate the impact of detector-to-detector correlations on instrument sensitivity, which is the primary goal of this paper. As shown in Fig. 9, detector outputs can correlate if their pixel pitch is ${D_{{\rm pix}}} \lt 1.2F\lambda$, and these correlations will slow noise averaging during coaddition and therefore degrade array sensitivity [47]. In this section, we introduce a formalism for the mapping speed, which measures the total sensitivity of the detector array, and we inspect the impact of HBT correlations on the mapping speed versus the pixel size, which is a key metric used for the focal plane design [48].

A. Mapping Speed

We assume that each detector in the imaging array has three noise components: photon shot noise, photon wave noise, and internal noise. The covariance between the detected modes $i$ and $j$ is

To both simplify and clarify the calculations that follow, we assume that all detectors in the array have the same noise properties. Internal detector noise and photon shot noise cannot correlate between outputs, which allows the covariance to be written as

Given the simplification in Eq. (65) and noting that $\gamma _{{ii}}^{(2)} = 1$, the total variance of the detector array can be written as

Internal detector noise ${\sigma _{{\rm int}}}$ depends on many factors, including the detector’s architecture, thermal noise properties, amplifier noise properties, linearity, and dynamic range. To remain agnostic to these experiment-specific characteristics, we set $\sigma _{{\rm int}}^2/\sigma _{{\rm ph}}^2 = 0.1$ hereafter, noting that modern mm-wave observatories aim to be photon noise dominated [49–52].

Finally, we quantify the experiment’s signal-to-noise using its mapping speed, which is defined as the square ratio of the input signal $S$ to the array-averaged noise ${\sigma _{{\rm arr}}}$:

B. Pixel Size Optimization

Using the optics and detector assumptions in Section 3 and the VCZT and HBT coefficients in Eqs. (59), (17), and (62), we now calculate the mapping speed versus the pixel size. Provided a fixed FOV, or equivalently a fixed focal plane size, decreasing pixel diameter ${D_{{\rm pix}}}$ increases the number of detectors as ${N_{{\det}}} \propto D_{{\rm pix}}^{- 2}$, but decreases the aperture spillover efficiency as ${\eta _{{\rm ap}}} \propto \exp [- {({\pi {D_{{\rm pix}}}/(F\lambda {w_{\rm f}})})^2}/2]$. These competing effects combine to form a peak in the mapping speed versus the pixel size, which reveals the optimal packing density.

 

Fig. 10. Example of a “classic” mapping speed calculation—which ignores the impact of HBT correlations—for a 90 GHz telescope with a 4 K aperture stop. The $MS$ peak arises from the opposing effects of more detectors to average versus less aperture efficiency with the decreasing pixel size. The optimum when ignoring correlations is ${\sim}0.7 F\lambda$ but, as shown below, the addition of the HBT coefficient modifies this classic curve.

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Fig. 11. Assumed observation bands plotted over the assumed CMB, atmosphere, and telescope temperatures (left), and the wave-noise fraction versus the sky temperature for each band at ${D_{{\rm pix}}} = 3F\lambda$, where ${\eta _{{\rm ap}}} \gt 0.95$ (right).

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An example “classic” mapping speed versus the pixel size curve [53–57]—one for which ${\gamma ^{(2)}} \equiv 0$—of a model 90 GHz instrument with a 4 K stop is shown in Fig. 10. Historically, ground-based CMB experiments have observed at 95, 150, and 220 GHz with ${D_{{\rm pix}}} = 1 \sim 2F\lambda$ [54,55,58–65], where the HBT correlation coefficient is small. However, as new readout architectures become available [66,67] and as CMB experiments push to lower frequencies for improved synchrotron characterization [68–70], focal planes with ${D_{{\rm pix}}} \lesssim 1.2F\lambda$ are becoming increasingly practical. Therefore, the impact of HBT correlations on the mapping speed is of interest to upcoming mm-wave experiments, such as CMB-S4 [71]. In this section, we calculate the HBT-modified mapping speed versus pixel size curves for our model telescope at several observation frequencies, and we discuss the results.

We assume that our telescope is ground-based with cryogenically cooled optics, infrared filters, and sub-Kelvin detectors. As shown in Eq. (46), the radiation at the aperture plane can be summed over all sky-side sources and represented by a single effective brightness temperature ${T_{({\rm ap})}}$. To simplify and generalize the following analysis, we assume only three sources viewed through the aperture: the CMB with ${T_{{\rm CMB}}}$, the atmosphere with ${T_{{\rm atm}}}$, and telescope optics with ${T_{{\rm tel}}}$. The telescope’s effective temperature ${T_{{\rm tel}}}$ can vary considerably depending on the specifics of the mirrors, ground shield, cryostat window, and anti-reflection coatings but, as an example, we assert a default configuration where ${T_{{\rm tel}}} = 10\;{\rm K}$. We also assert the stop’s physical temperature to be ${T_{{\rm stop}}} = 4\;{\rm K}$ and that each detector’s quantum efficiency is ${\eta _{{\det}}} = 0.8$. To model power due to atmospheric emission, we assume that the telescope observes from the Chajnantor plateau in the Atacama Desert of Chile, and we use the AM model [72] to generate the atmosphere’s effective brightness temperature and transmittance at 1 mm precipitable water vapor (PWV) and 50 deg elevation above the horizon.

 

Fig. 12. Stokes Q HBT coefficient $\gamma _{{ij}}^{Q,(2)} = \cos 2({\psi _i} - {\psi _j}) \gamma _{{ij}}^{(2)}$ given pixel pitches ${D_{{\rm pix}}} = ({0.2,0.5,0.7,1.0})$ $F\lambda$ from top to bottom for the radiation from within the aperture (left), from the stop (right), and their cross- term (middle). When the pixel pitch is $\lesssim \!F\lambda$, the detectors oversample the spatial modes on the focal plane, giving rise to intensity correlations between nearby pixels.

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Given this telescope + sky model, we consider four top-hat observation bands centered at 35, 95, 150, and 220 GHz with bandwidths of 17, 33, 39, and 44 GHz. The chosen bands, along with ${T_{{\rm CMB}}}$, ${T_{{\rm atm}}}$, and ${T_{{\rm tel}}}$, are shown in Fig. 11. The channels are chosen to fit within atmospheric windows and are similar to those of existing Atacama instruments. Additionally, Fig. 11 shows the bunching fraction for each frequency band versus the sky temperature at ${D_{{\rm pix}}} = 3F\lambda$, where ${\eta _{{\rm ap}}} \gt 0.95$. As discussed in Section 5.A, while the lower-frequency modes tend to have larger occupation numbers, the sky is brighter at higher frequencies; therefore, the correlations substantially impact sensitivity in all four bands.

Figure 12 shows an example of four pixel pitch scenarios ${D_{{\rm pix}}} = (0.2,0.5,0.7,1.0) F\lambda$ given the hex packing described in Section 3.B, and the contour shows the Stokes $Q$ HBT coefficient $\gamma _{{ij}}^{Q,(2)}$ due to both the aperture and stop radiation. As expected, the degree of correlation is a two-dimensional version of the VCZT curves in Fig. 9. The effect of pushing ${D_{{\rm pix}}} \lesssim F\lambda$ is for the detected modes (the pixel apertures) to oversample the input mode (the $\gamma _{{ij}}^{Q,(2)}$ contour), giving rise to correlated noise between nearby detectors.

 

Fig. 13. Stokes $Q$ mapping speed versus the pixel size in each frequency band for four sets of external sources illuminating the aperture: the CMB only with ${T_{{\rm CMB}}} = 2.725\;{\rm K}$ (top left), the atmosphere only assuming 1 mm PWV and 50 deg elevation at the Chajnantor plateau (top right), the telescope emission only assuming ${T_{{\rm tel}}} = 10\;{\rm K}$ (bottom left), and all three sources combined (bottom right). We assume that each detector’s quantum efficiency is ${\eta _{{\det}}} = 0.8$ and that $\sigma _{{\rm int}}^2 = 0.1(\sigma _{{\rm shot}}^2 + \sigma _{{\rm wave}}^2)$. Each band is normalized to the peak of its classic curve, which is shown as faded lines. Therefore, the opaque curves represent the achievable mapping speed when ${\gamma ^{(2)}} \ne 0$ with respect to the maximum of the ${\gamma ^{(2)}} \equiv 0$ case. The pixel size is plotted in units of $F\lambda$, where $\lambda$ is the mean wavelength in each band.

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When calculating the Stokes $Q$ mapping speed, we sum the HBT coefficients in Fig. 12 over a 4 $F\lambda$ radius to find

 

Fig. 14. Impact of stop temperature ${T_{{\rm stop}}}$ (left column), telescope temperature ${T_{{\rm tel}}}$ (middle column), and detector internal noise $\sigma _{{\rm int}}^2$ (right column) on the Stokes $Q$ mapping speed versus the pixel size in the presence of HBT correlations for each observation band (rows). These parameters are among many that vary between experiments, and we include them here as a reference for focal plane designers. The default parameters from Fig. 13 of ${T_{{\rm stop}}} = 4\;{\rm K}$, ${T_{{\rm tel}}} = 10\;{\rm K}$, and $\sigma _{{\rm int}}^2/\sigma _{{\rm ph}}^2 = 0.1$ are assumed when not being swept.

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There are several features in Fig. 13 that are worth noting explicitly. First, the impact of HBT correlations depends on the source temperature and is most pronounced in the presence of a brightly illuminated aperture. This effect is most clearly seen when contrasting the CMB and atmosphere, especially at 220 GHz where the CMB’s photon occupation number is falling while that of the atmosphere is rising. Secondly, while HBT correlations impact curve shape most prominently at low frequencies, the atmosphere is brighter at higher frequencies, inducing a similar HBT suppression across all bands. Thirdly, the mapping speed peak is located at a slightly larger ${D_{{\rm pix}}}$ than that of the classic curves at 35 and 95 GHz, but resides at a similar location to that of the classic curves at 150 and 220 GHz. This effect arises because the stop is significantly fainter than the sky at 150 and 220 GHz; therefore, as ${D_{{\rm pix}}}$ falls below $1.2F\lambda$, $(1 + {\gamma ^{(2)}})\sigma _{{\rm wave}}^2$ decreases less rapidly than ${N_{{\det}}}\eta _{{\rm ap}}^2$. Lastly, the impact of correlations on the mapping speed starts to become most important when ${D_{{\rm pix}}} \lt 1.2F\lambda$, but there also percent-level impacts at larger spacings, which correspond to local maxima in the aperture/stop VCZT patterns, as shown in Fig. 9. This effect gets smoothed out when averaging the HBT coefficients across each channel’s finite bandwidth. Regardless of the input assumptions in this section, the mapping speed gain by undersized pixels when ${\gamma ^{(2)}} \ne 0$ is suppressed compared to the classic ${\gamma ^{(2)}} \equiv 0$ case, especially for ground-based telescopes.

6. IMPLICATIONS FOR EXPERIMENT DESIGN

As shown in Fig. 13, HBT correlations both modify the optimal pixel packing density and suppress the achievable mapping speed with respect to the “classic” ${\gamma ^{(2)}} \equiv 0$ calculation. However, the degree of modification depends on a plethora instrument details, including internal detector noise, observation site and conditions, stop temperature, detector efficiency, telescope optical throughput, and extraneous noise sources, such as electromagnetic interference, vibrational pickup, and detector nonidealities. A more comprehensive handling of correlations within a more general experiment is available via the BoloCalc sensitivity calculator [14] but, in this section, we sweep a few parameters in our model telescope to serve as a quick reference for focal plane designers. The results of these calculations are shown in Fig. 14.

The first column of Fig. 14 shows the mapping speed versus the pixel size for various stop temperatures ${T_{{\rm stop}}} = (2,3,4,5){\rm K}$. As the stop temperature decreases, so does photon loading due to stop spillover which, in turn, favors smaller pixels. A colder stop also suppresses the relative contribution of ${\sigma _{{\rm wave}}}$, especially at higher frequencies, modulating the slope of the mapping speed curve below ${D_{{\rm pix}}} = 1.2F\lambda$. The second column of Fig. 14 shows the mapping speed versus the pixel size for various telescope temperatures ${T_{{\rm tel}}} = (10{,}20{,}30{,}40){\rm K}$. As the telescope temperature increases, so does the photon load within the aperture which, in turn, favors smaller pixels. In addition, the brighter aperture radiation increases ${\sigma _{{\rm wave}}}$ and hence also increases the HBT suppression. The third column of Fig. 14 shows the mapping speed versus the pixel size in the presence of a constant internal detector noise $\sigma _{{\rm int}}^2 = (0.1,0.5,0.7,1.0) \times \sigma _{{\rm ph}}^2$ at ${D_{{\rm pix}}} = 1.2F\lambda$, where $\sigma _{{\rm ph}}^2 \equiv (\sigma _{{\rm shot}}^2 + \sigma _{{\rm wave}}^2)$. As ${\sigma _{{\rm int}}}$ increases with respect to ${\sigma _{{\rm ph}}}$, larger pixel sizes are favored to improve signal strength via an increased ${\eta _{{\rm ap}}}$. Simultaneously, the impact of HBT correlations is reduced due to a smaller relative contribution of ${\sigma _{{\rm wave}}}$ and due to an optimum ${D_{{\rm pix}}} \gtrsim 1.2F\lambda$ where ${\gamma ^{(2)}}$ is small.

These few examples only graze the rich topic of focal plane optimization, and we leave a more comprehensive discussion of experiment-specific applications to other publications. Nonetheless, regardless of the context, HBT correlations should be considered when designing dense focal planes, especially for ground experiments where the sky and telescope brightness temperatures are substantially larger than that of the CMB.

7. CONCLUSION

We have presented a theoretical formalism for photon noise correlations by extending the quantum optical circuit-based model in Zmuidzinas [11] to a free-space classical model using the optical equivalence theorem of Glauber and Sudarshan [15]. We have used this formalism to estimate the Hanbury Brown-Twiss (HBT) coefficient [7–9] in a simplified telescope optical system, and we have shown that these simulations match the expectation of the van Cittert–Zernike theorem (VCZT) [12,13]. This equivalence allows the HBT coefficient to be calculated with only a knowledge of the radiation intensity profile at the aperture plane.

We then uniformly illuminated our model telescope with blackbody sources representative of the radiation from the CMB, atmosphere, and telescope, and we have calculated the impact of HBT correlations on the experiment mapping speed versus the pixel size within observation bands centered at 35, 95, 150, and 220 GHz. Acknowledging that sensitivity calculations have many inputs and assumptions, we have further discussed three useful variations to the simplified instrument—stop temperature, telescope temperature, and internal detector noise—and showed how each parameter modulates the HBT-modified mapping speed curves. This work builds on an initial discussion by Padin [10] and formalizes the calculation of photon noise correlations between detector pixels in millimeter and submillimeter telescopes for astronomy. The presented formalism and results are useful to gigahertz focal plane designers, especially as emerging readout technologies enable the deployment of dense detector arrays.

APPENDIX A: THERMAL PHOTON DENSITY MATRIX

The statistical state, or the mixed state, of photons can be described using the density matrix $\hat \rho$. We consider a single-mode photon state, where the state is single-mode in both spatial and frequency domains as well as in the polarization state. Using creation and annihilation operators ${a^\dagger}$ and $a$, respectively, the density matrix can be written as that of a Bose–Einstein distribution:

(A1)$$\hat \rho = \frac{{{e^{- \gamma {a^\dagger}a}}}}{{{\rm Tr}\left({{e^{- \gamma {a^\dagger}a}}} \right)}} = \sum\limits_n p(n;\bar n)\left| n \right\rangle \left\langle n \right| ,$$

with

(A2)$$\gamma \equiv \frac{{h\nu}}{{{k_{\rm B}}T}}\quad {\rm and}\quad p(n;\bar n) \equiv \frac{1}{{1 + \bar n}}{\left({\frac{{\bar n}}{{1 + \bar n}}} \right)^n} .$$

Here $\{| n \rangle \}$ is the Fock state, and $\bar n$ is a mean occupation number. We consider a detection process whose integration time $\tau$ (the inverse of sampling rate) is significantly longer than the coherence time ${\tau _c} \equiv 1/\Delta \nu$, where $\Delta \nu$ is the detection bandwidth. This is usually the case for a CMB instrument, where $\Delta \nu \sim {\cal O}(10 \;{\rm GHz})$ and $\tau \sim {\cal O}(10 \;{\rm msec})$. In this situation, where $\tau \gg {\tau _c}$, the mean occupation number $\bar n$ can be written as

(A3)$$\bar n = n(T,\nu) = \frac{1}{{{e^\gamma} - 1}} .$$

We can now rewrite the density matrix in terms of Glauber’s coherent state $|\alpha \rangle$. The coherent state is written as

(A4)$$\left| \alpha \right\rangle \equiv {e^{\alpha {a^\dagger} - {\alpha ^*}a}}\left| 0 \right\rangle ,$$

and satisfies

(A5)$$a\left| \alpha \right\rangle = \alpha \left| \alpha \right\rangle .$$

The density matrix can be rewritten in the Glauber–Sudarshan $P$ representation [2,15] as

(A6)$$\hat \rho = \int {d^2}\alpha \,{p_g}(\alpha ;\bar n)\left| \alpha \right\rangle \left\langle \alpha \right|,$$

with

(A7)$${p_g}(\alpha ;\bar n) \equiv \frac{1}{{\pi \bar n}}\exp \left({- \frac{{|\alpha {|^2}}}{{\bar n}}} \right) ,$$

where the integral is over the entire complex plane. Here the complex amplitude $\alpha$ follows a Gaussian distribution ${p_g}(\alpha ;\bar n)$, in agreement with the expectation in the classical limit. The photon counting of a coherent state follows a Poisson distribution as

(A8)$$|\!\left\langle n \right.\!|\left. \!\alpha \right\rangle \!{|^2} = \exp \left({- |\alpha {|^2}} \right)\frac{{|\alpha {|^{2n}}}}{{n!}} .$$

Equations (A6) and (A8) immediately lead to a special case of Mandel’s formula [73,74]:

(A9)$$\left\langle {n\left| {\hat \rho} \right|n} \right\rangle = \int_0^\infty {\rm d}W\,{e^{- W}}\frac{{{W^n}}}{{n!}}\,\,\frac{{{e^{- W/\bar n}}}}{{\bar n}} = p(n;\bar n) .$$

APPENDIX B: PARTIAL COHERENCE OF SOURCES

In this paper, we assume complete incoherence between two source elements that are physically apart from one another. However, it is known that blackbody sources have a finite correlation at the length scale of a wavelength [5,27,38,39]. The effect of the source coherence on the applicability of the VCZT for quasihomogeneous sources, whose spatial intensity variations are slow compared to its coherence length, is extensively discussed in literature [6,28,29,75]. Thus, it is worthwhile to clarify the assumptions in this paper regarding the source coherence. In the end, we find that the assumption of completely incoherent sources is a good approximation for telescope systems relevant to our discussion. We first consider the source coherence for the simple case presented in Section 2.B, which is readily comparable to examples in the literature. We then discuss the source coherence in the general formalism of Section 2.A.

Equation (20), in combination with (10), constitutes the VCZT of a completely incoherent source for the simple geometry in Fig. 2. For sources with a partial coherence (e.g., see Ref.  [46]), the first-order coherence is

(B1)$$\Gamma _{{ij}}^{(1)} \propto \iint_\sigma {{\rm d}^2}\vec r\iint_\sigma {{\rm d}^2}\vec r^\prime {\gamma _c}(|\vec r - \vec r^\prime |) {e^{2\pi i\bar \nu ({{R^\prime}_j} - {R_i})/c}} ,$$

with

(B2)$${R_i} \equiv |{\vec r_i} - \vec r|\quad \quad \quad {R_{{j^\prime}}} = |{\vec r_j} - \vec r^\prime | ,$$

where ${\vec r_i}$ and ${\vec r_j}$ are the positions of detectors $i$ and $j$, respectively, and where ${\gamma _c}(|\vec r - \vec r^\prime |)$ is the coherence of the field between locations $\vec r$ and $\vec r^\prime $ on the source surface $\sigma$. Here, we assume that the source’s coherence is statistically isotropic and that the source coherence function can be written as ${\gamma _c}(\vec r,\vec r^\prime) = {\gamma _c}(|\vec r - \vec r^\prime |)$. Complete incoherence of the source corresponds to the limit of ${\gamma _c}(|\vec r - \vec r^\prime |) \to \delta (\vec r - \vec r^\prime)$, and we immediately find that Eq. (B1) reduces to Eq. (20) in such a limit.

We define a coherence length ${R_c}$ such that ${\gamma _c}(|\vec r - \vec r^\prime |) \simeq 0$ for $|\vec r - \vec r^\prime | \gt {R_c}$. For blackbody radiators, ${R_c} \sim \lambda$, since we assume detectors with a limited detection band [76]. Focusing on cases where ${R_c}$ is significantly smaller than the source size, Eq. (B1) can be approximated as

(B3)$$\Gamma _{{ij}}^{(1)} \propto \iint_\sigma {{\rm d}^2}\vec r\iint_{|\vec \Delta | \le {R_c}} {{\rm d}^2}\vec \Delta {\gamma _c}(|\vec \Delta |) {e^{2\pi i\bar \nu ({{R_j^\prime}} - {R_i})/c}},$$

(B4)$$\simeq \iint_\sigma {{\rm d}^2}\vec r {e^{- 2\pi i\bar \nu ({R_j} - {R_i})/c}} {\cal J}({\theta _r}),$$

with

(B5)$$\begin{split} {{\cal J}({\theta _r})}&{\equiv \iint_{|\vec \Delta | \le {R_c}} {{\rm d}^2}\vec \Delta {\gamma _c}(|\vec \Delta |) {e^{2\pi i\frac{{\bar \nu}}{c}\frac{{\vec r - {{\vec r}_j}}}{{{R_j}}} \cdot \vec \Delta}}}\\& = {\int_0^{2\pi} {\rm d}\phi \int_0^{{R_c}} s {\rm d}s {\gamma _c}(s) {e^{- 2\pi i\sin {\theta _r} \cdot \cos \phi \cdot s/\bar \lambda}}\,,}\end{split}$$

where ${\theta _r}$ is the angle between $\vec r - {\vec r_j}$ and the source plane’s normal vector. The expression in Eq. (B4) becomes equivalent to Eq. (20) when ${\cal J}({\theta _r})$ can be regarded as a constant function of $\vec r$.

To evaluate ${\cal J}({\theta _r})$, we consider a blackbody surface source that may not be centered at $x = y = 0$, and we assume that the solid angle of the source is small compared to $\pi$. We consider two regions of the parameter space depending on the source’s position. Region 1 is when ${\theta _r} \ll 1$. In this case,

(B6)$$\left| {\sin {\theta _r}\frac{{|\vec \Delta |}}{\lambda}} \right| \le |\sin {\theta _r}|\frac{{{R_c}}}{\lambda} \ll 1 ,$$

and thus ${\cal J}({\theta _r}) \simeq {\cal J}(0)$, and Eq. (B4) becomes equivalent to Eq. (20). Region 2 is when ${\theta _r} \gtrsim 1$. In this case, ${\theta _r}$ is approximately constant across the source $\sigma$, and thus ${\cal J}({\theta _r}) \simeq {\cal J}({\bar \theta _r})$ where ${\bar \theta _r}$ is the mean of ${\theta _r}$ for the source $\sigma$. The factor ${\cal J}({\bar \theta _r})/{\cal J}(0)$ maps to the radiance reduction due to source coherence, which shows up in $\Gamma _{{ii}}^{(1)}$ and $\Gamma _{{jj}}^{(1)}$ as well. Thus, the normalized amplitude coherence ${\gamma _{{ij}}}$ remains identical to the case of a completely incoherent source, which is consistent with the results presented in the literature [6]. When the source composes Lambertian blackbody emitters, the radiance satisfies ${\cal J}({\bar \theta _r})/{\cal J}(0) \simeq 1$ by construction.

In summary, for a source with the geometry assumed in Section 2.B, a partial coherence between spatially independent blackbody radiators can be neglected, and Eq. (20) is a good approximation.

We now look at Section 2.A, which considers a more general, VCZT-free formalism. For quasihomogeneous sources with a partial coherence; in contrast to completely incoherent sources, the Kronecker delta ${\delta _{{km}}}$ in Eq. (3) is replaced by a source coherence function ${\gamma _{c,km}}$. The mutual intensity is then expressed as

(B7)$${B_{{ij}}}(\nu) = \sum\limits_\sigma \sum\limits_{k,m \in \sigma} S_{{ik}}^*(\nu){S_{{jm}}}(\nu) \gamma _{c,km}^\sigma n({T_\sigma},\nu) ,$$

where the index $\sigma$ denotes each quasihomogeneous source with a temperature ${T_\sigma}$, and $\gamma _{c,km}^\sigma$ is the coherence function of the source. Equation (B7) is the generalized version of Eq. (B1). Similar to Eq. (B4), we can decompose Eq. (B7) as

(B8)$${B_{{ij}}}(\nu) \simeq \sum\limits_\sigma \sum\limits_{k \in \sigma} S_{{ik}}^*(\nu){S_{{jk}}}(\nu) n({T_\sigma},\nu) {{\cal J}_k} ,$$

with

(B9)$${{\cal J}_k} \equiv \sum\limits_{m \in \rho (k)} \frac{{{S_{{jm}}}(\nu)}}{{{S_{{jk}}}(\nu)}} \gamma _{c,km}^\sigma ,$$

where $\rho (k)$ is a collection of modes close enough to the mode $k$ such that $\gamma _{c,km}^\sigma$ is non-zero. The expressions for a normalized coherence, both ${\gamma _{{ij}}}$ and $\gamma _{{ij}}^{(2)}$, become identical to the case with completely incoherent sources if ${{\cal J}_k}$ is constant across $k \in \sigma$ [77]. Whether ${{\cal J}_k}$ can be regarded as a constant should be evaluated on a case-by-case basis.

For the cases discussed in this paper, we can regard ${{\cal J}_k}$ as a constant given the following arguments. For the aperture radiation, $k$ and $m$ correspond to the modes emitted by infinitesimal sources at $({u_k},{v_k})$ and $({u_m},{v_m})$, respectively. Using Eqs. (47) and (48), ${{\cal J}_k}$ can be written as

(B10)$$\begin{split}{{{\cal J}_k}}& = {\iint_{m \in \rho (k)} {\rm d}{u_m} {\rm d}{v_m} \frac{{{G_j}({u_m},{v_m})}}{{{G_j}({u_k},{v_k})}}}\\{}&\quad \cdot {{e^{2\pi i[({u_m} - {u_k}){x_j} + ({v_m} - {v_k}){y_j}]/{D_{{\rm ap}}}F\lambda}}\gamma _{c,km}^\sigma \,,}\end{split}$$

with the integrated region $m \in \rho (k)$ being

$$s \equiv \sqrt {{{({u_m} - {u_k})}^2} + {{({v_m} - {v_k})}^2}} \le {R_c} .$$

Since ${R_c} \sim \lambda \ll {D_{{\rm ap}}}$, the aperture illumination function varies minimally within the integrated range and thus ${G_j}({u_m},{v_m}) \simeq {G_j}({u_k},{v_k})$. The source coherence function $\gamma _{c,km}^\sigma$ is isotropic and thus depends only on $s$ as $\gamma _{c,km}^\sigma = \gamma _c^\sigma (s)$, leading to

(B11)$${{\cal J}_k} \simeq \int_0^{2\pi} {\rm d}\phi \int_0^{{R_c}} s {\rm d}s \gamma _c^\sigma (s) {e^{2\pi i \cdot s {d_j} \cdot \cos \phi /{D_{{\rm ap}}}F\lambda}} ,$$

with

$${d_j} \equiv \sqrt {x_j^2 + y_j^2} .$$

Thus, ${{\cal J}_k}$ is constant as a function of $k$, and the source coherence can therefore be ignored.

For the stop radiation, the physical geometry may be more complicated than for the aperture radiation; therefore, ${{\cal J}_k}$ may sometimes vary across the source. However, because ${{\cal J}_k}$ typically varies slowly compared to $\lambda$, we can divide the source into sections that are significantly larger than $\lambda$ and in which ${{\cal J}_k}$ is effectively constant. Given this setup, we can ignore the source coherence by regarding each of these sections as independent sources indexed by $\sigma$ in Eq. (B7).

In summary, for the model optical system presented in Section 3, the assumption of completely incoherent sources is a good approximation for the context of this paper.

 

Fig. 15. Simulated correlation at the focal plane due to the radiation within the aperture for four representative illumination patterns: ${\sigma _{{\rm ap}}} = (\infty ,{D_{{\rm ap}}}/2)$, which demonstrates the goodness of the uniform illumination approximation for Gaussian beams, and ${w_f} = (2,3)$, which demonstrates the goodness of the pixel-size-dependent illuminations in Eq. (43). Similar to Fig. 9, the points show the results of Monte Carlo simulations, while the lines show the expectation of the VCZT.

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APPENDIX C: GOODNESS OF FLAT-ILLUMINATION APPROXIMATION

C1. Aperture Radiation

In this section, we show that Eq. (55) is a good approximation of Eq. (52), even for a general Gaussian illumination function ${G_i}(u,v)$. The key assumption is that the beam can be approximated by Eqs. (43) and (44) and that the pixel spacing satisfies

(C1)$${p_{{ij}}} \ge {D_{{\rm pix}}} .$$

As described in Section 4.C and Eq. (54), we approximate the illumination function as

(C2)$${G_i}(u,v) = \frac{1}{{\sqrt {\pi \sigma _{{\rm ap}}^2}}}\exp \left({- \frac{{{u^2} + {v^2}}}{{2\sigma _{{\rm ap}}^2}}} \right) ,$$

and, as described in Eq. (43), the circumference of the aperture corresponds to an angle of $\theta \simeq 1/2F$. It thus follows that

(C3)$${\sigma _{{\rm ap}}} = {D_{{\rm ap}}}\frac{{{w_f}}}{{\sqrt 2 \pi}}\frac{{F\lambda}}{{{D_{{\rm pix}}}}} \ge {D_{{\rm ap}}}\frac{{{w_f}}}{{\sqrt 2 \pi}}\frac{{F\lambda}}{{{p_{{ij}}}}} ,$$

assuming a linear mapping from detector–beam angle $(\theta ,\phi)$ to aperture position $(u,v)$.

To evaluate the approximation in Eq. (55), which corresponds to the limit of ${\sigma _{{\rm ap}}} \to \infty$ or ${D_{{\rm pix}}} \to 0$, we evaluate the opposite limit of ${p_{{ij}}} = {D_{{\rm pix}}}$ or

$${\sigma _{{\rm ap,min}}}({p_{{ij}}}) = {D_{{\rm ap}}}\frac{{{w_f}}}{{\sqrt 2 \pi}}\frac{{F\lambda}}{{{p_{{ij}}}}} ,$$

where the approximation is its worst. In Fig. 15, we show the HBT correlation coefficient $|\gamma ({p_{{ij}}}{)|^2}$ calculated using ${\sigma _{{\rm ap}}} = \infty$ and ${\sigma _{{\rm ap}}} = {\sigma _{{\rm ap,min}}}({p_{{ij}}})$ for two typical ${w_f}$ values, and all curves match well.

C2. Stop Radiation

As Fig. 12 suggests, the stop radiation contributes only minorly to the total correlation among pixels. To demonstrate this contribution explicitly, Fig. 16 shows $(1 - {\eta _{{\rm ap}}}) \gamma _{{\rm stop},ij}^{{\rm np}}$ for various ${\sigma _{{\rm ap}}}$ of the Gaussian illumination in Eq. (C2). When ${\sigma _{{\rm ap}}} \gg {D_{{\rm ap}}}$, the VCZT coherence $\gamma _{{\rm stop},ij}^{{\rm np}}$ asymptotes to a Dirac delta function, and thus contributes negligibly to the intensity correlation. In the other limit of ${\sigma _{{\rm ap}}} \ll {D_{{\rm ap}}}$, $(1 - {\eta _{{\rm ap}}}) = \exp (- D_{{\rm ap}}^2/4\sigma _{{\rm ap}}^2)$ asymptotes to zero, and the stop becomes irrelevant.

 

Fig. 16. Fractional contribution of the stop radiation, $(1 - {\eta _{{\rm ap}}}) \gamma _{{\rm stop},ij}^{{\rm np}}$, to the VCZT coefficient. The solid lines show the numerical calculation assuming a Gaussian beam, and the dashed lines show the approximation $(1 - {\eta _{{\rm ap}}}) {\gamma _{{\rm approx},ij}}(\nu)$.

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The contribution of the stop radiation may become non-negligible, though still small, when ${\sigma _{{\rm ap}}}$ is in neither of these limits, or when ${\sigma _{{\rm ap}}} \sim {D_{{\rm ap}}}/2$. In this parameter region, an approximation can be obtained by calculating the coherence of a flat-illuminated annulus with width ${\sigma _{{\rm ap}}}$:

(C4)$$\begin{split} {}&{\gamma _{{\rm stop},ij}^{{\rm np}}(\nu)}\\{}& = {\;\frac{1}{{(1 - {\eta _{{\rm ap}}})}}\iint_{{D_{{\rm ap}}}/2 \le \sqrt {{u^2} + {v^2}}} {\rm d}u {\rm d}v\;|{G_i}(u,v{{)|}^2} {e^{2\pi i {p_{{ij}}}u/{D_{{\rm ap}}}F\lambda}}\,,}\\{}&\simeq {\; {{\left\{{\pi {{\left({\frac{{{D_{{\rm ap}}}}}{2} + {\sigma _{{\rm ap}}}} \right)}^2} - \pi {{\left({\frac{{{D_{{\rm ap}}}}}{2}} \right)}^2}} \right\}}^{- 1}}}\\{}&\cdot {\quad \iint_{{D_{{\rm ap}}}/2 \le \sqrt {{u^2} + {v^2}} \lt {D_{{\rm ap}}}/2 + {\sigma _{{\rm ap}}}} {\rm d}u {\rm d}v\;{e^{2\pi i {p_{{ij}}}u/{D_{{\rm ap}}}F\lambda}}\,,}\\{}& = {\;\frac{{{F^2}{{F^\prime}^2}}}{{{F^2} - {{F^\prime}^2}}}\left\{{\frac{1}{{{{F^\prime}^2}}}\frac{{2{J_1}(\pi {p_{{ij}}}/F^\prime \lambda)}}{{\pi {p_{{ij}}}/F^\prime \lambda}} - \frac{1}{{{F^2}}}\frac{{2{J_1}(\pi {p_{{ij}}}/F\lambda)}}{{\pi {p_{{ij}}}/F\lambda}}} \right\}}\\{}&{\; \equiv {\gamma _{{\rm approx},ij}}(\nu)\;,}\end{split}$$

where

$$F^\prime \equiv \frac{{{D_{{\rm ap}}}}}{{{D_{{\rm ap}}} + 2{\sigma _{{\rm ap}}}}}F .$$

The dashed lines in Fig. 16 show this approximation for a few relevant examples.

Funding

Office of Science (DE-AC0205CH11231); Japan Society for the Promotion of Science (JP19K21873).

Acknowledgment

Work at LBNL is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics. We acknowledge the support by JSPS. We thank our Simons Array and Simons Observatory colleagues for fruitful discussions on CMB telescope designs, and we thank Masahito Ueda for teaching us some of the basics of quantum statistical mechanics.

Disclosures

CAH: 3M Corporation (I,E,P).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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