Abstract
Many modern millimeter and submillimeter (“mm-wave”) telescopes for astronomy are deploying more detectors by increasing the detector pixel density and, with the rise of lithographed detector architectures and high-throughput readout techniques, it is becoming increasingly practical to overfill the focal plane. However, when the pixel pitch ${p_{{\rm pix}}}$ is small compared to the product of the wavelength $\lambda$ and the focal ratio $F$, or ${p_{{\rm pix}}}\lesssim 1.2F\lambda$, the Bose term of the photon noise correlates between neighboring detector pixels due to the Hanbury Brown and Twiss (HBT) effect. When this HBT effect is non-negligible, the array-averaged sensitivity scales with the detector count ${N_{{\det}}}$ less favorably than the uncorrelated limit of $N_{{\det}}^{- 1/2}$. In this paper, we present a general prescription to calculate this HBT correlation based on a quantum optics formalism and extend it to polarization-sensitive detectors. We then estimate the impact of HBT correlations on the sensitivity of a model mm-wave telescope and discuss the implications for a focal plane design.
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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
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
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:
In other words, in the reverse-time sense, the electric field emitted from detector ${i_1}$ has the polarization direction ${\hat e_{i1}}$ as a function of $(\theta ,\phi)$ in the far field. Similarly, detector ${i_2}$ only responds to the polarization directionThe 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
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:
since $S_k^{(i)}(\nu)$ satisfies the normalization: In Eq. (34), we define ${T_{(i)}}$ as the effective brightness temperature of the photons impinging on detector $i$. This quantity ${T_{(i)}}$ is generally frequency-dependent; thus, $n({T_{(i)}},\nu)$ does not follow a blackbody spectrum. However, the photon statistics at each frequency $\nu$ do follow the mixed-state thermal density matrix of temperature ${T_{(i)}}$ (see Appendix A), providing the physical foundation for the outcome presented below.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
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 $A\Omega = {\lambda ^2}$.
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]):
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
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.
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.
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:
which leads toThe 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.
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.
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.
When calculating the Stokes $Q$ mapping speed, we sum the HBT coefficients in Fig. 12 over a 4 $F\lambda$ radius to find
noting that, because each pixel has two orthogonal polarimeters, ${| {{\gamma _{{ij}}}} |^2}$ vanishes for half of all $(i,j)$ output pairs. The impact of HBT correlations roughly doubles when considering the mapping speed for measurements of intensity or Stokes $I$. We then assume that this sum applies to all detectors on the focal plane such that ${\gamma ^{Q,(2)}} \simeq \gamma _i^{Q,(2)}$. This treatment ignores the fact that edge pixels have fewer neighbors than internal ones, which is a reasonable approximation for focal planes of a large area. For small or moderately sized detector arrays, the fraction of edge pixels may become important, but such details are experiment-dependent and are therefore beyond the scope of this paper. Figure 13 shows the HBT-impacted mapping speed versus pixel size curves for each observation band, normalized to the peak of the “classic” curve for which $\gamma _{{ij}}^{Q,(2)} \equiv 0$. Three additional mapping speed curves with in-aperture loading from only the CMB, the atmosphere, and the telescope are also plotted to demonstrate the dependence of HBT correlations on various source temperatures.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:
We can now rewrite the density matrix in terms of Glauber’s coherent state $|\alpha \rangle$. The coherent state is written as
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
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
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,
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
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
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.
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
As described in Section 4.C and Eq. (54), we approximate the illumination function as
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
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.
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 ${\sigma _{{\rm ap}}}$:
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.
REFERENCES AND NOTES
1. U. Fano, “Quantum theory of interference effects in the mixing of light from phase-independent sources,” Am. J. Phys. 29, 539–545 (1961). [CrossRef]
2. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963). [CrossRef]
3. R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10, 84–86 (1963). [CrossRef]
4. L. Mandel and E. Wolf, “Photon correlations,” Phys. Rev. Lett. 10, 276–277 (1963). [CrossRef]
5. W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources*,” J. Opt. Soc. Am. 65, 1067–1071 (1975). [CrossRef]
6. W. H. Carter and E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources*,” J. Opt. Soc. Am. 67, 785–796 (1977). [CrossRef]
7. R. H. Brown, R. C. Jennison, and M. K. D. Gupta, “Apparent angular sizes of discrete radio sources: observations at Jodrell Bank, Manchester,” Nature 170, 1061–1063 (1952). [CrossRef]
8. R. H. Brown and R. Q. Twiss, “LXXIV. A new type of interferometer for use in radio astronomy,” Philos. Mag. 45(366), 663–682 (1954). [CrossRef]
9. R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). [CrossRef]
10. S. Padin, “Mapping speed for an array of corrugated horns,” Appl. Opt. 49, 479–483 (2010). [CrossRef]
11. J. Zmuidzinas, “Thermal noise and correlations in photon detection,” Appl. Opt. 42, 4989–5008 (2003). [CrossRef]
12. P. H. van Cittert, “Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene,” Physica 1, 201–210 (1934). [CrossRef]
13. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938). [CrossRef]
14. C. A. Hill, S. M. M. Bruno, S. M. Simon, et al., “BoloCalc: a sensitivity calculator for the design of Simons Observatory,” Proc. SPIE 10708, 1070842 (2018). Equation (4) has a typo corrected in Erratum Proc. SPIE 10708, 107084C (2023). [CrossRef]
15. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963). [CrossRef]
16. There is no fundamental distinction between the inputs and outputs, and every port has both incoming and outgoing photons, even though we will relate the input ports to optical inputs and the output ports to detectors.
17. The Kronecker delta ${\delta _{\textit{km}}}$ in Eq. (3) indicates complete incoherence between the modes from input sources $k$ and $m$. This is a good approximation for the applications discussed in this paper. Further discussion on partial coherence of sources can be found in Appendix B.
18. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
19. In other words, while the approximation in Eq. (10) neglects decoherence, those in Eqs. (14) and (17) do not, and the second equality in Eq. (17) is exact.
20. A. Ludwig, “The definition of cross polarization,” IEEE Trans. Antennas Propag. 21, 116–119 (1973). [CrossRef]
21. Ideal polarimetry is often characterized as having low cross polarization. See, e.g., Refs. [22–25]. for further discussion.
22. P.-S. Kildal, Foundations of Antennas: A Unified Approach (Studentlitteratur AB, 2000).
23. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1997).
24. L. Zeng, C. L. Bennett, D. T. Chuss, et al., “A low cross-polarization smooth-walled horn with improved bandwidth,” IEEE Trans. Antennas Propag. 58, 1383–1387 (2010). [CrossRef]
25. A. Kusaka, E. J. Wollack, and T. R. Stevenson, “Angular and polarization response of multimode sensors with resistive-grid absorbers,”J. Opt. Soc. Am. A 31, 1557–1576 (2014). [CrossRef]
26. While the right-hand side of Eq. (40) does not have an explicit dependence on the pixel index $i$, an implicit dependence enters into ${\eta _\rho}(\nu)$, ${\epsilon _\rho}(\nu)$, and ${\delta _\rho}(\nu)$, since different pixels have slightly different viewing angles of and path length differences to each optical element. These differences are minor, however, and can be ignored in most of practical cases.
27. H. P. Baltes, B. Steinle, and G. Antes, “Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources,” Opt. Commun. 18, 242–246 (1976). [CrossRef]
28. E. Wolf and W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975). [CrossRef]
29. E. Wolf and W. H. Carter, “A radiometric generalization of the Van Cittert-Zernike theorem for fields generated by sources of arbitrary state of coherence,” Opt. Commun. 16, 297–302 (1976). [CrossRef]
30. E. Wolf, “Coherence and radiometry*,” J. Opt. Soc. Am. 68, 6–17 (1978). [CrossRef]
31. G. S. Agarwal, G. Gbur, and E. Wolf, “Coherence properties of sunlight,” Opt. Lett. 29, 459–461 (2004). [CrossRef]
32. An obvious exception is the aperture stop, which we treat separately from the aperture-filling radiation.
33. In theory, the necessary condition for an ideal aperture is for the mapping to be identical only on the stop circumference but, in practice, when this condition is met, the mapping becomes identical within the aperture stop as well.
34. Y. Mizugutch, M. Akagawa, and H. Yokoi, “Offset dual reflector antenna,” in IEEE International Symposium on Antennas and Propagation Digest (1976), pp. 2–5.
35. C. Dragone, “Offset multireflector antennas with perfect pattern symmetry and polarization discrimination,” Bell Syst. Tech. J. 57, 2663–2684 (1978). [CrossRef]
36. H. Tran, A. Lee, S. Hanany, et al., “Comparison of the crossed and the Gregorian Mizuguchi-Dragone for wide-field millimeter-wave astronomy,” Appl. Opt. 47, 103–109 (2008). [CrossRef]
37. A Ludwig-3 polarization beam pattern often results from an angular response function with $E$-plane/$H$-plane symmetry. See also the footnote in Section 2.C.
38. C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. I. Correlation tensors of the classical field,” Phys. Rev. 134, A1143–A1149 (1964). [CrossRef]
39. B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources*,” J. Opt. Soc. Am. 67, 241–247 (1977). [CrossRef]
40. We assume that the incident radiation is largely unpolarized, or $I \gg \sqrt {{Q^2} + {U^2}}$ in terms of Stokes $I,Q,U$. This is a good approximation for millimeter and submillimeter telescopes. Consideration of polarized sources, which is outside the scope of this paper, can be found elsewhere [41].
41. J. Tervo, T. Setälä, J. Turunen, et al., “Van Cittert-Zernike theorem with Stokes parameters,” Opt. Lett. 38, 2301–2303 (2013). [CrossRef]
42. While plane-wave modes with propagation vectors outside the FOV may have different radiation properties, they are irrelevant in our context of evaluating the photon correlation among detectors.
43. D. T. Chuss, E. J. Wollack, S. H. Moseley, et al., “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008). [CrossRef]
44. The chosen $\lambda$ grid spacing corresponds approximately to the coherence length of the simulated thermal radiation and therefore is small enough to represent the stop as a completely incoherent source. See Appendix B for more details.
45. One may regard this discrepancy as an artifact of the modal definition, as a fraction of the surface emitter’s mode projects to the $\hat z$ direction. For the case of ideal Ludwig-3 detectors, all modes are distributed between the two orthogonal detectors, which results in $E/H$-plane symmetry.
46. A. R. Thompson, J. M. Moran, J. Swenson, et al., “Van Cittert–Zernike theorem, spatial coherence, and scattering,” in Interferometry and Synthesis in Radio Astronomy, 3rd ed. (Springer, 2017), chap. 15, pp. 767–786.
47. Strictly speaking, a positive correlation between the neighboring pixels can technically improve sensitivity when the angular diameter of interest approaches the pixel spacing projected on the sky (e.g., for Sunyaev Zel’dovich galaxy cluster surveys).
48. The code used to generate plots in this section can be found at https://github.com/chill90/HBT-Correlations.
49. For the ubiquitous transition-edge sensor (TES) bolometric detectors [51,52], internal thermal noise is proportional to the detector’s saturation power ${P_{{\rm sat}}} = {P_{{\rm opt}}} + {P_{{\rm bias}}}$, where ${P_{{\rm opt}}}$ is the detected optical power, and ${P_{{\rm bias}}}$ is the detector bias power [53]. Therefore, scaling ${\sigma _{{\rm int}}}$ and ${\sigma _{{\rm ph}}}$ together during the experiment design and optimization, while not an exact metric, is well motivated.
50. P. L. Richards, “Bolometers for infrared and millimeter waves,”J. Appl. Phys. 76, 1–24 (1994). [CrossRef]
51. A. T. Lee, J. M. Gildemeister, S.-F. Lee, et al., “Voltage-biased high-T/sub c/superconducting infrared bolometers with strong electrothermal feedback,” IEEE Trans. Appl. Supercond. 7, 2378–2381 (1997). [CrossRef]
52. J. C. Mather, “Bolometer noise: nonequilibrium theory,” Appl. Opt. 21, 1125–1129 (1982). [CrossRef]
53. M. J. Griffin, J. J. Bock, and W. K. Gear, “The relative performance of filled and feedhorn-coupled focal-plane architectures,” Appl. Opt. 41, 6543 (2002). [CrossRef]
54. K. S. Arnold, “Design and deployment of the polarbear cosmic microwave background polarization experiment,” Ph.D. thesis (University of California, Berkeley, 2010).
55. A. Suzuki, “Multichroic bolometric detector architecture for cosmic microwave background polarimetry experiments,” Ph.D. thesis (University of California, Berkeley, 2013).
56. R. Datta, J. Hubmayr, C. Munson, et al., “Horn coupled multichroic polarimeters for the atacama cosmology telescope polarization experiment,” J. Low Temp. Phys. 176, 670–676 (2014). [CrossRef]
57. A. Cukierman, A. T. Lee, C. Raum, et al., “Hierarchical sinuous-antenna phased array for millimeter wavelengths,” Appl. Phys. Lett. 112, 132601 (2018). [CrossRef]
58. J. W. Henning, P. Ade, K. A. Aird, et al., “Feedhorn-coupled TES polarimeter camera modules at 150 GHz for CMB polarization measurements with SPTpol,” Proc. SPIE 8452, 84523A (2012). [CrossRef]
59. R. J. Thornton, P. A. R. Ade, S. Aiola, et al., “The ATACAMA cosmology telescope: the polarization-sensitive ACTPol instrument,” Astrophys. J. Suppl. Ser. 227, 21 (2016). [CrossRef]
60. S. M. Simon, J. Austermann, J. A. Beall, et al., “The design and characterization of wideband spline-profiled feedhorns for Advanced ACTPol,” Proc. SPIE 9914, 991416 (2016). [CrossRef]
61. S. M. Simon, J. A. Beall, N. F. Cothard, et al., “The Advanced ACTPol 27/39 GHz array,” J. Low Temp. Phys. 193, 1041–1047 (2018). [CrossRef]
62. C. M. Posada, P. A. R. Ade, Z. Ahmed, et al., “Fabrication of detector arrays for the SPT-3G receiver,” J. Low Temp. Phys. 193, 703–711 (2018). [CrossRef]
63. H. Hui, P. A. R. Ade, Z. Ahmed, et al., “BICEP3 focal plane design and detector performance,”Proc. SPIE 9914, 99140T (2016). [CrossRef]
64. A. Schillaci, P. A. R. Ade, Z. Ahmed, et al., “Design and performance of the first BICEP array receiver,” J. Low Temp. Phys. 199, 976–984 (2020). [CrossRef]
65. S. Dahal, M. Amiri, J. W. Appel, et al., “The CLASS 150/220 GHz polarimeter array: design, assembly, and characterization,” J. Low Temp. Phys. 199, 289–297 (2020). [CrossRef]
66. M. A. Dobbs, M. Lueker, K. A. Aird, et al., “Frequency multiplexed superconducting quantum interference device readout of large bolometer arrays for cosmic microwave background measurements,” Rev. Sci. Instrum. 83, 73113 (2012). [CrossRef]
67. B. Dober, D. T. Becker, D. A. Bennett, et al., “Microwave SQUID multiplexer demonstration for cosmic microwave background imagers,” Appl. Phys. Lett. 111, 243510 (2017). [CrossRef]
68. Y. Li, J. E. Austermann, J. A. Beall, et al., “Performance of the advanced ACTPol low frequency array,” Proc. SPIE 10708, 107080A (2018). [CrossRef]
69. Z. Xu, M. K. Brewer, P. F. Rojas, et al., “Two-year cosmology large angular scale surveyor (CLASS) observations: 40 GHz telescope pointing, beam profile, window function, and polarization performance,” Astrophys. J. 891, 134 (2020). [CrossRef]
70. C. Zhang, P. A. R. Ade, Z. Ahmed, et al., “Characterizing the sensitivity of 40 GHz TES bolometers for BICEP array,” J. Low Temp. Phys. 199, 968–975 (2020). [CrossRef]
71. M. H. Abitbol, Z. Ahmed, D. Barron, et al., “CMB-S4 Technology Book, First Edition,” arXiv, arXiv:1706.02464 (2017). [CrossRef]
72. S. Paine, The AM Atmospheric Model (Zenodo, 2019).
73. L. Mandel, “Fluctuations of photon beams and their correlations,” Proc. Phys. Soc. 72, 1037–1048 (1958). [CrossRef]
74. L. Mandel, “Fluctuations of photon beams: The distribution of the photo-electrons,” Proc. Phys. Soc. 74, 233–243 (1959). [CrossRef]
75. A. T. Friberg, “Radiation from partially coherent sources,” Opt. Eng. 21, 212362 (1982). [CrossRef]
76. See, e.g., Ref. [38] for consideration without a band limit.
77. Strictly speaking, reducing ${{\cal J}_k}$ modifies the normalized coherence from the case with complete incoherence. However, these changes in ${{\cal J}_k}$ lead to changes in the apparent brightness temperature, which can be absorbed into ${T_\sigma}$; therefore, the formal equivalence for the normalized coherence still holds.