Paraxial phasor-field physical optics

Phasor-field ($\mathcal{P}$-field) imaging is a promising recent solution to the task of non-line-of-sight (NLoS) imaging, colloquially referred to as"seeing around corners". It consists of treating the oscillating envelope of amplitude-modulated, spatially-incoherent light as if it were itself an optical wave, akin to the oscillations of the underlying electromagnetic field. This resemblance enables traditional optical imaging strategies, e.g., lenses, to be applied to NLoS imaging tasks. To date, however, this ability has only been applied computationally. In this paper, we provide a rigorous mathematical demonstration that $\mathcal{P}$-field imaging can be performed with physical optics, viz., that ordinary lenses can focus or project the $\mathcal{P}$ field through intervening diffusers, and that they can image scenes hidden by such diffusers. Hence NLoS imaging can be carried out via $\mathcal{P}$-field physical optics without the nontrivial computational burden of prior NLoS techniques.


I. INTRODUCTION
Non-line-of-sight (NLoS) imaging is a growing field of research concerned with the task of generating accurate reconstructions of scenes whose observation can only be accomplished by means of intervening diffuse scattering events, e.g., by penetrating diffuse transmissive scattering media like ground glass or fog, or reflecting off diffuse surfaces like conventional walls. Traditional approaches to this problem, which largely depend on time-of-flight (ToF) information and computational backprojection, have been facilitated by pulsed illumination and time-resolved detection [1,2]. More recent approaches have leveraged closed-form inversion techniques for ToF data [3,4], or the presence of occluders that obviate the need for ToF information [5,6]. In all of these schemes, the task of forming an image from collected data is computational, often requiring significant overhead.
Phasor-field (P-field) imaging is a new approach to NLoS imaging that exploits the wave-like propagation behavior of the oscillating envelope of amplitude-modulated, spatiallyincoherent light [7][8][9]. It relies on the physical correlates of diffuse phase disruption often being large compared to the wavelength of the optical carrier field but insignificant relative to the much longer wavelength of radio-frequency (or even microwave) amplitude modulation. As a result, walls that diffusely scatter light, owing to their roughness at the opticalwavelength scale, appear smooth to the P field, and thin transmissive diffusers appear transparent to the P field. The upshot is that traditional wave-optical imaging techniques, e.g., lenses, can be applied to the P field despite the presence of these optically disruptive elements. To date, however, this capability has only been applied computationally to form NLoS images from ToF datasets [10].
Recently, Reza et al. [11] reported experiments verifying the P field's physical wave-like properties. In one of them, they showed that a diffuse, concave reflector focuses the P field, even though it scatters the optical carrier. Reza et al.'s demonstrations stoke dreams of P-field physical-optics imagers that could see around corners in real time without the nontrivial computational burden of prior NLoS techniques. In this paper, we encourage that imagination through a rigorous mathematical demonstration that ordinary lenses can focus or project the P field through intervening diffusers, and image a scene hidden by such diffusers.

II. P-FIELD SETUP FOR COMPUTATIONAL IMAGING
We begin by reviewing our previously developed framework for computational P-field imaging [8]. We assume paraxial, scalar-wave optics wherein an optical carrier at frequency ω 0 is modulated by a baseband complex field envelope E z (ρ z , t) of bandwidth ∆ω ω 0 to produce an optical field U z (ρ z , t) = Re[E z (ρ z , t)e −iω 0 t ], where ρ z is the two-dimensional transverse spatial coordinate in the plane denoted by z. The amplitude modulation is characterized by the short-time-average (STA) irradiance, given by I z (ρ z , t) = |E z (ρ z , t)| 2 , which can be measured by direct photodetection assuming detectors with sufficient bandwidth.
The P field is defined to be the temporal Fourier transform of the STA irradiance, averaged over any diffusers present in the scenario, whose surface fluctuations are treated statistically: where E z (ρ z , ω) = dt E z (ρ z , t)e iωt is the frequency-domain complex field envelope.
FIG. 1. Unfolded geometry for three-bounce NLoS active imaging. The blue rectangles represent thin transmissive diffusers, which serve as analogs of diffuse reflections, and the black line represents a thin transmissivity mask whose intensity transmission pattern, T (ρ 1 ), represents the albedo pattern of a diffuse target in the hidden scene. Here, and elsewhere, we simplify subscripts involving z where the meaning is clear from context.
We assume a transmissive geometry, shown in Fig. 1, that is a proxy for three-bounce reflective NLoS imaging. Propagation of the frequency-domain complex field envelope through each diffuser in Fig. 1 is given by where the {h z (ρ z )} are the diffusers' thickness profiles, which we take to be a collection of independent, identically-distributed, zero-mean Gaussian random processes with standard deviation satisfying 2πc/ω 0 σ h 2πc/∆ω, where c is light speed, and correlation length obeying ρ h ∼ 2πc/ω 0 . Propagation through the transmissivity mask at z = L 1 , which is the analog of the albedo pattern of a diffuse planar target, is given by Free-space propagation of the complex field envelope is governed by Fresnel diffraction, e.g., where ω 0 + ω ≈ ω 0 may be used in the leading factor, but not in the phase term. The key result from our earlier work is that the P field obeys a modified form of Fresnel diffraction when propagating away from a pure diffuser. In particular, we have that which also demonstrates the effect of the transmissivity mask. This result is facilitated by approximating the diffuser correlation function in integrals involving the P field as where λ 0 = 2πc/ω 0 . In what follows, all of these results will be freely used.
The central purpose of this paper is to show that the foregoing computational approach has a P-field physical optics replacement.

III. PLANE-WAVE P-FIELD FOCUSING
The focusing capability of a convex lens is the natural starting point for that optical element's use in conventional physical optics. Thus we begin our development of P-field physical optics by showing how such a lens can focus the P field. Consider the configuration shown in Fig. 2, in which an infinite plane wave with complex field envelope illuminates a focal-length f = L in + L 1 lens with a Gaussian field-transmission pupil e −|ρ in | 2 /2D 2 . This field is propagating along a unit vector with transverse component s, the lens is set back a distance L in from the first diffuser, and our goal is to focus the P field onto the hidden target in the z = L 1 plane.  The temporal-frequency-domain complex field envelope at the first diffuser is given by [14] Performing the integral in Eq. (10), we get where To simplify this result, we impose the reasonable assumption that |Re[A(ω)]|

|Im[A(ω)]|.
For f = 2 m, L in = 1 m, and λ 0 = 532 nm this condition becomes D 0.4 mm. Using this assumption we find that which yields Using the P field's Fresnel-diffraction formula now gives us Because we have chosen f = L in + L 1 , the final exponential term disappears and other terms simplify. The integral that remains evaluates to The implication of this result is that the incident plane wave creates a P-field illumination that is focused by the lens onto a diffraction-limited (at the modulation wavelength) region in the target plane whose center is offset from the origin in accord with the input illumination's angle of arrival at the lens.
From the perspective of P-field imaging, focusing enables us to raster scan the target as if the initial diffuser were not there. That said, although the lens focuses the P field, it does not focus the optical power, which is still spread out by the diffuser, as demonstrated by Reza et al. [11] for their diffuse, concave reflector. Because the P field is ultimately supported by the optical field, its peak strength, like the STA irradiance's, is still subject to inverse-square law falloff, even in the presence of the lens. That Eq. (17) suggests otherwise, i.e., that increasing D can offset the inverse-square law attenuation, is because we have assumed infinite-plane-wave illumination for which the power passing through the lens is proportional to D 2 . Correcting for this scaling, it is clear that the peak P field has an inverse-square law falloff relative to the input power, regardless of the pupil diameter, i.e., regardless of how tightly the P field is confined in the target plane.
3. Geometry for post-diffuser P-field Fresnel propagation with an intervening lens.
In this section we derive a P-field propagation primitive for post-diffuser P-field Fresnel diffraction through an intervening focal-length-f thin lens that has a Gaussian fieldtransmission pupil e −|ρ| 2 /2D 2 , as depicted in Fig. 3. This primitive will prove useful for both the P-field projection and P-field imaging cases that follow. We have that where and from which it follows that y y X s e S F 9 w V s 0 a w l W 4 y y p 9 1 S n c p s q p 3 W 7 n u P z m 9 0 r u F 7 C 9 k / V u 9 + U 7 M r 3 g r  With the Eq. (23) primitive in hand, we now turn our attention to the task of projecting an arbitrary P-field pattern from an input diffuser to a hidden target plane. Considering the Fig. 1 scenario, we imagine that figure's initial diffuser being preceded by an instance of the lens primitive depicted in Fig. 3, as shown in Fig. 4. Modifying the Fig. 3 scenario's placeholder notation, we will label the transverse coordinate of Fig. 4's input plane as ρ in , its first distance as L in 1 , and its second distance as L in 2 . The lens primitive's output-plane transverse coordinate remains as ρ 0 , leading into the same notation as Fig. 1 for the rest of that figure's geometry. We take the lens to be configured to project the input P field onto the z = L 1 plane by choosing its focal length to obey 1/f = 1/L in 1 + 1/(L in 2 + L 1 ). For this configuration we have Π(f, L in 1 , L in 2 ) = L in 2 (L 1 + L in 2 )/L 1 , and so the lens primitive gives which, after Fresnel propagation, leads to where L prj ≡ L in 2 + L 1 and M ≡ L prj /L in 1 is the magnification/minification factor. Now  [15], which we discuss in Appendix Awould reduce this condition to a more reasonable D 1 cm. If we can achieve this condition, then the final phase term in Eq. (26) can be ignored and we get subject to image inversion and magnification/minification, with resolution diffraction limited at the modulation wavelength. This result enables structured illumination [16] and dual photography [17] techniques to be applied to NLoS P-field imaging. It turns out that the same kind of result just exhibited for P-field projection can be obtained for physical-optics P-field imaging of a hidden target plane. To do so, we place our lens primitive from Fig. 3 behind Fig. 1's final diffuser so that the transverse coordinate of the first plane of the primitive is ρ 2 , as depicted in Fig. 5. We denote the lens primitive's two distances as L im 1 and L im 2 , and we denote the transverse coordinate of the primitive's final plane as ρ im with the intention in mind that a multipixel detector array capable of measuring the P field will be present there. The hidden plane is imaged onto this hypothetical detector by taking the focal length of the lens to obey 1/f = 1/(L 2 + L im 1 ) + 1/L im 2 . For this configuration we have Π(f, L im 1 , L im 2 ) = L im 1 (L 2 + L im 1 )/L 2 , and so we get

VI. P-FIELD IMAGING OF A HIDDEN PLANE
where L out ≡ L 2 + L im 1 and M ≡ L out /L im 2 is the magnification/minification factor. Similar to the projection case, we enforce the assumption D cL im 1 L out /ω − L 2 so that we can ignore the final phase term, which leaves us with Again ignoring inessential terms, this is an inverted and magnified/minified P-field image of the hidden target plane with spatial resolution that is diffraction limited at the modulation wavelength. This confirms the potential for direct P-field imaging of NLoS scenes with little or no computational overhead.

VII. SUMMARY AND DISCUSSION
In this paper, we have mathematically demonstrated the ability for NLoS imaging tasks to be carried out in the P-field framework using physical optics in lieu of the more involved computational techniques employed to date. First, we demonstrated that plane-wave Pfield illumination can be focused onto a small target-plane region despite the presence of an intervening diffuser. Moreover, this P-field focusing occurs even though the optical carrier, and thus the power, is broadly scattered. Next, we developed a useful primitive for P-field propagation in scenarios involving lenses. We then applied that primitive to P-field projection and P-field imaging of a hidden plane's albedo pattern. We found that arbitrary P-field patterns can be projected, through an intervening diffuser, onto a target plane. Likewise, we showed that P-field imaging of a target plane's albedo pattern can be accomplished despite the presence of another intervening diffuser between that plane and the sensor. In these last two cases, the mathematical assumptions made in our analysis rely on rather large lenses. However, the qualifying lens size can be reduced by increasing the P-field frequency. For all three cases-P-field focusing, projection, and imaging-we found that the spatial resolution was at the P-field frequency's diffraction limit, further encouraging the pursuit of higher P-field frequencies.
Ultimately, the highest usable P-field frequency for physical-optics imaging will be set by the bandwidth of available detectors, which is not likely to exceed 100 GHz. Despite this detector limitation, we allege that higher P-field frequencies, perhaps up to 1 THz, can be achieved using Willomitzer et al.'s synthetic-wavelength holography [15], wherein coherently-detected outputs from sequentially illuminating an NLoS system with unmodulated light at two optical frequencies can be correlated and filtered to produce a synthetic P field at the difference frequency, without the need for ultrafast detectors. This technique offers a computational approach to P-field generation and detection that stands in contrast to this paper's physical-optics paradigm. However, the required computation is trivial in comparison to what is typically used, at present, for extracting images from NLoS datasets.
So, even for this synthetic approach, the conclusion of this paper stands: NLoS imaging can be accomplished with traditional optical techniques in the P-field framework using real, physical optics, obviating the need for the nontrivial computational inversion schemes that have been applied to date.

Appendix A: Synthetic-Wavelength Holography
In this appendix, we will develop Willomitzer et al.'s synthetic-wavelength holography [15] in our P-field framework. Within the limitations of current laser and detector technology, this approach offers a practical method to access higher P-field frequencies than are presently attainable with modulated illumination and direct detection. To begin, consider the P field associated with single-frequency irradiance modulation at angular frequency ∆ω: One way to generate this P field is by coherently summing optical fields at two angular frequencies, ω 0 and ω 1 ≡ ω 0 −∆ω, in which case the frequency-domain complex field envelope associated with such a signal is The P field associated with this complex field envelope is given by This is a single-frequency P field at frequency ∆ω. In particular, we have that For systems that are linear and time invariant with respect to the optical field-as is the case for typical NLoS scenarios-the frequency structure of the P field and its underlying complex field envelope are unaffected by that system. Consequently, propagating them through that system reduces to relating the input quantities to the output quantities at each frequency. It follows from Eq. (A5) that the input-output relations for the complex field envelopes at ω 0 and ω 1 suffice to characterize the input-output behavior of the P field. More importantly, for linear time-invariant systems, the input-output relations that take inputs E in,ω 0 (ρ in ) and E in,ω 1 (ρ in ) and yield outputs E out,ω 0 (ρ out ) and E out,ω 1 (ρ out ) are the same as the complex field envelopes' input-output relations for illuminating the system with unmodulated light at each frequency individually. The implication is that, provided we can accurately measure the output complex field envelopes and accomplish the required diffuser averaging, phasor-field imaging tasks can be carried out by sequentially illuminating the system with unmodulated inputs, meaning we are not burdened by needing direct detectors that are sufficiently fast to capture the P-field modulation frequency. We will turn now to each of these concerns.
To measure the output complex field envelope E out,ωn (ρ), for n = 0, 1, we will use balanced heterodyne detection [18]. The signal field is mixed on a 50-50 beam splitter with a strong, plane-wave local oscillator that is detuned from ω n by an intermediate frequency ω IF ∆ω and has phase θ LO . The light emerging from the beam-splitter's output arms are detected, at high spatial resolution, with detector arrays whose temporal bandwidths, including those of their post-detection electronics, exceed ω IF . For simplicity, we will treat these arrays as performing ideal continuum photodetection, i.e., they have unlimited spatial resolution. We will also ignore the effects of noise and assume the arrays have unity quantum efficiency.
In effect, we take the arrays to accurately detect the STA irradiances at the beam-splitter outputs-denoted I + (ρ, t) and I − (ρ, t)-at full spatial resolution. The difference of these outputs carries the desired complex field envelope at the intermediate frequency: when the illumination frequency is ω n , for n = 0, 1. Thus the quadratures of that signal can be extracted by standard communication electronics and we obtain E out,ωn (ρ).
Now we can computationally form E out,ω 0 (ρ)E * out,ω 1 (ρ), but we still need to perform diffuser averaging. It is inadvisable to use arrays whose detector elements average over many speckles, because that destroys the signal of interest, i.e., Hence, it is critical that we use detector arrays whose spatial resolution is high enough to resolve the speckle pattern. In practice, we want no more than a few speckle cells to fall on each pixel in our detector arrays to avoid the associated signal attenuation. So, to perform diffuser averaging, we propose low-pass spatial filtering E out,ω 0 (ρ)E * out,ω 1 (ρ) with a Gaussian kernel to generateP out,∆ω (ρ) ≡ 1 2πR 2 d 2ρ e −|ρ−ρ| 2 /2R 2 E out,ω 0 (ρ)E * out,ω 1 (ρ).
What remains then is to assess the variance ofP out,∆ω (ρ). To do so, we appeal to a speckle analysis developed elsewhere [12,13]. We assume the unfolded three-bounce geometry from Fig. 1, with monochromatic (frequency ω n ) illumination of power P with spatial profile E 0 (ρ 0 ) = 8P/πd 2 0 exp(−4|ρ 0 | 2 /d 2 0 ). For simplicity, the target at the second bounce is replaced by a Gaussian field-transmissivity pupil, exp(−4|ρ 1 | 2 /d 2 1 ). Another such pupil, exp(−4|ρ 2 | 2 /d 2 2 ), is placed at the third-bounce location to characterize the finite size of the visible wall, and the distances propagated after each bounce are all L. We shall also assume that the Fresnel number products, d 2 0 d 2 1 /λ 2 0 L 2 and d 2 1 d 2 2 /λ 2 0 L 2 , are large enough that the third-bounce speckle is well-approximated as single-bounce speckle from the final diffuser.
This condition leads to a third-bounce complex field envelope that is Gaussian distributed.
It is clear that the variance vanishes as R grows without bound. However, we would like the variance to be significantly less than the squared mean for a small enough value of R that the P field's spatial detail is preserved. Taking reasonable parameter values λ 0 = 532 nm, L = 1 m, d 2 = 2 m, we ambitiously take the difference frequency to be ∆ω/2π = 1 THz so that ∆λ = 300 µm. Even setting taking our Gaussian kernel to have R = 1 µm, we find Z(1 µm) ≈ 141, implying that the squared mean greatly exceeds the variance, and thuŝ P out,∆ω (ρ) ≈ P out,∆ω (ρ). Of course, we have traded the need for exceedingly good time resolution in detection for exceedingly good spatial resolution, as sub-micron resolution is required to avoid speckle averaging at the detector arrays. A possible approach to realizing such arrays is to combine magnifying optics with lock-in cameras [19]. Indeed, Willomitzer et al. [15] have used such a camera in proofs-of-concept NLoS experiments using syntheticwavelength holography.