Phasor field waves: A statistical treatment for the case of a partially coherent optical carrier
PPhasor field waves: A statistical treatment forthe case of a partially coherent optical carrier S YED A ZER R EZA , S EBASTIAN B AUER , AND A NDREAS V ELTEN Department of Physics and Optical Engineering, Rose-Hulman Institute of Technology, Terre Haute, IN47803, USA Department of Biostatistics and Medical Informatics, University of Wisconsin – Madison, Madison, WI53706, USA Department of Electrical and Computer Engineering, University of Wisconsin – Madison, Madison, WI53706, USA * [email protected] * [email protected] https://biostat.wisc.edu/~compoptics/ Abstract:
This paper presents a statistical treatment of phasor fields ( P -fields) – a wave-likequantity denoting the slow temporal variations in time-averaged irradiance (which was recentlyintroduced to model and describe non-line-of-sight (NLoS) imaging as well as imaging throughdiffuse or scattering apertures) – and quantifies the magnitude of a spurious signal which emergesdue to a partial spatial coherence of the underlying optical carrier. This spurious signal is notdescribed by the Huygens-like P -field imaging integral which assumes optical incoherence as anecessary condition to describe P -field imaging completely (as was shown by Reza et al. [1–3]).In this paper, we estimate the relationship between the expected magnitude of this spurious signaland the degree of partial roughness within the P -field imaging system. The treatment allows usto determine the accuracy of the estimate provided by the P -field integral for varying degrees ofpartial coherence and allows to define a P -field signal-to-noise ratio as a figure-of-merit for thecase of a partially coherent optical carrier. The study of partial coherence also enables to betterrelate aperture roughness to P -field noise. © 2020 Optical Society of America
1. Introduction
Non-line-of-sight (NLoS) around-the-corner imaging was first experimentally demonstrated byVelten et al. [4]. NLoS imaging is based on diffuse reflections off a surface which is usually calledthe relay wall. An exemplary NLoS setup is shown in Fig. 1. In the configuration discussed here,the relay wall is scanned by a picosecond laser, and the travel time of the photons returning to therelay wall at one specific spot is recorded using an ultrafast detector such as a Single-PhotonAvalanche Diode (SPAD). Over the years, various techniques including filtered backprojection,speckle correlations etc. [5–11] have been proposed for NLoS reconstructions. While each ofthese various approaches entail respective benefits and drawbacks, all of them treat NLoS imagingas a separate class of imaging to conventional line-of-sight (LoS) imaging which is describedby the solution to the wave equation; namely the Huygens’ integral. An NLoS reconstructionmethod based on virtual, so-called phasor field ( P -field) waves was introduced in [12], where therequired sinusoidal light intensity modulation was performed in post-processing. This enablesthe description of NLoS reconstruction as a (virtual) wave propagation problem where the relaywall can be treated as the aperture plane of a virtual camera.A more thorough theoretical analysis of phasor fields as well as practical experiments withreal modulated light sources have been presented as well [1, 3]. In [1], the propagation of P -fields was shown to be described by the P -field integral, an integral which is analogous to1 a r X i v : . [ phy s i c s . op ti c s ] J un ig. 1. NLoS imaging setup: The laser sends short light pulses towards the relay wallfrom where light scatters in all directions. Some photons hit the hidden scene objectand reflect back towards the relay wall. The ultrafast detector looks at a fixed spot onthe relay wall and determines the travel time of the few photons that make their wayback to this spot on the relay wall. Scanning the laser at all relay wall positions andacquiring the photon travel times at those positions provides the geometric informationneeded for 3D reconstruction of the hidden scene. the Huygens’ integral with spherical P -field contributions replacing the spherical Electric field(E-field) contributions. This P -field integral only holds exactly true if the underlying opticalcarrier can be considered as spatially incoherent [1]. In this case the P -field contributionsadd linearly as is the case for E-field contributions with the Huygens’ integral. The effect oftemporal coherence of the optical carrier on P -field propagation and interference has been brieflydiscussed by Teichman [13]. Furthering the theory of P -field imaging, Dove et al. [14] recentlyproposed P -field occlusion-aided NLoS imaging.All the aforementioned contributions share the assumption that the relay wall is rough on anoptical scale, meaning that the optical phase is shifted by the reflection such that it is uniformlydistributed over the range [ , π ] when reduced to modulo-2 π . In this case, the reflected light isincoherent and addition of the sinusoidally modulated intensities reflected off different relay wallpoints is sufficient to describe the resulting wave front in front of the relay wall. However, thereare cases where the relay wall is not optically rough, meaning that the reflected light cannot beconsidered incoherent any more. For this reason, in this paper, we discuss the effect of partialspatial coherence of the underlying optical carrier (E-field) on P -field imaging. This allowsus to calculate the magnitude of an additive P -field noise that is introduced by partial opticalcoherence and enables us to calculate an expected P -field signal-to-noise ratio (SNR) for varyingdegrees of partial spatial coherence. This statistical treatment can also be useful to describe therelation between a certain degree of partial-roughness of the P -field aperture and the expected P -field SNR. This allows for calculating the limits on minimum aperture roughness that satisfyany desired P -field SNR tolerance value. We want to remark that our work could be extended inthe future by incorporating the second-order statistics (auto-correlation) calculations developedfor speckle effects [15, 16]. These allow for example for the calculation of the 2D spatial intensityauto-correlation after the reflection based on the 2D auto-correlation of the surface profile.2 . P -field imaging for incoherent and partially coherent E-field P -fields The notion of the P -fields was introduced in [1, 2]. The goal of this is to describe an NLoSimaging system analogously to a LoS imaging system - which enables us to describe and evaluateNLoS system performance and limitations from the well-known treatment of LoS systems.It is well-known that the Huygens’ integral is a solution to the scalar wave equation anddescribes the E-field at a location ( x , y ) in a detection plane Σ ( z = Z ) as a sum of E-fieldspherical wavelet contributions from all locations ( x (cid:48) , y (cid:48) ) of a specular or non-rough aperture A defined by the plane z =
0. In the context of imaging, the Huygens’ integral E ( x , y , z = Z ) = j λ E ∫ A E ( x (cid:48) , y (cid:48) , ) e jk | r | | r | χ E dx (cid:48) d y (cid:48) (1)describes E-field propagation and interference resulting in image formation in Σ as is shown inFig. 2. In (1), | r | = (cid:113) ( x − x (cid:48) ) + ( y − y (cid:48) ) + Z is the absolute distance between any unique pairof locations ( x (cid:48) , y (cid:48) , ) ∈ A and ( x , y , Z ) ∈ Σ , z = Z is the separation distance between A and Σ , k is the E-field wave number expressed in terms of the E-field (optical) wavelength λ E as k = π / λ E and χ E is the E-field obliquity factor.For the case of a diffuse aperture plane A , the notion of P -field propagation was describedin [1] where a P -field signal is described as the baseband envelope of the amplitude-modulatedoptical irradiance of the carrier. It has been shown in [1] that this is equivalent to effectivelyamplitude modulating the optical carrier E-field to yield a modulated scalar time-harmonicreal-valued field Re [ E ( x (cid:48) , y (cid:48) , t ) e j ω t ] where E ( x (cid:48) , y (cid:48) , t ) = E ( x (cid:48) , y (cid:48) ) cos ( Ω t / ) is the basebandmodulating signal of angular frequency Ω / E scaled by the impedance ζ ofthe medium to effectively have units of (cid:112) W / m in the z = phasor field is defined as the slowly-varying signed envelope of the time-averagedoptical irradiance of frequency Ω – which depicts the signal that is used to directly amplitude-modulate the optical carrier. At a diffuse aperture (or origin) plane at z = P -field isdescribed as Re (cid:16) P , Ω ( x (cid:48) , y (cid:48) ) e j Ω t (cid:17) = ∆ I ( x (cid:48) , y (cid:48) , t ) = I ( x (cid:48) , y (cid:48) , t ) − (cid:104) I ( x (cid:48) , y (cid:48) , t )(cid:105) , (2)where I ( x (cid:48) , y (cid:48) , t ) = T ∫ t + T / t − T / | E ( x (cid:48) , y (cid:48) , τ ) | d τ = |E ( x (cid:48) , y (cid:48) , t ) | [ + cos ( Ω t )] (3)and (cid:104) I ( x (cid:48) , y (cid:48) , t )(cid:105) = lim T →∞ T ∫ t + T / t − T / | E ( x (cid:48) , y (cid:48) , t ) | dt = |E ( x (cid:48) , y (cid:48) , t ) | (4)are the short time and the long time averages of the optical irradiance contribution from ( x (cid:48) , y (cid:48) , ) ∈ A under the condition that Ω (cid:28) ω and that the integration time T of the detector ismuch longer than the time period of the optical carriers’ E-field oscillation and much smallerthan the time period of the P -field signal, i.e.,2 πω (cid:28) T (cid:28) π Ω . (5)The resulting P -field contribution Re (cid:0) P , Ω ( x (cid:48) , y (cid:48) ) e j Ω t (cid:1) = ∆ I ( x (cid:48) , y (cid:48) , t ) from ( x (cid:48) , y (cid:48) ) (the samequantity as stated in (2)) is hence a real signed time-harmonic function of frequency Ω and3mplitude P , Ω ( x (cid:48) , y (cid:48) ) = P | E ( x (cid:48) , y (cid:48) )| / ζ with E ( x (cid:48) , y (cid:48) ) representing the E-field amplitudecontribution from ( x (cid:48) , y (cid:48) ) and ζ the impedance of the propagation medium. Hence, ∆ I ( x (cid:48) , y (cid:48) , t ) is effectively expressed as ∆ I ( x (cid:48) , y (cid:48) , t ) = P , Ω ( x (cid:48) , y (cid:48) ) cos ( Ω t ) . (6)In (6), the operation of subtracting the long-time average from the short-time average is equivalentto physically detecting with an AC-coupled detector which removes the DC offset from thereceived slowly-varying optical carrier envelope. It was also shown in [1] that the expected valueof the total P -field – P Sum ( x , y , z = Z , t ) (denoted as I Tot − F ( x , y , Z ) in [1]) – received at a location ( x , y ) within the detection plane Σ located at z = Z for a spatially incoherent optical carrieris expressed as the sum of all P -field spherical contributions P( r ) from A in a Huygens’-likeintegral as P Sum ( x , y , Z , t ) ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | r ( x , y , Z )| Av ∬ A P , Ω ( x (cid:48) , y (cid:48) ) e j β | r | | r | (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) P( r ) χ P dx (cid:48) d y (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7)In (7), Re [P( r ) e j Ω t ] = ∆ I ( x (cid:48) , y (cid:48) , t ) /| r | is one time-harmonic real P -field contribution from ( x (cid:48) , y (cid:48) ) scaled down by | r | expressed in the complex phasor notation. AlsoRe (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) | r ( x , y , Z )| Av ∬ A P , Ω ( x (cid:48) , y (cid:48) ) e j β | r | | r | (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) P( r ) χ P dx (cid:48) d y (cid:48) e j Ω t (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = ∆ I Z ( x , y , t ) = I Z ( x , y , t )−(cid:104) I Z ( x , y , t )(cid:105) , (8)is again the difference between the short time averaged and the long time statistically averaged opti-cal irradiance quantities I Z ( x (cid:48) , y , t ) and (cid:104) I Z ( x (cid:48) , y , t )(cid:105) at ( x , y , z = Z ) , | r | = (cid:112) Z + ( x − x (cid:48) ) + ( y − y (cid:48) ) is the distance between any locations ( x (cid:48) , y (cid:48) , z = ) ∈ A and ( x , y , Z ) ∈ Σ , | r ( x , y , Z )| Av = (cid:112) Z + ( x − (cid:104) x (cid:48) (cid:105)) + ( y − (cid:104) y (cid:48) (cid:105)) is the average distance between a location ( x , y , Z ) ∈ Σ from anaverage location ((cid:104) x (cid:48) (cid:105) , (cid:104) y (cid:48) (cid:105) , z = ) in A , χ P ( x (cid:48) , x , y (cid:48) , y ) is the P -field obliquity factor, and β isthe P -field wave number expressed in terms of the P -field wavelength λ P as β = π / λ P . Thisintegral in (8) is referred to as the P -field integral which clearly depicts that spherical P -fieldcontributions from a diffuse surface A interfere analogously to spherical E-field contributionsfrom a specular surface as was originally described by the Huygens’ integral presented in (1). P -field sum For the case of partial optical coherence, we will show during the course of our mathematicaltreatment that the total P -field [P , Ω ( x , y , Z , t )] Tot at any location ( x , y , Z ) ∈ Σ is described as a P -field integral sum P Sum ( x , y , Z , t ) described in [1] with an additive spurious sum P Sp ( x , y , Z , t ) from the cross-interference of all P -field contributions from A . This cross-interference dueto the partial spatial coherence of the optical carrier is the underlying basis of E-field and P -field speckle and is a result of aperture roughness and uncertainties in phases accumulated inpropagation from A to Σ . In other words, we will demonstrate that the total P -field contribution [P , Ω ( x , y , Z , t )] Tot is equal to the Huygens-like P -field sum added with a spurious P -field noisesignal P Sp ( x , y , Z , t ) , i.e., [P , Ω ( x , y , Z , t )] Tot = P Sum ( x , y , Z , t ) + P Sp ( x , y , Z , t ) (9)4 ig. 2. Analogy between E-field and P -field propagation from an aperture plane A toan observation plane Σ separated by a distance Z . and relate the magnitude of this spurious P -field noise to the degree of roughness in the apertureplane A . In the case of a diffuse surface/aperture A in transmission or reflection, the opticalcarrier as well as its slowly varying envelope experience a phase shift which is spatially-dependenton the roughness profile of A , i.e., the roughness in each location within A determines themagnitudes of the phase shifts δφ k and δφ β imparted to the optical carrier and its slowly-varyingenvelope respectively. Let ω denote the frequency of the optical carrier and Ω the frequency ofthe slowly-varying envelope of amplitude-modulated carrier irradiance (a P -field contribution).In the case ω (cid:29) Ω , the random phase δφ k = φ R of the E-field accumulated due to diffusereflections or transmission is much larger than the random phase δφ β of the P -field becauseof the difference of a few orders of magnitude between the optical and P -field wavelengths; φ R (cid:29) δφ β . We will show in subsequent sections that in the case of a partially spatially coherentoptical carrier [P , Ω ( x , y , Z , t )] Tot is described as (cid:12)(cid:12) [P , Ω ( x , y , z = Z , t )] Tot (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:213) n = P n , Ω | r n | cos (cid:2) Ω t + φ β n (cid:3) dA (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) P− field Integral Sum |P Sum ( x , y , Z , t )| + σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Spurious P− field Signal Noise | P Sp ( x , y , Z , t ) | , (10)where σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] = √ (cid:115) θ + θ + ( θ ) − (cid:0) + θ (cid:1) cos ( θ ) − θ + θ + cos ( θ ) − θ + sinc ( θ )·· (cid:20) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) dA (cid:21) (11)is the standard deviation of the zero-mean cross interference from all pairs of the m th and n th (with m (cid:44) n ) P -field (and E-field) contributions from ( x m , y m , ) and ( x n , y n , ) ∈ A respectivelyout of a total of N contributions and [− θ, θ ] is the range of random phase introduced exclusivelyby the roughness at each location of A . In (11), it is assumed that these contributions stemfrom infinitesimally small relay wall (aperture in the context of NLoS imaging) patches and areindependent from one another. The phases φ km , φ kn and φ β m , φ β n denote the E-field and P -fieldphases accumulated by these respective contributions with corresponding magnitudes of P m , Ω P n , Ω . The variables | r m | and | r n | are the distances between ( x m , y m , ) , ( x n , y n , ) ∈ A and ( x , y , z = Z ) ∈ Σ .In Section 3, the spurious P -field signal P Sp ( x , y , Z , t ) will be derived as a function of theallowable range [− θ, θ ] of values that a random variable φ R where φ R denotes the phase difference(imparted solely by aperture roughness) between any pair of P -field contributions originatingfrom two different locations in A . Hence, we derive a direct relationship in (11) between themagnitude of the spurious P -field noise P Sp ( x , y , Z , t ) and the degree of randomness in A .Moreover – analogous to conventional signal theory – we calculate a P -field signal-to-noiseratio (SNR) as a figure-of-merit of the P -field NLoS imaging system as P SNR ( x , y , Z ) = |P Sum ( x , y , Z , t )| (cid:12)(cid:12) P Sp ( x , y , Z , t ) (cid:12)(cid:12) (12)and show that the P -field SNR saturates beyond a certain degree of surface roughness and theperformance of a P -field imaging system remains consistent beyond this threshold apertureroughness as indicated by a saturation in the P -field SNR.
3. Deriving the relationship between partial spatial E-field coherence and the P -field noise We begin by reminding the reader of how the interference of N discrete unmodulated E-fieldcontributions from A when observed at Σ can be described. The irradiance I ( x , y , Z ) at a genericlocation ( x , y , z = Z ) ∈ Σ is the square of the sum of all E-field contributions from A and it isstated as I ( x , y , Z ) = ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:213) n = E n | r n | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (13)In (13), | r n | = (cid:112) ( x − x (cid:48) n ) + ( y − y (cid:48) n ) + Z is the distance between any location ( x (cid:48) n , y (cid:48) n , ) ∈ A and ( x , y , Z ) and the E-field contribution from ( x (cid:48) n , y (cid:48) n , ) has magnitude | E n | and phase φ kn = k | r n | . Hence the n th E-field contribution – whose polarization is expressed by a unit vectorˆ e – can be stated as E n = | E n | cos ( ω t + φ kn ) ˆ e . From (13), the irradiance at ( x , y , Z ) can beexpressed as I ( x , y , Z ) = ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:213) n = | E || r | e j ( ω t + φ k ) + | E || r | e j ( ω t + φ k ) + · · · · + | E N || r N | e j ( ω t + φ k N ) + cc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (14)where ’ cc ’ denotes the complex conjugate sum of each of the preceding E-field terms. Atime-averaged irradiance (cid:104) I ( x , y , Z )(cid:105) is calculated by integrating I ( x , y , Z ) over the detectorintegration time T (cid:104) I ( x , y , Z )(cid:105) = I | r | + I | r | + I | r | + · · · · + I N | r N | + √ I √ I | r || r | cos ∆ φ , + √ I √ I | r || r | cos ∆ φ , + · · · √ I √ I N | r || r N | cos ∆ φ , N + √ I √ I N | r || r N | cos ∆ φ , N + · · · √ I N − √ I N | r N − || r N | cos ∆ φ N − , N , (15)as the average of all high frequency sinusoidal terms with a frequency of 2 ω is approximatelyzero if T (cid:29) π / ω . In (15), each irradiance term I n represents the time-averaged irradiancecontribution from ( x (cid:48) n , y (cid:48) n , ) ∈ A given by I n = | E n | / ζ . (cid:104) I ( x , y , Z )(cid:105) can be simply expressed6s (cid:104) I ( x , y , Z )(cid:105) = N (cid:213) n = I n | r n | (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 1 + N (cid:213) p = N (cid:213) q = q (cid:44) p (cid:112) I p (cid:112) I q | r p || r q | cos ∆ φ p , q (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 2 ( p (cid:44) q ) . (16)In Eqns. (15) and (16), ∆ φ p , q = φ kp − φ kq denotes the optical (E-field) phase difference betweenthe p th and q th E-field contributions from A while I p and I q denote the time-averaged irradiancecontributions from the p th and q th locations in A respectively. Note that the expression in (16)has been separated into two terms; the first term is the sum of individual irradiance contributions(i.e., Term 1 for p = q ) from each location ( x (cid:48) , y (cid:48) , ) ∈ A and a second term which is a sumof P -field cross-multiplication terms between any two non-identical locations ( x p , y p , ) and ( x q , y q , ) in A .In this paper, we will derive a similar general expression for the interference of N P -fieldcontributions for the case of a partially coherent optical carrier. This will allow us to establisha relationship between the partial coherence of the E-field and the magnitude of the additiveterm P Sp ( x , y , Z , t ) which describes the sum of P -field cross-interference between P -fieldcontributions from different locations in A . To re-emphasize, this sum of P -field cross-interference is not explained by the P -field integral because of speckle-averaging which resultsin the expected value of the cross-interference term to 0 for fully incoherent summation. P -field contributions When optical irradiance is directly amplitude-modulated using a low frequency signal P ( t ) = P [ + cos Ω t ] , (17)the time-averaged irradiance at a location ( x , y , Z ) ∈ Σ is simply calculated by multiplying thePoynting vector S ( x , y , Z , t ) at ( x , y , Z ) by the P -field modulating signal contribution from alocation ( x (cid:48) , y (cid:48) , ) ∈ A and calculating a time-average over the integration time of the detector.In order to calculate the time-averaged irradiance at ( x , y , Z ) as a result of multiple contributionsfrom various locations within A , all E-field contributions from A have to be added, the result issquared and integrated over the integration time window T of the detector. Moreover, in [1], it wasshown that when all the E-field contributions from A are incoherent, the resulting time-averagedAC-coupled value of irradiance (the P -field sum) at any location ( x , y , Z ) ∈ Σ is simply the linearsum of individual irradiance P -field contributions from all locations in A . Hence the subsequent P -field sum is expressed as a Huygens-like P -field integral.Here, we determine the time-averaged irradiance and the resulting P -field sum at any locationwithin the detector plane when the underlying E-field carrier contributions from A cannot beconsidered completely incoherent. If the phase accumulated due to propagation between A and Σ by an n th P -field contribution of amplitude P n can be denoted by φ β, n = β | r n | and therelatively smaller random phase shift due to aperture roughness or any other reason denoted by δφ β ( x (cid:48) , y (cid:48) , ) , then as was shown in [1], the n th P -field contribution at ( x , y , Z ) is simply statedas P n ( r n , t ) = P cos (cid:2) Ω t + β | r n | + δφ β, n ( x (cid:48) , y (cid:48) , ) (cid:3) . (18)It was also shown in [1], that irradiance contribution in Σ that is amplitude-modulated with P( t ) is equivalent to modulating the E-field of the underlying optical carrier by Q n ( r n , t ) = (cid:112) P cos (cid:2) Ω (cid:48) t + β | r n |/ + δφ β ( x (cid:48) , y (cid:48) , )/ (cid:3) , (19)where Ω (cid:48) = Ω /
2. Knowing this fundamental P -field framework developed earlier, we first derivean expression for the P -field sum (i.e., total AC coupled time-averaged optical irradiance) at any7ocation ( x , y , Z ) ∈ Σ due to only two P -field contributions from A (i.e., two modulated E-fieldspherical wavelet contributions) with no assumptions made regarding the coherence propertiesof the optical carrier. Then, based on the two- P -field wavelet interference result, we present ageneral expression for the P -field sum due to any number of N P -field contributions from A .Let the n th unmodulated E-field contribution incident at a location ( x , y , Z ) ∈ Σ be representedby E n ( r n , t ) = E n | r n | cos [ ω t + k | r n | + δφ k ( x (cid:48) , y (cid:48) , )] , (20)where k | r n | is the phase accumulated by the n th E-field contribution in propagating a distance | r n | between locations ( x (cid:48) , y (cid:48) , ) ∈ A and ( x , y , Z ) ∈ Σ , δφ k ( x (cid:48) , y (cid:48) , ) is the random phase added to this n th E-field contribution and E n = | E n | ˆ e represents a vector of magnitude | E n | in the directionof the unit vector ˆ e that denotes the polarization of the E-field. If the total phases accumulatedby the n th E-field and Q n ( r n , t ) contributions are stated as φ kn = k | r n | + δφ k ( x (cid:48) , y (cid:48) , ) and φ β n = β | r n |/ + δφ β, n ( x (cid:48) , y (cid:48) , )/ (cid:104) I ( x , y , Z , t )(cid:105) measured at a location ( x , y , Z ) ∈ Σ due to just two modulated E-field contributions is simplyexpressed as (cid:104) I ( x , y , Z , t )(cid:105) = ζ T ∫ t + T / t − T / (cid:20)(cid:18)(cid:112) P cos (cid:0) Ω (cid:48) τ + φ β (cid:1) E | r | cos ( ωτ + φ k ) + (cid:112) P cos (cid:0) Ω (cid:48) τ + φ β (cid:1) E | r | cos ( ωτ + φ k ) (cid:19) · (cid:18)(cid:112) P cos (cid:0) Ω (cid:48) τ + φ β (cid:1) E | r | cos ( ωτ + φ k ) + (cid:112) P cos (cid:0) Ω (cid:48) τ + φ β (cid:1) E | r | cos ( ωτ + φ k ) (cid:19)(cid:21) d τ. (21)Note that (cid:104) I ( x , y , Z , t )(cid:105) is still a function of time t as the modulation components of (cid:104) I ( x , y , Z , t )(cid:105) of frequency Ω (cid:48) do not average out for the chosen detector integration time window of length T . Assuming identical polarization of each of the E-field contributions, (21) can be furtherexpressed as (cid:104) I ( x , y , Z , t )(cid:105) = ζ T ∫ t + T / t − T / (cid:20) P | E | | r | cos (cid:0) Ω (cid:48) τ + φ β (cid:1) cos ( ωτ + φ k ) ++ P | E | | r | cos (cid:0) Ω (cid:48) τ + φ β (cid:1) cos ( ωτ + φ k ) ++ | E || E | P | r || r | cos (cid:0) Ω (cid:48) τ + φ β (cid:1) cos (cid:0) Ω (cid:48) τ + φ β (cid:1) cos ( ωτ + φ k ) cos ( ωτ + φ k ) (cid:21) d τ. (22)Using the product-of-cosines identitycos ( α ) cos ( α ) = (cid:20) cos ( α + α ) + cos ( α − α ) (cid:21) , (23)and the cosine double angle identitycos ( α ) = + cos ( α ) , (24)8e can express (22) as (cid:104) I ( x , y , Z , t )(cid:105) = ζ T ∫ t + T / t − T / (cid:18) P | E | | r | (cid:2) + cos (cid:0) Ω τ + φ β (cid:1)(cid:3) [ + cos ( ωτ + φ k )] ++ P | E | | r | (cid:2) + cos (cid:0) Ω τ + φ β (cid:1)(cid:3) [ + cos ( ωτ + φ k )] ++ P | E || E | | r || r | (cid:2) cos (cid:0) Ω τ + φ β + φ β (cid:1) + cos (cid:0) φ β − φ β (cid:1) (cid:3)(cid:2) cos ( ωτ + φ k + φ k ) + cos ( φ k − φ k ) (cid:3) (cid:19) d τ. (25)After performing time-integration over an integration window T that satisfies the condition in (5),i.e., 2 πω (cid:28) T (cid:28) π Ω , (26)we eliminate all high frequency terms oscillating at 2 ω as their mean over several cycles canbe approximated to zero. Moreover, considering the fact that T (cid:28) π / Ω , the values of the lowfrequency sinusoidal terms with frequency Ω almost remain approximately constant during eachof the detector integration time interval T and these sinusoidal terms are retrieved almost perfectlyover successive time integrations. Hence, for the condition in (26), we can express (25) as (cid:104) I ( x , y , Z , t )(cid:105) = P | E | ζ | r | (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 1 + P | E | ζ | r | (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 2 + P | E | ζ | r | cos (cid:2) Ω t + φ β (cid:3)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 3 + P | E | ζ | r | cos (cid:2) Ω t + φ β (cid:3)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 4 ++ P | E || E | ζ | r || r | cos [ φ k − φ k ] cos (cid:2) Ω t + φ β + φ β (cid:3)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 5 ++ P | E || E | ζ | r || r | cos [ φ k − φ k ] cos (cid:2) φ β − φ β (cid:3)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 6 . (27)When the detector is AC-coupled, the constant Terms 1, 2 and 6 from (27) are eliminated. Theoperation of short-time integration to eliminate high frequency terms in (25) in conjunction withthe AC-coupling operation in (27) is equivalent to subtracting the long-time average from theshort-time average as was stated in (8). If the P -field amplitude P n , Ω of the n th contribution isdefined as P n , Ω = P | E n | / ζ , we obtain ∆ I Z ( x , y , t ) = P , Ω | r | cos (cid:2) Ω t + φ β (cid:3)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 1 + P , Ω | r | cos (cid:2) Ω t + φ β (cid:3)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 2 ++ (cid:112) P , Ω P , Ω | r || r | cos [ φ k − φ k ] cos (cid:2) Ω t + φ β + φ β (cid:3)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Term 3 . (28)In the general case with N P -field contributions from A contributing to ∆ I Z ( x , y , t ) , (28) can be9xpanded and expressed as the following Riemann sum: ∆ I Z ( x , y , t ) = N (cid:213) n = P n , Ω | r n | cos (cid:2) Ω t + φ β n (cid:3) dA ++ N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos [ φ km − φ kn ] cos (cid:2) Ω t + φ β m + φ β n (cid:3) dA (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Additive Spurious P− field Noise Term (cid:104) I Cross ( x , y , Z , t )(cid:105) , (29)where dA = dx (cid:48) d y (cid:48) represents an infinitesimally small area in A over which φ k and φ β remainalmost constant. Each such small area dA is responsible for one of many P -field contributionsthat sum a detector plane location ( x , y , Z ) . In (29), the first summation term represents the P -field summation for an incoherent optical carrier as was presented in [1]. The next termrepresents the summation of all P -field cross-interference terms previously not considered in [1]. P -field speckled-based noise as a function of aperture roughness Now that we have split the total P -field at any location ( x , y , z = Z ) into two distinct sums -i.e. the desired P -field sum and an additive spurious cross-interference P -field sum - we canproceed to focus on the sum of cross-interference contributions (cid:104) I Cross ( x , y , Z , t )(cid:105) which is thebasis of the additive P -field speckle-based noise. The sum of the product of the mixed P -fieldinterference terms (cid:104) I Cross ( x , y , Z , t )(cid:105) from (29) (cid:104) I Cross ( x , y , Z , t )(cid:105) = N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos [ φ km − φ kn ] cos (cid:2) Ω t + φ β m + φ β n (cid:3) dA (30)is an additive term to Term 1 in (29) (which represents the P -field integral in continuous form).Our objective is to determine the expected value and the variance of I Cross depending on themagnitude of roughness in A . Let us first look into the expression in (30) and determine theeffect of a random phase added by each location ( x (cid:48) , y (cid:48) , ) ∈ A .We know from our definitions thatcos [ φ km − φ kn ] = cos (cid:20) ( k | r m | − k | r n |) + δφ k , m ( x (cid:48) m , y (cid:48) m , ) − δφ k , n ( x (cid:48) n , y (cid:48) n , ) (cid:21)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) φ R ( x (cid:48){ m , n } , y (cid:48){ m , n } , ) , cos (cid:2) Ω t + φ β m + φ β n (cid:3) = cos (cid:2) Ω t + β | r m |/ + β | r n |/ + δφ β, m / + δφ β, n / (cid:3) . (31)If additional phase shifts in the P -field and the E-field contributions can be simply attributedto aperture roughness and assuming that this roughness in A is much smaller than the P -fieldwavelength λ P (see [1]), we can assume that δφ β, m + δφ β, n ≈
0. On the other hand the phase term ( β | r m | + β | r n |) is deterministic because its assumed that the degree of roughness is much smallerthan the P -field carrier modulation wavelength. The term ( k | r m | − k | r n |) denoting differencein phase accumulated via propagation exclusively by E-field carrier contributions from A isalso considered to be a random variable owing to minute path length differences in the orderof the E-field wavelength compared to an ideal-world scenario where these would be perfectlyaccounted for. The uncertainty in ( k | r m | − k | r n |) gives rise to optical speckle in most E-fieldspatial distributions arising from any interference and propagation of E-fields.Lastly, but most importantly, the phase term φ R ( x (cid:48){ m , n } , y (cid:48){ m , n } , ) is most affected by minorrandom path changes (exclusively due to roughness in A ) as these path changes result in phase10hanges to the E-field contributions which are much larger than one E-field phase cycle. Ifrandom phase shifts δφ k , m ( x (cid:48) m , y (cid:48) m , ) and δφ k , n ( x (cid:48) n , y (cid:48) n , ) to the m th and n th E-field contributionsare each considered to be uniformly-distributed random variables that exist in the range [− θ, θ ] – where θ depends on the amount of roughness in the diffuse surface A that determines thedegree of partial coherence of the carrier – then φ R ( x (cid:48){ m , n } , y (cid:48){ m , n } , ) is a triangularly-distributedrandom variable in the range [− θ, θ ] (Recall that the probability distribution of a difference oftwo uniformly-distributed random variables is the convolution of these two distributions). Forsimplicity, we denote this triangularly-distributed random variable as φ R . We can express from(31)cos ( φ km − φ kn ) cos (cid:0) Ω t + φ β m + φ β n (cid:1) = cos ( Ω t + β | r m |/ + β | r n |/ ) (cid:20) cos ( k | r m | − k | r n |) cos ( φ R ) − sin ( k | r m | − k | r n |) sin ( φ R ) (cid:21) . (32)This results incos ( φ km − φ kn ) cos (cid:0) Ω t + φ β m + φ β n (cid:1) = (cid:20) cos ([ k + β / ]| r m | − [ k − β / ]| r n | + Ω t ) + cos ([ k − β / ]| r m | − [ k + β / ]| r n | − Ω t ) (cid:21) cos ( φ R )− (cid:20) sin ([ k + β / ]| r m | − [ k − β / ]| r n | + Ω t ) + sin ([ k − β / ]| r m | − [ k + β / ]| r n | − Ω t ) (cid:21) sin ( φ R ) . (33)For the condition β / (cid:28) k , we assume that k + β / ≈ k and k − β / ≈ k and given that cosine and sine are even and odd functions respectively, (33) can be expressed ascos ( φ km − φ kn ) cos (cid:0) Ω t + φ β m + φ β n (cid:1) ≈ cos ( Ω t ) cos ( k | r m | − k | r n |) cos ( φ R ) − cos ( Ω t ) sin ( k | r m | − k | r n |) sin ( φ R ) . (34)It is now that we are in a position to calculate the expected value of (cid:104) I Cross ( x , y , Z )(cid:105) as afunction of the range of allowable values [− θ, θ ] which the continuous random variable φ R can assume. From (34), we can express (cid:104) I Cross ( x , y , Z , t )(cid:105) as (cid:104) I Cross ( x , y , Z , t )(cid:105) = N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | × cos ( Ω t ) (cid:20) cos ( k | r m | − k | r n |) cos ( φ R ) − sin ( k | r m | − k | r n |) sin ( φ R ) (cid:21) . (35)Consider γ and γ to be two random variables that represent the quantities γ = cos ( k | r m | − k | r n |) ∀ m (cid:44) n , (36)and γ = sin ( k | r m | − k | r n |) ∀ m (cid:44) n . (37)Moreover, let η and η represent two random variables which describe the quantities η = cos ( φ R ) η = sin ( φ R ) . (38)11or clarity, a distinction between η , η and γ , γ has to be made. The difference in carrierpropagation phases (for reasons other than aperture roughness) from the m th and n th contributionsfrom A is denoted by random variables γ and γ which are zero-mean variables with a finitevariance for most practical imaging purposes. Assuming that uncertainties (or randomness)in propagation phases accumulated by contributions from A are uniformly distributed in themodulo 2 π range [− α, α ] , the resulting random variables γ and γ denoting phase differencesaccumulated due to propagation path difference between pairs of contributions from A to alocation x , y , z = Z ∈ Σ are triangularly distributed in the modulo 2 π range [− α, α ] becausevariations in each of the constituent propagation phases k | r m | and k | r n | are uniformly distributedover the modulo-2 π ‘ range [− α, α ] . These random differences in the propagation phases describethe well-known optical speckle. γ and γ do not account for additional phase changes induceddue to surface roughness at A .On the other hand, recalling from (31), the random variable φ R (and its derived randomvariables η and η ) denotes the difference in the phase induced solely by the aperture A for anygiven pair of P -field contributions. Assuming that each phase change induced due to apertureroughness is uniformly distributed within the range [− θ, θ ] , the random phase difference φ R istriangularly distributed from [− θ, θ ] .As the random variables η and γ are mutually independent (as well as η and γ ), theexpected value of (cid:104) I Cross ( x , y , Z , t )(cid:105) over the entire aperture plane can now be expressed as E [(cid:104) I Cross ( x , y , Z , t )(cid:105)] = E (cid:20) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:34) γ η (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) − γ η (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) (cid:35) dA (cid:21) , (39) = ⇒ E [(cid:104) I Cross ( x , y , Z , t )(cid:105)] = (cid:18) E [ γ ] E [ η ]− E [ γ ] E [ η ] (cid:19) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) dA . (40)Knowing from Appendix A that E [ η ] = E [(cid:104) I Cross ( x , y , Z , t )(cid:105)] can be expressed as E [(cid:104) I Cross ( x , y , Z , t )(cid:105)] = (cid:18) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) E [ γ ] E [ η ] dA (cid:19) . (41)Furthermore, as is commonly the case in optical wave propagation α = π in the modulo-2 π sense. This is because the propagation path differences between different optical componentsare multiple times the optical wavelength. As E [ γ ] = sinc ( α ) from (64), this results in E [ γ ] = α = π for a triangularly-distributed γ . Therefore the expected value of thecross-interference sum E [(cid:104) I Cross ( x , y , Z , t )(cid:105)] =
0. This is a well-known result of first orderoptical speckle and as P fields are slowly-varying envelopes of the underlying optical carrier,this resulting cross-interference expected value is zero regardless of whether optical carrier ismodulated or not.As is the case for studying optical speckle, the quantity of interest for describing P -fieldspeckle is the variance (and the resulting standard deviation) of (cid:104) I Cross ( x , y , Z , t )(cid:105) . From (35),we obtain σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] = σ (cid:20) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:34) γ η (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) − γ η (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) (cid:35) dA (cid:21) , (42)12 ⇒ σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] = σ [ γ η − γ η ] (cid:20) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) dA (cid:21) , (43)If we define random variables µ = γ η and µ = γ η , which may necessarily not be consideredindependent for the moment. We can define σ µ = (cid:16) σ γ + E [ γ ] (cid:17) · (cid:16) σ η + E [ η ] (cid:17) − E [ γ ] E [ η ] , (44)and σ µ = (cid:16) σ γ + E [ γ ] (cid:17) · (cid:16) σ η + E [ η ] (cid:17) − E [ γ ] E [ η ] . (45)For E [ γ ] = E [ γ ] = α = π in the modulo-2 π sense and E [ η ] =
0, we can express (44)and (45) as σ µ = (cid:16) σ γ (cid:17) · (cid:16) σ η + E [ η ] (cid:17) , (46)and σ µ = (cid:16) σ γ (cid:17) · (cid:16) σ η (cid:17) . (47)Moreover, we can express σ [ γ η − γ η ] = σ [ µ − µ ] = σ µ + σ µ − [ µ , µ ] , (48)where ’Cov’ denotes the covariance of µ and µ . For Cov [ µ , µ ] = E [ µ µ ] − E [ µ ] E [ µ ] σ [ γ η − γ η ] = σ µ + σ µ − ( E [ µ µ ] − E [ µ ] E [ µ ]) . (49)The term E [ µ µ ] can be expressed as E [ µ µ ] = E [ cos ( k | r m | − k | r n |) sin ( k | r m | − k | r n |) cos ( φ R ) sin ( φ R )] = E (cid:20)
14 sin ( [ k | r m | − k | r n |]) sin ( φ R ) (cid:21) = . (50)Also E [ µ ] = E [ η ] E [ γ ] = , (51) E [ µ ] = E [ η ] E [ γ ] = . (52)This results in σ [ γ η − γ η ] = σ µ + σ µ . (53)Substituting σ µ and σ µ from (46) and (47) into (53) yields σ [ γ η − γ η ] = (cid:16) σ γ (cid:17) · (cid:16) σ η + E [ η ] (cid:17) + (cid:16) σ γ (cid:17) · (cid:16) σ η (cid:17) . (54)This results in σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] = (cid:113)(cid:0) σ γ (cid:1) · (cid:0) σ η + E [ η ] (cid:1) + (cid:0) σ γ (cid:1) · (cid:0) σ η (cid:1) (cid:20) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) dA (cid:21) , (55)Now we only need to substitute for all the quantities in (55) in terms of θ (taking α = π ) toobtain an expression for the the spurious P -field noise sum which is additive to the P -field sumprovided by the P -field integral in [1]. Substituting all the terms in (55) from Appendix A, we13btain a rather complicated expression for the P -field speckle noise term σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] .Substituting for α = π in (65) and (66), we obtain σ γ = . σ γ = .
5. Also using (64),(65), and (66), we substitute for σ η , E [ η ] , and σ η in (55) to obtain σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] as afunction of aperture roughness θ as σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] = √ (cid:113)(cid:0) σ η + E [ η ] (cid:1) + (cid:0) σ η (cid:1) (cid:20) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) dA (cid:21) = √ (cid:115) θ + θ + ( θ ) − (cid:0) + θ (cid:1) cos ( θ ) − θ + θ + cos ( θ ) − θ + sinc ( θ )·· (cid:20) N (cid:213) n = N (cid:213) m = m (cid:44) n (cid:112) P m , Ω P n , Ω | r m || r n | cos ( Ω t ) dA (cid:21) (56)This terms represents P -field speckle in light of the carrier speckle. We note that σ [ γ η − γ η ] ≈ (cid:104)(cid:16) σ η + E [ η ] (cid:17) + (cid:16) σ η (cid:17)(cid:105) , (57)for α = π . Moreover, as θ even slightly large, E [ η ] = sinc ( θ ) → σ [ γ η − γ η ] ≈ (cid:104)(cid:16) σ η (cid:17) + (cid:16) σ η (cid:17)(cid:105) , (58)which indicates that the P -field speckle directly depends on carrier speckle only which is due tothe sum of variances in phases accumulated as a result of propagation and aperture roughness. Inother words, P -field speckle is no worse than carrier speckle in most practical scenarios with α = π .The general expression for the magnitude of the total P -field sum (cid:12)(cid:12) [P , Ω ( x , y , z = Z , t )] Tot (cid:12)(cid:12) (where (cid:12)(cid:12) [P , Ω ( x , y , z = Z , t )] Tot (cid:12)(cid:12) = |(cid:104) I ( x , y , Z , t )(cid:105)| ) is given by (cid:12)(cid:12) [P , Ω ( x , y , z = Z , t )] Tot (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:213) n = P n , Ω | r n | cos (cid:2) Ω t + φ β n (cid:3) dA (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) P− field Integral Sum |P Sum ( x , y , Z , t )| + σ [(cid:104) I Cross ( x , y , Z , t )(cid:105)] (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Spurious P− field Signal Noise | P Sp ( x , y , Z , t ) | (59)The total P -field in (59) comprises of what has already been described as the Huygens-like P -field sum in [1] and the additional additive P -field interference sum of contributions fromdifferent locations within A . This we denote as the spurious term P Sp ( x , y , Z , t ) and for the caseof an incoherent optical carrier (i.e. θ = π ), this term reaches a saturation value (cid:12)(cid:12) P Sp ( x , y , Z , t ) (cid:12)(cid:12) Sat resulting in a quasi-steady P -field SNR which we discuss next. A complete spatial randomizationof the carrier phase due to roughness in A is explained when θ = π and consequently | φ R | ≤ π .Under this condition (cid:12)(cid:12) [P , Ω ( x , y , z = Z , t )] Tot (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)∬ A P , Ω ( x (cid:48) , y (cid:48) , ) cos ( Ω t + β | r |)| r | dx (cid:48) d y (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) P Sp ( x , y , Z , t ) (cid:12)(cid:12) Sat , = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∬ A P , Ω ( x (cid:48) , y (cid:48) , ) e β | r | | r | dx (cid:48) d y (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) P− field Integral + (cid:12)(cid:12) P Sp ( x , y , Z , t ) (cid:12)(cid:12) Sat , (60)14nd the P -field integral completely describes P -field imaging. Another observation is that P Sp ( x , y , Z , t ) has a frequency of Ω which is the same as the P -field modulating signal frequencyas well as the P -field sum P Sum ( x , y , Z , t ) that the P -field integral yields. Therefore, the spurious P -field noise exists within the same frequency bandwidth as the actual P -field modulating signaland cannot be simply filtered out and remains as additive noise to the desired P Sum ( x , y , Z , t ) quantity.The model presented here can also be extended to incorporate the spatial structure of theinterference patterns to come up with NLoS reconstruction algorithms that can work with partiallyspecular surfaces. The speckle analysis discussed in [15, 16] can provide a starting point for this. P -field signal-to-noise ratio We are now also in a position to mathematically express the ratio between the magnitudes ofdesired P -field sum from the P -field integral and the spurious P -field signal as a functionof θ . We are tempted to call this ratio the P -field SNR (signal-to-noise ratio) represented by P SNR ( x , y , Z ) and expressed as P SNR ( x , y , Z ) = |P Sum ( x , y , Z , t )| (cid:12)(cid:12) P Sp ( x , y , Z , t ) (cid:12)(cid:12) (61)It is pertinent to plot 1 / (cid:12)(cid:12) P Sp ( x , y , Z , t ) (cid:12)(cid:12) versus θ to describe the change in the P -field SNR with an Fig. 3. Plot of 1 / (cid:12)(cid:12) P Sp ( x , y , Z , t ) (cid:12)(cid:12) versus aperture roughness θ to indicate the trend of P -field SNR with increasing roughness. increasing aperture roughness. We do so in Fig. 3 where we make two key observations. Firstly,the SNR for low aperture roughness is remarkably low as is expected. At these low apertureroughness levels, the P -field integral |P Sum ( x , y , Z , t )| fails and does not model NLoS imaginganalogous to how Huygens integral describes conventional LoS imaging. Spatial incoherenceis a fundamental underlying condition which is required for the P -field integral to accuratelydescribe NLoS imaging.Secondly, we observe that the P -field SNR reaches a steady state value when the apertureroughness allows induces a random phase shift of up to φ R = π /
2. It remains steady and showsminimal change for increasing aperture roughness thereafter.15 .5. Useful insights from estimating a relationship between P -field noise and apertureroughness If we attribute the partial coherence of the E-field solely to the roughness in the aperture plane A ,we are able to establish a relationship between this roughness and the magnitude of the P -fieldnoise P Sp ( x , y , Z , t ) . For this purpose, we model an aperture as a collection of infinitesimallysmall statistically independent regions each imparting a random phase to the carrier E-field. Forthe case of a single random aperture plane, we can simply calculate the P -field noise from (56)by knowing the maximum roughness variation in A and setting the limits of the random variable φ R as [− θ, θ ] accordingly.For a random aperture which is modeled as a collection of N mutually-independent smallregions, we can also determine the error in the estimate of the P -field noise from (56). Eqn.(56)can also be used to determine the maximum roughness of the wall to achieve the saturation levelof the P -field signal-to-noise ratio.Due to the saturation of the P -field SNR, most NLoS experiments conducted so far usediffuse relay walls (which exhibit surface roughness resulting in a phase shift by multipleperiods). Reconstruction methods have been developed for this highly-diffuse aperture scenario.A saturation of the P -field SNR for such highly diffuse surfaces, as we show here, explains thehigh quality P -field reconstruction such as the ones demonstrated in [12]. While these methodsyield very good results in this case, there are also more specularly reflective surfaces in the realworld. Hopefully, this work provides the necessary theoretical insight that allows for the futuredevelopment of reconstruction methods that can handle more specular surfaces (but not mirrors –otherwise, NLoS imaging would be pointless).Optical NLoS imaging around corners generally involves relay surfaces which are typicallyrough enough - owing to the small wavelength scales at optical frequencies - to obtain a stable P -field SNR. Outside of the realm of optical carrier-based NLoS imaging, our analysis can proveto be a valuable tool for future imaging around corner applications under consideration involvinglow carrier frequencies such as applications involving ultrasound-based NLoS imaging aroundcorners where the P -field SNR performance can be evaluated for semi-rough relay wall aperturesat the ultrasound signal wavelength range and help design aperture roughness profiles to obtain adesired minimal P -field imaging SNR.
4. Conclusion
This paper expands on the topic of phasor field propagation and interference for NLoS imagingwhen, unlike previous treatment in [1] where the E-field was considered as spatially incoherent,the E-field is considered as partially coherent. We determine the magnitude of an increasing P -field additive noise with an increase in E-field coherence. This deviation from a complete P -field integral-based solution to NLoS imaging is very important to study the effects of apertureroughness on NLoS imaging scene reconstruction as well as quantifying the degree of spatialcoherence and its effects on occlusion-aided imaging. The statistical treatment presented in thispaper is critical in establishing a unified P -field behavior under various different levels of opticalcoherence for NLoS imaging and other types of LoS imaging techniques such as P -field imagingthrough fog. Via this statistical treatment of P -fields, we also validate optical incoherence asa necessary condition for the P -field integral to completely describe NLoS imaging as wasdescribed in [1]. The magnitude of the P -field noise that is introduced by virtue of deviatingfrom this condition also allows us to determine a P -field signal-to-noise ratio.16 ppendix A: Expected value and standard deviation of cos ( φ R ) and sin ( φ R ) for atriangularly distributed random variable φ R In this section, we summarize and enlist (omitting proofs) the fundamental expressions for theexpected values of cos ( φ R ) and sin ( φ R ) and the variance of cos θ , where φ R is treated as atriangularly distributed random variable in the modulo 2 π range [− θ, θ ] for the possible valuesof θ . We require the analytical forms of these expressions for determining the expected value ofthe sum of all P -field cross-multiplication terms in (29). As a reminder, the probability densityfunction of φ R is considered triangularly distributed because φ R denotes the difference betweentwo uniformly-distributed random variables (that each denote phase difference added throughrandom propagation path lengths which are considered uniformly distributed respectively) inthe range modulo 2 π [− θ, θ ] . The triangularly distributed random phase φ R has the probabilitydensity function P D ( φ R ) = (cid:16) θ (cid:17) ( φ R + θ ) − θ ≤ φ R ≤ , (cid:16) θ (cid:17) ( θ − φ R ) ≤ φ R ≤ θ, . (62) Expected value of cos ( φ R ) When g ( U ) denotes the function of the random variable and P D ( U ) its probability densityfunction, the Fundamental theorem of expectation [17] E { g ( U )} = ∫ ∞−∞ g ( U ) P D ( U ) dU , (63)can be used to determine the expected values of cos ( φ R ) and sin ( φ R ) . If η = cos ( φ R ) , then theexpected value E ( η ) = E ( cos ( φ R )) is given by E ( η ) = ∫ − θ cos ( φ R ) (cid:18) θ (cid:19) ( φ R + θ ) d φ R + ∫ θ cos ( φ R ) (cid:18) θ (cid:19) ( θ − φ R ) d φ R (64) = sinc ( θ ) . Expected value of sin ( φ R ) Since sine is an odd function, it is quite evident that E [ sin ( φ R )] = Standard deviation of cos ( φ R ) Similarly, we can determine the standard deviation of η = cos ( φ R ) . The full proof is omitted,but the result for triangularly distributed φ R is given by σ η = (cid:115) θ + θ + ( θ ) − (cid:0) + θ (cid:1) cos ( θ ) − θ . (65) Standard deviation of sin ( φ R ) We can also determine the standard deviation of η = sin ( φ R ) . We state it here without proof that σ η = (cid:114) θ + cos ( θ ) − θ . (66)17 eferences
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