Abstract

Speckle is the spatial fluctuation of irradiance seen when coherent light is reflected from a rough surface. It is due to light reflected from the surface’s many nooks and crannies accumulating vastly discrepant time delays, spanning much more than an optical period, en route to an observation point. Although speckle with continuous-wave (cw) illumination is well understood, the emerging interest in non-line-of-sight (NLoS) imaging using coherent light has created the need to understand the higher-order speckle that results from multiple rough-surface reflections, viz., speckled speckle and speckled speckled speckle. Moreover, the recent introduction of phasor-field (${{\mathcal {P}}}$-field) NLoS imaging—which relies on amplitude-modulated coherent illumination—requires pushing beyond cw scenarios for speckle and higher-order speckle. In this paper, we take first steps in addressing the foregoing needs using a three-diffuser transmissive geometry that is a proxy for three-bounce NLoS imaging. In the small-diffusers limit, we show that the irradiance variances of cw and modulated nth-order speckle coincide and are (2n − 1)-times those of ordinary (first-order) speckle. The more important case for NLoS imaging, however, involves extended diffuse reflectors. For our transmissive geometry with extended diffusers, we treat third-order cw speckle and first-order modulated speckle. Our results there imply that speckle is unlikely to impede successful operation of coherent-illumination cw imagers, and they suggest that the same might be true for ${{\mathcal {P}}}$-field imagers.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. W. Goodman, Speckle Phenomena in Optics, (Roberts & Company, 2007).
  2. S. A. Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].
  3. J. Dove and J. H. Shapiro, “Paraxial theory of phasor-field imaging,” Opt. Express 27(13), 18016–18037 (2019).
    [Crossref]
  4. J. A. Teichman, “Phasor field waves: a mathematical treatment,” Opt. Express 27(20), 27500–27506 (2019).
    [Crossref]
  5. J. Dove and J. H. Shapiro, “Paraxial phasor-field physical optics,” Opt. Express 28(14), 21095–21109 (2020).
    [Crossref]
  6. J. Dove, “Theory of phasor-field imaging,” Ph.D. thesis, Massachusetts Institute of Technology (2020).
  7. The STA irradiance $I_z(\boldsymbol {\rho}_z, t)$Iz(ρz,t) is the power density, averaged over the time interval [t − Ta, t] where 2π/ω0 ≪ Ta ≪ 2π/Δω, that illuminates the transverse coordinates $\boldsymbol {\rho}_z $ρz in the z-plane.
  8. For cw illumination Δλ is infinite, hence in that case we need only assume that the diffuser is rough at the optical wavelength.
  9. In the NLoS scenario, both $h_0(\boldsymbol {\rho}_0) $h0(ρ0) and $h_2(\boldsymbol {\rho}_2) $h2(ρ2) represent the visible wall. Nevertheless, they can be taken to be statistically independent in our transmissive-geometry proxy if the NLoS imager’s visible-wall illumination falls on a different portion of that wall than what lies in its camera’s field of view.
  10. Interestingly, having λ0 ≪ σh ≪ Δλ and ρh ∼ λ0 results in the first and second moments of ${E_0}^\prime(\boldsymbol {\rho}_0) $E0′(ρ0) being independent of both σh and ρh.
  11. That the diffuser-averaged STA irradiance is independent of the transverse coordinate is an artifact of Fresnel diffraction and total spatial incoherence produced by our delta-function approximation to $\langle e^{i\omega _0[h_0({\boldsymbol \rho }_0)-h_0(\tilde {{\boldsymbol \rho }}_0)]/c}\rangle$⟨eiω0[h0(ρ0)−h0(ρ~0)]/c⟩. Note that 〈I1〉 < 〈I0〉 holds because d0 ≪ L is required for Fresnel diffraction to be valid.
  12. E. W. Weisstein, “Meijer G-function,” http://mathworld.wolfram.com/MeijerG-Function.html .
  13. J. H. Shapiro, B. A. Capron, and R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20(19), 3292–3313 (1981).
    [Crossref]
  14. J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1547–1569 (2009).
    [Crossref]
  15. We used this condition in our analysis of synthetic-wavelength holography [5].
  16. F. Willomitzer, P. V. Rangarajan, F. Li, M. M. Balaji, M. P. Christensen, and O. Cossairt, “Synthetic wavelength holography: an extension of Gabor’s holographic principle to imaging with scattered wavefronts,” arXiv:1912.11438 [physics.optics].
  17. X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
    [Crossref]

2020 (1)

2019 (3)

J. Dove and J. H. Shapiro, “Paraxial theory of phasor-field imaging,” Opt. Express 27(13), 18016–18037 (2019).
[Crossref]

J. A. Teichman, “Phasor field waves: a mathematical treatment,” Opt. Express 27(20), 27500–27506 (2019).
[Crossref]

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

2009 (1)

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1547–1569 (2009).
[Crossref]

1981 (1)

Balaji, M. M.

F. Willomitzer, P. V. Rangarajan, F. Li, M. M. Balaji, M. P. Christensen, and O. Cossairt, “Synthetic wavelength holography: an extension of Gabor’s holographic principle to imaging with scattered wavefronts,” arXiv:1912.11438 [physics.optics].

Capron, B. A.

Christensen, M. P.

F. Willomitzer, P. V. Rangarajan, F. Li, M. M. Balaji, M. P. Christensen, and O. Cossairt, “Synthetic wavelength holography: an extension of Gabor’s holographic principle to imaging with scattered wavefronts,” arXiv:1912.11438 [physics.optics].

Cossairt, O.

F. Willomitzer, P. V. Rangarajan, F. Li, M. M. Balaji, M. P. Christensen, and O. Cossairt, “Synthetic wavelength holography: an extension of Gabor’s holographic principle to imaging with scattered wavefronts,” arXiv:1912.11438 [physics.optics].

Dove, J.

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics, (Roberts & Company, 2007).

Guillén, I.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

Gutierrez, D.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

Harney, R. C.

Jarabo, A.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

La Manna, M.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

S. A. Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].

Le, T. H.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

Li, F.

F. Willomitzer, P. V. Rangarajan, F. Li, M. M. Balaji, M. P. Christensen, and O. Cossairt, “Synthetic wavelength holography: an extension of Gabor’s holographic principle to imaging with scattered wavefronts,” arXiv:1912.11438 [physics.optics].

Liu, X.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

Nam, J. H.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

Rangarajan, P. V.

F. Willomitzer, P. V. Rangarajan, F. Li, M. M. Balaji, M. P. Christensen, and O. Cossairt, “Synthetic wavelength holography: an extension of Gabor’s holographic principle to imaging with scattered wavefronts,” arXiv:1912.11438 [physics.optics].

Reza, S. A.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

S. A. Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].

Shapiro, J. H.

Teichman, J. A.

Velten, A.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

S. A. Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].

Weisstein, E. W.

E. W. Weisstein, “Meijer G-function,” http://mathworld.wolfram.com/MeijerG-Function.html .

Willomitzer, F.

F. Willomitzer, P. V. Rangarajan, F. Li, M. M. Balaji, M. P. Christensen, and O. Cossairt, “Synthetic wavelength holography: an extension of Gabor’s holographic principle to imaging with scattered wavefronts,” arXiv:1912.11438 [physics.optics].

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1547–1569 (2009).
[Crossref]

Nature (1)

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, A. Jarabo, D. Gutierrez, and A. Velten, “Non-line-of-sight imaging using phasor-field virtual wave optics,” Nature 572(7771), 620–623 (2019).
[Crossref]

Opt. Express (3)

Other (11)

We used this condition in our analysis of synthetic-wavelength holography [5].

F. Willomitzer, P. V. Rangarajan, F. Li, M. M. Balaji, M. P. Christensen, and O. Cossairt, “Synthetic wavelength holography: an extension of Gabor’s holographic principle to imaging with scattered wavefronts,” arXiv:1912.11438 [physics.optics].

J. W. Goodman, Speckle Phenomena in Optics, (Roberts & Company, 2007).

S. A. Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].

J. Dove, “Theory of phasor-field imaging,” Ph.D. thesis, Massachusetts Institute of Technology (2020).

The STA irradiance $I_z(\boldsymbol {\rho}_z, t)$Iz(ρz,t) is the power density, averaged over the time interval [t − Ta, t] where 2π/ω0 ≪ Ta ≪ 2π/Δω, that illuminates the transverse coordinates $\boldsymbol {\rho}_z $ρz in the z-plane.

For cw illumination Δλ is infinite, hence in that case we need only assume that the diffuser is rough at the optical wavelength.

In the NLoS scenario, both $h_0(\boldsymbol {\rho}_0) $h0(ρ0) and $h_2(\boldsymbol {\rho}_2) $h2(ρ2) represent the visible wall. Nevertheless, they can be taken to be statistically independent in our transmissive-geometry proxy if the NLoS imager’s visible-wall illumination falls on a different portion of that wall than what lies in its camera’s field of view.

Interestingly, having λ0 ≪ σh ≪ Δλ and ρh ∼ λ0 results in the first and second moments of ${E_0}^\prime(\boldsymbol {\rho}_0) $E0′(ρ0) being independent of both σh and ρh.

That the diffuser-averaged STA irradiance is independent of the transverse coordinate is an artifact of Fresnel diffraction and total spatial incoherence produced by our delta-function approximation to $\langle e^{i\omega _0[h_0({\boldsymbol \rho }_0)-h_0(\tilde {{\boldsymbol \rho }}_0)]/c}\rangle$⟨eiω0[h0(ρ0)−h0(ρ~0)]/c⟩. Note that 〈I1〉 < 〈I0〉 holds because d0 ≪ L is required for Fresnel diffraction to be valid.

E. W. Weisstein, “Meijer G-function,” http://mathworld.wolfram.com/MeijerG-Function.html .

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Figures (3)

Fig. 1.
Fig. 1. Geometry for third-order speckle analysis. Thin blue rectangles represent idealized, thin diffusers. The black frames in front of the diffusers in planes 1 and 2 represent Gaussian pupils that capture the essence of the target and visible-wall sizes, respectively. The dashed line represents the detection plane.
Fig. 2.
Fig. 2. Logarithmic plots of the pdfs for $\tilde {I}_n \equiv I_n(\textbf {0})/\langle I_n\rangle$ , the normalized $n$ th-order speckle in the small-diffusers limit.
Fig. 3.
Fig. 3. Saturation signal-to-noise ratio, $\textrm {SNR}_\textrm {sat}$ , in dB versus ratio of the detector’s diameter to target plane’s diameter, $D/d_1$ , for four representative $(\Omega _{01},\Omega _{12})$ pairs.

Equations (72)

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E 0 ( ρ 0 ) = I 0 e 4 | ρ 0 | 2 / d 0 2 ,
E 0 ( ρ 0 ) = E 0 ( ρ 0 ) e i ω 0 h 0 ( ρ 0 ) / c ,
K h h ( ρ n , ρ ~ n ) h n ( ρ n ) h n ( ρ ~ n ) = σ h 2 e | ρ n ρ ~ n | 2 / ρ h 2 ,
E 0 ( ρ 0 ) 0 = E 0 ( ρ 0 ) e ω 0 2 σ h 2 / 2 c 2 0 ,
E 0 ( ρ 0 ) E 0 ( ρ ~ 0 ) 0 = E 0 ( ρ 0 ) E 0 ( ρ ~ 0 ) exp [ ω 0 2 σ h 2 ( 1 + e | ρ 0 ρ ~ 0 | 2 / ρ h 2 ) / c 2 ] 0 ,
E 0 ( ρ 0 ) E 0 ( ρ ~ 0 ) 0 = E 0 ( ρ 0 ) E 0 ( ρ ~ 0 ) exp [ ω 0 2 σ h 2 ( 1 e | ρ 0 ρ ~ 0 | 2 / ρ h 2 ) / c 2 ]
E 0 ( ρ 0 ) E 0 ( ρ ~ 0 ) λ 0 2 δ ( ρ 0 ρ ~ 0 ) ,
E 1 ( ρ 1 ) = e i ω 0 L / c i λ 0 L d 2 ρ 0 E 0 ( ρ 0 ) e i ω 0 | ρ 1 ρ 0 | 2 / 2 c L ,
E 1 ( ρ 1 ) E 1 ( ρ ~ 1 ) 0 = e i ω 0 ( | ρ 1 | 2 | ρ ~ 1 | 2 ) / 2 c L L 2 d 2 ρ 0 | E 0 ( ρ 0 ) | 2 e i ω 0 ( ρ ~ 1 ρ 1 ) ρ 0 / c L ,
E 2 ( ρ 2 ) E 2 ( ρ ~ 2 ) 1 = e i ω 0 ( | ρ 2 | 2 | ρ ~ 2 | 2 ) / 2 c L L 2 d 2 ρ 1 | E 1 ( ρ 1 ) | 2 e 8 | ρ 1 | 2 / d 1 2 e i ω 0 ( ρ ~ 2 ρ 2 ) ρ 1 / c L ,
E 2 ( ρ 2 ) E 2 ( ρ ~ 2 ) 0 , 1 = e i ω 0 ( | ρ 2 | 2 | ρ ~ 2 | 2 ) / 2 c L L 2 d 2 ρ 1 | E 1 ( ρ 1 ) | 2 0 e 8 | ρ 1 | 2 / d 1 2 e i ω 0 ( ρ ~ 2 ρ 2 ) ρ 1 / c L ,
| E 1 ( ρ 1 ) | 2 e 8 | ρ 1 | 2 / d 1 2 | E 1 ( 0 ) | 2 e 8 | ρ 1 | 2 / d 1 2 .
p I 2 ( I 2 ) = 0 d I 1 exp ( I 1 / I 1 ) I 1 exp [ I 2 / ( π d 1 2 I 1 / 8 L 2 ) ] ( π d 1 2 I 1 / 8 L 2 ) u ( I 2 ) = [ 2 K 0 ( 2 I 2 / I 2 ) / I 2 ] u ( I 2 ) ,
| E 2 ( ρ 2 ) | 2 e 8 | ρ 2 | 2 / d 2 2 | E 2 ( 0 ) | 2 e 8 | ρ 2 | 2 / d 2 2 ,
E 3 ( ρ 3 ) E 3 ( ρ ~ 3 ) 2 = e i ω 0 ( | ρ 3 | 2 | ρ ~ 3 | 2 ) / 2 c L L 2 | E 2 ( 0 ) | 2 d 2 ρ 2 e 8 | ρ 2 | 2 / d 2 2 e i ω 0 ( ρ ~ 3 ρ 3 ) ρ 2 / c L .
p I 3 ( I 3 ) = 0 d I 2 2 K 0 ( 2 I 2 / I 2 ) I 2 exp [ I 3 / ( π d 2 2 I 2 / 8 L 2 ) ] ( π d 2 2 I 2 / 8 L 2 ) u ( I 3 ) .
p I 3 ( I 3 ) = [ G 0 , 3 3 , 0 ( I 3 / I 3 | 0 , 0 , 0 ) / I 3 ] u ( I 3 ) ,
p I n ( I n ) = [ G 0 , n n , 0 ( I n / I n | 0 , , 0 ) / I n ] u ( I n ) ,
NVar I n Var [ I n ( 0 ) ] / I n 2 = ( 2 n 1 ) .
E 1 ( ρ 1 , t ) e 4 | ρ 1 | 2 / d 1 2 = e i ω 0 L / c ( d 2 ρ 0 E 0 ( ρ 0 ) i λ 0 L ) S ( t L / c ) e 4 | ρ 1 | 2 / d 1 2
= E 1 ( 0 ) S ( t L / c ) e 4 | ρ 1 | 2 / d 1 2 ,
E 2 ( ρ 2 , t ) e 4 | ρ 2 | 2 / d 2 2 = e i ω 0 L / c ( d 2 ρ 1 E 1 ( ρ 1 ) i λ 0 L ) S ( t 2 L / c ) e 4 | ρ 2 | 2 / d 2 2
= E 2 ( 0 ) S ( t 2 L / c ) e 4 | ρ 2 | 2 / d 2 2 ,
SNR N 2 Var ( N ) = ( η P 3 T / ω 0 ) 2 η P 3 T / ω 0 + η 2 Var ( P 3 ) T 2 / ( ω 0 ) 2 ,
P 3 = | ρ 3 | D / 2 d 2 ρ 3 I 3 ( ρ 3 )
SNR = SNR sat SNR sat / N + 1 ,
SNR sat P 3 2 Var ( P 3 ) = ( π D 2 I 3 / 4 ) 2 | ρ 3 | D / 2 d 2 ρ 3 | ρ ~ 3 | D / 2 d 2 ρ ~ 3 Covar [ I 3 ( ρ 3 ) , I 3 ( ρ ~ 3 ) ] ,
NCovar I 3 ( ρ 3 ρ ~ 3 ) Covar [ I 3 ( ρ 3 ) , I 3 ( ρ ~ 3 ) ] I 3 2 ,
Covar [ I 1 ( ρ 1 ) , I 1 ( ρ ~ 1 ) ] = | E 1 ( ρ 1 ) | 2 | E 1 ( ρ ~ 1 ) | 2 0 I 1 2
= | E 1 ( ρ 1 ) E 1 ( ρ ~ 1 ) 0 | 2
= | e i ω 0 ( | ρ 1 | 2 | ρ ~ 1 | 2 ) / 2 c L L 2 d 2 ρ 0 | E 0 ( ρ 0 ) | 2 e i ω 0 ( ρ ~ 1 ρ 1 ) ρ 0 / c L | 2 ,
NCovar I 1 ( ρ 1 ρ ~ 1 ) = e 4 Ω 01 | ρ 1 ρ ~ 1 | 2 / d 1 2 ,
I 2 ( ρ 2 ) I 2 ( ρ ~ 2 ) 1 = E 2 ( ρ 2 ) E 2 ( ρ 2 ) E 2 ( ρ ~ 2 ) E 2 ( ρ ~ 2 ) 1
= I 2 ( ρ 2 ) 1 I 2 ( ρ ~ 2 ) 1 + | E 2 ( ρ 2 ) E 2 ( ρ ~ 2 ) 1 | 2 .
I 2 ( ρ 2 ) I 2 ( ρ ~ 2 ) 1 = 1 L 4 [ d 2 ρ 1 d 2 ρ ~ 1 I 1 ( ρ 1 ) I 1 ( ρ ~ 1 ) e 8 ( | ρ 1 | 2 + | ρ ~ 1 | 2 ) / d 1 2 × ( 1 + e i ω 0 ( ρ 1 ρ ~ 1 ) ( ρ 2 ρ ~ 2 ) / c L ) ] .
I 2 ( ρ 2 ) I 2 ( ρ ~ 2 ) 0 , 1 = 1 L 4 [ d 2 ρ 1 d 2 ρ ~ 1 I 1 ( ρ 1 ) I 1 ( ρ ~ 1 ) 0 e 8 ( | ρ 1 | 2 + | ρ ~ 1 | 2 ) / d 1 2 × ( 1 + e i ω 0 ( ρ 1 ρ ~ 1 ) ( ρ 2 ρ ~ 2 ) / c L ) ] ,
NCovar I 2 ( ρ 2 ρ ~ 2 ) = e 4 Ω 01 | ρ 2 ρ ~ 2 | 2 / d 0 2 + 1 1 + Ω 01 ( 1 + e 4 Ω 01 | ρ 2 ρ ~ 2 | 2 / d 0 2 ( 1 + Ω 01 ) ) ,
NCovar I 2 ( ρ 2 ρ ~ 2 ) = e ( π d 1 / 2 λ 0 L ) 2 | ρ 2 ρ ~ 2 | 2
NCovar I 1 ( ρ 1 ρ ~ 1 ) = e ( π d 0 / 2 λ 0 L ) 2 | ρ 1 ρ ~ 1 | 2 ,
NCovar I 3 ( ρ 3 ρ ~ 3 ) = e 4 Ω 12 | ρ 3 ρ ~ 3 | 2 / d 1 2 + 1 1 + Ω 01 ( 1 + e 4 Ω 12 | ρ 3 ρ ~ 3 | 2 / d 1 2 ) + 1 1 + Ω 12 ( 1 + e 4 Ω 12 | ρ 3 ρ ~ 3 | 2 / d 1 2 ( 1 + Ω 12 ) ) + 1 1 + Ω 01 + Ω 12 ( 1 + e 4 Ω 12 ( 1 + Ω 01 ) | ρ 3 ρ ~ 3 | 2 / d 1 2 ( 1 + Ω 01 + Ω 12 ) ) ,
NCovar I 3 ( ρ 3 ρ ~ 3 ) = e ( π d 2 / 2 λ 0 L ) 2 | ρ 3 ρ ~ 3 | 2 ,
SNR sat = ( π D 2 / 4 ) 2 | ρ 3 | D / 2 d 2 ρ 3 | ρ ~ 3 | D / 2 d 2 ρ ~ 3 NCovar I 3 ( ρ 3 ρ ~ 3 ) .
SNR sat = ( π D 2 / 4 ) 2 | ρ | D d 2 ρ NCovar I 3 ( ρ ) O ( ρ , D )
O ( ρ , D ) = D 2 2 [ cos 1 ( | ρ | D ) | ρ | D 1 | ρ | 2 D 2 ] ,  for 0 | ρ | D ,
SNR sat = [ 1 1 + Ω 01 + 1 1 + Ω 12 + 1 1 + Ω 01 + Ω 12 + d 1 2 D 2 Ω 12 ( 1 + Ω 12 ) ( 1 + 1 1 + Ω 01 ) d 1 2 ( 2 + Ω 01 ) D 2 Ω 12 ( 1 + Ω 01 ) B ( 2 D 2 Ω 12 d 1 2 ) d 1 2 D 2 Ω 12 B ( 2 D 2 Ω 12 d 1 2 ( 1 + Ω 12 ) ) d 1 2 D 2 Ω 12 ( 1 + Ω 01 ) B ( 2 D 2 Ω 12 ( 1 + Ω 01 ) d 1 2 ( 1 + Ω 01 + Ω 12 ) ) ] 1 ,
B ( x ) e x [ BI 0 ( x ) + BI 1 ( x ) ] ,
P z ( ρ z , ω ) d t I z ( ρ z , t ) e i ω t = d ω + 2 π E z ( ρ z , ω + + ω / 2 ) E z ( ρ z , ω + ω / 2 ) ,
P ~ z ( ρ z , ω ) d ω + 2 π E z ( ρ z , ω + + ω / 2 ) E z ( ρ z , ω + ω / 2 ) ,
| P ~ 1 ( ρ 1 , ω ) | 2 0 = d ω + 2 π d ω ~ + 2 π E 1 ( ρ 1 , ω + + ω / 2 ) E 1 ( ρ 1 , ω + ω / 2 ) E 1 ( ρ 1 , ω ~ + + ω / 2 ) E 1 ( ρ 1 , ω ~ + ω / 2 ) 0 = d ω + 2 π d ω ~ + 2 π [ E 1 ( ρ 1 , ω + + ω / 2 ) E 1 ( ρ 1 , ω + ω / 2 ) 0 E 1 ( ρ 1 , ω ~ + + ω / 2 ) E 1 ( ρ 1 , ω ~ + ω / 2 ) 0 + E 1 ( ρ 1 , ω + + ω / 2 ) E 1 ( ρ 1 , ω ~ + + ω / 2 ) 0 E 1 ( ρ 1 , ω + ω / 2 ) E 1 ( ρ 1 , ω ~ + ω / 2 ) 0 ] .
Var [ P ~ 1 ( ρ 1 , ω ) ] = | P ~ 1 ( ρ 1 , ω ) | 2 0 | P 1 ( ρ 1 , ω ) | 2 = d ω + 2 π d ω ~ + 2 π E 1 ( ρ 1 , ω + + ω / 2 ) E 1 ( ρ 1 , ω ~ + + ω / 2 ) 0 E 1 ( ρ 1 , ω + ω / 2 ) E 1 ( ρ 1 , ω ~ + ω / 2 ) 0 = 1 L 4 d ω + 2 π d ω ~ + 2 π d 2 ρ 0 d 2 ρ ~ 0 E 0 ( ρ 0 , ω + + ω / 2 ) E 0 ( ρ 0 , ω ~ + + ω / 2 ) E 0 ( ρ ~ 0 , ω + ω / 2 ) E 0 ( ρ ~ 0 , ω ~ + ω / 2 )
× e i ( ω + ω ~ + ) ( | ρ 1 ρ 0 | 2 | ρ 1 ρ ~ 0 | 2 ) / 2 c L .
Var [ P ~ 1 ( ρ 1 , ω ) ] = 1 L 4 d ω + 2 π d ω ~ + 2 π d 2 ρ 0 d 2 ρ ~ 0 | E 0 ( ρ 0 ) | 2 | E 0 ( ρ ~ 0 ) | 2 e i ( ω + ω ~ + ) ( | ρ 1 ρ 0 | 2 | ρ 1 ρ ~ 0 | 2 ) / 2 c L × S ( ω + + ω / 2 ) S ( ω ~ + + ω / 2 ) S ( ω + ω / 2 ) S ( ω ~ + ω / 2 ) ,
Var [ P ~ 1 ( ρ 1 , 0 ) ] = 1 L 4 d ω + 2 π d ω ~ + 2 π d 2 ρ 0 d 2 ρ ~ 0 | E 0 ( ρ 0 ) | 2 | E 0 ( ρ ~ 0 ) | 2 | S ( ω + ) | 2 | S ( ω ~ + ) | 2 × e i ( ω + ω ~ + ) ( | ρ 1 ρ 0 | 2 | ρ 1 ρ ~ 0 | 2 ) / 2 c L 1 L 4 d ω + 2 π d ω ~ + 2 π d 2 ρ 0 d 2 ρ ~ 0 | E 0 ( ρ 0 ) | 2 | E 0 ( ρ ~ 0 ) | 2 | S ( ω + ) | 2 | S ( ω ~ + ) | 2
= | 1 L 2 d 2 ρ 0 P 0 ( ρ 0 , 0 ) | 2 = | P 1 ( ρ 1 , 0 ) | 2 ,
S ( t ) = e t 2 / T 2 cos ( Ω t )
S ( ω ) = T π 2 ( e T 2 ( ω Ω ) 2 / 4 + e T 2 ( ω + Ω ) 2 / 4 ) ,
S ( ω + + Ω ) S ( ω ~ + + Ω ) S ( ω + Ω ) S ( ω ~ + Ω ) π 2 T 4 16 e T 2 ( ω + 2 + ω ~ + 2 ) / 2
Var [ P ~ 1 ( ρ 1 , 2 Ω ) ] = ( π I 0 T 2 4 L 2 ) 2 d ω + 2 π d ω ~ + 2 π d 2 ρ 0 d 2 ρ ~ 0 e 8 ( | ρ 0 | 2 + | ρ ~ 0 | 2 ) / d 0 2 × e T 2 ( ω + 2 + ω ~ + 2 ) / 2 e i ( ω + ω ~ + ) ρ 1 ( ρ 0 ρ ~ 0 ) / c L
= π P 0 2 T 2 32 L 4 1 1 + ( d 0 / 2 c L T ) 2 | ρ 1 | 2 ,
( d 0 / 2 c L T ) 2 | ρ 1 | 2 < ( 2 π / Ω T ) 2 L / Λ 1 ,
Var [ P ~ 1 ( ρ 1 , 2 Ω ) ] = π P 0 2 T 2 / 32 L 4 .
| P 1 ( ρ 1 , 2 Ω ) | 2 = ( π I 0 T 2 4 L 2 ) 2 d ω + 2 π d ω ~ + 2 π d 2 ρ 0 d 2 ρ ~ 0 × e 8 ( | ρ 0 | 2 + | ρ ~ 0 | 2 ) / d 0 2 e T 2 ( ω + 2 + ω ~ + 2 ) / 2 e i 2 Ω ρ 1 ( ρ 0 ρ ~ 0 ) / c L
= π P 0 2 T 2 32 L 4 e ( d 0 Ω / 2 c L ) 2 | ρ 1 | 2 .
e ( d 0 Ω / 2 c L ) 2 | ρ 1 | 2 < e ( 2 π d 0 / Λ ) 2 1.08 ,
| S ( ω + ) | 2 | S ( ω ~ + ) | 2 π 2 T 4 16 ( e T 2 ( ω + Ω ) 2 / 2 + e T 2 ( ω + + Ω ) 2 / 2 ) ( e T 2 ( ω ~ + Ω ) 2 / 2 + e T 2 ( ω ~ + + Ω ) 2 / 2 ) ,
Var [ P ~ 1 ( ρ 1 , 0 ) ] = π P 0 2 T 2 16 L 4 1 + α 2 | ρ 1 | 2 ( 1 + e Ω 2 T 2 α 2 | ρ 1 | 2 / ( 1 + α 2 | ρ 1 | 2 ) + 2 e Ω 2 T 2 + 4 e Ω 2 T 2 ( 2 + 3 α 2 | ρ 1 | 2 ) / 4 ( 1 + α 2 | ρ 1 | 2 ) ) ,
Var [ P ~ 1 ( 0 , 0 ) ] = π P 0 2 T 2 8 L 4 ( 1 + e Ω 2 T 2 + 2 e Ω 2 T 2 / 2 ) .
| P 1 ( ρ 1 , 0 ) | 2 = π P 0 2 T 2 8 L 4 ( 1 + e Ω 2 T 2 + 2 e Ω 2 T 2 / 2 ) ,
Covar [ P ~ 1 ( ρ 1 , ω ) , P ~ 1 ( ρ ~ 1 , ω ) ] = P ~ 1 ( ρ 1 , ω ) P ~ 1 ( ρ ~ 1 , ω ) 0 P 1 ( ρ 1 , ω ) P 1 ( ρ ~ 1 , ω ) = I 0 2 L 4 d ω + 2 π d ω ~ + 2 π d 2 ρ 0 d 2 ρ ~ 0 e 8 ( | ρ 0 | 2 + | ρ ~ 0 | 2 ) / d 0 2 × S ( ω + + ω / 2 ) S ( ω ~ + + ω / 2 ) S ( ω + ω / 2 ) S ( ω ~ + ω / 2 ) × e i [ ( ω 0 + ω + + ω / 2 ) | ρ 1 ρ 0 | 2 ( ω 0 + ω ~ + + ω / 2 ) | ρ ~ 1 ρ 0 | 2 ( ω 0 + ω + ω / 2 ) | ρ 1 ρ ~ 0 | 2 + ( ω 0 + ω ~ + ω / 2 ) | ρ ~ 1 ρ ~ 0 | 2 ] / 2 c L .
Covar [ P ~ 1 ( ρ 1 , 2 Ω ) , P ~ 1 ( ρ ~ 1 , 2 Ω ) ] = ( π I 0 T 2 4 L 2 ) 2 e i Ω ( | ρ 1 | 2 | ρ ~ 1 | 2 ) / c L × d ω + 2 π d ω ~ + 2 π d 2 ρ 0 d 2 ρ ~ 0 e 8 ( | ρ 0 | 2 + | ρ ~ 0 | 2 ) / d 0 2 e T 2 ( ω + 2 + ω ~ + 2 ) / 2 × e i [ ρ 0 ρ 1 ( ω 0 + ω + + Ω ) + ρ 0 ρ ~ 1 ( ω 0 + ω ~ + + Ω ) + ρ ~ 0 ρ 1 ( ω 0 + ω + Ω ) ρ ~ 0 ρ ~ 1 ( ω 0 + ω ~ + Ω ) ] / c L .
Covar [ P ~ 1 ( ρ 1 , 2 Ω ) , P ~ 1 ( ρ ~ 1 , 2 Ω ) ] = π P 0 2 T 2 32 L 4 e i Ω ( | ρ 1 | 2 | ρ ~ 1 | 2 ) / c L e α 2 Ω 2 T 2 | ρ | 2 / 4 ( 1 + α 2 | ρ + | 2 ) ( 1 + α 2 | ρ | 2 / 4 ) α 4 ( ρ + ρ ) 2 / 4 × exp { ω 0 2 T 2 4 α 2 | ρ | 2 + α 4 [ | ρ + | 2 | ρ | 2 ( ρ + ρ ) 2 ] ( 1 + α 2 | ρ + | 2 ) ( 1 + α 2 | ρ | 2 / 4 ) α 4 ( ρ + ρ ) 2 / 4 } ,
lim T | Covar [ P ~ 1 ( ρ 1 , 2 Ω ) , P ~ 1 ( ρ ~ 1 , 2 Ω ) ] | Var [ P ~ 1 ( ρ 1 , 2 Ω ) ] Var [ P ~ 1 ( ρ ~ 1 , 2 Ω ) ] = e ( ω 0 2 + Ω 2 ) d 0 2 | ρ | 2 / 16 c 2 L 2 .