Abstract

A short pulse propagating through a medium consisting of randomly distributed scatterers, large compared to the wavelength, is expected to develop an “early-time diffusion” (ETD) behavior: a sharply rising structure in the time-resolved intensity, immediately following the coherent (ballistic) component. Since the ETD signal is attenuated at a rate substantially lower than the coherent wave, it offers a possibility of application in imaging through diverse scattering media, such as atmospheric obscurants (clouds, fog, mist), dust, aerosols, fuel sprays, or biological tissues. We describe here a two-way (reflection) imaging scenario utilizing the ETD phenomenon, and propose a specific image formation technique. We evaluate, by using the radiative transport theory, the resulting point-spread function (PSF) characterizing the image resolution. We show that the directly formed image has an angular resolution comparable to the width of the forward peak in the ensemble-averaged scattering cross section of the medium constituents. Subsequently, we show that, through the application of a regularized deconvolution technique enhancing higher Fourier components of the PSF, the resolution can be further significantly improved—at least by a factor of ${\sim}4$ for a medium layer of optical thickness of the order of 20. Such an improvement can be reached even if the noise level is a few orders of magnitude higher than the coherent (ballistic) image component.

© 2021 Optical Society of America

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References

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  1. E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Early-time diffusion in pulse propagation through dilute random media,” Opt. Lett. 39, 5862–5865 (2014).
    [Crossref]
  2. E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Enhancing early-time diffusion through beam collimation in pulse propagation in sparse discrete random media,” Opt. Lett. 43, 3762–3765 (2018).
    [Crossref]
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    [Crossref]
  7. M. Bertero, P. Boccacci, and V. Ruggiero, Inverse Imaging with Poisson Data: From Cells to Galaxies (IOP, 2018).
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  9. W. C. Karl, “Regularization in image restoration and reconstruction,” in Handbook of Image and Video Processing (Elsevier, 2005), pp. 183–202.
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    [Crossref]
  11. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM 9, 84–97 (1962).
    [Crossref]
  12. A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl. 4, 1624–1627 (1963).
  13. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V. H. Winston & Sons, 1977).
  14. V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer, 1984).
  15. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. A 57, 297–303 (1967).
    [Crossref]
  16. W. K. Pratt, Digital Image Processing (Wiley, 2007).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  22. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13–L19 (2004).
    [Crossref]
  23. G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
    [Crossref]
  24. T. Kato, Perturbation Theory for Linear Operators (Springer, 2013).
  25. W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
    [Crossref]
  26. M.-A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
    [Crossref]

2019 (1)

M.-A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

2018 (2)

2016 (1)

A. M. Grigoryan, E. R. Dougherty, and S. S. Agaian, “Optimal Wiener and homomorphic filtration: review,” Signal Process. 121, 111–138 (2016).
[Crossref]

2014 (1)

2006 (1)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[Crossref]

2004 (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13–L19 (2004).
[Crossref]

2000 (1)

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

1996 (1)

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

1975 (1)

K. Furutsu, “Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation,” Radio Sci. 10, 29–44 (1975).
[Crossref]

1967 (1)

C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. A 57, 297–303 (1967).
[Crossref]

1963 (1)

A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl. 4, 1624–1627 (1963).

1962 (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM 9, 84–97 (1962).
[Crossref]

1934 (1)

A. Y. Khintchine, “Korrelationstheorie der stationaren stochastischen Prozesse,” Math. Ann. 109, 604–615 (1934).
[Crossref]

1930 (1)

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117–258 (1930).
[Crossref]

Agaian, S. S.

A. M. Grigoryan, E. R. Dougherty, and S. S. Agaian, “Optimal Wiener and homomorphic filtration: review,” Signal Process. 121, 111–138 (2016).
[Crossref]

Alu, A.

M.-A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V. H. Winston & Sons, 1977).

Benning, M.

M. Benning and M. Burger, “Modern regularization methods for inverse problems,” Acta Numer. 27, 1–111 (2018).
[Crossref]

Bertero, M.

M. Bertero, P. Boccacci, and V. Ruggiero, Inverse Imaging with Poisson Data: From Cells to Galaxies (IOP, 2018).

Bleszynski, E.

Bleszynski, M.

Boccacci, P.

M. Bertero, P. Boccacci, and V. Ruggiero, Inverse Imaging with Poisson Data: From Cells to Galaxies (IOP, 2018).

Burger, M.

M. Benning and M. Burger, “Modern regularization methods for inverse problems,” Acta Numer. 27, 1–111 (2018).
[Crossref]

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Series in Nuclear Science and Engineering (Addison-Wesley, 1967).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Dougherty, E. R.

A. M. Grigoryan, E. R. Dougherty, and S. S. Agaian, “Optimal Wiener and homomorphic filtration: review,” Signal Process. 121, 111–138 (2016).
[Crossref]

Furutsu, K.

K. Furutsu, “Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation,” Radio Sci. 10, 29–44 (1975).
[Crossref]

Grigoryan, A. M.

A. M. Grigoryan, E. R. Dougherty, and S. S. Agaian, “Optimal Wiener and homomorphic filtration: review,” Signal Process. 121, 111–138 (2016).
[Crossref]

Heiss, W. D.

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

Helstrom, C. W.

C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. A 57, 297–303 (1967).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Wiley, 1999).

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

Jaroszewicz, T.

Karl, W. C.

W. C. Karl, “Regularization in image restoration and reconstruction,” in Handbook of Image and Video Processing (Elsevier, 2005), pp. 183–202.

Kato, T.

T. Kato, Perturbation Theory for Linear Operators (Springer, 2013).

Keller, J. B.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

Khintchine, A. Y.

A. Y. Khintchine, “Korrelationstheorie der stationaren stochastischen Prozesse,” Math. Ann. 109, 604–615 (1934).
[Crossref]

Markel, V. A.

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[Crossref]

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13–L19 (2004).
[Crossref]

Miri, M.-A.

M.-A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

Morozov, V. A.

V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer, 1984).

Panasyuk, G.

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[Crossref]

Papanicolaou, G.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

Phillips, D. L.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM 9, 84–97 (1962).
[Crossref]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, 2007).

Ruggiero, V.

M. Bertero, P. Boccacci, and V. Ruggiero, Inverse Imaging with Poisson Data: From Cells to Galaxies (IOP, 2018).

Ryzhik, L.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

Schotland, J. C.

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[Crossref]

Tikhonov, A. N.

A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl. 4, 1624–1627 (1963).

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V. H. Winston & Sons, 1977).

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002), Vol. 23.

Wiener, N.

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117–258 (1930).
[Crossref]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Series in Nuclear Science and Engineering (Addison-Wesley, 1967).

Acta Math. (1)

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117–258 (1930).
[Crossref]

Acta Numer. (1)

M. Benning and M. Burger, “Modern regularization methods for inverse problems,” Acta Numer. 27, 1–111 (2018).
[Crossref]

J. ACM (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM 9, 84–97 (1962).
[Crossref]

J. Opt. Soc. Am. A (1)

C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. A 57, 297–303 (1967).
[Crossref]

J. Phys. A (1)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A 39, 115–137 (2006).
[Crossref]

Math. Ann. (1)

A. Y. Khintchine, “Korrelationstheorie der stationaren stochastischen Prozesse,” Math. Ann. 109, 604–615 (1934).
[Crossref]

Opt. Lett. (2)

Phys. Rev. E (1)

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

Radio Sci. (1)

K. Furutsu, “Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation,” Radio Sci. 10, 29–44 (1975).
[Crossref]

Science (1)

M.-A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

Signal Process. (1)

A. M. Grigoryan, E. R. Dougherty, and S. S. Agaian, “Optimal Wiener and homomorphic filtration: review,” Signal Process. 121, 111–138 (2016).
[Crossref]

Sov. Math. Dokl. (1)

A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl. 4, 1624–1627 (1963).

Wave Motion (1)

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

Waves Random Media (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14, L13–L19 (2004).
[Crossref]

Other (11)

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V. H. Winston & Sons, 1977).

V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer, 1984).

M. Bertero, P. Boccacci, and V. Ruggiero, Inverse Imaging with Poisson Data: From Cells to Galaxies (IOP, 2018).

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002), Vol. 23.

W. C. Karl, “Regularization in image restoration and reconstruction,” in Handbook of Image and Video Processing (Elsevier, 2005), pp. 183–202.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Series in Nuclear Science and Engineering (Addison-Wesley, 1967).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Wiley, 1999).

T. Kato, Perturbation Theory for Linear Operators (Springer, 2013).

W. K. Pratt, Digital Image Processing (Wiley, 2007).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

Data Availability

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

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Figures (11)

Fig. 1.
Fig. 1. Schematic representation of the considered imaging scenario, showing propagation paths contributing to (a) the RTE Green’s function evaluated to the first order in scattering on the target and (b) the remainder of the solution, due to scattering on the medium constituents only.
Fig. 2.
Fig. 2. Propagation paths contributing to the radiance (20).
Fig. 3.
Fig. 3. Geometry of pulse propagation from the source to the target and back in the imaging configuration of Fig. 1. In (a) a light ray of the beam, emitted at an angle $\theta ^\prime $ relative to the beam axis, enters the medium layer at a point ${\boldsymbol R}^\prime ({\hat {\boldsymbol s}}^\prime)$ and travels, undergoing multiple scattering, to the target. In (b) the light diffusively scattered on the target propagates, undergoing scattering, to a point ${\boldsymbol R}({\hat {\boldsymbol s}})$ on the slab surface and from there to the detector, arriving at an angle $\theta$ . The circles around the target location ${\textbf{0}}$ indicate integration over flux directions. The angles $\chi$ and $\chi ^\prime $ refer to Eq. (31).
Fig. 4.
Fig. 4. Propagation paths resulting in negligible contributions to the ETD signal.
Fig. 5.
Fig. 5. (a) Radiance ${{\mathscr I}_\infty}(t,L,\cos \chi)$ on the upper slab surface (height $L = 24{\ell _{\text{t}}}$ ), due to a pulse emitted from the point ${\textbf{0}}$ at $t = 0$ ; (b) radiance ${{\mathscr I}_\infty}(t - \tau (1 - \cos \chi),L,\cos \chi)$ for the same source, measured at the height $L + {h_{\text{S}}}$ , ${h_{\text{S}}} = 50{\ell _{\text{t}}}$ , showing angle-dependent time-delay effects proportional to $\tau = L{h_{\text{S}}}/[c(L + {h_{\text{S}}})]$ . The isolines represent function values $0.05 \cdot {10^{- 6}}$ , $2 \cdot {10^{- 6}}$ , $4 \cdot {10^{- 6}}$ , $6 \cdot {10^{- 6}}$ , and $8 \cdot {10^{- 6}}$ .
Fig. 6.
Fig. 6. Intensity ${{\mathscr I}_{\text{T}}}(t,L,{h_{\text{S}}})$ illuminating the target, plotted for heights ${h_{\text{S}}} = 0, 50{\ell _{\text{t}}}, 250{\ell _{\text{t}}}$ , and for several beam widths, $\Theta = 0.005$ , 0.01 and 0.02; other parameters are as in Eqs. (8) and (42). In all the plots, the solid black curve refers to the ${h_{\text{S}}} = 0$ result and the dashed red curve to ${h_{\text{S}}} = 50{\ell _{\text{t}}}$ . In (a), the additional solid blue line corresponds to ${h_{\text{S}}} = 250{\ell _{\text{t}}}$ , and the black dashed curve is the radiance for ${h_{\text{S}}} = 0$ and a temporal delta-function pulse.
Fig. 7.
Fig. 7. The filtered radiances (a)  ${I^\Phi}(t,L,0,\cos \chi)$ for ${h_{\text{S}}} = 0$ and (b)  ${I^\Phi}(t,L,{h_{\text{S}}},\cos \theta)$ for ${h_{\text{S}}} = 50{\ell _{\text{t}}}$ , computed for $L = 24{\ell _{\text{t}}}$ and for the indicated angles $\chi$ or $\theta$ (in radians), plotted as functions of time. The beam width is $\Theta = 0.01\;{\text{rad}} $ .
Fig. 8.
Fig. 8. PSF obtained from the filtered radiance in Fig. 7 for the two choices of the time-integration interval width: ${\upsilon _0}\Delta {t_\Phi}/{\ell _{\text{t}}} = 0.05$ and 0.10.
Fig. 9.
Fig. 9. MTF obtained from the PSF in Fig. 8 with the two curves corresponding to two different choices of the time-integration interval $\Delta {t_\Phi}$ . The $u$ -independent coherent contributions ${M_{\text{c}}} = |{\tilde \Lambda _{\text{c}}}/\tilde \Lambda (0)|$ [Section 4.D, Eq. (68)], also corresponding to two different choices of $\Delta {t_\Phi}$ , are shown as thick dashed lines. The other dashed lines indicate regularization parameters $\nu {M_{\text{c}}}$ corresponding to the values $\nu = 10^3 ,10^4 ,10^5$ of the noise strength parameter [Section 5, Eq. (80)].
Fig. 10.
Fig. 10. Original (dashed lines) and deblurred (solid lines) PSFs for the three indicated values of the noise-strength parameter $\nu$ of Eq. (79), i.e., the ratio of the noise RMS value to the coherent image component. All the calculations were done, as before, for the medium thickness, the pulse, and filter parameters specified in (37) through (42). The PSFs were calculated for ${\upsilon _0}\Delta {t_\Phi}/{\ell _{\text{t}}} = 0.05$ and 0.10.
Fig. 11.
Fig. 11. Attenuation coefficients and propagation velocities (A9) of the RTE modes, relative to the corresponding quantities for the coherent wave propagation, computed with the truncation $N = 800$ . Dashed curves represent the low-pass filter in the integration variable $P$ , due to the transmitted pulse (39) of duration ${T_{\text{p}}} = 0.002{\ell _{\text{t}}}/{\upsilon _0}$ .

Equations (104)

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ETD image > noise coherent image .
θ n n θ 0 .
Δ θ N θ 0 = R t θ 0 .
z n = t cos θ n t ( 1 1 2 θ n 2 )
Δ z t 2 n = 1 N θ n 2 t θ 0 2 2 n = 1 N n t θ 0 2 N 2 4 R 4 ( Δ θ ) 2 ,
Δ t = Δ z υ 0 = R 4 υ 0 ( Δ θ ) 2 ,
Δ θ ETD ( R ) R t κ θ k 0 a
Δ t ETD ( R ) R 4 υ 0 ( Δ θ ETD ( R ) ) 2 R 2 4 υ 0 t κ θ 2 ( k 0 a ) 2 R 2 κ θ 2 4 t 2 ( k 0 a ) 2 t υ 0 ,
λ 0 = 0.633 µ m , a = 5 µ m 8 λ 0 , hence k 0 a 50 , n 0 = 10 9 m 3 , t = 1 n 0 σ t 7 m
κ θ = 0.42 .
Δ θ ETD ( R ) 0.041 rad and Δ t ETD ( R ) 0.010 t υ 0 0.23 ns .
{ μ t L ( R ) + υ 0 1 L ( R ) t + c 1 [ 1 L ( R ) ] t + s ^ R } × Γ ( t , R , R ; s ^ , s ^ ) L ( R ) d 2 s ^ Σ ( s ^ s ^ ) Γ ( t , R , R ; s ^ , s ^ ) δ 3 ( R R T ) d 2 s ^ Σ T ( s ^ s ^ ) Γ ( t , R , R ; s ^ , s ^ ) = δ ( t ) δ 3 ( R R ) δ 2 ( s ^ s ^ )
I ( t , R S ; s ^ ) = d t d 2 s ^ Γ ( t t , R S , R S ; s ^ , s ^ ) A ( t ) B ( z ^ s ^ ) ,
Γ = Γ M + Γ T ,
( D 0 Σ Σ T ) Γ = I
( D 0 Σ ) Γ M = I ,
( D 0 Σ Σ T ) Γ T = Σ T Γ M .
Γ T = Γ M Σ T Γ M + Γ M Σ T Γ M Σ T Γ M + ,
Γ = Γ M + Γ M Σ T Γ M + Γ M Σ T Γ M Σ T Γ M + .
Γ T ( 1 ) = Γ M Σ T Γ M ,
I ( t , L , h S , z ^ s ^ ) = d t d 2 s ^ Γ T ( 1 ) ( t t , R S , R S ; s ^ , s ^ ) A ( t ) B ( z ^ s ^ ) .
I ( t , L , h S , z ^ s ^ ) = d t d t d 2 s ^ T d 2 s ^ T d 2 s ^ × Γ M ( t t , R S , 0 ; s ^ , s ^ T ) Σ T ( s ^ T s ^ T ) Γ M ( t t , 0 , R S ; s ^ T , s ^ ) A ( t ) B ( z ^ s ^ ) ,
I ( t , L , h S , z ^ s ^ ) = Σ T d t d t d 2 s ^ I UP ( t t , R S ; 0 , s ^ ) I DOWN ( t t , 0 , R S ; s ^ ) A ( t ) B ( z ^ s ^ ) ,
I DOWN ( t t , 0 , R S , s ^ ) = d 2 s ^ T Γ M ( t t , 0 , R S ; s ^ T , s ^ ) = d 2 s ^ T Γ M ( t t , R S , 0 ; s ^ , s ^ T ) = d 2 s ^ T Γ M ( t t , R S , 0 ; s ^ , s ^ T ) = I ( t t , L , h S , s ^ )
I UP ( t t , R S , 0 , s ^ ) = d 2 s ^ T Γ M ( t t , R S , 0 ; s ^ , s ^ T ) = I ( t t , L , h S , s ^ )
I ( t , L , h S , z ^ s ^ ) = Σ T d t I ( t t , L , h S , s ^ ) I T ( t , L , h S ) .
I T ( t , L , h S ) = d t d 2 s ^ I ( t t , L , h S , s ^ ) A ( t ) B ( z ^ s ^ ) ,
I ( t , L , h S , s ^ ) = d 2 s ^ T Γ M ( t , R S , 0 ; s ^ , s ^ T ) ,
DOWN : I ( t t , L , h S , s ^ ) = d 2 s ^ T Γ L ( t t | R S R ( s ^ ) | / c , R ( s ^ ) , 0 ; s ^ , s ^ T ) ,
UP : I ( t t , L , h S , s ^ ) = d 2 s ^ T Γ L ( t t | R S R ( s ^ ) | / c , R ( s ^ ) , 0 ; s ^ , s ^ T ) .
DOWN : I ( t t , L , h S , s ^ ) d 2 s ^ T Γ ( t t | R S R ( s ^ ) | / c , R ( s ^ ) ; s ^ , s ^ T ) = I ( t t | R S R ( s ^ ) | / c , | R ( s ^ ) | ; R ^ ( s ^ ) s ^ ) ,
I ( t , | R | , R ^ s ^ ) := d 2 s ^ Γ ( t , R ; s ^ , s ^ ) d 2 s ^ Γ ( t , R ; s ^ , s ^ )
UP : I ( t t , L , h S , s ^ ) d 2 s ^ T Γ ( t t | R S R ( s ^ ) | / c , R ( s ^ ) ; s ^ , s ^ T ) = I ( t t | R S R ( s ^ ) | / c , | R ( s ^ ) | , R ^ ( s ^ ) s ^ ) .
R ^ ( s ^ ) s ^ cos χ cos H θ L , R ^ ( s ^ ) s ^ cos χ cos H θ L ,
| R S R ( s ^ ) | h S [ 1 + L 2 H 2 ( 1 cos χ ) ] , | R ( s ^ ) | L [ 1 + h S 2 H 2 ( 1 cos χ ) ] ,
| R S R ( s ^ ) | h S [ 1 + L 2 H 2 ( 1 cos χ ) ] , | R ( s ^ ) | L [ 1 + h S 2 H 2 ( 1 cos χ ) ] .
I ( t , R + r , R ^ s ^ ) = I ( t r / υ 0 , R , R ^ s ^ ) × [ 1 + O ( | r | / t ) ] for | r | t .
DOWN : I ( t t , L , h S , s ^ ) = I ( t t | R S R ( s ^ ) | / c , | R ( s ^ ) | , R ^ ( s ^ ) s ^ ) I ( t t h S c [ 1 + L 2 H 2 ( 1 cos χ ) ] , L [ 1 + h S 2 H 2 ( 1 cos χ ) ] , cos χ ) I ( t t h S c ( L 2 h S c H 2 + L h S 2 υ 0 H 2 ) ( 1 cos χ ) , L , cos χ L 2 h S c H 2 ) I ( t t h S c L h S c H ( 1 cos χ ) , L , cos χ ) = I ( t t h S c τ ( 1 cos χ ) L , cos χ )
UP : I ( t t , L , h S , s ^ ) I ( t t | R S R ( s ^ ) | / c , | R ( s ^ ) | , R ^ ( s ^ ) s ^ ) I ( t t h S c τ ( 1 cos χ ) , L , cos χ ) .
τ τ ( L , h S ) := L h S c H = L h S c ( L + h S ) L c ,
I ( t , L , h S , cos L χ L + h S ) Σ T d t I ( t t h S c τ ( L , h S ) ( 1 cos χ ) , L , cos χ ( t , L , h S , cos L χ L + h S ) ) I T ( t , L , h S ) ,
I T ( t , L , h S ) = 2 π L 2 ( L + h S ) 2 d t A ( t ) d cos χ B ( cos L χ L + h S ) I ( t t h S c τ ( L , h S ) ( 1 cos χ ) , L , cos χ ) ,
L = 24 t 168 m .
B ( t , cos θ ) = A ( t ) B ( cos θ ) ,
A ( t ) = e t 2 / 2 T p 2 2 π T p , such that d t A ( t ) = 1 ,
B ( cos θ ) = e ( 1 cos θ ) / Θ 2 2 π Θ 2 ( 1 e 2 / Θ 2 ) e θ 2 / 2 Θ 2 2 π Θ 2 , such that 2 π 1 1 d cos θ B ( cos θ ) = 1 ,
Φ ( t ) = δ ( t ) 1 2 π T f e t 2 / 2 T f 2 .
T p = 0.002 t / υ 0 0.047 ns , T f = 0.030 t / υ 0 0.70 ns , Θ = 0.01 rad .
τ ( 1 cos χ ) L h S ( χ ) 2 2 c ( L + h S ) L h S ( Δ θ ETD ( L ) ) 2 2 c ( L + h S ) 2 h S L + h S Δ t ETD ( L ) ,
h S ( L + h S ) 2 c L Θ 2 < Δ t ETD ( L ) ,
Θ < L 2 h S ( L + h S ) Δ θ ETD ( L ) Θ max ( L , h S ) { 0.011 for h S = 50 t , 0.0027 for h S = 250 t ,
Θ < Θ max ( L , h S ) (Fig. 6(a)) ,
Θ Θ max ( L , h S ) (Fig. 6(b)) ,
Θ > Θ max ( L , h S ) (Fig. 6(c)) .
I T ( t , L , h S ) 2 π L 2 ( L + h S ) 2 d t A ( t ) d cos χ × B ( cos L χ L + h S ) I ( t t h S c , L , 1 ) = d t A ( t ) I ( t t h S c , L , 1 ) ,
I T ( t , L , h S ) 2 π L 2 ( L + h S ) 2 d t A ( t ) d cos χ B ( cos L χ L + h S ) × I ( t t h S c , L , cos χ ) 2 π d t A ( t ) d cos χ B ( cos χ ) × I ( t t h S c , L , cos ( L + h S ) χ L ) .
I T ( t , L , h S ) I T ( t h S c , L , 0 ) ,
I ( t , L , h S , cos θ ) I ( t 2 h S c τ ( 1 cos ( L + h S ) θ L ) , L , 0 , cos ( L + h S ) θ L ) .
I Φ ( t , L , h S , cos θ ) := 1 Σ T d t Φ ( t t ) I ( t , L , h S , cos θ ) ;
I Φ ( t , L , h S , cos θ ) I Φ ( t 2 h S c τ ( 1 cos ( L + h S ) θ L ) , L , 0 , cos ( L + h S ) θ L ) ,
I Φ ( t , L , h S , cos L χ L + h S ) I Φ ( t 2 h S c τ ( 1 cos χ ) , L , 0 , cos χ ) ,
t 0 ( L , h S , θ ) = 2 ( L + h S ) c + τ ( L , h S ) ( 1 cos ( L + h S ) θ L ) .
Λ ( L , h S , cos θ ) = t ( θ ) t + ( θ ) d t | I Φ ( t , L , h S , cos θ ) | .
t ± ( θ ) = t peak ( θ ) ± Δ t Φ ,
Λ ( L , h S , cos θ ) = Λ ( L , 0 , cos ( L + h S ) θ L ) = t ( θ ) t + ( θ ) d t | I Φ ( t , L , 0 , cos ( L + h S ) θ L ) | ,
Λ ( L , h S , cos L χ L + h S ) = Λ ( L , 0 , cos χ ) = t ( θ ) t + ( θ ) d t | I Φ ( t , L , 0 , cos χ ) | .
Λ ( χ ) Λ ( L , 0 , cos χ ) ,
I f Φ ( t , χ ) = d 2 χ I Φ ( t , L , 0 , cos | χ | ) f ( χ χ ) ( I Φ f ) ( t , χ ) ,
F ( χ ) = t ( χ ) t + ( χ ) d t | I f Φ ( t , χ ) | ,
F ( χ ) ( Λ f ) ( χ )
Λ ~ ( u ) Λ ~ ( | u | ) := d 2 χ e i χ u Λ ( χ ) = 2 π 0 π d χ χ J 0 ( χ u ) Λ ( χ ) ,
M ( u ) := | Λ ~ ( u ) Λ ~ ( 0 ) |
Γ c ( t , R ; s ^ , s ^ ) = υ 0 H ( t ) e μ t | R | δ 3 ( R υ 0 t s ^ ) δ 2 ( s ^ s ^ ) .
I c ( t , L , 0 , cos χ ) = 1 L 2 e μ t L δ ( t L υ 0 ) δ 2 ( s ^ ( χ ) z ^ ) = 1 2 π L 2 e L / t δ ( t L υ 0 ) δ ( cos χ 1 ) .
I T c ( t , L , 0 ) = 2 π d t A ( t ) d cos χ B ( cos χ ) × I c ( t t , L , cos χ ) = 1 2 π L 2 Θ 2 e L / t A ( t 2 L υ 0 ) ,
I c ( t , L , 0 , cos χ ) = Σ T d t I c ( t t , L , cos χ ) I T c ( t , L , 0 ) = Σ T ( 2 π ) 2 L 4 Θ 2 e 2 L / t A ( t 2 L υ 0 ) δ ( cos χ 1 ) .
Λ c ( χ ) = 1 Σ T d t I c ( t , L , 0 , cos χ ) = 1 ( 2 π ) 2 L 4 Θ 2 e 2 L / t δ ( cos χ 1 )
Λ c ( χ ) = Λ c 0 δ 2 ( χ ) with Λ c 0 = 1 2 π L 4 Θ 2 e 2 L / t = 6.84 10 24 ,
Λ ~ c ( u ) Λ ~ c = Λ c 0 and M c ( u ) := | Λ ~ c ( u ) Λ ~ ( 0 ) | = | Λ c 0 Λ ~ ( 0 ) | M c .
F ~ ( u ) = Λ ~ ( u ) f ~ ( u ) + η ~ ( u ) F ~ 0 ( u ) + η ~ ( u ) ,
f ~ β ( u ) = Λ ~ ( u ) Λ ~ 2 ( u ) + β 2 F ~ ( u ) R β ( u ) F ~ ( u ) ,
f ~ β = arg min f ~ { F ~ Λ ~ f ~ 2 + β 2 f ~ 2 }
D d 2 χ | f ( χ ) ( W f , η ( Λ f + η ) ) ( χ ) | 2 = | D | d 2 u ( 2 π ) 2 { | 1 W ~ f , η ( u ) Λ ~ ( u ) | 2 S f ( u ) + | W ~ f , η ( u ) | 2 S η ( u ) } ,
S f ( u ) = d 2 χ e i χ u f ( 0 ) f ( χ ) and S η ( u ) = d 2 χ e i χ u η ( 0 ) η ( χ )
W ~ f , η ( u ) = Λ ~ ( u ) Λ ~ 2 ( u ) + S η ( u ) / S f ( u ) .
f ( 0 ) f ( χ ) = δ 2 ( χ ) and S f ( u ) = 1 .
Λ c ( 0 ) Λ c ( χ ) = Λ c 0 2 δ 2 ( χ ) and S Λ c ( u ) = Λ c 0 2 .
η ( 0 ) η ( χ ) = η 0 2 δ 2 ( χ ) and S η ( u ) = η 0 2 .
W ~ f , η ( u ) = Λ ~ ( u ) Λ ~ 2 ( u ) + η 0 2 R η 0 ( u ) ,
η 0 = ν Λ c 0 with ν > 1 ,
Λ ~ ν Λ c 0 ( u ) = R ν Λ c 0 ( u ) Λ ~ ( u ) = Λ ~ 2 ( u ) Λ ~ 2 ( u ) + ν 2 Λ c 0 2 = M 2 ( u ) M 2 ( u ) + ν 2 M c 2 ,
( μ t + υ 0 1 t + s ^ R ) Γ ( t , R ; s ^ , s ^ ) d 2 s ^ Σ ( s ^ s ^ ) Γ ( t , R ; s ^ , s ^ ) = δ ( t ) δ 3 ( R ) δ 2 ( s ^ s ^ ) .
Γ ( t , R ; s ^ , s ^ ) = d Ω 2 π d 3 P ( 2 π ) 3 e i Ω t e i P R Γ ~ ( Ω , P ; s ^ , s ^ ) ,
( μ t i υ 0 1 Ω + i P z ^ s ^ ) Γ ~ ( Ω , P ; s ^ , s ^ ) d 2 s ^ Σ ( s ^ s ^ ) Γ ~ ( Ω , P ; s ^ , s ^ ) = δ 2 ( s ^ s ^ ) .
Γ ~ ( Ω , P ; s ^ , s ^ ) = m , l , l Y l , m ( s ^ ) Y l , m ( s ^ ) Γ l , l m ( Ω , P ) ,
l [ M l , l m ( P ) i υ 0 1 Ω δ l , l ] Γ l , l m ( Ω , P ) = δ l , l ,
M m ( P ) w j m ( P ) = i υ 0 1 Ω j m ( P ) w j m ( P )
Γ l , l m ( Ω , P ) = i υ 0 j = 1 N w j , l m ( P ) w j , l m ( P ) Ω Ω j m ( P ) .
Γ ( t , R ; s ^ , s ^ ) = υ 0 m , j d 3 P ( 2 π ) 3 e i Ω j m ( P ) t e i P R l , l w j , l m ( P ) w j , l m ( P ) Y l , m ( s ^ ) Y l , m ( s ^ ) .
μ j m ( P ) = Im Ω j m ( P ) υ 0 , υ j m ( P ) = Re Ω j m ( P ) P .
I ( t , R , cos θ ) = l = 0 N 1 2 l + 1 P l ( cos θ ) I l ( t , R ) ,
I l ( t , R ) = υ 0 2 π 2 0 P cut d P P 2 j = 1 N e i Ω j 0 ( P ) t × i l j l ( P R ) w j , 0 0 ( P ) w j , l 0 ( P ) ,
N > 1 Δ θ 1 Θ .
I ( t τ ( 1 cos χ ) , L , cos χ ) = l = 0 N 1 2 l + 1 P l ( cos χ ) I l ( t τ ( 1 cos χ ) , L ) ,