Abstract

This paper uses wave-optics and signal-to-noise models to explore the estimation accuracy of digital-holographic detection in the off-axis pupil plane recording geometry for deep-turbulence wavefront sensing. In turn, the analysis examines three important parameters: the number of pixels across the width of the focal-plane array, the window radius in the Fourier plane, and the signal-to-noise ratio. By varying these parameters, the wave-optics and signal-to-noise models quantify performance via a metric referred to as the field-estimated Strehl ratio, and the analysis leads to a method for optimal windowing of the turbulence-limited point spread function. Altogether, the results will allow future research efforts to assess the number of pixels, pixel size, pixel-well depth, and read-noise standard deviation needed from a focal-plane array when using digital-holographic detection in the off-axis pupil plane recording geometry for estimating the complex-optical field when in the presence of deep turbulence and detection noise.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

Corrections

19 January 2018: A correction was made to the copyright.


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References

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2016 (5)

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2016).
[Crossref]

M. T. Banet, M. F. Spencer, R. A. Raynor, and D. K. Marker, “Digital holography wavefront sensing in the pupil-plane recording geometry for distributed-volume atmospheric aberrations,” Proc. SPIE 9982, 998208 (2016).
[Crossref]

J. R. Kraczek, P. F. McManamon, and E. A. Watson, “High resolution non-iterative aperture synthesis,” Opt. Express 24, 6229–6239 (2016).
[Crossref]

M. A. Vorontsov, S. L. Lachinova, and A. K. Majumdar, “Target-in-the-loop remote sensing of laser beam and atmospheric turbulence characteristics,” Appl. Opt. 55, 5172–5179 (2016).
[Crossref]

G. DiComo, M. Helle, J. Peñano, A. Ting, A. Schmitt-Sody, and J. Elle, “Implementation of a long range, distributed-volume, continuously variable turbulence generator,” Appl. Opt. 55, 5192–5197 (2016).
[Crossref]

2015 (1)

M. F. Spencer, I. V. Dragulin, D. S. Cargill, and M. J. Steinbock, “Digital holography wave-front sensing in the presence of strong atmospheric turbulence and thermal blooming,” Proc. SPIE 9617, 961705 (2015).
[Crossref]

2014 (2)

2011 (1)

2010 (1)

2009 (1)

2002 (2)

1998 (1)

Alenin, A. S.

J. S. Tyo and A. S. Alenin, Field Guide to Linear Systems in Optics (SPIE, 2015).

Andrews, L. C.

L. C. Andrews, Field Guide to Atmospheric Optics (SPIE, 2004).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Banet, M. T.

M. T. Banet, M. F. Spencer, R. A. Raynor, and D. K. Marker, “Digital holography wavefront sensing in the pupil-plane recording geometry for distributed-volume atmospheric aberrations,” Proc. SPIE 9982, 998208 (2016).
[Crossref]

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2016).
[Crossref]

Barchers, J. D.

Boreman, G. D.

E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).

Brennan, T. J.

T. J. Brennan and P. H. Roberts, AOTools: The Adaptive Optics Toolbox for use with MATLAB, User’s Guide Version 1.4 (Optical Sciences, 2010).

T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: A Wave Optics Simulation System for use with MATLAB, User’s Guide Version 1.3 (Optical Sciences, 2010).

M. F. Spencer and T. J. Brennan, “Branch-cut accumulation using LSPV+7,” in Imaging and Applied Optics (Optical Society of America, 2017), paper PTh2D.2.

Cargill, D. S.

M. F. Spencer, I. V. Dragulin, D. S. Cargill, and M. J. Steinbock, “Digital holography wave-front sensing in the presence of strong atmospheric turbulence and thermal blooming,” Proc. SPIE 9617, 961705 (2015).
[Crossref]

Cusumano, S. J.

G. P. Perram, S. J. Cusumano, R. L. Hengehold, and S. T. Fiorino, An Introduction to Laser Weapon Systems (Directed Energy Professional Society, 2010).

Dereniak, E. L.

E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).

DiComo, G.

Dragulin, I. V.

M. F. Spencer, I. V. Dragulin, D. S. Cargill, and M. J. Steinbock, “Digital holography wave-front sensing in the presence of strong atmospheric turbulence and thermal blooming,” Proc. SPIE 9617, 961705 (2015).
[Crossref]

Elle, J.

Fiorino, S. T.

G. P. Perram, S. J. Cusumano, R. L. Hengehold, and S. T. Fiorino, An Introduction to Laser Weapon Systems (Directed Energy Professional Society, 2010).

Frazier, B. W.

R. K. Tyson and B. W. Frazier, Field Guide to Adaptive Optics (SPIE, 2004).

Fried, D. L.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

Grow, T. D.

Helle, M.

Hengehold, R. L.

G. P. Perram, S. J. Cusumano, R. L. Hengehold, and S. T. Fiorino, An Introduction to Laser Weapon Systems (Directed Energy Professional Society, 2010).

Höft, T. A.

Hyde, M. W.

Kendrick, R. L.

Kraczek, J. R.

Lachinova, S. L.

Lebow, P. S.

Link, D. J.

Liu, J. P.

T. C. Poon and J. P. Liu, Introduction to Modern Digital Holography (Cambridge University, 2014).

Louthain, J. A.

Majumdar, A. K.

Mann, D. C.

T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: A Wave Optics Simulation System for use with MATLAB, User’s Guide Version 1.3 (Optical Sciences, 2010).

Marker, D. K.

M. T. Banet, M. F. Spencer, R. A. Raynor, and D. K. Marker, “Digital holography wavefront sensing in the pupil-plane recording geometry for distributed-volume atmospheric aberrations,” Proc. SPIE 9982, 998208 (2016).
[Crossref]

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2016).
[Crossref]

M. F. Spencer, R. A. Raynor, T. A. Rhoadarmer, and D. K. Marker, “Deep-turbulence simulation in a scaled-laboratory environment using five phase-only spatial light modulators,” in 18th Coherent Laser Radar Conference (2015).

Marron, J. C.

McManamon, P. F.

Nielson, P. E.

P. E. Nielson, Effects of Directed Energy Weapons (Directed Energy Professional Society, 2009).

Osche, G. R.

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).

Peñano, J.

Perram, G. P.

G. P. Perram, S. J. Cusumano, R. L. Hengehold, and S. T. Fiorino, An Introduction to Laser Weapon Systems (Directed Energy Professional Society, 2010).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Poon, T. C.

T. C. Poon and J. P. Liu, Introduction to Modern Digital Holography (Cambridge University, 2014).

Raynor, R. A.

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2016).
[Crossref]

M. T. Banet, M. F. Spencer, R. A. Raynor, and D. K. Marker, “Digital holography wavefront sensing in the pupil-plane recording geometry for distributed-volume atmospheric aberrations,” Proc. SPIE 9982, 998208 (2016).
[Crossref]

M. F. Spencer, R. A. Raynor, T. A. Rhoadarmer, and D. K. Marker, “Deep-turbulence simulation in a scaled-laboratory environment using five phase-only spatial light modulators,” in 18th Coherent Laser Radar Conference (2015).

Rhoadarmer, T. A.

J. D. Barchers and T. A. Rhoadarmer, “Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function,” Appl. Opt. 41, 7499–7509 (2002).
[Crossref]

M. F. Spencer, R. A. Raynor, T. A. Rhoadarmer, and D. K. Marker, “Deep-turbulence simulation in a scaled-laboratory environment using five phase-only spatial light modulators,” in 18th Coherent Laser Radar Conference (2015).

Roberts, P. H.

T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: A Wave Optics Simulation System for use with MATLAB, User’s Guide Version 1.3 (Optical Sciences, 2010).

T. J. Brennan and P. H. Roberts, AOTools: The Adaptive Optics Toolbox for use with MATLAB, User’s Guide Version 1.4 (Optical Sciences, 2010).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Schmidt, J. D.

Schmitt-Sody, A.

Seldomridge, N.

Spencer, M. F.

M. T. Banet, M. F. Spencer, R. A. Raynor, and D. K. Marker, “Digital holography wavefront sensing in the pupil-plane recording geometry for distributed-volume atmospheric aberrations,” Proc. SPIE 9982, 998208 (2016).
[Crossref]

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2016).
[Crossref]

M. F. Spencer, I. V. Dragulin, D. S. Cargill, and M. J. Steinbock, “Digital holography wave-front sensing in the presence of strong atmospheric turbulence and thermal blooming,” Proc. SPIE 9617, 961705 (2015).
[Crossref]

M. F. Spencer and T. J. Brennan, “Branch-cut accumulation using LSPV+7,” in Imaging and Applied Optics (Optical Society of America, 2017), paper PTh2D.2.

M. F. Spencer, R. A. Raynor, T. A. Rhoadarmer, and D. K. Marker, “Deep-turbulence simulation in a scaled-laboratory environment using five phase-only spatial light modulators,” in 18th Coherent Laser Radar Conference (2015).

Steinbock, M. J.

M. F. Spencer, I. V. Dragulin, D. S. Cargill, and M. J. Steinbock, “Digital holography wave-front sensing in the presence of strong atmospheric turbulence and thermal blooming,” Proc. SPIE 9617, 961705 (2015).
[Crossref]

M. J. Steinbock, M. W. Hyde, and J. D. Schmidt, “LSPV+7, a branch-point-tolerant reconstructor for strong turbulence adaptive optics,” Appl. Opt. 53, 3821–3831 (2014).
[Crossref]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Tellez, J. A.

Ting, A.

Tyo, J. S.

J. S. Tyo and A. S. Alenin, Field Guide to Linear Systems in Optics (SPIE, 2015).

Tyson, R. K.

R. K. Tyson and B. W. Frazier, Field Guide to Adaptive Optics (SPIE, 2004).

Vorontsov, M. A.

Watnik, A. T.

Watson, E. A.

Appl. Opt. (7)

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

Opt. Eng. (1)

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2016).
[Crossref]

Opt. Express (3)

Proc. SPIE (2)

M. F. Spencer, I. V. Dragulin, D. S. Cargill, and M. J. Steinbock, “Digital holography wave-front sensing in the presence of strong atmospheric turbulence and thermal blooming,” Proc. SPIE 9617, 961705 (2015).
[Crossref]

M. T. Banet, M. F. Spencer, R. A. Raynor, and D. K. Marker, “Digital holography wavefront sensing in the pupil-plane recording geometry for distributed-volume atmospheric aberrations,” Proc. SPIE 9982, 998208 (2016).
[Crossref]

Other (16)

T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: A Wave Optics Simulation System for use with MATLAB, User’s Guide Version 1.3 (Optical Sciences, 2010).

T. J. Brennan and P. H. Roberts, AOTools: The Adaptive Optics Toolbox for use with MATLAB, User’s Guide Version 1.4 (Optical Sciences, 2010).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

J. S. Tyo and A. S. Alenin, Field Guide to Linear Systems in Optics (SPIE, 2015).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).

J. D. Schmidt, Numerical Simulation of Optical Wave propagation (SPIE, 2010).

R. K. Tyson and B. W. Frazier, Field Guide to Adaptive Optics (SPIE, 2004).

L. C. Andrews, Field Guide to Atmospheric Optics (SPIE, 2004).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).

T. C. Poon and J. P. Liu, Introduction to Modern Digital Holography (Cambridge University, 2014).

M. F. Spencer and T. J. Brennan, “Branch-cut accumulation using LSPV+7,” in Imaging and Applied Optics (Optical Society of America, 2017), paper PTh2D.2.

M. F. Spencer, R. A. Raynor, T. A. Rhoadarmer, and D. K. Marker, “Deep-turbulence simulation in a scaled-laboratory environment using five phase-only spatial light modulators,” in 18th Coherent Laser Radar Conference (2015).

P. E. Nielson, Effects of Directed Energy Weapons (Directed Energy Professional Society, 2009).

G. P. Perram, S. J. Cusumano, R. L. Hengehold, and S. T. Fiorino, An Introduction to Laser Weapon Systems (Directed Energy Professional Society, 2010).

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Figures (9)

Fig. 1.
Fig. 1. Above is a diagram of digital-holographic detection in the off-axis PPRG. In (a), the coherent light from a master-oscillator (MO) laser actively illuminates a distant cooperative object (e.g., an unresolved ball bearing) creating a point-source beacon that propagates through deep turbulence to the entrance-pupil plane of an imaging system. Through collimation, the entrance-pupil plane creates a signal beam, which we interfere with a reference beam in the exit-pupil plane of the imaging system. Note that the reference beam manifests itself from an off-axis local oscillator (LO), which we derive from the MO laser. Also note that the exit-pupil plane is where we place the FPA, so that we can record a digital hologram. In (b), we can then use signal-processing techniques to manipulate the digital hologram into a wrapped-phase estimate.
Fig. 2.
Fig. 2. In (a) and (b), we show the normalized irradiance of the signal beam, whereas in (c) and (d), we show the resultant digital holograms for two different values of the number of detector pixels, N P , across the exit-pupil diameter, D 2 (white circles). The black-dashed circles have diameters equal to the demagnified spherical-wave Fried coherence diameter, that is, M T r 0 s w for turbulence Scenario 5 in Table 1. It is important to note that we display the associated turbulence-limited sampling quotient for a spherical wave Q P s w in the bottom-left corner of each subplot.
Fig. 3.
Fig. 3. This figure displays the simulated Fourier plane for two different values of the number of detector pixels, N P , across the exit-pupil diameter, D 2 . From Eq. (21), the autocorrelation term and the Dirac-delta term appear at the center of each plot; however, the strengths of these terms are negligible because we used a strong reference beam and subtracted off the average photoelectron count from each digital hologram. With that said, the turbulence-limited PSF and its complex conjugate are readily visible in each plot, and we isolate the correct turbulence-limited PSF with a window function. Note that the white circles represent the window function, and the green-dotted circles demarcate the turbulence-limited bucket diameter for a spherical wave D T s w for Scenario 5 in Table 1. Also note that the Fourier-plane side length, L F , and radius, ρ W , of the window function vary from subplot to subplot. Here, we express the size of ρ W and L F in terms of the approximate diffraction-limited bucket radius ρ I .
Fig. 4.
Fig. 4. This plot displays the percentage errors between the analytically and numerically calculated SNRs for 10 different SNRs as given by Eq. (39) for turbulence Scenarios 1 and 5 in Table 1. We calculated the SNRs by using the maximum allowed radius, ρ W , of the window function in the Fourier plane (i.e., ρ W was set to one fourth of the Fourier-plane side length L F ). To vary the SNRs, we adjusted the per-pixel mean number of photoelectrons from the signal beam. Here, the error bars depict the standard-deviation offsets for 20 independent realizations of turbulence and 20 independent realizations of detection noise, that is, 400 realizations.
Fig. 5.
Fig. 5. In this plot, we see the trade-space trends that arise from varying the number of FPA pixels, N P , across the exit-pupil diameter, D 2 , in (a) and the SNR in (b) for the turbulence scenarios in Table 1. All of the solid colored lines are averages of 20 independent realizations of turbulence and 20 independent realizations of detection noise, that is, 400 realizations. The dashed lines are standard-deviation offsets from these averages. In both plots, the radius, ρ W , of the window function in the Fourier plane is always four times the approximate diffraction-limited bucket radius, ρ I .
Fig. 6.
Fig. 6. In this plot, we see the trade-space trends that arise from varying the radius ρ W of the window function in the Fourier plane in (a) and the SNR in (b) for the turbulence scenarios in Table 1. All of the solid colored lines are averages of 20 independent realizations of turbulence and 20 independent realizations of detection noise, that is, 400 realizations. The dashed lines are standard-deviation offsets from these averages. It is important to note that the peaks in the curves in (a) are pushed out to higher values of ρ W / ρ I as the strength of turbulence increases ( ρ W / ρ I = 31.5 for Scenario 1 and ρ W / ρ I = 38.0 for Scenario 5). Again, ρ I is the approximate diffraction-limited bucket radius. In both plots, the number of FPA pixels, N P , across the diameter of the exit pupil, D 2 , is 256.
Fig. 7.
Fig. 7. Here, we show the wrapped-phase truth in (a) and the wrapped-phase estimates in (b)–(d) determined via a single independent realization of digital-holographic detection in the off-axis PPRG. The number of FPA pixels, N P , across the exit-pupil diameter, D 2 , is 256 for these results, and the turbulence strength corresponds to Scenario 5 in Table 1. Note that the black-dashed circle in (a) has a diameter equal to the Fried coherence diameter for a spherical wave r 0 s w .
Fig. 8.
Fig. 8. This figure displays the azimuthally averaged and normalized [with respect to the maximum of (a)] turbulence-limited PSFs with and without noise as well as the simulated-noise power and expected-noise power σ n 2 in the Fourier plane for Scenarios 1 and 5 [cf. Table 1] in (a) and (b), respectively. Note that the values of ρ / ρ I at the crossings of the blue and red curves for each turbulence scenario are roughly equal to the values of ρ W / ρ I at the peaks in performance for Fig. 6(a) ( ρ W / ρ I = 31.5 for Scenario 1 and ρ W / ρ I = 38.0 for Scenario 5). Also note that the y axes for both plots are normalized and displayed on a log scale to help visualize the high-spatial-frequency content of the turbulence-limited PSFs. Both plots show the averages of 20 independent realizations of turbulence and 20 independent realizations of detection noise, that is, 400 realizations.
Fig. 9.
Fig. 9. This figure displays the differences between the window radii associated with maximum performance and the crossover radii where the turbulence-limited PSF dips below the noise floor in the Fourier plane for all of the turbulence scenarios in Table 1. As a function of SNR, all of the points waver around a mean value of 0.0498 with a standard deviation of 0.6449 in units of length normalized to the approximate diffraction-limited bucket radius ρ I (which is a small quantity compared with the radius ρ W of the window functions in the Fourier plane). The crossing points of the turbulence-limited PSFs with the noise floor were determined by taking the average of the first 10 points that were within 0.0005    pe 2 of the expected-noise power σ n 2 . The data show the averages of 20 independent realizations of turbulence and 20 independent realizations of detection noise.

Tables (1)

Tables Icon

Table 1. Turbulence Scenarios Used in the Trade-Space Analysis

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

U S ( x 2 , y 2 ) = U S ( x 2 , y 2 ) cyl ( x 2 2 + y 2 2 D 2 ) ,
cyl ( x 2 + y 2 ) = { 1 0 x 2 + y 2 < 0.5 0.5 x 2 + y 2 = 0.5 0 x 2 + y 2 > 0.5
U R ( x 2 , y 2 ) = A R exp [ j 2 π x R x 2 M T λ z ] exp [ j 2 π y R y 2 M T λ z ] ,
M T = D 2 D 1 ;
I H ( x 2 , y 2 ) = | U S ( x 2 , y 2 ) + U R ( x 2 , y 2 ) | 2 = | U S ( x 2 , y 2 ) | 2 + | U R ( x 2 , y 2 ) | 2 + U S ( x 2 , y 2 ) U R * ( x 2 , y 2 ) + U R ( x 2 , y 2 ) U S * ( x 2 , y 2 ) ,
I ^ H ( n x s , m y s ) = [ I H ( x 2 , y 2 ) × 1 w x rect ( x 2 n x s w x ) 1 w y rect ( y 2 m y s w y ) ] d x 2 d y 2 ,
rect ( x ) = { 1 0 | x | < 0.5 0.5 | x | = 0.5 0 | x | < 0.5
m ¯ H ( n x s , m y s ) = η T h ν I ^ H ( n x s , m y s ) w x w y ,
d H ( x 2 , y 2 ) = m ¯ H ( x 2 , y 2 ) × 1 x s comb ( x 2 x s ) 1 y s comb ( y 2 y s ) × rect ( x 2 N x s ) rect ( y 2 M y s ) ,
m ¯ H ( x 2 , y 2 ) = η T h ν [ I H ( x 2 , y 2 ) × rect ( x 2 x 2 w x ) rect ( y 2 y 2 w y ) ] d x 2 y 2 = η T h ν I H ( x 2 , y 2 ) ** rect ( x 2 w x ) rect ( y 2 w y )
1 | w | comb ( x w ) = n = δ ( x n w )
δ ( x ) = 1 | w | lim w 0 p ( x w )
V ( x , y ) ** W ( x , y ) = V ( x , y ) W ( x x , y y ) d x d y ,
V ˜ ( ν x , ν y ) = V ( x , y ) e j 2 π ( x ν x + y ν y ) d x d y .
d ˜ H ( ν x , ν y ) = η T h ν α ˜ ( ν x , ν y ) I ˜ H ( ν x , ν y ) ** w x comb ( x s ν x ) w y comb ( y s ν y ) ** N x s sinc ( N x s ν x ) M y s sinc ( M y s ν y ) ,
α ˜ ( ν x , ν y ) = sinc ( w x ν x ) sinc ( w y ν y )
I ˜ H ( ν x , ν y ) = Γ ˜ ( ν x , ν y ) + | A R | 2 δ ( ν x ) δ ( ν y ) + A R * U ˜ S ( ν x + x R M T λ z , ν y + y R M T λ z ) + A R U ˜ S * ( ν x x R M T λ z , ν y y R M T λ z ) ,
Γ ˜ ( ν x , ν y ) = U ˜ S ( ν x , ν y ) ** U ˜ S * ( ν x , ν y ) .
d ˜ H ( x 0 M T λ z , y 0 M T λ z ) = η T h ν α ˜ ( x 0 M T λ z , y 0 M T λ z ) × { 1 ( M T λ z ) 2 Γ ˜ ( x 0 M T λ z , y 0 M T λ z ) + | A R | 2 δ ( x 0 M T λ z ) δ ( y 0 M T λ z ) + A R * U ˜ S ( x 0 + x R M T λ z , y 0 + y R M T λ z ) + A R U ˜ S * ( x 0 x R M T λ z , y 0 y R M T λ z ) } ** p 2 ( M T λ z ) 2 comb ( p x 0 M T λ z ) comb ( p y 0 M T λ z ) ** N P 2 p 2 ( M T λ z ) 2 sinc ( N P p x 0 M T λ z ) sinc ( N P p y 0 M T λ z ) .
L F = M T λ z p = D 2 D 1 λ z p = λ z D 1 N P = ρ I N P .
d ˜ H ( x 0 M T λ z , y 0 M T λ z ) = η T h ν α ˜ ( x 0 M T λ z , y 0 M T λ z ) × { 1 ( M T λ z ) 2 Γ ˜ ( x 0 M T λ z , y 0 M T λ z ) + | A R | 2 δ ( x 0 M T λ z ) δ ( y 0 M T λ z ) + A R * U ˜ S ( x 0 + x R M T λ z , y 0 + y R M T λ z ) + A R U ˜ S * ( x 0 x R M T λ z , y 0 y R M T λ z ) } ** 1 ρ I 2 N P 2 comb ( x 0 ρ I N P ) comb ( y 0 ρ I N P ) ** 1 ρ I 2 sinc ( x 0 ρ I ) sinc ( y 0 ρ I ) .
w ( x 0 , y 0 ) = cyl ( x 0 2 + y 0 2 2 ρ W ) ,
L F = M T r 0 p λ z r 0 = Q P λ z r 0 ,
D T = 2.44 λ z r 0 ,
Q P 2 λ z r 0 2.44 λ z r 0 ,
Q P 4.88 .
( x R M T λ z , y R M T λ z ) .
ψ ˜ ^ S ( x 0 M T λ z , y 0 M T λ z ) d ˜ H ( x 0 x R M T λ z , y 0 y R M T λ z ) w ( x 0 , y 0 ) η T h ν α ˜ ( x 0 x R M T λ z , y 0 y R M T λ z ) A R * A S = ψ ˜ S ( x 0 M T λ z , y 0 M T λ z ) w ( x 0 , y 0 ) ,
V ( x , y ) = V ˜ ( ν x , ν y ) e j 2 π ( x ν x + y ν y ) d ν x d ν y .
ψ ^ S ( M T x 1 , M T y 1 ) ψ S ( M T x 1 , M T y 1 ) ** π ρ W 2 ( λ z ) 2 somb ( 2 ρ W x 1 2 + y 1 2 λ z ) ,
somb ( x 2 + y 2 ) = 2 J 1 ( π x 2 + y 2 ) π x 2 + y 2
r 0 p w = 0.185 ( λ 2 Z C n 2 ) 3 / 5
r 0 s w = 0.33 ( λ 2 Z C n 2 ) 3 / 5 ,
σ χ p w 2 = 0.307 k 7 / 6 Z 11 / 6 C n 2
σ χ s w 2 = 0.124 k 7 / 6 Z 11 / 6 C n 2 ,
S F = | E { U S ( x 1 , y 1 ) U ^ S * ( x 1 , y 1 ) } | 2 E { | U S ( x 1 , y 1 ) | 2 } E { | U ^ S ( x 1 , y 1 ) | 2 } ,
σ n 2 = m ¯ S + m ¯ R + σ r 2 ,
SNR = E { | U ˜ S | 2 } Var { U ˜ N } ,
SNR A = m ¯ S m ¯ R γ σ n 2 = m ¯ S m ¯ R γ ( m ¯ S + m ¯ R + σ r 2 ) ,
γ = π ρ W 2 ρ I 2 N P 2
Var { U ˜ N } m ¯ R + σ r 2 .
SNR N = E { | U ˜ ^ S + N | 2 | U ˜ ^ N | 2 } Var { U ˜ ^ N } ,

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