Abstract

We have improved the resolution of our laser optical feedback imaging (LOFI) setup by using a synthetic aperture (SA) process. We report a two-dimensional (2D) SA LOFI experiment where the unprocessed image (i.e., the classical LOFI image) is obtained point by point, line after line using full 2D galvanometric scanning. The 2D superresolved image is then obtained by successively computing two angular SA operations while a one-dimensional angular synthesis is preceded by a frequency synthesis to obtain a 2D superresolved image conventionally in the synthetic aperture radar (SAR) method and their corresponding laser method called synthetic aperture ladar. The numerical and experimental results are compared.

© 2008 Optical Society of America

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  1. J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).
  2. R. O. Harger, Synthetic Aperture Radar Systems: Theory and Design (Academic, 1970).
  3. A. Ja. Pasmurov and J. S. Zimoview, Radar Imaging and Holography (IEEE, 2005).
    [CrossRef]
  4. T. S. Lewis and H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781-1782 (1970).
    [CrossRef]
  5. C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).
  6. S. Markus, B. D. Colella, and T. J. Green, Jr., “Solid-state laser synthetic aperture radar,” Appl. Opt. 33, 960-964 (1994).
  7. A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031-3033 (2006).
    [CrossRef] [PubMed]
  8. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983-1985 (2002).
    [CrossRef]
  9. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging laser radar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621-7629 (2005).
    [CrossRef] [PubMed]
  10. E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744-746 (1999).
    [CrossRef]
  11. K. Otsuka, “Ultrahigh sensitivity laser Doppler velocimetry with a microchip solid-state laser,” Appl. Opt. 33, 1111-1114 (1994).
    [PubMed]
  12. E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
    [CrossRef]
  13. O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
    [PubMed]
  14. V. Muzet, E. Lacot, O. Hugon, and Y. Guillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793-799 (2005).
    [CrossRef]
  15. When αx=0 and αy=0, if there is no misalignment, the center of the Gaussian laser beam successively reaches the two galvanometric mirrors on their rotation axis and finally the target on the origin of our experimental coordinate system (x=0, y=0, z=0). If we consider a laser misalignment, the final position of the Gaussian laser beam center is given by (Δx, Δy, z=0).
  16. A. Papoulis, Signal Analysis (McGraw-Hill, 1977), pp. 262-272.
  17. D. Park and J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” Proc. SPIE 999, 100-116(1989).

2007

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

2006

2005

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging laser radar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621-7629 (2005).
[CrossRef] [PubMed]

V. Muzet, E. Lacot, O. Hugon, and Y. Guillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793-799 (2005).
[CrossRef]

2002

2001

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

1999

1994

1989

D. Park and J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” Proc. SPIE 999, 100-116(1989).

1987

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

1970

T. S. Lewis and H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781-1782 (1970).
[CrossRef]

Abshier, J. O.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

Accetta, J. S.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

Aleksoff, C. C.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

Bashkansky, M.

Beck, S. M.

Buck, J. R.

Buell, W. F.

Colella, B. D.

Curlander, J. C.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

Day, R.

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744-746 (1999).
[CrossRef]

Dickinson, R. P.

Fee, M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

Funk, E.

Green, T. J.

Guillard, Y.

V. Muzet, E. Lacot, O. Hugon, and Y. Guillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793-799 (2005).
[CrossRef]

Harger, R. O.

R. O. Harger, Synthetic Aperture Radar Systems: Theory and Design (Academic, 1970).

Hugon, O.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031-3033 (2006).
[CrossRef] [PubMed]

V. Muzet, E. Lacot, O. Hugon, and Y. Guillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793-799 (2005).
[CrossRef]

Hutchins, H. S.

T. S. Lewis and H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781-1782 (1970).
[CrossRef]

Jacquin, O.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031-3033 (2006).
[CrossRef] [PubMed]

Klossler, A.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

Kozlowski, D. A.

Lacot, E.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031-3033 (2006).
[CrossRef] [PubMed]

V. Muzet, E. Lacot, O. Hugon, and Y. Guillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793-799 (2005).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744-746 (1999).
[CrossRef]

Lewis, T. S.

T. S. Lewis and H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781-1782 (1970).
[CrossRef]

Lucke, R. L.

Majwski, R. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

Marechal, N. J.

Markus, S.

McDonough, R. N.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

Muzet, V.

V. Muzet, E. Lacot, O. Hugon, and Y. Guillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793-799 (2005).
[CrossRef]

Otsuka, K.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, 1977), pp. 262-272.

Park, D.

D. Park and J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” Proc. SPIE 999, 100-116(1989).

Pasmurov, A. Ja.

A. Ja. Pasmurov and J. S. Zimoview, Radar Imaging and Holography (IEEE, 2005).
[CrossRef]

Paun, I. A.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

Peterson, L. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

Reintjes, J.

Ricard, C.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

Rickard, L. J.

Schroeder, K. S.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

Shapiro, J. H.

D. Park and J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” Proc. SPIE 999, 100-116(1989).

Stoeckel, F.

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744-746 (1999).
[CrossRef]

Tai, A. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

van der Sanden, B.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

Witomski, A.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031-3033 (2006).
[CrossRef] [PubMed]

Wright, T. J.

Zimoview, J. S.

A. Ja. Pasmurov and J. S. Zimoview, Radar Imaging and Holography (IEEE, 2005).
[CrossRef]

Appl. Opt.

Opt. Lett.

Phys. Rev. A

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

Proc. IEEE

T. S. Lewis and H. S. Hutchins, “A synthetic aperture at 10.6 Microns,” Proc. IEEE 58, 1781-1782 (1970).
[CrossRef]

Proc. SPIE

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klossler, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 laser,” Proc. SPIE 783, 29-40 (1987).

V. Muzet, E. Lacot, O. Hugon, and Y. Guillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793-799 (2005).
[CrossRef]

D. Park and J. H. Shapiro, “Performance analysis of optical synthetic aperture radars,” Proc. SPIE 999, 100-116(1989).

Ultramicroscopy

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent back-scattering microscopy using frequency-shifted optical feedback in a microchip laser,” Ultramicroscopy (2007).
[PubMed]

Other

When αx=0 and αy=0, if there is no misalignment, the center of the Gaussian laser beam successively reaches the two galvanometric mirrors on their rotation axis and finally the target on the origin of our experimental coordinate system (x=0, y=0, z=0). If we consider a laser misalignment, the final position of the Gaussian laser beam center is given by (Δx, Δy, z=0).

A. Papoulis, Signal Analysis (McGraw-Hill, 1977), pp. 262-272.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

R. O. Harger, Synthetic Aperture Radar Systems: Theory and Design (Academic, 1970).

A. Ja. Pasmurov and J. S. Zimoview, Radar Imaging and Holography (IEEE, 2005).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Description of the LOFI experimental setup. The target is located in the vertical plane (x, y, z = 0 ). L1, L2, L3, and L 3 ; lenses; BS, beam splitter; F e is the total optical frequency shift; and p and q are the quadrature components voltages delivered by the lock-in amplifier at the demodulation frequency F e . M x and M y , respectively, are the rotating mirrors that allow to us scan the target in the horizontal direction x and the vertical direction y. The angular orientation of the galvanometric mirrors M x and M y , respectively, is given by the angles α x and α y . (a) Classical LOFI experiment where the laser is focused in the target plane. (b) SA LOFI experiment. The laser is focused in front of the target plane. L is the laser-focal spot distance, L x is the focal spot M x distance, l is the M x M y distance, L y is the M y target plane distance, and D is the Gaussian laser beam diameter in the target plane.

Fig. 2
Fig. 2

Typical LOFI signal spectrum (solid curve) obtained by performing an experimental galvanometric scanning of a quasi-punctual light-scattering target in the along line direction (y direction) and the across line direction (x direction). The corresponding adapted filters ( H y ( f y ) and H x ( f y ) ) are also plotted (dotted line). (a) Along line direction (y direction), angular velocity α ˙ y = 8 × 10 - 2 rad / s , Doppler central frequency F y α ˙ y = 1000 Hz , and Doppler bandwidth W y α ˙ y = 270 Hz . (b) Across line direction (x direction), angular velocity α ˙ x = 8 × 10 - 4 rad / s , Doppler central frequency F x α ˙ x = 4.5 Hz , and Doppler bandwidth W x α ˙ x = 2.5 Hz . Experimental parameters; r 0 = 550 μm , λ = 1064 nm , L x = 145 cm , l = 2.5 cm , L y = 40 cm , Δ x = 1.5 mm , and Δ y = 3.3 mm .

Fig. 3
Fig. 3

Two-dimensional SA image obtained by frequency-shifted optical feedback combined with galvanometric scanning. (a) Classical (i.e., unprocessed) LOFI image. (b) LOFI image after SA processing in the across line direction y. (c) LOFI image after SA processing in the along line direction x. (d) LOFI image after 2D SA processing (x and y directions). Numerical parameters; r 0 = 550 μm , λ = 1064 nm , L x = 50 cm , l = 1 cm , L y = 50 cm , Δ x = 0.04 mm , and Δ y = 0.08 mm . The target is composed of four punctual reflectors at the following positions: ( x 1 = 0 , y 1 = 0 ), ( x 2 = 2 r 0 , y 2 = 0 ), ( x 3 = - 2 r 0 , y 3 = 2 r 0 ), and ( x 4 = 2 r 0 , y 4 = - 2 r 0 ).

Fig. 4
Fig. 4

Effect of the length ratio L y / L x on the optical resolution of a SAL experiment using galvanometric scanning. Left column: classical (i.e., unprocessed) LOFI image. Right column: LOFI image after SA processing. (a)  L x = 2 3 × 100 cm , l = 1 cm , and L y = 1 3 × 100 cm . (b)  L x = 1 2 × 100 cm , l = 1 cm , and L y = 1 2 × 100 cm . (c)  L x = 1 3 × 100 cm , l = 1 cm , and L y = 2 3 × 100 cm . Numerical parameters: r 0 = 550 μm , λ = 1064 nm , Δ x = 0.04 mm , and Δ y = 0.08 mm .

Fig. 5
Fig. 5

Left column: classical (i.e., unprocessed) LOFI image of the letter X. Right column: image after SA processing. Up: experimental results. Down: numerical results. Experimental parameters: r 0 = 550 μm , λ = 1064 nm , L x = 145 cm , l = 2.5 cm , L y = 40 cm , Δ x = 1.5 mm , Δ y = 3.3 mm , D 2.2 mm , and dimensions of the letter X: 1 mm × 1 mm .

Equations (28)

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Δ P out ( α x , α y ) = 2 G R ( Ω e ) i R e , i P out ( α x , α y , x i , y i ) cos [ Ω e t + Φ ( α x , α y , x i , y i ) + Φ R ( Ω e ) ] ,
G R ( Ω e ) = γ c ( η γ 1 ) 2 + Ω e 2 ( Ω R 2 - Ω e 2 ) 2 + ( η γ 1 Ω e ) 2 ,
Φ R ( Ω e ) = arctan [ Ω e ( Ω R 2 Ω e 2 - ( η γ 1 ) 2 ) η γ 1 Ω R 2 ] ,
p ( α x , α y ) = 2 G R ( Ω e ) i R e , i P out ( α x , α y , x i , y i ) cos [ Φ ( α x , α y , x i , y i ) ] ,
q ( α x , α y ) = 2 G R ( Ω e ) i R e , i P o u t ( α x , α y , x i , y i ) sin [ Φ ( α x , α y , x i , y i ) ] ,
s ( α x , α y ) = p ( α x , α y ) + j q ( α x , α y ) = 2 G R ( Ω e ) i R e , i P out ( α x , α y , x i , y i ) exp [ j Φ ( α x , α y , x i , y i ) ] = | s ( α x , α y ) | exp [ j Φ s ( α x , α y ) ] ,
h x ( α x , 0 ) = exp [ - ( α x - α ^ x ) 2 Δ α x 2 ] exp [ j Φ h ( α x , 0 ) ] ,
h y ( 0 , α y ) = exp [ - ( α y - α ^ y ) 2 Δ α y 2 ] exp [ j Φ h ( 0 , α y ) ] ,
Φ h ( α x , α y ) = Φ 0 + Δ Φ + 2 π [ K x α x 2 2 + D x α x ] + 2 π [ K y α y 2 2 + D y α y ] .
K x = - 8 λ L x ( l + L y ) ( L x + l + L y ) , K y = - 8 λ ( L x + l ) L y ( L x + l + L y ) ,
D x = - 4 λ Δ x ( l + L y ) ( L x + l + L y ) , D y = - 4 λ Δ y L y ( L x + l + L y ) .
α ^ x = Δ x 2 ( l + L y ) , Δ α x = λ π r 0 ( L x + l + L y ) 2 ( l + L y ) , α ^ y = Δ y 2 L y , Δ α y = λ π r 0 ( L x + l + L y ) 2 L y ) ,
H x ( f x ) = FT [ h x ( α x , 0 ) ] ,
H x ( f x ) j K x exp [ - ( f x - F x ) 2 W x 2 ] exp [ - j π ( f x - F x ) 2 K x ] exp [ j Φ h ( α ^ x , 0 ) ] .
F x = | D x + K x α ^ x | = | - 4 Δ x λ | ,
W x = | K x Δ α x | = | - 4 L x π r 0 | .
F y = | D y + K y α ^ y | = | - 4 Δ y λ | ,
W y = | K y Δ α y | = | - 4 ( L x + l ) π r 0 | .
SAL 1 D x ( α x , α y ) = s ( α x , α y ) h x * ( - α x , 0 ) ,
SAL 1 D y ( α x , α y ) = s ( α x , α y ) h y * ( 0 , - α y ) .
SAL 2 D ( α x , α y ) = SAL 1 D x ( α x , α y ) h y * ( 0 , - α y ) = SAL 1 D y ( α x , α y ) h x * ( - α x , 0 ) .
g x ( α x ) = h x ( α x , 0 ) h x * ( - α x , 0 ) = T F [ | H x ( f x ) | 2 ] π 2 Δ α x exp ( - π 2 W x 2 α x 2 2 ) exp ( j 2 π F x α x ) .
δ α x = 2 π W x = r 0 2 L x ,
δ α y = 2 π W y = r 0 2 ( L x + l ) .
δ x SAL ( l + L y ) 2 δ α x = r 0 ( l + L y ) L x ,
δ y SAL L y 2 δ α y = r 0 L y L x + l .
δ x SAL δ y SAL r 0 L y L x < D λ π r 0 ( L x + L y ) .
δ x SAL δ y SAL r 0 L y L x < r 0 .

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