Abstract

Digital speckle photography can be used in the analysis of surface motion in combination with an optical linear canonical transform (LCT). Previously [ D. P. Kelly et al.Appl. Opt. 44, 2720 (2005) ] it has been shown that optical fractional Fourier transforms (OFRTs) can be used to vary the range and sensitivity of speckle-based metrology systems, allowing the measurement of both the magnitude and direction of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. This requires two bulk optical systems. We extend the OFRT analysis to more general LCT systems with a single limiting aperture. The effect of a limiting aperture in LCT systems is examined in more detail by deriving a generalized Yamaguchi correlation factor. We demonstrate the benefits of using an LCT approach to metrology design. Using this technique, we show that by varying the curvature of the illuminating field, we can effectively change the output domain. From a practical perspective this means that estimation of the motion of a target can be achieved by using one bulk optical system and different illuminating conditions. Experimental results are provided to support our theoretical analysis.

© 2006 Optical Society of America

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2006 (1)

2005 (6)

2004 (2)

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, "Speckle correlation and the fractional Fourier transform," in Optical Information Systems II, B Javidi, ed., Proc. SPIE 5557, 255-266 (2004).
[CrossRef]

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, "Measurement of surface roughness by speckle correlation," Opt. Eng. 43, 2753-2761 (2004).
[CrossRef]

2003 (3)

2002 (1)

L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).

2001 (1)

2000 (2)

J. T. Sheridan and R. Patten, "Fractional Fourier speckle photography: motion detection and the optical fractional Fourier transformation," Optik (Stuttgart) 111, 329-331 (2000).

J. T. Sheridan and R. Patten, "Holographic interferometry and the fractional Fourier transformation," Opt. Lett. 25, 448-450 (2000).
[CrossRef]

1998 (1)

1995 (3)

1994 (3)

1993 (3)

1992 (2)

Q. Li and F. Chiang, "Three-dimensional dimension of laser speckle," Appl. Opt. 31, 6287-6291 (1992).
[CrossRef] [PubMed]

J. Widjaja, J. Uozumi, and T. Asakura, "Real-time evaluation of local displacement of objects by means of the Wigner distribution function," J. Opt. (Paris) 23, 13-18 (1992).
[CrossRef]

1991 (1)

1990 (1)

1987 (1)

1972 (1)

H. Tiziani, "A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately," Opt. Commun. 5, 271-274 (1972).
[CrossRef]

1970 (1)

1932 (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Abe, S.

S. Abe and J. T. Sheridan, "Generalization of the fractional Fourier transformation to an arbitrary linear loss-less transformation: an operator approach," J. Phys. A 27, 4179-4187 (1994).
[CrossRef]

S. Abe and J. T. Sheridan, "Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation," Opt. Lett. 19, 1801-1803 (1994).
[CrossRef] [PubMed]

Alieva, T.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional transforms in optical information processing," EURASIP J. Appl. Signal Process. 10, 1498-1519 (2005).

Asakura, T.

J. Widjaja, J. Uozumi, and T. Asakura, "Real-time evaluation of local displacement of objects by means of the Wigner distribution function," J. Opt. (Paris) 23, 13-18 (1992).
[CrossRef]

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional transforms in optical information processing," EURASIP J. Appl. Signal Process. 10, 1498-1519 (2005).

Bastians, M. J.

M. J. Bastians, "Application of the Wigner distribution function in optics," in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbrauker and F.Hlawatsch, eds. (Elsevier Science, 1997), Chap. 7.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965).

Cai, L. Z.

L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).

Calvo, M. L.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional transforms in optical information processing," EURASIP J. Appl. Signal Process. 10, 1498-1519 (2005).

Chiang, F.

Collins, S. A.

Dainty, J. C.

J. C. Dainty, "The statistics of speckle patterns," in Progress in Optics, Vol. XIV, E.Wolf, ed. (North-Holland, 1976).

Diazdelacruz, J. M.

Ding, J. J.

Dorsch, R. G.

Fricke-Begemann, T.

Goodman, J. W.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gopinathan, U.

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O'Neill, and J. T. Sheridan, "The generalized Yamaguchi correlation factor for quadratic phase speckle metrology systems with apertures," Opt. Lett. (to be published).

Gundu, P.

Hack, E.

Hanson, S. G.

Hennelly, B. M.

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, "Speckle photography: mixed domain fractional Fourier motion detection," Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, "Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform," Appl. Opt. 44, 2720-2727 (2005).
[CrossRef] [PubMed]

B. M. Hennelly and J. T. Sheridan, "Fast numerical algorithms for the linear canonical transform," J. Opt. Soc. Am. A 22, 928-938 (2005).
[CrossRef]

D. P. Kelly, R. F. Patten, B. M. Hennelly, Y. Liu, J. E. Ward, and J. T. Sheridan, "Linear canonical transforms, and speckle based metrology," in Recent Developments in Traceable Dimensional Measurements III, J. E. Decken and G.-S. Peng, eds., Proc. SPIE 5879, 223-234 (2005).

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, "Speckle correlation and the fractional Fourier transform," in Optical Information Systems II, B Javidi, ed., Proc. SPIE 5557, 255-266 (2004).
[CrossRef]

J. T. Sheridan, B. M. Hennelly, and D. Kelly, "Motion detection, the Wigner distribution function, and the optical fractional Fourier transform," Opt. Lett. 28, 884-886 (2003).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O'Neill, and J. T. Sheridan, "The generalized Yamaguchi correlation factor for quadratic phase speckle metrology systems with apertures," Opt. Lett. (to be published).

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. (to be published).

Jones, R.

R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, 1989).

Kelly, D.

Kelly, D. P.

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, "Speckle photography: mixed domain fractional Fourier motion detection," Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, "Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform," Appl. Opt. 44, 2720-2727 (2005).
[CrossRef] [PubMed]

D. P. Kelly, R. F. Patten, B. M. Hennelly, Y. Liu, J. E. Ward, and J. T. Sheridan, "Linear canonical transforms, and speckle based metrology," in Recent Developments in Traceable Dimensional Measurements III, J. E. Decken and G.-S. Peng, eds., Proc. SPIE 5879, 223-234 (2005).

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, "Speckle correlation and the fractional Fourier transform," in Optical Information Systems II, B Javidi, ed., Proc. SPIE 5557, 255-266 (2004).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O'Neill, and J. T. Sheridan, "The generalized Yamaguchi correlation factor for quadratic phase speckle metrology systems with apertures," Opt. Lett. (to be published).

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. (to be published).

Kirchner, M.

Kobayashi, K.

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, "Measurement of surface roughness by speckle correlation," Opt. Eng. 43, 2753-2761 (2004).
[CrossRef]

Kutay, M. A.

M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Leushacke, L.

Li, Q.

Liu, Y.

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, "Speckle photography: mixed domain fractional Fourier motion detection," Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, R. F. Patten, B. M. Hennelly, Y. Liu, J. E. Ward, and J. T. Sheridan, "Linear canonical transforms, and speckle based metrology," in Recent Developments in Traceable Dimensional Measurements III, J. E. Decken and G.-S. Peng, eds., Proc. SPIE 5879, 223-234 (2005).

Lohmann, A. W.

Mendlovic, D.

O'Neill, E. L.

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, 1962).

O'Neill, F. T.

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, "Speckle photography: mixed domain fractional Fourier motion detection," Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O'Neill, and J. T. Sheridan, "The generalized Yamaguchi correlation factor for quadratic phase speckle metrology systems with apertures," Opt. Lett. (to be published).

Owner-Petersen, M.

Ozaktas, M.

D. Mendlovic, M. Ozaktas, and A. W. Lohmann, "Graded-index media, Wigner-distribution functions, and the fractional Fourier transform," Appl. Opt. 33, 6188-6193 (1994).
[CrossRef] [PubMed]

M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Ozatkas, H. M.

Patten, R.

J. T. Sheridan and R. Patten, "Holographic interferometry and the fractional Fourier transformation," Opt. Lett. 25, 448-450 (2000).
[CrossRef]

J. T. Sheridan and R. Patten, "Fractional Fourier speckle photography: motion detection and the optical fractional Fourier transformation," Optik (Stuttgart) 111, 329-331 (2000).

Patten, R. F.

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, "Speckle photography: mixed domain fractional Fourier motion detection," Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, R. F. Patten, B. M. Hennelly, Y. Liu, J. E. Ward, and J. T. Sheridan, "Linear canonical transforms, and speckle based metrology," in Recent Developments in Traceable Dimensional Measurements III, J. E. Decken and G.-S. Peng, eds., Proc. SPIE 5879, 223-234 (2005).

Pei, S. C.

Rastogi, P.

Rastogi, P. K.

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

P. K. Rastogi, "Techniques of displacement and deformation measurements in speckle metrology," in Speckle Metrology, R.S.Sirohi, ed. (Marcel Dekker, 1993).

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. (to be published).

Rose, B.

Sheridan, J. T.

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, "Speckle photography: mixed domain fractional Fourier motion detection," Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, "Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform," Appl. Opt. 44, 2720-2727 (2005).
[CrossRef] [PubMed]

B. M. Hennelly and J. T. Sheridan, "Fast numerical algorithms for the linear canonical transform," J. Opt. Soc. Am. A 22, 928-938 (2005).
[CrossRef]

D. P. Kelly, R. F. Patten, B. M. Hennelly, Y. Liu, J. E. Ward, and J. T. Sheridan, "Linear canonical transforms, and speckle based metrology," in Recent Developments in Traceable Dimensional Measurements III, J. E. Decken and G.-S. Peng, eds., Proc. SPIE 5879, 223-234 (2005).

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, "Speckle correlation and the fractional Fourier transform," in Optical Information Systems II, B Javidi, ed., Proc. SPIE 5557, 255-266 (2004).
[CrossRef]

J. T. Sheridan, B. M. Hennelly, and D. Kelly, "Motion detection, the Wigner distribution function, and the optical fractional Fourier transform," Opt. Lett. 28, 884-886 (2003).
[CrossRef] [PubMed]

J. T. Sheridan and R. Patten, "Fractional Fourier speckle photography: motion detection and the optical fractional Fourier transformation," Optik (Stuttgart) 111, 329-331 (2000).

J. T. Sheridan and R. Patten, "Holographic interferometry and the fractional Fourier transformation," Opt. Lett. 25, 448-450 (2000).
[CrossRef]

S. Abe and J. T. Sheridan, "Generalization of the fractional Fourier transformation to an arbitrary linear loss-less transformation: an operator approach," J. Phys. A 27, 4179-4187 (1994).
[CrossRef]

S. Abe and J. T. Sheridan, "Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation," Opt. Lett. 19, 1801-1803 (1994).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O'Neill, and J. T. Sheridan, "The generalized Yamaguchi correlation factor for quadratic phase speckle metrology systems with apertures," Opt. Lett. (to be published).

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. (to be published).

Sjodahl, M.

Tiziani, H.

H. Tiziani, "A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately," Opt. Commun. 5, 271-274 (1972).
[CrossRef]

Uozumi, J.

J. Widjaja, J. Uozumi, and T. Asakura, "Real-time evaluation of local displacement of objects by means of the Wigner distribution function," J. Opt. (Paris) 23, 13-18 (1992).
[CrossRef]

Wang, Y. Q.

L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).

Ward, J. E.

D. P. Kelly, R. F. Patten, B. M. Hennelly, Y. Liu, J. E. Ward, and J. T. Sheridan, "Linear canonical transforms, and speckle based metrology," in Recent Developments in Traceable Dimensional Measurements III, J. E. Decken and G.-S. Peng, eds., Proc. SPIE 5879, 223-234 (2005).

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O'Neill, and J. T. Sheridan, "The generalized Yamaguchi correlation factor for quadratic phase speckle metrology systems with apertures," Opt. Lett. (to be published).

Widjaja, J.

J. Widjaja, J. Uozumi, and T. Asakura, "Real-time evaluation of local displacement of objects by means of the Wigner distribution function," J. Opt. (Paris) 23, 13-18 (1992).
[CrossRef]

Wigner, E.

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Wykes, C.

R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, 1989).

Yamaguchi, I.

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, "Measurement of surface roughness by speckle correlation," Opt. Eng. 43, 2753-2761 (2004).
[CrossRef]

Yaroslavsky, L.

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, "Measurement of surface roughness by speckle correlation," Opt. Eng. 43, 2753-2761 (2004).
[CrossRef]

Yura, H. T.

Zalevsky, Z.

M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Appl. Opt. (8)

EURASIP J. Appl. Signal Process. (1)

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional transforms in optical information processing," EURASIP J. Appl. Signal Process. 10, 1498-1519 (2005).

J. Opt. (Paris) (1)

J. Widjaja, J. Uozumi, and T. Asakura, "Real-time evaluation of local displacement of objects by means of the Wigner distribution function," J. Opt. (Paris) 23, 13-18 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

J. Phys. A (1)

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Opt. Commun. (2)

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[CrossRef]

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Opt. Eng. (2)

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, "Measurement of surface roughness by speckle correlation," Opt. Eng. 43, 2753-2761 (2004).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. (to be published).

Opt. Express (1)

Opt. Laser Technol. (1)

L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 34, 249-252 (2002).

Opt. Lett. (4)

Optik (Stuttgart) (1)

J. T. Sheridan and R. Patten, "Fractional Fourier speckle photography: motion detection and the optical fractional Fourier transformation," Optik (Stuttgart) 111, 329-331 (2000).

Phys. Rev. (1)

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[CrossRef]

Proc. SPIE (2)

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[CrossRef]

D. P. Kelly, R. F. Patten, B. M. Hennelly, Y. Liu, J. E. Ward, and J. T. Sheridan, "Linear canonical transforms, and speckle based metrology," in Recent Developments in Traceable Dimensional Measurements III, J. E. Decken and G.-S. Peng, eds., Proc. SPIE 5879, 223-234 (2005).

Other (15)

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O'Neill, and J. T. Sheridan, "The generalized Yamaguchi correlation factor for quadratic phase speckle metrology systems with apertures," Opt. Lett. (to be published).

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Figures (7)

Fig. 1
Fig. 1

Optical arrangement for a speckle-photography-based metrology system.

Fig. 2
Fig. 2

Two-lens optical LCT system with an aperture plane.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Correlation results for translation of 100 μ m for an LCT system with plane wave illumination. Measured displacement, 99.96 μ m .

Fig. 5
Fig. 5

Correlation results for translation of 100 μ m for an LCT system with diverging wave illumination. Measured displacement, 100.5 μ m .

Fig. 6
Fig. 6

Correlation results for rotation of 0.55 mrad for an LCT system where the target is illuminated with a plane wave. Measured rotation, 0.534 mrad .

Fig. 7
Fig. 7

Correlation results for simultaneous in-plane translation of 50 μ m and rotation of 0.55 mrad . Measured rotation, 0.55 mrad ; in-plane translation, 46.1 μ m .

Equations (32)

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u ̃ ( x ̃ 0 ) = u ( x 0 + ξ ) exp ( j 2 π λ κ x 0 ) ,
u a ( x a ) = ( 1 j λ B 1 ) u ( x 0 ) exp [ j π λ B 1 ( D 1 x a 2 + A 1 x 0 2 2 x a x 0 ) ] d x 0 ,
( x a k a ) = [ A 1 B 1 C 1 D 1 ] ( x 0 k 0 ) ,
u ̃ a ( x ̃ a ) = u ̃ ( x ̃ 0 ) exp [ j π λ B 1 ( D 1 x ̃ a 2 + A 1 x ̃ 0 2 2 x ̃ a x ̃ 0 ) ] d x ̃ 0 ,
u ̃ a ( x ̃ a ) = u a ( x a + ξ a ) exp ( j 2 π κ a x a λ ) ,
v a ( ) = u a ( ) p ( ) v ̃ a ( ) = u ̃ a ( ) p ( ) .
v c ( x c ) = v a ( x a ) exp [ j π λ B 2 ( D 2 x c 2 + A 2 x a 2 2 x a x c ) ] d x a .
c I I ̃ ( s ) = v c * ( r ) v ̃ c ( r + s ) 2 σ I σ I ̃ ,
v c * ( x c ) v ̃ c ( x c + s ) = u a * ( x a ) u ̃ a ( x ̃ a ) p * ( x a ) p ( x ̃ a ) exp [ j π λ B 2 ( D 2 x c 2 + A 2 x a 2 2 x a x c ) ] exp { j π λ B 2 [ D 2 ( x c + s ) 2 + A 2 x ̃ a 2 2 x ̃ a ( x c + s ) ] } d x a d x ̃ a .
u a * ( x a ) u ̃ a ( x ̃ c ) = u * ( x 0 ) u ̃ ( x ̃ 0 ) exp [ j π λ B 1 ( D 1 x a 2 + A 1 x 0 2 2 x a x 0 ) ] exp [ j π λ B 1 ( D 1 x ̃ a 2 + A 1 x ̃ 0 2 2 x ̃ a x ̃ 0 ) ] d x 0 d x ̃ 0 .
u * ( x 0 ) u ̃ ( x ̃ 0 ) = u * ( x 0 ) u ( x 0 + ξ ) exp ( j 2 π λ κ x 0 ) = σ u 2 δ ( x ̃ 0 x 0 ξ ) exp ( j 2 π λ κ x ̃ 0 ) .
u a * ( x a ) u ̃ a ( x ̃ a ) = σ u 2 exp [ j π λ B 1 ( D 1 x a 2 + A 1 x 0 2 2 x a x 0 ) ] × exp [ j 2 π λ κ ( x 0 + ξ ) ] exp { j π λ B 1 [ D 1 x ̃ a 2 + A 1 ( x 0 + ξ ) 2 2 x ̃ a ( x 0 + ξ ) ] } d x 0 .
u a * ( x a ) u ̃ a ( x ̃ a ) = σ u 2 exp { j π λ B 1 [ D 1 ( x ̃ a 2 x a 2 ) 2 x ̃ a ξ ] } exp ( j π 2 x 0 Δ λ B 1 ) d x 0 ,
v c * ( x c ) v ̃ c ( x c + s ) = C σ u 2 δ ( x ̃ a x a A 1 ξ B 1 κ ) exp { j π λ B 1 [ D 1 ( x ̃ a 2 x a 2 ) 2 x ̃ a ξ ] } p * ( x a ) p ( x ̃ a ) exp { j π λ B 2 [ D 2 ( s 2 + 2 s x c ) + A 2 ( x ̃ a 2 x a 2 ) 2 x c ( x ̃ a x a ) 2 s x ̃ a ] } d x a d x ̃ a .
v c * ( x c ) v ̃ c ( x c + s ) = C σ u 2 exp { j π λ B 1 [ 2 D 1 ( A 1 ξ + B 1 κ ) x a 2 x a ξ ] } p * ( x a ) p ( x a + A 1 ξ + B 1 κ ) × exp { j π λ B 2 [ D 2 ( s 2 + 2 s x c ) + 2 A 2 ( A 1 ξ + B 1 κ ) x a 2 x c ( A 1 ξ + B 1 κ ) 2 s ( x a + A 1 ξ + B 1 κ ) ] } d x a .
v c * ( x c ) v ̃ c ( x c + s ) = C σ u 2 p * ( x a ) p ( x a + A 1 ξ + B 1 κ ) × exp ( j π λ { [ 2 D 1 ( A 1 ξ + B 1 κ ) 2 ξ ] B 1 + [ 2 A 2 ( A 1 ξ + B 1 κ ) 2 s ] B 2 } x a ) d x a .
v c * ( x c ) v ̃ c ( x c + s ) 2 σ u 4 C 2 p * ( x a ) p ( x a + A 1 ξ + B 1 κ ) d x a 2 ,
s = [ A 1 A 2 + B 2 C 1 ] ξ + [ A 2 B 1 + B 2 D 1 ] κ .
v a * ( x a ) v a ( x a ) 2 C σ u 2 p ( x a ) 2 d x a ,
v ̃ a * ( x a ) v ̃ a ( x a ) 2 C σ u 2 p ( x a ) 2 d x a .
C I I ̃ = p * ( x a ) p ( x a + A 1 ξ + B 1 κ ) d x a p ( x a ) 2 d x a 2 .
[ 0 f 1 f 0 ] , [ cos ( θ ) f sin ( θ ) sin ( θ ) f cos ( θ ) ] ,
[ 1 z 0 1 ] , [ 1 0 1 f 1 ] .
[ A B C D ] = [ 1 d 3 0 1 ] [ 1 0 1 f 2 1 ] [ 1 d 2 0 1 ] [ 1 0 1 f 1 1 ] [ 1 d 1 0 1 ] = [ 1 d 3 + d 2 ( 1 d 3 f 2 ) f 1 d 3 f 2 d 3 + d 2 + d 1 d 2 d 3 ( d 1 + d 2 ) d 3 f 1 ( d 2 + d 3 f 1 ) d 1 f 2 f 1 f 2 d 2 f 2 1 f 1 1 f 2 1 + d 1 ( d 2 f 2 1 f 1 1 f 2 ) d 2 f 2 ] .
M 1 = [ 1 z a f 2 f ( 1 z a f ) + z a 1 f 1 ] .
u ( x 0 ) = exp [ j k h ( x 0 ) ] ,
u sph ( x , z ) = exp [ j k x 2 2 ( ε z ) ] ,
u sw ( x 0 ) = exp { j k [ x 0 2 2 ( f z RS ) + h ( x 0 ) ] } = u ( x 0 ) exp [ j π x 0 2 λ ( f z RS ) ] .
v sw ( x c 1 ) = u sw ( x 0 ) exp [ j π λ B ( A x 0 2 + D x c 1 2 2 x c 1 x 0 ) ] d x 0 ,
v sw ( x c 2 ) = u ( x 0 ) exp [ j π λ B sw ( A sw x c 2 2 + D sw x 0 2 2 x c 2 x 0 ) ] d x 0 ,
[ A sw B sw C sw D sw ] = [ A B C D ] [ 1 0 1 ( f z RS ) 1 ] = [ A B f z RS B C D f z RS D ]
ξ = ξ c 1 ξ c 2 A A sw , κ = A sw ξ c 1 A ξ c 2 B ( A sw A ) ,

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