Abstract

The two-dimensional (2D) nonseparable linear canonical transform (NSLCT) is a generalization of the fractional Fourier transform (FRFT) and the LCT. It is useful in signal analysis and optics. The eigenfunctions of both the FRFT and the LCT have been derived. In this paper, we extend the previous work and derive the eigenfunctions of the 2D NSLCT. Although the 2D NSLCT is very complicated and has 16 parameters, with the proposed methods, we can successfully find the eigenfunctions of the 2D NSLCT in all cases. Since many optical systems can be represented by the 2D NSLCT, our results are useful for analyzing the self-imaging phenomena of optical systems.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11-26 (2002).
    [CrossRef]
  15. T. Alieva and A. M. Barbe, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11-15 (1998).
    [CrossRef]
  16. G. B. Folland, Harmonic Analysis in Phase Space (Princeton University, 1989).
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    [CrossRef]
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2010 (3)

2008 (1)

2007 (2)

2006 (2)

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35-37 (2006).
[CrossRef] [PubMed]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

2002 (1)

S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11-26 (2002).
[CrossRef]

2001 (1)

S. C. Pei and J. J. Ding, “Two-dimensional affine generalized fractional Fourier transform,” IEEE Trans. Signal Process. 49, 878-897 (2001).
[CrossRef]

1998 (2)

T. Alieva and A. M. Barbe, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11-15 (1998).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD-Bessel transformation,” Opt. Commun. 147, 39-41 (1998).
[CrossRef]

1996 (2)

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207-212 (1996).
[CrossRef]

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732-740 (1996).
[CrossRef]

1995 (1)

1994 (2)

1990 (1)

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths. Appl. 25, 241-265 (1980).
[CrossRef]

1970 (1)

S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168-1177(1970).
[CrossRef]

Abe, S.

Agarwal, G. S.

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207-212 (1996).
[CrossRef]

Alieva, T.

Barbe, A. M.

T. Alieva and A. M. Barbe, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11-15 (1998).
[CrossRef]

Bastiaans, M. J.

Bernardo, L. M.

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732-740 (1996).
[CrossRef]

Calvo, M. L.

Collins, S. A.

S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168-1177(1970).
[CrossRef]

Ding, J. J.

S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11-26 (2002).
[CrossRef]

S. C. Pei and J. J. Ding, “Two-dimensional affine generalized fractional Fourier transform,” IEEE Trans. Signal Process. 49, 878-897 (2001).
[CrossRef]

S. C. Pei and J. J. Ding, “Matlab programs for generating the eigenfunctions of the 2-D non-separable linear canonical transform,” http://djj.ee.ntu.edu.tw/NSLCTevc.htm.

S. C. Pei and J. J. Ding, “Properties, digital implementation, applications, and self image phenomena of the Gyrator transform,” presented at the 17th European Signal Processing Conference, Glasgow, Scotland, 24-28 Aug. 2009.

Folland, G. B.

G. B. Folland, Harmonic Analysis in Phase Space (Princeton University, 1989).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Healy, J. J.

Hennelly, B. M.

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599-2601 (2008).
[CrossRef] [PubMed]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Hesselink, L.

James, D. F. V.

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207-212 (1996).
[CrossRef]

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Koç, A.

Kutay, M. A.

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35-37 (2006).
[CrossRef] [PubMed]

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley, 2000).

Lohmann, A. W.

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths. Appl. 25, 241-265 (1980).
[CrossRef]

Oktem, F. S.

Ozaktas, H. M.

Pei, S. C.

S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11-26 (2002).
[CrossRef]

S. C. Pei and J. J. Ding, “Two-dimensional affine generalized fractional Fourier transform,” IEEE Trans. Signal Process. 49, 878-897 (2001).
[CrossRef]

S. C. Pei and J. J. Ding, “Properties, digital implementation, applications, and self image phenomena of the Gyrator transform,” presented at the 17th European Signal Processing Conference, Glasgow, Scotland, 24-28 Aug. 2009.

S. C. Pei and J. J. Ding, “Matlab programs for generating the eigenfunctions of the 2-D non-separable linear canonical transform,” http://djj.ee.ntu.edu.tw/NSLCTevc.htm.

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. 45, 088201 (2006).
[CrossRef]

Rodrigo, J. A.

Sari, I.

Sheridan, J. T.

Thomas, J. A.

Wolf, K. B.

K. B. Wolf, “Canonical transforms,” in Integral Transforms in Science and Engineering, (Plenum, 1979), pp. 381-416.

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD-Bessel transformation,” Opt. Commun. 147, 39-41 (1998).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley, 2000).

Appl. Opt. (2)

IEEE Trans. Signal Process. (2)

S. C. Pei and J. J. Ding, “Two-dimensional affine generalized fractional Fourier transform,” IEEE Trans. Signal Process. 49, 878-897 (2001).
[CrossRef]

S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11-26 (2002).
[CrossRef]

J. Inst. Maths. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths. Appl. 25, 241-265 (1980).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (3)

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207-212 (1996).
[CrossRef]

T. Alieva and A. M. Barbe, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11-15 (1998).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD-Bessel transformation,” Opt. Commun. 147, 39-41 (1998).
[CrossRef]

Opt. Eng. (2)

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732-740 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Other (6)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

S. C. Pei and J. J. Ding, “Properties, digital implementation, applications, and self image phenomena of the Gyrator transform,” presented at the 17th European Signal Processing Conference, Glasgow, Scotland, 24-28 Aug. 2009.

S. C. Pei and J. J. Ding, “Matlab programs for generating the eigenfunctions of the 2-D non-separable linear canonical transform,” http://djj.ee.ntu.edu.tw/NSLCTevc.htm.

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley, 2000).

K. B. Wolf, “Canonical transforms,” in Integral Transforms in Science and Engineering, (Plenum, 1979), pp. 381-416.

G. B. Folland, Harmonic Analysis in Phase Space (Princeton University, 1989).

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Figures (4)

Fig. 1
Fig. 1

Relations between the 2D NSLCT and its special cases and their eigenfunctions.

Fig. 2
Fig. 2

Methods for deriving the eigenfunctions of the 2D NSLCT for different cases.

Fig. 3
Fig. 3

2D optical system with two lenses and three free spaces. The widths of the two lenses are t 1 ( x , y ) and t 2 ( x , y ) .

Fig. 4
Fig. 4

Two eigenfunctions of the 2D NSLCT corresponding to the optical system in Fig. 3. These eigenfunctions will cause the self-imaging phenomena for the optical system in Fig. 3.

Tables (1)

Tables Icon

Table 1 Classifying the Distributions of the Eigenvalues of the ABCD Matrix into Five Cases a

Equations (115)

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FT [ g ( x , y ) ] = 1 2 π exp [ j ( u x + v y ) ] g ( x , y ) d x d y .
O FRFT α , β [ g ( x , y ) ] = ( 1 j cot α 2 π ) 1 / 2 ( 1 j cot β 2 π ) 1 / 2 exp ( j 2 u 2 cot α j u x csc α + j 2 x 2 cot α ) exp ( j 2 v 2 cot β j v y csc β + j 2 y 2 cot β ) g ( x , y ) d x d y .
O LCT ( a x , b x , c x , d x , a y , b y , c y , d y ) [ g ( x , y ) ] = ( 1 j 2 π b x ) 1 / 2 ( 1 j 2 π b y ) 1 / 2 exp ( j 2 d x b x u 2 j u b x x + j 2 a x b x x 2 ) exp ( j 2 d y b y v 2 j v b y y + j 2 a y b y y 2 ) g ( x , y ) d x d y ,
a x d x b x c x = 1 and a y d y b y c y = 1
O NSLCT ( A , B , C , D ) [ g ( x , y ) ] = 1 2 π [ det ( B ) ] 1 / 2 exp [ j 2 det ( B ) ( k 1 u 2 + k 2 u v + k 3 v 2 ) ] exp { j det ( B ) [ ( b 22 u + b 12 v ) x + ( b 21 u b 11 v ) y + 1 2 ( p 1 x 2 + p 2 x y + p 3 y 2 ) ] } g ( x , y ) d x d y ,
k 1 = d 11 b 22 d 12 b 21 , k 2 = 2 ( d 11 b 12 + d 12 b 11 ) , k 3 = d 21 b 12 + d 22 b 11 , p 1 = a 11 b 22 a 21 b 12 , p 2 = 2 ( a 12 b 22 a 22 b 12 ) , p 3 = a 12 b 21 + a 22 b 11 ,
A = [ a 11 a 12 a 21 a 22 ] , B = [ b 11 b 12 b 21 b 22 ] , C = [ c 11 c 12 c 21 c 22 ] , D = [ d 11 d 12 d 21 d 22 ] .
O NSLCT ( A , B , C , D ) [ g ( x , y ) ] = [ det ( D ) ] 1 / 2 exp { j 2 [ ( c 11 d 11 + c 12 d 12 ) x 2 + 2 ( c 11 d 21 + c 12 d 22 ) x y + ( c 21 d 21 + c 22 d 22 ) y 2 ] } g ( d 11 x + d 21 y , d 12 x + d 22 y ) .
A T C = C T A , B T D = D T B , A T D C T B = I
AB T = BA T , CD T = DC T , AD T BC T = I .
a 11 c 12 + a 21 c 22 = a 12 c 11 + a 22 c 21 , b 11 d 12 + b 21 d 22 = b 12 d 11 + b 22 d 21 , a 11 d 11 + a 21 d 21 ( c 11 b 11 + c 21 b 21 ) = 1 , a 12 d 12 + a 22 d 22 ( c 12 b 12 + c 22 b 22 ) = 1 , a 11 d 12 + a 21 d 22 = c 11 b 12 + c 21 b 22 , a 12 d 11 + a 22 d 21 = c 12 b 11 + c 22 b 21 .
O NSLCT ( P , Q , R , S ) [ g ( x , y ) ] = O NSLCT ( A , B , C , D ) { O NSLCT ( A , B , C , D ) [ g ( x , y ) ] } ,
[ P Q R S ] = [ A B C D ] [ A B C D ] .
[ A B C D ]
ϕ m , n ( x , y ) = exp [ ( x 2 + y 2 ) / 2 ] H m ( x ) H n ( y ) ,
A exp ( i ρ x 2 2 σ 2 x 2 2 σ 2 ) H m ( x σ ) ,
σ 2 = 2 | b | [ 4 ( a + d ) 2 ] 1 / 2 , and ρ = sgn ( b ) ( a d ) [ 4 ( a + d ) 2 ] 1 / 2 .
exp ( j 2 ρ x 2 ) exp [ j 2 h ( x q ) 2 ] g ( q ) d q ,
O p [ f ( x , y ) ] = O S ( O Q { O S 1 [ f ( x , y ) ] } ) ,
O P { O S [ e ( x , y ) ] } = λ O S [ e ( x , y ) ] if O Q [ e ( x , y ) ] = λ e ( x , y ) .
[ A B C D ] = [ A S B S C S D S ] [ A Q B Q C Q D Q ] [ A S B S C S D S ] 1 ,
[ A 1 B 1 C 1 D 1 ] = [ I Φ 1 0 I ] [ I 0 E a I ] [ A B C D ] [ I 0 E a I ] [ I Φ 1 0 I ] ,
I = [ 1 0 0 1 ] , 0 = [ 0 0 0 0 ] , E a = [ η 1 η 2 η 2 η 3 ] , Φ 1 = [ τ 1 0 0 0 ] ,
η 1 = η 2 2 ( b 12 b 21 ) + η 2 ( a 11 + d 11 a 22 d 22 ) + η 3 ( d 12 + a 21 ) + c 12 c 21 η 3 ( b 12 b 21 ) + a 12 + d 21 , τ 1 = b 12 b 21 a 21 + d 12 + η 1 ( b 21 b 12 ) ,
η 3 ( b 12 b 21 ) + a 12 + d 21 0 , a 21 + d 12 + η 1 ( b 21 b 12 ) 0 .
[ A 1 B 1 C 1 D 1 ] = [ A + BE a Φ 1 C 0 B + ( A + BE a ) Φ 1 Φ 1 C 0 Φ 1 Φ 1 ( D E a B ) C 0 D E a B + C 0 Φ 1 ] ,
C 0 = E a A + C E a BE a + DE a .
C 1 = C 1 T and B 1 = B 1 T .
A 1 = E 2 V 1 E 2 1 ,
[ A 1 B 1 C 1 D 1 ] = [ E 2 0 0 ( E 2 T ) 1 ] [ A 2 B 2 C 2 D 2 ] [ E 2 1 0 0 E 2 T ] .
V 1 = [ σ 1 0 0 σ 2 ] .
B 2 = E 2 1 B 1 ( E 2 T ) 1 , B 2 T = E 2 1 B 1 T ( E 2 T ) 1 = E 2 1 B 1 ( E 2 T ) 1 = B 2 .
[ σ 1 B 2 ( 1 , 1 ) σ 1 B 2 ( 2 , 1 ) σ 2 B 2 ( 1 , 2 ) σ 2 B 2 ( 2 , 2 ) ] = [ σ 1 B 2 ( 1 , 1 ) σ 2 B 2 ( 1 , 2 ) σ 1 B 2 ( 2 , 1 ) σ 2 B 2 ( 2 , 2 ) ] ,
σ 2 B 2 ( 1 , 2 ) = σ 1 B 2 ( 2 , 1 ) = σ 1 B 2 ( 1 , 2 ) ( from B 2 T = B 2 ) .
[ A B C D ] = K 1 [ A 2 B 2 C 2 D 2 ] K 1 1 ,
K 1 = [ I 0 E a I ] [ I Φ 1 0 I ] [ E 2 0 0 ( E 2 T ) 1 ] = [ E 2 Φ 1 ( E 2 T ) 1 E a E 2 ( E a Φ 1 + I ) ( E 2 T ) 1 ] .
O s [ g ( μ , ρ ) ] = K exp [ j 2 ( η 1 x 2 + η 2 x y + η 3 y 2 ) ] exp [ j 2 τ 1 ( x μ ) 2 ] g ( ε 11 μ + ε 21 y , ε 12 μ + ε 22 y ) d μ ,
( E 2 T ) 1 = [ ε 11 ε 12 ε 21 ε 22 ] .
A 2 = [ a 1 0 0 a 2 ] , B 2 = [ b 1 0 0 b 2 ] , C 2 = [ c 1 0 0 c 2 ] , D 2 = [ d 1 0 0 d 2 ] .
O LCT ( a 1 , b 1 , c 1 , d 1 ) [ e 1 ( x ) ] = λ 1 e 1 ( x ) , O LCT ( a 2 , b 2 , c 2 , d 2 ) [ e 2 ( x ) ] = λ 2 e 2 ( x ) ,
E ( x , y ) = K exp [ j 2 ( η 1 x 2 + η 2 x y + η 3 y 2 ) ] exp [ j 2 τ 1 ( x μ ) 2 ] e 1 ( ε 11 μ + ε 21 y ) e 2 ( ε 12 μ + ε 22 y ) d μ
O AGFFT ( A , B , C , D ) [ E ( x , y ) ] = λ 1 λ 2 E ( x , y ) .
a 12 + d 21 0 or b 12 b 21 0 .
b 12 b 21 = 0 and a 12 + d 12 = 0 but c 12 c 21 0 or a 21 + d 12 = 0 ,
[ A 1 B 1 C 1 D 1 ] = [ I 0 E b I ] [ I Φ b 0 I ] [ A B C D ] [ I Φ b 0 I ] [ I 0 E b I ] ,
Φ b = [ τ 4 τ 5 τ 5 τ 6 ] , E b = [ η 0 0 0 ] , E b = [ η 0 0 0 ] , η = c 12 c 21 a 12 + d 21 + τ 4 ( c 21 c 12 ) ,
τ 4 = τ 5 2 ( c 12 c 21 ) + τ 5 ( a 11 + d 11 a 22 d 22 ) + τ 6 ( a 12 + d 21 ) + b 12 b 21 a 21 + d 12 + τ 6 ( c 12 c 21 ) .
E ( x , y ) = exp { j 2 ( τ 4 τ 6 τ 5 2 ) [ τ 6 ( x μ ) 2 + τ 4 ( y ρ ) 2 2 τ 5 ( x μ ) ( y ρ ) ] } exp ( j 2 η μ 2 ) e 1 ( ε 11 μ + ε 21 ρ ) e 2 ( ε 12 μ + ε 22 ρ ) d μ d ρ ,
b 12 b 21 = a 12 + d 21 = c 12 c 21 = a 21 + d 12 = 0 ,
[ A 1 B 1 C 1 D 1 ] = [ A B C D ]
E ( x , y ) = e 1 ( ε 11 x + ε 21 y ) e 2 ( ε 12 x + ε 22 y ) ,
[ A Q B Q C Q D Q ] = [ A 1 0 C 1 ( A 1 T ) 1 ] .
[ A 0 B 0 C 0 D 0 ] = [ I 0 E a I ] [ A B C D ] [ I 0 E a I ] , where E a = [ η 1 η 2 η 2 η 3 ] , η 1 = η 2 2 ( b 12 b 21 ) + η 2 ( a 11 + d 11 a 22 d 22 ) + η 3 ( d 12 + a 21 ) + c 12 c 21 η 3 ( b 12 b 21 ) + a 12 + d 21 .
η 3 ( b 12 b 21 ) + a 12 + d 21 0 and det ( C 0 ) 0
C 0 = C 0 T .
[ A 1 B 1 C 1 D 1 ] = [ I Φ 0 0 I ] [ A 0 B 0 C 0 D 0 ] = [ I Φ 0 0 I ] = [ A 0 Φ 0 C 0 A 0 Φ 0 Φ 0 C 0 Φ 0 + B Φ 0 D 0 C 0 D 0 + C 0 Φ 0 ] ,
Φ 0 = [ τ 1 τ 2 τ 2 τ 3 ] .
A 0 + D 0 T = E 2 W 1 E 2 1 .
E 2 ( 1 , 2 ) = E 2 * ( 1 , 1 ) , E 2 ( 2 , 2 ) = E 2 * ( 2 , 1 ) .
U 1 = [ ϕ 1 0 0 ϕ 2 ] , where ϕ 1 = ψ 1 ± ( ψ 1 2 4 ) 1 / 2 2 , ϕ 2 = ψ 2 ± ( ψ 2 2 4 ) 1 / 2 2 .
W 1 = U 1 + U 1 1
Φ 0 = C 0 1 [ ( E 2 1 ) T U 1 1 E 2 T D 0 ] .
C 0 D 0 T = D 0 C 0 T , and C 1 D 1 T = D 1 C 1 T where D 1 = D 0 + C 0 Φ 0
D 0 T ( C 0 T ) 1 = C 0 1 D 0 , D 1 T ( C 1 T ) 1 = C 1 1 D 1 , Φ 0 = C 0 1 ( D 1 D 0 ) , Φ 0 T = [ D 1 T D 0 T ] ( C 0 1 ) T = C 0 1 D 1 C 0 1 D 0 = Φ 0 .
D 1 = D 0 + C 0 C 0 1 [ ( E 2 1 ) T U 1 1 E 2 T D 0 ] = ( E 2 1 ) T U 1 1 E 2 T .
A 1 T = A 0 T C 0 T Φ 0 T = A 0 T C 0 Φ 0 = A 0 T C 0 C 0 1 [ ( E 2 1 ) T U 1 1 E 2 T D 0 ] = A 0 T + D 0 ( E 2 1 ) T U 1 1 E 2 T = ( E 2 1 ) T W 1 E 2 T ( E 2 1 ) T U 1 1 E 2 T = ( E 2 1 ) T ( U 1 + U 1 1 ) E 2 T ( E 2 1 ) T U 1 1 E 2 T = ( E 2 1 ) T U 1 E 2 T ,
A 1 T D 1 = ( E 2 1 ) T U 1 E 2 T ( E 2 1 ) T U 1 1 E 2 T = I .
C 0 T B 1 = 0 ( since C 1 = C 0 ) , B 1 = ( C 0 T ) 1 0 = 0 .
[ A 2 0 C 2 D 2 ] = [ I 0 E 3 I ] [ A 1 0 C 1 ( A 1 T ) 1 ] [ I 0 E 3 I ] = [ A 1 0 C 1 E 3 A 1 + ( A 1 T ) 1 E 3 ( A 1 T ) 1 ] ,
E 3 = [ η 4 η 5 η 5 η 6 ] and C 2 = C 1 E 3 A 1 + D 1 E 3 = 0
A 1 ( 1 , 1 ) η 4 + A 1 ( 2 , 1 ) η 5 A 1 ( 2 , 2 ) det ( A 1 ) η 4 + A 1 ( 2 , 1 ) det ( A 2 ) η 5 = C 1 ( 1 , 1 ) , A 1 ( 1 , 2 ) η 4 + A 1 ( 2 , 2 ) η 5 A 1 ( 2 , 2 ) det ( A 1 ) η 5 + A 1 ( 2 , 1 ) det ( A 2 ) η 6 = C 1 ( 1 , 2 ) , A 1 ( 1 , 2 ) det ( A 1 ) η 4 + A 1 ( 1 , 1 ) η 5 A 1 ( 1 , 1 ) det ( A 1 ) η 5 + A 1 ( 2 , 1 ) η 6 = C 1 ( 2 , 1 ) , A ( 1 , 2 ) η 5 + A ( 2 , 2 ) η 6 + A 1 ( 1 , 2 ) det ( A 1 ) η 5 A 1 ( 1 , 1 ) det ( A 1 ) η 6 = C 1 ( 2 , 2 ) .
[ A 2 0 C 2 D 2 ] = [ A 1 0 0 ( A 1 T ) 1 ] .
A 1 = E 4 W 2 E 4 1 , [ A 1 0 0 ( A 1 T ) 1 ] = [ E 4 0 0 ( E 4 T ) 1 ] [ W 2 0 0 W 2 1 ] [ E 4 1 0 0 E 4 T ] .
( E 4 T ) 1 = [ ε 11 ε 12 ε 21 ε 22 ] , W 2 = [ κ 1 0 0 κ 2 ] ,
q ( ε 11 x + ε 21 y , ε 12 x + ε 22 y ) ,
q ( κ 1 1 x , κ 2 1 y ) = λ q ( x , y )
[ A B C D ] = [ I 0 E 1 I ] [ I Φ 0 0 I ] [ I 0 E 3 I ] [ A 1 0 0 ( A 1 T ) 1 ] [ I 0 E 3 I ] [ I Φ 0 0 I ] [ I 0 E 1 I ] ,
E ( x , y ) = exp [ j 2 ( η 1 x 2 + 2 η 2 x y + η 3 y 2 ) ] × exp { j τ 1 τ 3 τ 2 2 [ τ 3 2 ( x μ ) 2 + τ 1 2 ( y ρ ) 2 τ 2 ( x μ ) ( y ρ ) ] } exp [ j ( η 4 μ 2 + 2 η 5 μ ρ + η 6 ρ 2 ) / 2 ] q ( ε 11 μ + ε 21 ρ , ε 12 μ + ε 22 ρ ) d μ d ρ ,
A 1 = E 4 Γ 1 Γ W 2 Γ 1 Γ E 4 1 = E 5 W 3 E 5 1 , where Γ = [ 1 + j 2 1 j 2 1 j 2 1 + j 2 ] ,
W 3 = [ κ 1 + κ 1 * 2 j κ 1 κ 1 * 2 j κ 1 * κ 1 2 κ 1 + κ 1 * 2 ] , E 5 = [ E 4 ( 1 , 1 ) 1 j 2 + E 4 * ( 1 , 1 ) 1 + j 2 E 4 ( 1 , 1 ) 1 + j 2 + E 4 * ( 1 , 1 ) 1 j 2 E 4 ( 2 , 1 ) 1 j 2 + E 4 * ( 2 , 1 ) 1 + j 2 E 4 ( 2 , 1 ) 1 + j 2 + E 4 * ( 2 , 1 ) 1 j 2 ] ,
[ A 1 0 0 ( A 1 T ) 1 ] = [ E 5 0 0 ( E 5 T ) 1 ] [ W 3 0 0 ( W 3 T ) 1 ] [ E 5 1 0 0 E 5 T ] .
W 3 = [ σ cos ϕ σ sin ϕ σ sin ϕ σ cos ϕ ] , where σ = | κ 1 | , ϕ = cos 1 ( κ 1 κ 1 * 2 σ ) = sin 1 ( j κ 1 κ 1 * 2 σ ) .
O NSLCT [ W 3 , 0 , 0 , ( W 3 T ) 1 ] [ g ( x , y ) ] = | σ | 1 g ( x σ 1 cos ϕ y σ 1 sin ϕ , x σ 1 sin ϕ + y σ 1 cos ϕ ) .
e ( r cos θ 1 , r sin θ 1 ) = e ( r cos θ 2 , r sin θ 2 ) for any θ 1 , θ 2 ,
e ( κ 1 x , κ 1 y ) = λ e ( x , y ) , λ is some constant ,
e ( ζ 11 x + ζ 21 y , ζ 12 x + ζ 22 y ) , where ( E 5 T ) 1 = [ ζ 11 ζ 12 ζ 21 ζ 22 ] ,
E ( x , y ) = exp [ j 2 ( η 1 x 2 + 2 η 2 x y + η 3 y 2 ) ] exp { j τ 1 τ 3 τ 2 2 [ τ 3 2 ( x μ ) 2 + τ 1 2 ( y ρ ) 2 τ 2 ( x μ ) ( y ρ ) ] } exp [ j ( η 4 μ 2 + 2 η 5 μ ρ + η 6 ρ 2 ) / 2 ] e ( ζ 11 μ + ζ 21 ρ , ζ 12 μ + ζ 22 ρ ) d μ d ρ ,
O NSLCT ( A 1 , 0 , C 1 , ( A 1 T ) 1 ) [ g ( x , y ) ] = [ det ( A 1 ) ] 1 2 exp [ j ( q 1 x 2 + q 2 x y + q 3 y 2 ) / 2 ] g ( σ 11 x + σ 21 y , σ 12 x + σ 22 y ) ,
( A 1 T ) 1 = [ σ 11 σ 12 σ 21 σ 22 ] , q 1 = C 1 [ 1 , 1 ] σ 11 + C 1 [ 1 , 2 ] σ 12 , q 2 = 2 C 1 [ 1 , 1 ] σ 21 + 2 C 1 [ 1 , 2 ] σ 22 , q 3 = C 1 [ 2 , 1 ] σ 21 + C 1 [ 2 , 2 ] σ 22 .
p ( x , y ) = n = exp ( j ϕ n ) δ ( x x n ) δ ( y y n ) ,
[ x n y n ] = A 1 n [ x 0 y 0 ] , ϕ n = ϕ 0 + n ψ + m = 1 n [ x 0 y 0 ] ( A 1 T ) m C 1 A 1 m 1 [ x 0 y 0 ] ,
[ A B C D ] = [ I 0 E 1 I ] [ I Φ 0 0 I ] [ A 1 0 0 ( A 1 T ) 1 ] [ I 0 E 3 I ] [ I Φ 0 0 I ] ,
E ( x , y ) = exp [ j 2 ( η 1 x 2 + 2 η 2 x y + η 3 y 2 ) ] exp { j τ 1 τ 3 τ 2 2 [ τ 3 2 ( x μ ) 2 + τ 1 2 ( y ρ ) 2 τ 2 ( x μ ) ( y ρ ) ] } p ( μ , ρ ) d μ d ρ ,
a 12 + d 21 0 or b 12 b 21 0 .
b 12 b 21 = a 12 + d 21 = 0 but c 12 c 21 0 or a 21 + d 12 0 ,
[ A 0 B 0 C 0 D 0 ] = [ I 0 E a I ] [ 0 I I 0 ] [ A B C D ] [ 0 I I 0 ] [ I 0 E a I ] = [ I 0 E a I ] [ D C B A ] [ I 0 E a I ] ,
E a = [ η 1 η 2 η 2 η 3 ] , η 1 = η 2 2 ( c 21 c 12 ) + η 2 ( d 11 + a 11 d 22 a 22 ) + η 3 ( a 12 + d 21 ) + b 21 b 12 η 3 ( c 21 c 12 ) + d 12 + a 21 ,
E ( x , y ) = exp [ j ( x p + y q ) ] exp [ j ( η 1 p 2 + 2 η 2 p q + η 3 q 2 ) / 2 ] exp { j τ 1 τ 3 τ 2 2 [ τ 3 2 ( p μ ) 2 + τ 1 2 ( q ρ ) 2 τ 2 ( p μ ) ( q ρ ) ] } exp [ j 2 ( η 4 μ 2 + 2 η 5 μ ρ + η 6 ρ 2 ) ] q ( ε 11 μ + ε 21 ρ , ε 12 μ + ε 22 ρ ) d μ d ρ d p d q .
η 2 = 0 , η 1 and η 3 are free to be chosen, but det ( C 0 ) 0 should be satisfied .
[ A B C D ] K 1 [ A 2 B 2 C 2 D 2 ] K 1 1 ,
L 1 = [ a 1 b 1 c 1 d 1 ] , L 2 = [ a 2 b 2 c 2 d 2 ] .
λ 2 = λ 1 1 , λ 4 = λ 3 1 .
λ 2 = λ 1 * , λ 4 = λ 3 * .
[ A B C D ] = K 1 [ A 1 0 C 1 ( A 1 T ) 1 ] K 1 1 .
[ I Z 0 I ] , where Z = [ ( 2 π ) 1 λ z 0 0 ( 2 π ) 1 λ z ] ,
t ( x , y ) = p 1 x 2 + p 2 x y + p 3 y 2 + T ,
[ I 0 Φ I ] , where Φ = [ 4 π λ ( n 1 ) p 1 2 π λ ( n 1 ) p 2 2 π λ ( n 1 ) p 2 4 π λ ( n 1 ) p 3 ]
[ A B C D ] = [ A K B K C K D K ] [ A 2 B 2 C 2 D 2 ] [ A 1 B 1 C 1 D 1 ] .
z 1 = 5 × 10 3 mm , z 2 = 10 4 mm , z 3 = 5 × 10 3 mm , λ = 5 × 10 5 mm , n = 1.6 , t 1 ( x , y ) = T 1 4 × 10 6 ( x 2 y ) 2 6 × 10 6 ( x + y ) 2 , t 2 ( x , y ) = T 2 3 × 10 6 x 2 4 × 10 6 ( x y ) 2 ,
[ A B C D ] = [ 0 . 7836 0 . 0527 0 . 1472 0 . 0040 0 . 0565 0 . 5869 0 . 0042 0 . 1408 2 . 4224 0 . 7202 0 . 8196 0 . 0767 0 . 8179 3 . 7470 0 . 0805 0 . 8029 ] .
λ 1 = 0.8336 + j 0.5524 , λ 2 = 0.8336 j 0.5524 , λ 3 = 0.6630 + j 0.7487 , λ 4 = 0.6630 j 0.7487 .
η 1 = 0.7336 , η 2 = η 3 = 0 , τ 1 = 0.0012 , [ ε 11 ε 12 ε 21 ε 22 ] = [ 0.9014 0.4326 0.4329 0.9016 ] , [ a 1 b 1 c 1 d 1 ] = [ 0.6984 0.1494 2.1642 0.9688 ] , [ a 2 b 2 c 2 d 2 ] = [ 0.5612 0.1389 4.1108 0.7647 ] .
E ( x , y ) = exp ( j 0.3668 x 2 ) exp [ j ( x τ ) 2 0.0023 ] e 1 ( 0 . 9014 τ + 0 . 4329 y ) e 2 ( 0 . 4326 τ + 0 . 9016 y ) d τ ,
E ( x , y ) = exp ( 0.3668 x 2 ) exp [ j 0.0023 ( x τ ) 2 ] exp ( 0.4999 τ 2 0.0001 τ y 0.5001 y 2 ) H 2 ( 0 . 9014 τ + 0 . 4329 y ) H 1 ( 0 . 4326 τ + 0 . 9016 y ) d τ .
n 2 ( x , y ) = n 0 2 [ 1 n 1 n 0 ( g 1 x + g 2 y ) 2 n 2 n 0 ( g 3 x + g 4 y ) 2 ]

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