Abstract

We provide a general expression and different classification schemes for the general two-dimensional canonical integral transformations that describe the propagation of coherent light through lossless first-order optical systems. Main theorems for these transformations, such as shift, scaling, derivation, etc., together with the canonical integral transforms of selected functions, are derived.

© 2007 Optical Society of America

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References

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  6. T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional transforms in optical information processing," EURASIP (European Association for Signal and Image Processing) J. Appl. Signal Process. 2005: 10, 1498-1519 (2005).
    [CrossRef]
  7. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  8. E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
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  9. R. Simon and K. B. Wolf, "Fractional Fourier transforms in two dimensions," J. Opt. Soc. Am. A 17, 2368-2381 (2000).
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  16. M. J. Bastiaans and T. Alieva, "Classification of lossless first-order optical systems and the linear canonical transformation," J. Opt. Soc. Am. A 24, 1053-1062 (2007).
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  17. T. Alieva and M. J. Bastiaans, "Mode mapping in paraxial lossless optics," Opt. Lett. 30, 1461-1463 (2005).
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  18. M. J. Bastiaans and T. Alieva, "First-order optical systems with unimodular eigenvalues," J. Opt. Soc. Am. A 23, 1875-1883 (2006).
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  19. M. J. Bastiaans and T. Alieva, "First-order optical systems with real eigenvalues," Opt. Commun. 272, 52-55 (2007).
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  20. A. W. Lohmann, Z. Zalevsky, and D. Mendlovic, "Synthesis of pattern recognition filters for fractional Fourier processing," Opt. Commun. 128, 199-204 (1996).
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  21. O. Akay and G. F. Boudreaux-Bartels, "Fractional convolution and correlation via operator methods and an application to detection of linear FM signals," IEEE Trans. Signal Process. 49, 979-993 (2001).
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2007 (2)

2006 (3)

2005 (4)

2004 (1)

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[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 (2)

O. Akay and G. F. Boudreaux-Bartels, "Fractional convolution and correlation via operator methods and an application to detection of linear FM signals," IEEE Trans. Signal Process. 49, 979-993 (2001).
[CrossRef]

B. Zhu and S. Liu, "Optical image encryption based on the generalized fractional convolution operation," Opt. Commun. 195, 371-381 (2001).
[CrossRef]

2000 (1)

1996 (2)

1993 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

1982 (1)

1971 (1)

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

1970 (1)

Appl. Opt. (1)

IEEE Trans. Signal Process. (2)

O. Akay and G. F. Boudreaux-Bartels, "Fractional convolution and correlation via operator methods and an application to detection of linear FM signals," IEEE Trans. Signal Process. 49, 979-993 (2001).
[CrossRef]

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

J. Appl. Signal Process. (1)

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional transforms in optical information processing," EURASIP (European Association for Signal and Image Processing) J. Appl. Signal Process. 2005: 10, 1498-1519 (2005).
[CrossRef]

J. Math. Phys. (1)

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

B. Zhu and S. Liu, "Optical image encryption based on the generalized fractional convolution operation," Opt. Commun. 195, 371-381 (2001).
[CrossRef]

M. J. Bastiaans and T. Alieva, "First-order optical systems with real eigenvalues," Opt. Commun. 272, 52-55 (2007).
[CrossRef]

A. W. Lohmann, Z. Zalevsky, and D. Mendlovic, "Synthesis of pattern recognition filters for fractional Fourier processing," Opt. Commun. 128, 199-204 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Other (3)

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

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

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Equations (88)

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f i ( r i ) f o ( r o ) = R T [ f i ( r i ) ] ( r o ) .
( r o q o ) = [ A B C D ] ( r i q i ) = T ( r i q i ) ,
[ A B C D ] 1 = [ D t B t C t A t ] ,
AB t = BA t , CD t = DC t , AD t BC t = I ;
A t C = C t A , B t D = D t B , A t D C t B = I ;
[ A B C D ] = [ I 0 G I ] [ S 0 0 S 1 ] [ X Y Y X ] ,
G = ( CA t + DB t ) ( AA t + BB t ) 1 = G t ,
S = ( AA t + BB t ) 1 2 = S t ,
U = X + i Y = ( AA t + BB t ) 1 2 ( A + i B ) .
f o ( r o ) = R T [ f i ( r i ) ] ( r o ) = ( det i B ) 1 2 f i ( r i ) exp [ i π ( r i t B 1 Ar i 2 r i t B 1 r o + r o t DB 1 r o ) ] d r i .
T ̂ = [ A B C D ] 1 ,
f o ( r ) = ( det A ) 1 2 exp ( i π r t CA 1 r ) f i ( A 1 r ) .
F [ f ( r ) ] ( q ) = f ( r ) exp ( i 2 π q t r ) d r .
( T t ) 1 = [ D C B A ] .
W f ( r , q ) = f ( r + r 2 ) f * ( r r 2 ) exp ( i 2 π q t r ) d r .
W f ( r , q ) = W R T [ f ( ) ] ( Ar + Bq , Cr + Dq ) ,
A f ( r , q ) = f ( r + r 2 ) f * ( r r 2 ) exp ( i 2 π r t q ) d r ,
A f ( r , q ) = A R T [ f ( ) ] ( Ar + Bq , Cr + Dq ) .
F [ f o ( r ) ] ( q ) = ( det D ) 1 2 exp ( i π q t BD 1 q ) F [ f i ( r ) ] ( D 1 q ) = ( det A ) 1 2 exp ( i π q t BA t q ) F [ f i ( r ) ] ( A t q ) .
A f = D f = [ cos γ x 0 0 cos γ y ] X f ( γ x , γ y ) ,
B f = C f = [ sin γ x 0 0 sin γ y ] Y f ( γ x , γ y ) ,
A r = D r = [ cos α sin α sin α cos α ] X r ( α ) ,
B r = C r = 0 Y r ( α ) ,
A g = D g = [ cos ϑ 0 0 cos ϑ ] X g ( ϑ ) ,
B g = C g = [ 0 sin ϑ sin ϑ 0 ] Y g ( ϑ ) ,
A s = [ 1 u 0 1 ] , D s = [ 1 0 u 1 ] , B s = C s = 0 ,
S s = [ ( 1 + sin 2 θ ) cos θ sin θ sin θ cos θ ]
f o ( r o ) = ( det S ) 1 2 exp ( i π r o t Gr o ) R T f ( γ x , γ y ) [ f i ( X r ( α ) r i ) ] [ X r ( β ) S 1 r o ] .
T 2 T 1 : R T 2 [ R T 1 [ f ( ) ] ( ) ] ( r ) = R T 2 T 1 [ f ( ) ] ( r ) .
{ R T [ f ( ) ] ( r ) } * = R T ̂ 1 [ f * ( ) ] ( r ) ,
f ( r i ) g * ( r i ) d r i = R T [ f ( r i ) ] ( r o ) { R T [ g ( r i ) ] ( r o ) } * d r o
f ( r i ) 2 d r i = R T [ f ( r i ) ] ( r o ) 2 d r o .
R T [ f i ( r i v ) ] ( r o ) = exp [ i π ( 2 r o Av ) t Cv ] R T [ f i ( r i ) ] ( r o Av ) ,
R T [ f ( r i v ) ] ( r o ) = F T ( r o Av ) .
R T [ f ( r i v ) ] ( r o ) = F T ( r o Av )
R T [ f ( r i v ) ] ( r o ) = exp ( i π 2 r o t Cv ) F T ( r o ) .
( f h ) ( r ) = f ( r v ) h ( v ) d v = h ( r v ) f ( v ) d v
R T [ ( f h ) ( r i ) ] ( r o ) = exp [ i π ( 2 r o Av ) t Cv ] F T ( r o Av ) h ( v ) d v = exp [ i π ( 2 r o Av ) t Cv ] H T ( r o Av ) f ( v ) d v .
R T [ ( f h ) ( r i ) ] ( r o ) = ( det i B ) 1 2 exp ( i π r o t DB 1 r o ) F T ( r o ) H T ( r o ) ,
( f 1 f 2 ) ( r ) = F 1 { F [ f 1 ( ) ] ( u ) F [ f 2 ( ) ] ( u ) } ( r ) ,
R T 3 [ R T 1 [ f 1 ( ) ] ( u ) R T 2 [ f 2 ( ) ] ( u ) ] ( r ) ,
T ̃ = T [ W 1 0 0 W t ] .
R T [ ( det W ) 1 2 f ( Wr i ) ] ( r o ) = ( det S ) 1 2 R T [ f ( r i ) ] ( Sr o ) .
T [ W 1 0 0 W t ] = [ S 1 0 0 S t ] T ,
[ AW 1 BW t CW 1 DW t ] = [ S 1 A S 1 B S t C S t D ] ,
W = [ w x 0 0 w y ] , S = [ s x 0 0 s y ] .
T 1 = [ A 1 0 0 D 1 ] ,
T 2 = [ 0 B 2 C 2 0 ] ,
T 1 = [ S 1 X r ( α ) 0 0 S 1 1 X r ( α ) ] ,
T 2 = [ 0 S 2 X r ( α ) S 2 1 X r ( α ) 0 ] ,
S 3 = [ s 11 0 0 s 22 ] , G 3 = [ 0 g g 0 ] .
A 3 = S 3 X g ( ϑ ) ,
B 3 = S 3 Y g ( ϑ ) ,
C 3 = G 3 S 3 X g ( ϑ ) S 3 1 Y g ( ϑ ) ,
D 3 = G 3 S 3 Y g ( ϑ ) + S 3 1 X g ( ϑ ) .
A 4 = S 4 Y g ( ϑ ) ,
B 4 = S 4 X g ( ϑ ) ,
C 4 = G 4 S 4 Y g ( ϑ ) S 4 1 X g ( ϑ ) ,
D 4 = G 4 S 4 X g ( ϑ ) S 4 1 Y g ( ϑ ) ,
T 5 = [ a 11 a 12 0 0 0 0 b 21 b 22 0 0 d 11 d 12 c 21 c 22 0 0 ] ,
o f o ( r o ) = o { R T [ f i ( r i ) ] ( r o ) } = i 2 π ( B t ) 1 { D t r o f o ( r o ) R T [ r i f i ( r i ) ] ( r o ) } ,
R T [ r i f i ( r i ) ] ( r o ) = { D t r o B t i 2 π o } f o ( r o ) .
R T 1 [ o f o ( r o ) ] ( r i ) = i 2 π ( B t ) 1 { D t R T 1 [ r o f o ( r o ) ] ( r i ) r i f i ( r i ) } ,
R T [ i f i ( r i ) ] ( r o ) = i 2 π ( B ) 1 { A R T [ r i f i ( r i ) ] ( r o ) r o f o ( r o ) } .
R T [ i f i ( r i ) ] ( r o ) = ( i 2 π C t r o + A t o ) f o ( r o ) .
R T [ z i f i ( r i ) ] ( r o ) = z o f o ( r o ) ,
z i = ( i 2 π r i i ) , z o = T 1 ( i 2 π r o o ) .
R T [ z i { k l m n } f i ( r i ) ] ( r o ) = z o { k l m n } f o ( r o ) .
R T [ i { m n } f i ( r i ) ] ( r o ) = ( i 2 π C t r o ) { m n } R T [ f i ( ) ] ( r o ) ,
R T [ i { m n } f i ( r i ) ] ( r o ) = ( A t o ) { m n } R T [ f i ( r i ) ] ( r o ) .
f i ( r ) = exp ( i 2 π k i t r π r t L i r ) ,
f o ( r ) = R T [ f i ( ) ] ( r ) = [ det ( A + i BL i ) ] 1 2 exp [ i π k i t ( A + i BL i ) 1 Bk i + i 2 π k o t r π r t L o r ] ,
k o t = k i t ( A + i BL i ) 1 ,
i L o = ( C + i DL i ) ( A + i BL i ) 1 .
f o ( r ) = ( det A ) 1 2 exp ( i π k i t A 1 Bk i + i 2 π k i t A 1 r + i π r t CA 1 r ) .
f o ( r ) = [ det ( A + i BL i ) ] 1 2 exp ( π r t L o r ) ,
H o = ( C + DH i ) ( A + BH i ) 1 ,
R T [ δ ( r i v ) ] ( r o ) = ( det i B ) 1 2 exp [ i π ( v t B 1 Av 2 v t B 1 r o + r o t DB 1 r o ) ] .
f i ( r ) = m , n = a m n exp ( i 2 π k m n t r ) ,
f o ( r ) = ( det A ) 1 2 m , n = a m n exp ( i π k m n t A 1 Bk m n + i 2 π k m n t A 1 r + i π r t CA 1 r ) .
k m n t A 1 Bk m n = 2 l ,
f o ( r ) = ( det A ) 1 2 exp ( i π r t CA 1 r ) f i ( A 1 r ) ,
q 11 p x 2 m 2 + 2 q 12 p x p y m n + q 22 p y 2 n 2 = 2 l ,
[ cos 2 α tan γ x + sin 2 α tan γ y cos α sin α ( tan γ x tan γ y ) cos α sin α ( tan γ x tan γ y ) sin 2 α tan γ x + cos 2 α tan γ y ] .
( m p x ) 2 ( cos 2 α tan γ x + sin 2 α tan γ y ) + m m p x p y sin 2 α ( tan γ x tan γ y ) + ( n p y ) 2 ( sin 2 α tan γ x + cos 2 α tan γ y ) = 2 l .
tan γ [ ( m p x ) 2 + ( n p y ) 2 ] = 2 l
cos 2 α tan γ [ ( m p x ) 2 ( n p y ) 2 ] + m n p x p y 2 sin 2 α tan γ = 2 l .
Q f = [ tan γ x 0 0 tan γ y ] , Q g = [ 0 tan ϑ tan ϑ 0 ] .

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