Abstract

It is shown that a lossless first-order optical system whose real symplectic ray transformation matrix can be diagonalized and has only unimodular eigenvalues is similar to a separable fractional Fourier transformer in the sense that the ray transformation matrices of the unimodular system and the separable fractional Fourier transformer are related by means of a similarity transformation. Moreover, it is shown that the system that performs this similarity transformation is itself a lossless first-order optical system. Based on the fact that Hermite–Gauss functions are the eigenfunctions of a fractional Fourier transformer, the eigenfunctions of a unimodular first-order optical system can be formulated and belong to the recently introduced class of orthonormal Hermite–Gaussian-type modes. Two decompositions of a unimodular first-order optical system are considered, and one of them is used to derive an easy optical realization in more detail.

© 2006 Optical Society of America

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References

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  1. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in optics," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (North-Holland, 1998), pp. 263-342.
    [CrossRef]
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    [CrossRef]
  3. T. Alieva and M. L. Calvo, "Fractionalization of the linear cyclic transforms," J. Opt. Soc. Am. A 17, 2330-2338 (2000).
    [CrossRef]
  4. T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional cyclic transforms in optics: theory and applications," in Recent Research and Developments in Optics, S.G.Pandalai, ed. (Research Signpost, 2001), Vol. 1, pp. 105-122.
  5. T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractionalization of cyclic transformations in optics," in Proceedings of ICOL 2005, the International Conference on Optics and Optoelectronics, Dehradun, India, December 12-15, 2005, Paper IT-OIP-2, pp. 1-15.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
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    [PubMed]
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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]
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    [CrossRef]
  22. T. Alieva and M. J. Bastiaans, "Mode mapping in paraxial lossless optics," Opt. Lett. 30, 1461-1463 (2005).
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  24. T. Alieva and M. J. Bastiaans, "Alternative representation of the linear canonical integral transform," Opt. Lett. 30, 3302-3304 (2005).
    [CrossRef]

2005

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

M. J. Bastiaans and T. Alieva, "Generating function for Hermite-Gaussian modes propagating through first-order optical systems," J. Phys. A 38, L73-L78 (2005).
[CrossRef]

T. Alieva and M. J. Bastiaans, "Mode mapping in paraxial lossless optics," Opt. Lett. 30, 1461-1463 (2005).
[CrossRef] [PubMed]

T. Alieva and M. J. Bastiaans, "Alternative representation of the linear canonical integral transform," Opt. Lett. 30, 3302-3304 (2005).
[CrossRef]

2000

1998

1993

1987

A. C. McBride and F. H. Kerr, "On Namias' fractional Fourier transforms," IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

1980

V. Namias, "The fractional order Fourier transform and its applications to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

1971

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

1970

1961

V. Bargmann, "On a Hilbert space of analytic functions and an associated integral transform, Part I," Commun. Pure Appl. Math. 14, 187-214 (1961).
[CrossRef]

1958

I. E. Segal, "Distributions in Hilbert spaces and canonical systems of operators," Trans. Am. Math. Soc. 88, 12-42 (1958).
[CrossRef]

Alieva, T.

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

M. J. Bastiaans and T. Alieva, "Generating function for Hermite-Gaussian modes propagating through first-order optical systems," J. Phys. A 38, L73-L78 (2005).
[CrossRef]

T. Alieva and M. J. Bastiaans, "Mode mapping in paraxial lossless optics," Opt. Lett. 30, 1461-1463 (2005).
[CrossRef] [PubMed]

T. Alieva and M. J. Bastiaans, "Alternative representation of the linear canonical integral transform," Opt. Lett. 30, 3302-3304 (2005).
[CrossRef]

T. Alieva and M. L. Calvo, "Fractionalization of the linear cyclic transforms," J. Opt. Soc. Am. A 17, 2330-2338 (2000).
[CrossRef]

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional cyclic transforms in optics: theory and applications," in Recent Research and Developments in Optics, S.G.Pandalai, ed. (Research Signpost, 2001), Vol. 1, pp. 105-122.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractionalization of cyclic transformations in optics," in Proceedings of ICOL 2005, the International Conference on Optics and Optoelectronics, Dehradun, India, December 12-15, 2005, Paper IT-OIP-2, pp. 1-15.

Bargmann, V.

V. Bargmann, "On a Hilbert space of analytic functions and an associated integral transform, Part I," Commun. Pure Appl. Math. 14, 187-214 (1961).
[CrossRef]

Bastiaans, M. J.

T. Alieva and M. J. Bastiaans, "Mode mapping in paraxial lossless optics," Opt. Lett. 30, 1461-1463 (2005).
[CrossRef] [PubMed]

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

T. Alieva and M. J. Bastiaans, "Alternative representation of the linear canonical integral transform," Opt. Lett. 30, 3302-3304 (2005).
[CrossRef]

M. J. Bastiaans and T. Alieva, "Generating function for Hermite-Gaussian modes propagating through first-order optical systems," J. Phys. A 38, L73-L78 (2005).
[CrossRef]

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional cyclic transforms in optics: theory and applications," in Recent Research and Developments in Optics, S.G.Pandalai, ed. (Research Signpost, 2001), Vol. 1, pp. 105-122.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractionalization of cyclic transformations in optics," in Proceedings of ICOL 2005, the International Conference on Optics and Optoelectronics, Dehradun, India, December 12-15, 2005, Paper IT-OIP-2, pp. 1-15.

Calvo, M. L.

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

T. Alieva and M. L. Calvo, "Fractionalization of the linear cyclic transforms," J. Opt. Soc. Am. A 17, 2330-2338 (2000).
[CrossRef]

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional cyclic transforms in optics: theory and applications," in Recent Research and Developments in Optics, S.G.Pandalai, ed. (Research Signpost, 2001), Vol. 1, pp. 105-122.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractionalization of cyclic transformations in optics," in Proceedings of ICOL 2005, the International Conference on Optics and Optoelectronics, Dehradun, India, December 12-15, 2005, Paper IT-OIP-2, pp. 1-15.

Collins, S. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Kerr, F. H.

A. C. McBride and F. H. Kerr, "On Namias' fractional Fourier transforms," IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (McGraw-Hill, 1968).
[PubMed]

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (McGraw-Hill, 1968).
[PubMed]

Kutay, M. A.

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

Lohmann, A. W.

A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993).
[CrossRef]

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in optics," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (North-Holland, 1998), pp. 263-342.
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

McBride, A. C.

A. C. McBride and F. H. Kerr, "On Namias' fractional Fourier transforms," IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

Mendlovic, D.

Moshinsky, M.

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

Mukunda, N.

Namias, V.

V. Namias, "The fractional order Fourier transform and its applications to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

Ozaktas, H. M.

Quesne, C.

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

Segal, I. E.

I. E. Segal, "Distributions in Hilbert spaces and canonical systems of operators," Trans. Am. Math. Soc. 88, 12-42 (1958).
[CrossRef]

Simon, R.

Wolf, K. B.

Zalevsky, Z.

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

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in optics," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (North-Holland, 1998), pp. 263-342.
[CrossRef]

Commun. Pure Appl. Math.

V. Bargmann, "On a Hilbert space of analytic functions and an associated integral transform, Part I," Commun. Pure Appl. Math. 14, 187-214 (1961).
[CrossRef]

EURASIP J. Appl. Signal Process.

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

IMA J. Appl. Math.

A. C. McBride and F. H. Kerr, "On Namias' fractional Fourier transforms," IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, "The fractional order Fourier transform and its applications to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

J. Math. Phys.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

M. J. Bastiaans and T. Alieva, "Generating function for Hermite-Gaussian modes propagating through first-order optical systems," J. Phys. A 38, L73-L78 (2005).
[CrossRef]

Opt. Lett.

Trans. Am. Math. Soc.

I. E. Segal, "Distributions in Hilbert spaces and canonical systems of operators," Trans. Am. Math. Soc. 88, 12-42 (1958).
[CrossRef]

Other

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

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (McGraw-Hill, 1968).
[PubMed]

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

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional cyclic transforms in optics: theory and applications," in Recent Research and Developments in Optics, S.G.Pandalai, ed. (Research Signpost, 2001), Vol. 1, pp. 105-122.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractionalization of cyclic transformations in optics," in Proceedings of ICOL 2005, the International Conference on Optics and Optoelectronics, Dehradun, India, December 12-15, 2005, Paper IT-OIP-2, pp. 1-15.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in optics," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (North-Holland, 1998), pp. 263-342.
[CrossRef]

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Equations (65)

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[ r o p o ] = T [ r i p i ] = [ A B C D ] [ r i p i ] .
[ A B C D ] 1 = [ D t B t C t A t ] , or T 1 = JT t J
J = i [ 0 I I 0 ] , J = J 1 = J = J t ,
[ 1 λ z 0 1 ] , [ 1 0 1 λ f 1 ] , [ 0 w 2 w 2 0 ]
T f ( θ ) = [ cos θ w 2 sin θ w 2 sin θ cos θ ] ,
f i ( r i ) f o ( r o ) = R [ f i ( r i ) ] ( r o ) .
f o ( r o ) = exp ( i ϕ ) det i B 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 .
f o ( x o ) = { exp ( i θ 2 ) w i sin θ exp [ i π ( x o 2 + x i 2 ) cos θ w 2 sin θ ] × exp [ i 2 π x o x i w 2 sin θ ] f i ( x i ) d x i ( θ 0 ) , f i ( x o ) ( θ = 0 ) ]
T u = Q u Λ f Q u 1 ,
det ( T 1 λ I ) = det ( JT t J λ I ) = det [ J ( T t λ I ) J ] = det ( T t λ I ) = det ( T λ I ) ,
λ n q n t Jq m λ m = ( Tq n ) t J ( Tq m ) = q n t ( T t JT ) q m = q n t Jq m ,
( λ n λ m 1 ) q n t Jq m = 0 .
Λ = [ Δ 0 0 Δ 1 ] ,
Λ = [ Δ 0 0 Δ 1 ] = [ Δ 0 0 Δ * ] Λ f ,
Q u = 1 2 [ a + i bw 2 i ( a + i bw 2 ) * w 2 c + i dw 2 i ( c + i dw 2 ) * w 2 ] ,
Q u = [ a b c d ] 1 2 [ I i w 2 i w 2 I ] MF ,
M = [ a b c d ] ,
F = 1 2 [ I i w 2 i w 2 I ] .
T u = MF Λ f F 1 M 1 MT f M 1 ,
T f = F Λ f F 1 = [ Δ + Δ * 2 Δ Δ * 2 i w 2 Δ Δ * 2 i w 2 Δ + Δ * 2 ] .
T f = [ w 0 0 w 1 ] [ cos Θ sin Θ sin Θ cos Θ ] [ w 1 0 0 w ] .
[ a b c d ] = [ cos θ + g w 2 sin θ w 2 sin θ ( g 2 w 2 + w 2 ) sin θ cos θ g w 2 sin θ ] = [ 1 0 g 1 ] [ cos θ w 2 sin θ w 2 sin θ cos θ ] [ 1 0 g 1 ]
[ a b c d ] = [ 1 0 g 1 ] [ ± cosh θ ± w 2 sinh θ ± w 2 sinh θ ± cosh θ ] [ 1 0 g 1 ] ,
T = [ w 0 0 w 1 ] [ X Y Y X ] [ w 1 0 0 w ] ,
T = [ P i P * i P P * ] [ Δ 0 0 Δ * ] [ P i P * i P P * ] 1 Q Λ f Q 1 .
U f ( θ x , θ y ) = [ exp ( i θ x ) 0 0 exp ( i θ y ) ] ,
T r ( θ ) = [ cos θ sin θ 0 0 sin θ cos θ 0 0 0 0 cos θ sin θ 0 0 sin θ cos θ ] ,
T g ( θ ) = [ cos θ 0 0 sin θ 0 cos θ sin θ 0 0 sin θ cos θ 0 sin θ 0 0 cos θ ]
U r ( θ ) = [ cos θ sin θ sin θ cos θ ] ,
U g ( θ ) = [ cos θ i sin θ i sin θ cos θ ] ,
P r = 1 2 [ 1 i i 1 ] [ exp ( i α ) 0 0 exp ( i β ) ] = U g ( π 4 ) U f ( α , β ) = P r ( α , β ) ,
P g = 1 2 [ 1 1 1 1 ] [ exp ( i α ) 0 0 exp ( i β ) ] = U r ( π 4 ) U f ( α , β ) = P g ( α , β ) ,
Δ r = Δ g = [ exp ( i θ ) 0 0 exp ( i θ ) ] = U f ( θ , θ ) .
M r = M r ( α , β ) = 1 2 [ cos α sin β sin α cos β sin α cos β cos α sin β sin α cos β cos α sin β cos α sin β sin α cos β ] ,
M g = M g ( α , β ) = 1 2 [ cos α cos β sin α sin β cos α cos β sin α sin β sin α sin β cos α cos β sin α sin β cos α cos β ]
T f = T f ( θ , θ ) = [ cos θ 0 sin θ 0 0 cos θ 0 sin θ sin θ 0 cos θ 0 0 sin θ 0 cos θ ] .
T r ( θ ) = T g ( π 4 ) T f ( θ , θ ) T g ( π 4 ) ,
T g ( θ ) = T r ( π 4 ) T f ( θ , θ ) T r ( π 4 ) ,
U f ( θ x , θ y ) = P f Δ f P f 1 U f ( α , β ) U f ( θ x , θ y ) U f 1 ( α , β ) ,
T f ( θ x , θ y ) = M f T f M f 1 T f ( α , β ) T f ( θ x , θ y ) T f 1 ( α , β ) .
T [ w 1 0 0 w ] T [ w 0 0 w 1 ] .
T f [ cos Θ sin Θ sin Θ cos Θ ]
[ a b c d ] [ w 1 0 0 w ] [ a b c d ] [ w 0 0 w 1 ] .
Φ n ( r o ) = μ n R u [ Φ n ( r i ) ] ( r o ) ,
H n ( x ) = 2 1 4 ( 2 n n ! ) 1 2 H n ( x 2 π ) exp ( π x 2 ) ,
2 1 2 det ( a + i b ) exp [ s t ( a + i b ) 1 ( a i b ) s + 2 s t ( a + i b ) 1 r 2 π π r t ( d i c ) ( a + i b ) 1 r ] = n = 0 m = 0 H n , m M ( r ) ( 2 n + m n ! m ! ) 1 2 s x n s y m ,
H n , m M ( r ) = 2 1 2 2 n + m π n + m n ! m ! det ( a + i b ) P x n P y m exp [ π r t ( d i c ) ( a + i b ) 1 r ] ,
[ P x P y ] = 2 π V [ x y ] Z [ x y ] ,
V = ( a i b ) t [ ( d i c ) ( a + i b ) 1 ] * ,
Z = ( a i b ) t .
H n , m M ( r ) = 2 1 2 P x n P y m exp ( π r t r ) 2 n + m π n + m n ! m ! det U ,
M = [ 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 ,
S = ( aa t + bb t ) 1 2 ,
U = X + i Y = ( aa t + bb t ) 1 2 ( a + i b ) .
T u = [ I 0 G I ] [ S 0 0 S 1 ] [ X Y Y X ] [ cos Θ sin Θ sin Θ cos Θ ] [ X t Y t Y t X t ] [ S 1 0 0 S ] [ I 0 G I ] .
[ X Y Y X ] [ cos Θ sin Θ sin Θ cos Θ ] [ X t Y t Y t X t ] ,
T u = [ I 0 G I ] [ S 0 0 S 1 ] [ X Y Y X ] [ S 1 0 0 S ] [ I 0 G I ] .
T u = [ a b c d ] [ cos Θ sin Θ sin Θ cos Θ ] [ d t b t c t a t ] .
U f ( γ x , γ y ) = [ exp ( i γ x ) 0 0 exp ( i γ y ) ] ,
exp [ i ( γ x + γ y ) ] = det U ,
cos ( γ x γ y ) = det X + det Y ,
θ 1 , 2 = ( γ x + γ y ) 2 ± arcos { cos ( α + β ) cos [ ( γ x γ y ) 2 ] } .
G = [ g 11 g 12 g 12 g 22 ] = U r ( φ g ) [ g 1 0 0 g 2 ] U r ( φ g ) ,
S = [ s 11 s 12 s 12 s 22 ] = U r ( φ s ) [ s 1 0 0 s 2 ] U r ( φ s )

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