Abstract

Based on the eigenvalues of the ray transformation matrix, a classification of ABCD systems is proposed and some nuclei (i.e., elementary members) in each class are described. In the one-dimensional case, possible nuclei are the magnifier, the lens, and the fractional Fourier transformer. In the two-dimensional case we have—in addition to the obvious concatenations of one-dimensional nuclei—the four combinations of a magnifier or a lens with a rotator or a shearing operator, where the rotator and the shearer are obviously inherently two-dimensional. Any ABCD system belongs to one of the classes described in this paper and is similar (in the sense of matrix similarity of the ray transformation matrices) to the corresponding nucleus. Knowledge of a nucleus may be helpful in finding eigenfunctions of the corresponding class of first-order optical systems: one only has to find eigenfunctions of the nucleus and to determine how these functions propagate through a first-order optical system.

© 2007 Optical Society of America

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    [CrossRef]

2007

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

2006

2005

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]

2004

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

2002

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

2001

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

2000

1999

W. W. Lin, V. Mehrmann, and H. Xu, "Canonical forms for Hamiltonian and symplectic matrices and pencils," Linear Algebr. Appl. 302-303, 469-533 (1999).
[CrossRef]

1998

1996

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

1993

1987

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

1982

1980

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

1979

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

1977

1971

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

1970

1966

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

A. Vander Lugt, "Operational notation for the analysis and synthesis of optical data-processing systems," Proc. IEEE 54, 1055-1063 (1966).
[CrossRef]

Alieva, T.

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

M. J. Bastiaans and T. Alieva, "First-order optical systems with unimodular eigenvalues," J. Opt. Soc. Am. A 23, 1875-1883 (2006).
[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]

Bastiaans, M. J.

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

M. J. Bastiaans and T. Alieva, "First-order optical systems with unimodular eigenvalues," J. Opt. Soc. Am. A 23, 1875-1883 (2006).
[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]

Butterweck, H. J.

Collins, S. A.

Ding, J. J.

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

Gantmacher, F. R.

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1974-1977).

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]

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).

Lin, W. W.

W. W. Lin, V. Mehrmann, and H. Xu, "Canonical forms for Hamiltonian and symplectic matrices and pencils," Linear Algebr. Appl. 302-303, 469-533 (1999).
[CrossRef]

Lohmann, A. W.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. 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]

Mehrmann, V.

W. W. Lin, V. Mehrmann, and H. Xu, "Canonical forms for Hamiltonian and symplectic matrices and pencils," Linear Algebr. Appl. 302-303, 469-533 (1999).
[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]

Nazarathy, M.

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]

Quesne, C.

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

Shamir, J.

Simon, R.

Vander Lugt, A.

A. Vander Lugt, "Operational notation for the analysis and synthesis of optical data-processing systems," Proc. IEEE 54, 1055-1063 (1966).
[CrossRef]

Wolf, K. B.

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

R. Simon and K. B. Wolf, "Structure of the set of paraxial optical systems," J. Opt. Soc. Am. A 17, 342-355 (2000).
[CrossRef]

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

Xu, H.

W. W. Lin, V. Mehrmann, and H. Xu, "Canonical forms for Hamiltonian and symplectic matrices and pencils," Linear Algebr. Appl. 302-303, 469-533 (1999).
[CrossRef]

Zalevsky, Z.

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

IEEE Trans. Signal Process.

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

Linear Algebr. Appl.

W. W. Lin, V. Mehrmann, and H. Xu, "Canonical forms for Hamiltonian and symplectic matrices and pencils," Linear Algebr. Appl. 302-303, 469-533 (1999).
[CrossRef]

Opt. Commun.

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

Opt. Lett.

Proc. IEEE

A. Vander Lugt, "Operational notation for the analysis and synthesis of optical data-processing systems," Proc. IEEE 54, 1055-1063 (1966).
[CrossRef]

Other

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

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

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

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

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

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1974-1977).

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Figures (1)

Fig. 1
Fig. 1

Different areas for the two-variable cases 1-1, 1-3, 3-3 (boundaries included), and 4. For the one-variable cases we have case 1-2, lines 1 and 2 ( a 2 > 6 or a 2 < 2 ); case 2-3, lines 1 and 2 ( 2 a 2 6 ) ; case 5, parabola ( a 2 6 ) ; case 6, parabola ( a 2 > 6 ) . For the isolated points we have case 2-2, ( a 1 , a 2 ) = ( ± 4 , 6 ) or ( a 1 , a 2 ) = ( 0 , 2 ) and case 7, ( a 1 , a 2 ) = ( ± 4 , 6 ) . Cases 2-2, 2-3, 3-3, 5, and 7 correspond to unimodular eigenvalues, while the other cases have two (cases 1-2 and 1-3) or four (cases 1-1, 4, and 6) nonunimodular eigenvalues.

Tables (2)

Tables Icon

Table 1 Coefficients for the Different Two-Dimensional Cases a

Tables Icon

Table 2 Seven Classes of Eigenvalue Distributions a

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 ,
T L ( f ) = [ 1 0 1 λ o f 1 ] , T S ( z ) = [ 1 λ o z 0 1 ] ,
T F ( θ ; w ) = [ cos θ w 2 sin θ w 2 sin θ cos θ ] ,
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 M ( s ) = [ s 0 0 s 1 ] .
J + ( λ ) = [ λ 1 0 λ ] or J ( λ ) = [ λ 0 1 λ ] .
Λ F ( θ ) = [ exp ( i θ ) 0 0 exp ( i θ ) ] .
[ a 1 b 1 c 1 d 1 ] [ a 2 b 2 c 2 d 2 ] = [ a 1 0 b 1 0 0 a 2 0 b 2 c 1 0 d 1 0 0 c 2 0 d 2 ] ;
[ s 0 0 s 1 ] [ λ 1 0 λ ] = [ s 0 0 0 0 λ 0 1 0 0 s 1 0 0 0 0 λ ] .
[ s exp ( i θ ) 0 0 0 0 s exp ( i θ ) 0 0 0 0 s 1 exp ( i θ ) 0 0 0 0 s 1 exp ( i θ ) ] .
[ J + ( exp ( i θ ) ) 0 0 J + ( exp ( i θ ) ) ]
or [ J ( exp ( i θ ) ) 0 0 J ( exp ( i θ ) ) ] .
[ J + ( s ) 0 0 J + ( s 1 ) ] or [ J ( s ) 0 0 J ( s 1 ) ] .
[ λ 1 0 0 0 λ 1 0 0 0 λ 1 0 0 0 λ ] or [ λ 0 0 0 1 λ 0 0 0 1 λ 0 0 0 1 λ ] .
a 2 > 6 occupied by case 1 - 2 , a 2 < 2 occupied by case 1 - 2 , 2 a 2 6 occupied by case 2 - 3 ;
a 2 > 6 occupied by case 1 - 2 , a 2 < 2 occupied by case 1 - 2 , 2 a 2 6 occupied by case 2 - 3 ;
a 2 > 6 occupied by case 6 , 2 a 2 6 occupied by case 5 ;
( a 1 , a 2 ) = ( 4 , 6 ) occupied by cases 2 - 2 and 7 , ( a 1 , a 2 ) = ( 4 , 6 ) occupied by cases 2 - 2 and 7 , ( a 1 , a 2 ) = ( 0 , 2 ) occupied by case 2 - 2 .
T M ( s ) = Λ M ( s ) = [ s 0 0 s 1 ] ,
T F ( π 4 ; w ) T M ( exp σ ) T F ( π 4 ; w ) = [ cosh σ w 2 sinh σ w 2 sinh σ cosh σ ] T H ( σ ; w ) ,
T = [ A B C D ] = A + D A + D [ 1 0 g 1 ] [ cosh σ w 2 sinh σ w 2 sinh σ cosh σ ] [ 1 0 g 1 ] ,
T = [ A B C D ] = [ a b c d ] ( ± 1 ) [ 1 0 g 1 ] [ d b c a ] = ( ± 1 ) [ 1 b d g b 2 g d 2 g 1 + b d g ] ,
[ 0 w 2 w 2 0 ] [ 1 0 g 1 ] [ 0 w 2 w 2 0 ] = [ 1 g w 4 0 1 ] .
T = [ A B C D ] = [ a b c d ] ( ± 1 ) [ 1 h 0 1 ] [ d b c a ] = ( ± 1 ) [ 1 a c h a 2 h c 2 h 1 + a c h ] ,
T F ( θ ; w ) = 1 2 [ 1 i w 2 i w 2 1 ] [ exp ( i θ ) 0 0 exp ( i θ ) ] 2 [ 1 i w 2 i w 2 1 ] 1 Q F ( w ) Λ F ( θ ) Q F 1 ( w ) ,
T = [ A B C D ] = [ 1 0 g 1 ] [ cos θ w 2 sin θ w 2 sin θ cos θ ] [ 1 0 g 1 ] ,
T M ( s x , s y ) = T M ( s x ) T M ( s y ) ,
T L ( f x , f y ) = T L ( f x ) T L ( f y ) ,
T F ( θ x , θ y ; w x , w y ) = T F ( θ x ; w x ) T F ( θ y ; w y ) ,
T M ( s x ) T L ( f y ) ,
T M ( s x ) T F ( θ y ; w y ) ,
T L ( f x ) T F ( θ y ; w y ) ,
[ 1 0 0 0 0 cos θ y 0 w y 2 sin θ y 1 λ f x 0 1 0 0 w y 2 sin θ y 0 cos θ y ] .
T MR ( θ , s ) = [ s R ( θ ) 0 0 s 1 R ( θ ) ] ,
R ( θ ) = [ cos θ sin θ sin θ cos θ ] .
T MR ( θ , s ) = [ s cos θ s sin θ 0 0 s sin θ s cos θ 0 0 0 0 s 1 cos θ s 1 sin θ 0 0 s 1 sin θ s 1 cos θ ] , = 1 2 [ 1 i 0 0 i 1 0 0 0 0 1 i 0 0 i 1 ] [ s exp ( i θ ) 0 0 0 0 s exp ( i θ ) 0 0 0 0 s 1 exp ( i θ ) 0 0 0 0 s 1 exp ( i θ ) ] 2 [ 1 i 0 0 i 1 0 0 0 0 1 i 0 0 i 1 ] 1 .
T R ( θ ) = [ R ( θ ) 0 0 R ( θ ) ] ,
T MR ( θ , s ) = T M ( s , s ) T R ( θ ) = T R ( θ ) T M ( s , s ) .
f o ( r ) = f i ( A 1 r ) exp ( i π r t CA 1 r ) det A ,
s f o ( s x , s y ) = f i ( x cos θ y sin θ , x sin θ + y cos θ ) ,
s f o ( s r , φ ) = f i ( r , φ + θ ) .
T LR ( θ ; f ) = [ R ( θ ) 0 ( 1 λ o f ) R ( θ ) R ( θ ) ] ,
T LR ( θ ; f ) = T L ( f , f ) T R ( θ ) = T R ( θ ) T L ( f , f ) .
f o ( x , y ) = f i ( x cos θ y sin θ , x sin θ + y cos θ ) exp [ i π ( 1 λ o f ) ( x 2 + y 2 ) ] ,
f o ( r , φ ) = f i ( r , φ + θ ) exp [ i π ( 1 λ o f ) r 2 ] .
T MZ ( s ) = [ s J + 0 0 s 1 J ] = [ s I 0 0 s 1 I ] [ J + 0 0 J ] ,
T Z = [ J + 0 0 J ] ,
T MZ ( s ) = T M ( s , s ) T Z = T Z T M ( s , s ) .
s f o ( s x , s y ) = f i ( x y , y ) ,
T Z = [ J + 0 0 J ] = [ ( J + J + t ) 1 2 0 0 ( J + J + t ) 1 2 ] [ ( J + J + t ) 1 2 J + 0 0 ( J + J + t ) 1 2 J + ] .
J + J + t = [ ϕ 1 1 ϕ ] [ ϕ 2 0 0 ϕ 2 ] [ ϕ 1 1 ϕ ] 1 .
( J + J + t ) 1 2 = [ ϕ 1 1 ϕ ] [ ϕ 0 0 ϕ 1 ] [ ϕ 1 1 ϕ ] 1 = 1 5 [ 3 1 1 2 ] [ s 11 s 12 s 12 s 22 ] ,
( J + J + t ) 1 2 J + = 1 5 [ 3 1 1 2 ] 1 [ 1 1 0 1 ] = 1 5 [ 2 1 1 2 ] R ( α ) ,
Z = R ( arccot ϕ ) M ( ϕ , ϕ 1 ) R ( arccot ϕ + arccot 2 ) .
T LZ ( f ) = [ J + 0 ( 1 λ o f ) J + J ] = [ I 0 ( 1 λ o f ) I I ] [ J + 0 0 J ] ,
T LZ = T L ( f , f ) T Z .
f o ( x , y ) = f i ( x y , y ) exp [ i π ( 1 λ o f ) ( x 2 + y 2 ) ] .
f o ( r o ) = T T [ f i ( r i ) ] ( r o ) = 1 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 ) = f i ( x ) exp [ i π ( 1 λ o f ) x 2 ] ,
f o ( x o ) = 1 i λ o z f i ( x i ) exp [ i π ( λ o z ) ( x o x i ) 2 ] d x i ,
s 1 2 f o ( s x ) = f i ( x ) ,
0 0 2 π δ ( r i ρ ) exp ( i m φ i ) exp [ i π b 1 ( a r i 2 2 r i r cos ( φ i φ ) + d r 2 ) ] r i d r i d φ i = ρ exp [ i π b 1 ( a ρ 2 + d r 2 ) ] exp ( i m φ ) 0 2 π exp [ i 2 π b 1 ρ r cos ( φ i φ ) + i m ( φ i φ ) ] d φ i = ρ exp [ i π b 1 ( a ρ 2 + d r 2 ) ] exp ( i m φ ) 2 π ( i ) m J m ( 2 π b 1 ρ r ) ,
0 0 2 π r i k exp ( i m φ i ) exp [ i π b 1 ( a r i 2 2 r i r cos ( φ i φ ) + d r 2 ) ] r i d r i d φ i = exp ( i π b 1 d r 2 ) exp ( i m φ ) 2 π ( i ) m 0 r i k + 1 exp ( i π b 1 a r i 2 ) J m ( 2 π b 1 r i r ) d r i .

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