Abstract

The properties of first-order optical systems are described paraxially by a ray transfer matrix, also called the ABCD matrix. Here we consider the inverse problem: an ABCD matrix is given, and we look for the minimal optical system that consists of only lenses and pieces of free-space propagation. Similar decompositions have been studied before but without the restriction to these two element types or without an attempt at minimalization. As the main results of this paper, we found that general lossless one- dimensional optical systems can be synthesized with a maximum of four elements and two-dimensional optical systems can be synthesized with six elements at most.

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    [CrossRef]
  13. H. Bartelt and K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260-262 (1980).
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    [CrossRef]

2006

2004

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004), Chap. 9, p. 164.

1998

1997

1994

1985

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realisation of first order optical systems using thin lenses,” Opt. Acta 32, 855-872 (1985).
[CrossRef]

1983

1982

M. Nazarathy and J. Shamir, “First-order optics--a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356-364 (1982).
[CrossRef]

K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310-314 (1982).
[CrossRef]

1981

1980

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 48, 397-399 (1980).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

H. Bartelt and K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260-262 (1980).

Abe, S.

Alieva, T.

Arsenault, H. H.

Bagini, V.

Bartelt, H.

H. Bartelt and K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260-262 (1980).

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

Bastiaans, M. J.

Brenner, K.-H.

K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310-314 (1982).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

H. Bartelt and K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260-262 (1980).

Casperson, L. W.

Lohmann, A. W.

K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310-314 (1982).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

Macukow, B.

Mukunda, N.

Nazarathy, M.

Palma, C.

Shamir, J.

Sheridan, J. T.

Simon, R.

Sudarshan, E. C. G.

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realisation of first order optical systems using thin lenses,” Opt. Acta 32, 855-872 (1985).
[CrossRef]

Wolf, K. B.

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004), Chap. 9, p. 164.

Am. J. Phys.

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 48, 397-399 (1980).
[CrossRef]

Appl. Opt.

Isr. J. Technol.

H. Bartelt and K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260-262 (1980).

J. Opt. Soc. Am.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realisation of first order optical systems using thin lenses,” Opt. Acta 32, 855-872 (1985).
[CrossRef]

Opt. Commun.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310-314 (1982).
[CrossRef]

Opt. Lett.

Other

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004), Chap. 9, p. 164.

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Tables (4)

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Table 1 Various Matrix Decompositions Using the Operators: Lens L, Free Space P, Scale S and Rotation Ψ a

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Table 2 Overview of Theoretical Minimal Decompositions for 1-D Optics a

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Table 3 Overview of the Optical Minimal Decomposition for 1-D Optics a

Tables Icon

Table 4 Overview of 4 × 4 Matrix Decomposition Using the Operators Astigmatic Lens L, Anisotropic Free Space P, Scale S and Rotation R, and Ψ a

Equations (98)

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( x o s o ) = M ( x i s i ) = ( A B C D ) ( x i s i ) .
L ( f ) = ( 1 0 1 f 1 ) , P ( z ) = ( 1 z 0 1 ) , S ( m ) = ( m 0 0 1 m ) ,
F f ( m , α ) = S ( m ) · Ψ ( α ) ,
Ψ ( α ) = ( cos α sin α sin α cos α ) .
( A B C D ) = L ( f 2 ) P ( z ) L ( f 1 ) ,
z = B , f 1 = B 1 A , f 2 = B 1 D .
z = m sin α , f 1 = m sin α 1 m cos α , f 2 = m sin α 1 m 1 cos α .
( A B C D ) = P ( z 2 ) L ( f ) P ( z 1 ) ,
f = 1 C , z 1 = D 1 C , z 2 = A 1 C .
f = m sin α , z 1 = m cos α sin α , z 2 = 1 m cos α m 1 sin α .
P ( b · ( a + b ) a ) L ( b ) P ( a + b ) L ( a ) = ( b a 0 0 a b ) ,
L ( b ) P ( a + b ) L ( a ) P ( a · ( a + b ) b ) = ( b a 0 0 a b ) .
L ( b 2 a + b ) P ( b ) L ( a b a + b ) P ( a ) = ( b a 0 0 a b ) ,
P ( b ) L ( a b a + b ) P ( a ) L ( a 2 a + b ) = ( b a 0 0 a b ) .
Ψ ( π ) = ( 1 0 0 1 ) .
( A B > 0 C D ) = ( 1 0 D 1 B 1 ) ( 1 B 0 1 ) ( 1 0 A 1 B 1 ) = LPL .
( A B < 0 C D ) = ( 1 0 0 1 ) coordinate mirroring ( A B C D ) as step   2 = ( 1 0 0 1 ) ( 1 0 D + 1 B 1 ) ( 1 B 0 1 ) ( 1 0 A + 1 B 1 ) = Ψ ( π ) LPL .
( 1 z 0 1 ) = ( 1 0 0 1 ) ( 1 z 0 1 ) = ( 1 0 0 1 ) ( 1 0 2 z 1 ) ( 1 z 0 1 ) ( 1 0 2 z 1 ) .
( 1 0 C 1 ) = L
( 1 0 C 1 ) = ( 1 0 0 1 ) coordinate mirroring ( 1 0 C 1 ) = Ψ ( π ) L .
( A < 0 0 C < 0 D < 0 ) = ( 1 A 1 C 0 1 ) ( 1 0 C 1 ) ( 1 D 1 C 0 1 ) = PLP .
( A > 0 0 C > 0 D > 0 ) = ( 1 0 0 1 ) coordinate mirroring ( A 0 C D ) as step   5 = Ψ ( π ) PLP .
( A < 0 0 C D < 0 ) = ( 1 0 C A 1 ) ( A 0 0 D ) scale matrix = L LPLP Eq.   ( 11 b)   or   ( 11 c) = LPLP ,
( A < 0 0 C D < 0 ) = ( A 0 0 D ) scale matrix ( 1 0 C D 1 ) = PLPL Eq. ( 11 a)   or   ( 11 d) L = PLPL .
( A > 0 0 C D > 0 ) = ( 1 0 0 1 ) coordinate mirroring ( A 0 C D ) as step   7 = Ψ ( π ) LPLP or Ψ ( π ) PLPL .
( x o y o s x o s y o ) = M ( x i y i s x i s y i ) = ( A B C D ) ( x i y i s x i s y i ) ,
M T ( 0 I I 0 ) M = ( 0 I I 0 ) A T C = C T A , B T D = D T B , A T D C T B = I ,
M ( 0 I I 0 ) M T = ( 0 I I 0 ) BA T = AB T , DC T = CD T , AD T BC T = I .
R ( φ ) = ( cos φ sin φ 0 0 sin φ cos φ 0 0 0 0 cos φ sin φ 0 0 sin φ cos φ ) .
L ( f x , f y ) = ( 1 0 0 0 0 1 0 0 1 f x 0 1 0 0 1 f y 0 1 ) ,
P ( z , z ) = ( 1 0 z 0 0 1 0 z 0 0 1 0 0 0 0 1 ) ,
L g ( f x , f y , φ ) = ( I 0 G I ) = R ( φ ) L ( f x , f y ) R ( φ ) , G = G T .
P g ( z x , z y , φ ) = ( I H 0 I ) = R ( φ ) P ( z x , z y ) R ( φ ) ,
H = H T , P ( z x , z y ) = ( 1 0 z x 0 0 1 0 z y 0 0 1 0 0 0 0 1 ) .
M = ( A B C D ) = R ( φ 2 ) ( A B C D ) · R ( φ 1 ) .
M = L 2 g P g L 1 g ,
L 2 g = ( I 0 G 2 I ) , G 2 = ( D I ) B 1 , P g = ( I B 0 I ) , L 1 g = ( I 0 G 1 I ) , G 1 = B 1 ( A I ) .
L 2 g = R 31 T L 31 R 31 , P g = R π L 21 P 2 L 22 P 2 L 21 , L 1 g = R 11 T L 11 R 11 .
M = R 31 T L 31 R 31 R π L 21 P 2 L 22 P 2 L 21 R 11 T L 11 R 11 .
M = R 4 · L 3 · R 3 · P · L 2 · P · R 2 · L 1 · R 1 ,
P = P 2 , L 2 = L 22 , R 2 · L 1 · R 1 = L 21 R 11 T L 11 R 11 R ( φ 1 ) , R 4 · L 3 · R 3 = R ( φ 2 ) R 31 T L 31 R 31 R π L 21 .
R 2 L 1 R 1 = L 21 R 11 T L 11 R 11 combine according to Eq. (A5) R ( φ 1 ) = L 3 g · R ( φ 1 ) ,
L 3 g = ( I 0 U I ) , U = ( u 11 u u u 22 ) ,
R 1 = R ( φ 1 ) , φ 1 = θ φ 1 ,
R 2 = R ( φ 2 ) , φ 2 = θ ,
L 1 = L ( f x , f y ) , f x 1 = u 11 · cos 2 ( θ ) + u 22 · sin 2 ( θ ) + u · sin ( 2 θ ) , f y 1 = u 11 · sin 2 ( θ ) + u 22 · cos 2 ( θ ) u · sin ( 2 θ ) ,
θ = 1 2 arctan ( 2 u u 11 u 22 ) .
R 4 L 3 R 3 = R ( φ 2 ) R 31 T L 31 R 31 R π L 21 combine according to Eq. (A5) = R ( φ 2 ) R π · L 4 g ,
L 4 g = ( I 0 V I ) , V = ( v 11 v v v 22 ) ,
R 3 = R ( φ 3 ) , φ 3 = ϑ ,
R 4 = R ( φ 4 ) , φ 4 = ϑ φ 2 + π ,
L 3 = L ( f x , f y ) , f x 1 = v 11 · cos 2 ( ϑ ) + v 22 · sin 2 ( ϑ ) + v · sin ( 2 ϑ ) , f y 1 = v 11 · sin 2 ( ϑ ) + v 22 · cos 2 ( ϑ ) v · sin ( 2 ϑ ) ,
ϑ = 1 2 arctan ( 2 v v 11 v 22 ) .
M = R 2 ( A B C D ) · R 1 ,
M = R 4 M · P ( z ) R 3 = R 4 M R 3 ,
P ( z ) = ( I z · I 0 I ) , z > 0 arbitrary ,   but z b 11 a 11 .
P ( z ) = ( I z · I 0 I ) , z > 0 arbitrary ,   but z b 22 a 22 .
P ( z ) = ( I z · I 0 I ) , z > 0 arbitrary.
M = L 2 g P g L 1 g ,
L 2 g = ( I 0 G 2 I ) , G 2 = ( D I ) B 1 , P g = ( I B 0 I ) , L 1 g = ( I 0 G 1 I ) , G 1 = B 1 ( A I ) .
L 2 g = R 31 T L 31 R 31 , P g = R π L 21 P 2 L 22 P 2 L 21 , L 1 g = R 11 T L 11 R 11 .
M = R 4 · L 3 · R 3 · P · L 2 · P · R 2 · L 1 · R 1 · P ( z )
P = P 2 , L 2 = L 22 , R 2 L 1 R 1 = L 21 R 11 T L 11 R 11 R ( φ 1 φ 3 ) , R 4 L 3 R 3 = R ( φ 2 φ 4 ) R 31 T L 31 R 31 R π L 21 .
R 2 L 1 R 1 = L 21 R 11 T L 11 R 11 combine according to Eq. (A5) R ( φ 1 φ 3 ) = L 3 g · R ( φ 1 φ 3 )
L 3 g = ( I 0 U I ) , U = ( u 11 u u u 22 ) ,
R 1 = R ( φ 1 ) , φ 1 = θ φ 1 φ 3 ,
R 2 = R ( φ 2 ) , φ 2 = θ ,
L 1 = L ( f x , f y ) , f x 1 = u 11 · cos 2 ( θ ) + u 22 · sin 2 ( θ ) + u · sin ( 2 θ ) , f y 1 = u 11 · sin 2 ( θ ) + u 22 · cos 2 ( θ ) u · sin ( 2 θ ) ,
θ = 1 2 arctan ( 2 u u 11 u 22 ) ,
R 4 L 3 R 3 = R ( φ 2 φ 4 ) R 31 T L 31 R 31 R π L 21 combine according to Eq. (A5) = R ( φ 2 φ 4 ) R π · L 4 g
L 4 g = ( I 0 V I ) , V = ( v 11 v v v 22 ) ,
R 3 = R ( φ 3 ) , φ 3 = ϑ ,
R 4 = R ( φ 4 ) , φ 4 = ϑ φ 2 φ 4 + π ,
L 3 = L ( f x , f y ) , f x 1 = v 11 · cos 2 ( ϑ ) + v 22 · sin 2 ( ϑ ) + v · sin ( 2 ϑ ) , f y 1 = v 11 · sin 2 ( ϑ ) + v 22 · cos 2 ( ϑ ) v · sin ( 2 ϑ ) ,
ϑ = 1 2 arctan ( 2 v v 11 v 22 ) .
P ( z x , z y ) ,
P ( z x , z y ) = R ( π ) L ( z x z y , z x z y ) P ( z x z y , z x z y ) L ( ( z x z y ) 2 3 z x 2 z y , ( z x z y ) 2 2 z x z y ) P ( z x z y , z x z y ) L ( z x z y , z x z y ) ;
P ( z x , z y ) = R ( π ) L ( z y z x , z y z x ) P ( z y z x , z y z x ) L ( ( z y z x ) 2 2 z y z x , ( z y z x ) 2 3 z y 2 z x ) P ( z y z x , z y z x ) L ( z y z x , z y z x ) .
L 2 g L 1 g = ( I 0 G 2 I ) ( I 0 G 1 I ) = ( I 0 G 2 + G 1 I ) = L g .
L 2 g R ( φ ) L 1 g = R ( φ 1 ) L ( f 1 x , f 1 y ) R ( φ 1 ) factorizing of   L 2 g   as Appendix A .7 R ( φ ) L 1 g = R ( φ ) R ( φ 1 φ ) L ( f 1 x , f 1 y ) R ( φ 1 + φ ) L 2 g L 1 g = R ( φ ) L 2 g L 1 g = R ( φ ) L 3 g .
R ( φ ) L 3 g = R ( φ ) R ( φ 3 ) L ( f 3 x , f 3 y ) R ( φ 3 ) f actorizing of  L 3 g as Appendix A .7 .
L 2 g R ( φ ) L 1 g = R 2 L 3 R 1 ,
R 1 = R ( φ 3 ) ,
R 2 = R ( φ φ 3 ) ,
L 3 = L ( f 3 x , f 3 y ) .
L 3 g = ( I 0 G 3 I ) ,
G 3 = ( γ 11 γ γ γ 22 ) ,
φ 3 = 1 2 arctan ( 2 γ γ 11 γ 22 ) , f 3 x 1 = γ 11 · cos 2 ( φ 3 ) + γ 22 · sin 2 ( φ 3 ) + γ · sin ( 2 φ 3 ) , f 3 y 1 = γ 11 · sin 2 ( φ 3 ) + γ 22 · cos 2 ( φ 3 ) γ · sin ( 2 φ 3 ) .
( A B C D ) = ( A B C D ) · R ( φ 1 ) ,
B = ( b 11 b 12 b 21 b 22 ) , φ 1 = arctan ( b 21 b 12 b 11 + b 22 ) , B = B T .
( A B C D ) = ( A B C D ) · R ( π + φ 1 ) ,
( A B C D ) = R ( ϕ 2 ) T · ( A B C D ) · R ( ϕ 2 ) ,
B = ( b 11 b b b 22 ) , B = ( b 11 0 0 b 22 ) , ϕ 2 = 1 2 arctan ( 2 b b 22 b 11 ) .
( A B C D ) = R ( β ) · ( A B C D ) · R ( α ) ,
B = ( b 11 b 12 b 21 b 22 ) , B = ( b 11 0 0 b 22 ) , β = 1 2 [ arctan ( b 21 b 12 b 11 + b 22 ) arctan ( b 12 + b 21 b 22 b 11 ) ] , α = 1 2 [ arctan ( b 12 + b 21 b 22 b 11 ) + arctan ( b 21 b 12 b 11 + b 22 ) ] .
( I 0 G I ) = R T ( φ ) L ( f x , f y ) R ( φ ) ,
G T = G = ( g 11 g g g 22 ) .
φ = 1 2 arctan ( 2 g g 11 g 22 ) , f x 1 = g 11 · cos 2 ( φ ) + g 22 · sin 2 ( φ ) + g · sin ( 2 φ ) , f y 1 = g 11 · sin 2 ( φ ) + g 22 · cos 2 ( φ ) g · sin ( 2 φ ) .

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