Abstract

Starting with the Iwasawa-type decomposition of a first-order optical system (or ABCD system) as a cascade of a lens, a magnifier, and an orthosymplectic system (a system that is both symplectic and orthogonal), a further decomposition of the orthosymplectic system in the form of a separable fractional Fourier transformer embedded between two spatial-coordinate rotators is proposed. The resulting decomposition of the entire first-order optical system then shows a physically attractive representation of the linear canonical integral transformation, which, in contrast to Collins integral, is valid for any ray transformation matrix.

© 2005 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).
  2. S. A. Collins, J. Opt. Soc. Am. 60, 1168 (1970).
    [CrossRef]
  3. M. Moshinsky and C. Quesne, J. Math. Phys. 12, 1772 (1971).
    [CrossRef]
  4. R. Simon and N. Mukunda, J. Opt. Soc. Am. A 15, 2146 (1998).
    [CrossRef]
  5. K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).
  6. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  7. V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
    [CrossRef]
  8. A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
    [CrossRef]

1998 (1)

1987 (1)

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

1980 (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

1971 (1)

M. Moshinsky and C. Quesne, J. Math. Phys. 12, 1772 (1971).
[CrossRef]

1970 (1)

Collins, S. A.

Kerr, F. H.

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (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).

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, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Moshinsky, M.

M. Moshinsky and C. Quesne, J. Math. Phys. 12, 1772 (1971).
[CrossRef]

Mukunda, N.

Namias, V.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Ozaktas, H. M.

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

Quesne, C.

M. Moshinsky and C. Quesne, J. Math. Phys. 12, 1772 (1971).
[CrossRef]

Simon, R.

Wolf, K. B.

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

Zalevsky, Z.

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

IMA J. Appl. Math. (1)

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

J. Math. Phys. (1)

M. Moshinsky and C. Quesne, J. Math. Phys. 12, 1772 (1971).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Other (3)

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).

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

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Tables (1)

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Table 1 Quadrants for α [ 0 , π ) and β [ 0 , 2 π ) , for Different Values of the Signs of Y 11 , Y 12 , Y 21 , and Y 22

Equations (39)

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( r o q o ) = ( A B C D ) ( r i q i ) T ( r i q i ) .
A B t = B A t , C D t = D C t , A D t B C t = I ,
A t C = C t A , B t D = D t B , A t D C t B = I .
f o ( r o ) = exp ( i ϕ ) det i B f i ( r i ) exp [ i π ( r i t B 1 A r i 2 r i t B 1 r o + r o t D B 1 r o ) ] d r i ,
f o ( r ) = f i ( A 1 r ) exp ( i π r t C A 1 r ) det A .
( A B C D ) = ( I B 0 I ) ( A B C B B D C D ) ,
( A B C D ) = ( I 0 G I ) ( S 0 0 S 1 ) ( X Y Y X ) ,
G = ( C A t + D B t ) ( A A t + B B t ) 1 ,
S = ( A A t + B B t ) 1 2 ,
U = X + i Y = ( A A t + B B t ) 1 2 ( A + i B ) .
U r ( ϑ ) = X r ( ϑ ) = ( cos ϑ sin ϑ sin ϑ cos ϑ )
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 ) ,
U f ( γ x , γ y ) = X f ( γ x , γ y ) + i Y f ( γ x , γ y ) = ( exp ( i γ x ) 0 0 exp ( i γ y ) )
T f ( γ x , γ y ) = ( cos γ x 0 sin γ x 0 0 cos γ y 0 sin γ y sin γ x 0 cos γ x 0 0 sin γ y 0 cos γ y ) ,
U = U r ( β ) U f ( γ x , γ y ) U r ( α ) .
X 11 = cos α cos β cos γ x sin α sin β cos γ y ,
X 12 = sin α cos β cos γ x + cos α sin β cos γ y ,
X 21 = cos α sin β cos γ x sin α cos β cos γ y ,
X 22 = sin α sin β cos γ x + cos α cos β cos γ y ,
Y 11 = cos α cos β sin γ x sin α sin β sin γ y ,
Y 12 = sin α cos β sin γ x + cos α sin β sin γ y ,
Y 21 = cos α sin β sin γ x sin α cos β sin γ y ,
Y 22 = sin α sin β sin γ x + cos α cos β sin γ y .
U r ( ϑ + π ) = U r ( ϑ ) ,
U f ( γ x + π , γ y + π ) = U f ( γ x , γ y ) ,
U r ( π 2 ) U f ( γ x , γ y ) U r ( π 2 ) = U f ( γ y , γ x ) ,
cos ( γ x + γ y ) = Re { det U } ,
sin ( γ x + γ y ) = Im { det U } ,
det X + det Y = cos ( γ x γ y )
X 11 + X 22 Y 12 + Y 21 = 2 cos ( α + β + γ 1 ) cos γ 2 ,
X 12 X 21 + Y 11 + Y 22 = 2 sin ( α + β + γ 1 ) cos γ 2 ,
X 11 + X 22 + Y 12 + Y 21 = 2 sin ( α β + γ 1 ) sin γ 2 ,
X 12 + X 21 + Y 11 Y 22 = 2 cos ( α β + γ 1 ) sin γ 2 .
( A B C D ) = ( I 0 G I ) ( S 0 0 S 1 ) ( X r ( β ) 0 0 X r ( β ) ) × ( X f ( γ x , γ y ) Y f ( γ x , γ y ) Y f ( γ x , γ y ) X f ( γ x , γ y ) ) ( X r ( α ) 0 0 X r ( α ) ) .
B = S X r ( β ) Y f ( γ x , γ y ) X r ( α ) ,
R γ x [ f ( x i ) ] ( x o )
= { exp ( i γ x 2 ) i sin γ x exp [ i π ( x o 2 + x i 2 ) cos γ x sin γ x ] × exp [ i 2 π x o x i sin γ x ] f ( x i ) d x i ( γ x 0 ) , f ( x o ) ( γ x = 0 ) , }
f γ x , γ y ( x o , y o ) = R γ x , γ y [ f ( x i cos α + y i sin α , x i sin α + y i cos α ) ] ( x o , y o ) .
f o ( r o ) = exp ( i π r o t G r o ) det S × R γ x , γ y [ f i ( X r ( α ) r i ) ] ( X r ( β ) S 1 r o ) .

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