Abstract

If the performance of an optical system A can be executed by a cascade of n identical optical systems B, we term the system B the nth root of A. At the same time A is the nth power of B. It is shown that, in principle, any optical system can be decomposed into its roots of any order. The procedure is facilitated by a merger of the ray matrix representation and the canonical operator representation of first-order optical systems. The results are demonstrated by several examples, including the fractional Fourier transform, which is just one special case in a complete group structure. Moreover, it is shown that the root and power transformations themselves represent special cases of a much more general family of transformations. Application in optical design, optical signal processing, and resonator theory can be envisaged.

© 1995 Optical Society of America

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References

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  1. M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [Crossref]
  2. A. Siegman, Lasers, 2nd ed. (Oxford U. Press, London, 1986).
  3. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [Crossref]
  4. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [Crossref]
  5. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [Crossref] [PubMed]
  6. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [Crossref]
  7. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [Crossref]
  8. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
    [Crossref] [PubMed]
  9. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
    [Crossref]
  10. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [Crossref]
  11. W. Bower, E. L. O’Neill, A. Walther, “The role of eikonal and matrix methods in contrast transfer calculus,” Appl. Opt. 2, 1239–1245 (1963).
    [Crossref]
  12. S. A. Collins, “Lens system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [Crossref]
  13. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
    [Crossref]
  14. R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985), p. 86.
  15. N. P. Erugin, Linear Systems of Ordinary Differential Equations (Academic, London, 1966), Chap. 1.
  16. P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, New York, 1985), Chap. 9.
  17. Ref. 13, Sec. 3.1.
  18. M. Nazarathy, J. Shamir, “First-order optics—operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1408 (1982).
    [Crossref]
  19. M. Nazarathy, A. Hardy, J. Shamir, “Misaligned first-order optics: canonical operator theory,” J. Opt. Soc. Am. A 3, 1360–1369 (1986).
    [Crossref]

1995 (1)

1994 (5)

1993 (2)

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[Crossref]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[Crossref]

1986 (1)

1982 (2)

1980 (1)

1970 (1)

1963 (1)

Barshan, B.

Bernardo, L. M.

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[Crossref]

Bower, W.

Collins, S. A.

Erugin, N. P.

N. P. Erugin, Linear Systems of Ordinary Differential Equations (Academic, London, 1966), Chap. 1.

Hardy, A.

Horn, R. A.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985), p. 86.

Johnson, C. R.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985), p. 86.

Lancaster, P.

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, New York, 1985), Chap. 9.

Lohmann, A. W.

Mendlovic, D.

Nazarathy, M.

O’Neill, E. L.

Onural, L.

Ozaktas, H. M.

Pellat-Finet, P.

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[Crossref] [PubMed]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[Crossref]

Shamir, J.

Siegman, A.

A. Siegman, Lasers, 2nd ed. (Oxford U. Press, London, 1986).

Soares, O. D. D.

Tismenetsky, M.

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, New York, 1985), Chap. 9.

Walther, A.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

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

Opt. Commun. (2)

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[Crossref]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[Crossref]

Opt. Lett. (2)

Other (5)

A. Siegman, Lasers, 2nd ed. (Oxford U. Press, London, 1986).

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985), p. 86.

N. P. Erugin, Linear Systems of Ordinary Differential Equations (Academic, London, 1966), Chap. 1.

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, New York, 1985), Chap. 9.

Ref. 13, Sec. 3.1.

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Figures (3)

Fig. 1
Fig. 1

Single-lens optical system with lens of focal length f. In the conventional 2f FT system d1 = d2 = f.

Fig. 2
Fig. 2

Fourier-transforming (2f) system with lenses at the input and output planes, implementing the roots of the FT (d1 = d2 = d).

Fig. 3
Fig. 3

Two-lens system (lenses at the input and output planes, implementing the roots of the FT).

Equations (92)

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BB . . . B n times = B n = A
u 0 ( x 0 ) = T [ M ] u i ( x i ) = 1 j λ B d x i × exp [ j k 2 B ( D x 0 2 2 x 0 x i + A x i 2 ) ] u i ( x i ) ,
M = [ A B C D ] , det ( M ) = A D B C = 1 .
Q [ a ] = T [ 1 0 a 1 ] ;
[ d ] = T [ 1 d 0 1 ] ;
V ¯ [ a ] = a V [ a ] = T [ 1 / a 0 0 a ] ;
¯ = 1 i λ V [ 1 λ ] T [ 0 0 1 0 ] .
V ¯ [ a ] ¯ ¯ [ a ] = T [ 0 1 / a a 0 ] ,
[ A B C D ] = [ 1 0 C / A 1 ] [ A 0 0 1 / A ] [ 1 B / A 0 1 ] , A 0 ,
[ A B C D ] = [ 1 0 D / B 1 ] [ B 0 0 1 / B ] [ 0 1 1 0 ] [ 1 0 A / B 1 ] , B 0 ,
[ A B C D ] = [ 1 A / C 0 1 ] [ 1 / C 0 0 C ] [ 0 1 1 0 ] × [ 1 D / C 0 1 ] , C 0 ,
[ A B C D ] = [ 1 B / D 0 1 ] [ 1 / D 0 0 D ] [ 1 0 C / D 1 ] , D 0 .
T [ M ] = Q [ C / A ] V ¯ [ 1 / A ] [ B / A ] , A 0 ,
T [ M ] = Q [ D / B ] V ¯ [ 1 / B ] ¯ Q [ A / B ] , B 0 ,
T [ M ] = [ A / C ] V ¯ [ C ] ¯ [ D / C ] , C 0 ,
T [ M ] = [ B / D ] V ¯ [ D ] Q [ C / D ] , D 0 ,
f ( A ) = f ( λ 1 ) f ( λ 2 ) λ 1 λ 2 A + f ( λ 2 ) λ 1 f ( λ 1 ) λ 2 λ 1 λ 2 I .
f ( A ) = f ( λ ) A + [ f ( λ ) λ f ( λ ) ] I ,
ξ k = λ 1 k 1 + λ 1 k 2 λ 2 + + λ 1 λ 2 k 2 + λ 2 k 1 .
A n = ξ n A α n I ,
α n = λ 1 λ 2 ξ n 1 .
μ 1 n = λ 1 , μ 2 n = λ 2 .
ξ k = μ 1 k 1 + μ 1 k 2 μ 2 + + μ 1 μ 2 k 2 + μ 2 k 1 .
α n = μ 1 μ 2 ξ n 1 .
B = 1 ξ n ( A + α n I )
ξ k = μ 1 k μ 2 k μ 1 μ 2 .
ξ n 1 = λ 1 μ 1 λ 2 μ 2 μ 1 μ 2 = 1 μ 1 μ 2 λ 1 μ 2 λ 2 μ 1 μ 1 μ 2 ,
ξ n = λ 1 λ 2 μ 1 μ 2 ,
α n = λ 1 μ 2 λ 2 μ 1 μ 1 μ 2 .
ξ k = k μ k 1 , α n = ( n 1 ) μ n ,
ξ n = n λ / μ , α n = ( n 1 ) λ .
( a 1 / n ) 1 / k = a 1 / n k ,
a 1 / n a 1 / k = a 1 / n + 1 / k .
M ; d = [ 1 d 0 1 ] , d 0 .
μ 1 = 1 1 / n , μ 2 = 1 1 / n .
( M ; d ) μ 1 / n = 1 + μ ( n 1 ) n [ 1 d 1 + μ ( n 1 ) 0 1 ] .
( M ; d ) 1 1 / n = [ 1 d / n 0 1 ] .
( M ; d ) 1 n = [ 1 d n 0 1 ] .
( M Q ; a ) μ 1 / n = 1 + μ ( n 1 ) n [ 1 0 a n μ ( n 1 ) 1 ] .
( M Q ; a ) μ 1 / n = [ 1 0 a / n 1 ] ,
( M Q ; a ) μ n = [ 1 0 a n 1 ] .
M ¯ ; a = [ 0 1 / a a 0 ]
ξ n = 2 i μ 1 μ 2 = 1 Im ( μ ) , α n = i μ 1 + μ 2 μ 1 μ 2 = Re ( μ ) Im ( μ ) ,
( M ¯ ; a ) μ 1 / n = Im ( μ ) [ 0 1 / a a 0 ] + Re ( μ ) [ 1 0 0 1 ] .
μ n m = exp ( i ϕ n m ) ,
ϕ n m = 1 n ( π 2 + 2 m π ) , m = 0 , 1 , , n 1 .
α n m = cos ϕ n m sin ϕ n m , ξ n m = 1 sin ϕ n m .
( M ¯ ; a ) μ 1 / n = ( sin ϕ n m ) [ cos ϕ n m sin ϕ n m 1 / a a cos ϕ n m sin ϕ n m ] [ cos ϕ n m sin ϕ n m a a sin ϕ n m cos ϕ n m ] .
u 0 ( x 0 ) = a j λ sin ϕ n m d x i exp { j k a 2 sin ϕ n m × [ ( cos ϕ n m ) x 0 2 2 x 0 x i + ( cos ϕ n m ) x i 2 ] } u i ( x i ) .
T [ ( M ¯ ; a ) μ 1 / n ] = Q [ a sin ϕ n M cos ϕ n m ] V ¯ [ 1 cos ϕ n m ] × [ sin ϕ n m a cos ϕ n m ] ,
T [ ( M ¯ ; a ) μ 1 / n ] = Q [ a cos ϕ n m sin ϕ n m ] V ¯ [ a sin ϕ n m ] × ¯ Q [ a cos ϕ n m sin ϕ n m ] ,
T [ ( M ¯ ; a ) μ 1 / n ] = [ cos ϕ n m a sin ϕ n m ] V ¯ [ a sin ϕ n m ] × ¯ [ cos ϕ n m a sin ϕ n m ] ,
T [ ( M ¯ ; a ) μ 1 / n ] = [ sin ϕ n m a sin ϕ n m ] V ¯ [ cos ϕ n m ] × Q [ a sin ϕ n m cos ϕ n m ] .
f = 1 a sin ϕ n m , d = f cos ϕ n m a sin ϕ n m .
d n m = f ( 1 cos ϕ n m ) .
T [ ( M ¯ ; a ) μ 1 / n ] = Q [ a ( cos ϕ n m 1 ) sin ϕ n m ] [ sin ϕ n m a ] × Q [ a ( cos ϕ n m 1 ) sin ϕ n m ] .
ξ n = i n ( i ) n 2 i , α n = i n 1 ( i ) n 1 2 i , ( M ¯ ; a ) n = i n ( i ) n 2 i M ¯ ; a i n 1 ( i ) n 1 2 i I .
M V ; a = [ 1 / a 0 0 a ] ,
ξ n = 1 a 2 a ( μ 1 μ 2 ) , α n = μ 2 a 2 μ 1 a ( μ 1 μ 2 ) ,
( M V ; a ) μ 1 / n = [ μ 1 0 0 μ 2 ] .
( M V ; a ) n = [ ( 1 / a ) n 0 0 a n ] .
M Q ; c V ; a = [ 1 0 c 1 ] [ 1 / a 0 0 a ] = [ 1 / a 0 c / a a ] .
( M Q ; c V ; a ) μ 1 / n = [ μ 1 0 c μ 1 μ 2 1 a 2 μ 2 ] .
( M Q ; c V ; a ) μ 1 / n = [ μ 0 c μ 2 1 μ ( 1 a 2 ) 1 / μ ] ,
( M Q ; c V ; a ) n = [ ( 1 / a ) n 0 c ( a 2 n 1 ) a n + 1 ( a 2 1 ) a n ] .
Q [ a ] = exp ( j k 2 a ρ 2 ) ,
ρ = x x ̂ + y ŷ , ρ = | ρ | .
V [ a ] f ( x , y ) = f ( a x , a y ) V [ a ] ,
f ( x , y ) = f ( x , y ) exp [ 2 π j ( x x + y y ) ] d x d y .
Q [ a ] Q [ b ] = Q [ a + b ] ,
V [ a ] V [ b ] = V [ a b ] ,
V [ a ] Q [ b ] = Q [ a 2 b ] V [ a ] ,
V [ b ] = V [ 1 / b ] ,
[ d ] = 1 Q [ λ 2 d ] = Q [ λ 2 d ] 1 = V [ 1 / λ d ] 1 Q [ 1 / d ] V [ 1 / λ d ] ,
[ d ] = Q [ 1 / d ] V [ 1 / λ d ] Q [ 1 / d ] .
[ a ] [ b ] = [ a + b ] .
[ f ] Q [ 1 / f ] [ d ] = Q [ 1 f ( 1 d f ) ] V [ 1 λ f ] ,
1 / a + 1 / b = 1 / f ,
[ a ] Q [ 1 / f ] [ b ] = Q [ 1 b ( 1 + a b ) ] V [ a / b ] .
[ d ] Q [ 1 / q ] = Q [ 1 / ( d + q ) ] V [ 1 / ( 1 + d / q ) ] × [ ( 1 / d + 1 / q ) 1 ] = Q [ 1 / ( d + q ) ] [ d ( 1 + d / q ) ] × V [ ( 1 / ( 1 + d / q ) ] ,
V [ b ] [ d ] = [ d / b 2 ] V [ b ] .
f ( J ) = f ( λ 1 ) f ( λ 2 ) λ 1 λ 2 J + f ( λ 2 ) λ 1 f ( λ 1 ) λ 2 λ 1 λ 2 I .
f ( A ) = f ( S 1 JS ) = S 1 f ( J ) S .
f ( A ) = S 1 [ f ( λ 1 ) f ( λ 2 ) λ 1 λ 2 J + f ( λ 2 ) λ 1 f ( λ 1 ) λ 2 λ 1 λ 2 I ] S = f ( λ 1 ) f ( λ 2 ) λ 1 λ 2 S 1 JS + f ( λ 2 ) λ 1 f ( λ 1 ) λ 2 λ 1 λ 2 I ,
δ = det ( B ) = μ 1 μ 2 , τ = trace ( B ) = μ 1 + μ 2 ,
δ = det ( A ) = λ 1 λ 2 , t = trace ( A ) = λ 1 + λ 2 .
A 2 = t A d , B 2 = τ B δ .
ξ k + 2 = τ ξ k + 1 δ ξ k , ξ 0 = 0 , ξ 1 = 1 .
ξ k = μ 1 k 1 + μ 1 k 2 μ 2 + + μ 1 μ 1 k 2 + μ 1 k 1 .
B n + 1 = B B n = B ( ξ n B μ 1 μ 2 ξ n 1 I ) .
B n + 1 = ξ n ( τ B δ ) δ ξ n 1 B = ( τ ξ n δ ξ n 1 ) B μ 1 μ 2 ξ n I .
μ 1 n = λ 1 , μ 2 n = λ 2

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