Abstract

There exists a fractional Fourier-transform relation between the amplitude distributions of light on two spherical surfaces of given radii and separation. The propagation of light can be viewed as a process of continual fractional Fourier transformation. As light propagates, its amplitude distribution evolves through fractional transforms of increasing order. This result allows us to pose the fractional Fourier transform as a tool for analyzing and describing optical systems composed of an arbitrary sequence of thin lenses and sections of free space and to arrive at a general class of fractional Fourier-transforming systems with variable input and output scale factors.

© 1995 Optical Society of America

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References

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  1. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [Crossref]
  2. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
    [Crossref]
  3. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.
  4. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [Crossref]
  5. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [Crossref]
  6. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [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. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084 (1994).
    [Crossref]
  9. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [Crossref]
  10. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [Crossref] [PubMed]
  11. A. W. Lohmann, B. H. Soffer, “Relationship between two transforms: Radon–Wigner and fractional Fourier,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.
  12. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
    [Crossref]
  13. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
    [Crossref]
  14. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680.
    [PubMed]
  15. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [Crossref]
  16. A similar limiting process is discussed in P. J. Readon, R. A. Chipman, “Maximum power of refractive lenses: a fundamental limit,” Opt. Lett. 15, 1409–1411 (1990).
    [Crossref]
  17. M. J. Bastiaans, “The Wigner distribution applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [Crossref]
  18. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
    [Crossref]
  19. M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
    [Crossref]
  20. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).
  21. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).
  22. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [Crossref] [PubMed]
  23. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [Crossref] [PubMed]
  24. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [Crossref]

1994 (7)

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[Crossref] [PubMed]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
[Crossref]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[Crossref]

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]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084 (1994).
[Crossref]

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]

1993 (5)

1991 (1)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

1990 (1)

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

1980 (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[Crossref]

1979 (2)

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[Crossref]

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[Crossref]

1978 (1)

M. J. Bastiaans, “The Wigner distribution applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084 (1994).
[Crossref]

Barshan, B.

Bastiaans, M. J.

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[Crossref]

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[Crossref]

M. J. Bastiaans, “The Wigner distribution applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

Bitran, Y.

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[Crossref]

Bonnet, G.

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

Chipman, R. A.

Dorsch, R. G.

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[Crossref]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

Lohmann, A. W.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
[Crossref]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[Crossref]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[Crossref] [PubMed]

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, B. H. Soffer, “Relationship between two transforms: Radon–Wigner and fractional Fourier,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

Mendlovic, D.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
[Crossref]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[Crossref]

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]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[Crossref] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[Crossref]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[Crossref]

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

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680.
[PubMed]

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[Crossref]

Onural, L.

Ozaktas, H. M.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[Crossref] [PubMed]

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]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
[Crossref]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[Crossref]

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

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[Crossref]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[Crossref]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680.
[PubMed]

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]

Readon, P. J.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[Crossref]

Soffer, B. H.

A. W. Lohmann, B. H. Soffer, “Relationship between two transforms: Radon–Wigner and fractional Fourier,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[Crossref]

Appl. Opt. (2)

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[Crossref] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[Crossref]

IEEE Trans. Signal Process. (1)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084 (1994).
[Crossref]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

J. Inst. Math. Its Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[Crossref]

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

Opt. Commun. (5)

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[Crossref]

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

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
[Crossref]

M. J. Bastiaans, “The Wigner distribution applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

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

Opt. Lett. (4)

Optik (2)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

Other (3)

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[Crossref]

A. W. Lohmann, B. H. Soffer, “Relationship between two transforms: Radon–Wigner and fractional Fourier,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.

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Figures (4)

Fig. 1
Fig. 1

Two spherical surfaces. This figure is drawn such that R1 < 0 and R2 > 0. The distance d is always taken to be positive.

Fig. 2
Fig. 2

Canonical realization Type II.

Fig. 3
Fig. 3

Canonical realization Type I.

Fig. 4
Fig. 4

Single-lens imaging.

Equations (76)

Equations on this page are rendered with MathJax. Learn more.

( F a q ̂ ) ( u ) B a ( u , u ) q ̂ ( u ) d u , B a ( u , u ) exp [ i ( π ϕ ̂ / 4 ϕ / 2 ) ] | sin ϕ | 1 / 2 × exp [ i π ( u 2 cot ϕ 2 u u csc ϕ + u 2 cot ϕ ) ] ,
ϕ a π 2
p 2 ( x ) = q 2 ( x ) exp ( i π x 2 / λ R 2 ) ,
p 1 ( x ) = q 1 ( x ) exp ( i π x 2 / λ R 1 ) ,
p 2 ( x ) = exp ( i 2 π d / λ ) i λ d exp [ i π ( x x ) 2 / λ d ] p 1 ( x ) d x .
q ̂ 2 ( u ) = exp ( i 2 π d / λ ) s 1 i λ d × exp [ i π λ d ( g 2 s 2 2 u 2 2 s 1 s 2 u u + g 1 s 1 2 u 2 ) ] × q ̂ 1 ( u ) d u ,
g 1 1 + d / R 1 ,
g 2 1 d / R 2 .
q ̂ 2 ( u ) = { exp ( i 2 π d / λ ) s 1 exp [ i ( π ϕ ̂ / 4 ϕ / 2 ) ] | sin ϕ | 1 / 2 i λ d } × ( F a q ̂ 1 ) ( u ) ,
g 2 s 2 2 λ d = cot ϕ ,
g 1 s 1 2 λ d = cot ϕ ,
s 1 s 2 λ d = csc ϕ .
g 1 s 1 2 = g 2 s 2 2 .
s 2 4 = ( λ d ) 2 ( g 2 / g 1 g 2 2 ) 1 ,
s 1 4 = ( λ d ) 2 ( g 1 / g 2 g 1 2 ) 1 .
0 g 1 g 2 1 .
tan ϕ = ± [ ( 1 / g 1 g 2 ) 1 ] 1 / 2 ,
d = ( s 1 s 2 / λ ) sin ϕ .
1 + d / R 1 g 1 = ( s 2 / s 1 ) cos ϕ ,
1 d / R 2 g 2 = ( s 1 / s 2 ) cos ϕ .
tan ϕ = λ d g 1 s 1 2 ,
s 2 2 = g 1 2 s 1 2 + ( λ d ) 2 s 1 2 ,
1 d / R 2 g 2 = g 1 s 1 4 g 1 2 s 1 4 + ( λ d ) 2 .
q ̂ 2 ( u ) = [ exp ( i 2 π d / λ ) exp ( i a π / 4 ) s 1 / s 2 ] ( F a q ̂ 1 ) ( u ) .
p 2 ( x ) = [ exp ( i 2 π d / λ ) exp ( i a π / 4 ) s 1 / s 2 ] exp [ ( i π x 2 / λ R 2 ) ] × ( F a t ̂ ) ( x / s 2 ) ,
a π 2 ϕ = arctan ( λ d s 1 2 ) ,
s 2 = s 1 [ 1 + ( λ d ) 2 s 1 4 ] 1 / 2 ,
R 2 = d [ 1 + ( s 1 4 ) ( λ d ) 2 ] ,
g ( s 2 / λ d ) = cot ϕ ,
s 2 / λ d = csc ϕ ,
g = cos ϕ .
d = ( s 2 / λ ) sin ϕ ,
f = ( s 2 / λ ) cot ( ϕ / 2 ) .
p ̂ out ( u ) = exp ( i 2 π l / λ ) exp ( i a π / 4 ) ( F a p ̂ in ) ( u ) ,
d = ( s 2 / λ ) sin ϕ ,
f = ( s 2 / λ ) cot ( ϕ / 2 ) .
ϕ = arccos ( 1 d / f ) ,
s 4 = λ 2 d f 2 d / f .
d = ( s 2 / λ ) tan ( ϕ / 2 ) ,
f = ( s 2 / λ ) csc ϕ .
ϕ = arccos ( 1 d / f ) ,
s 4 = λ 2 d f ( 2 d / f ) .
1 / R + = ( 1 / R ) ( 1 / f ) .
s + = s .
ϕ o + ϕ i = ± π .
cot ϕ o = g ( s 2 / λ d o ) ,
cot ϕ i = g + ( s + 2 / λ d i ) .
1 / f = ( 1 / d 0 ) + ( 1 / d i ) .
s o s / λ d o = csc ϕ o ,
s + s i / λ d i = csc ϕ i .
M s i / s o = d i / d o ,
g i + = 1 + d i / R i + ,
cot ϕ i = g i + s i + 2 / λ d i ,
s ( i + 1 ) = ( λ d i / s i + ) csc ϕ i ,
s ( i + 1 ) + = s ( i + 1 ) ,
g ( i + 1 ) = ( λ d i / s ( i + 1 ) 2 ) cot ϕ i ,
R ( i + 1 ) = d i 1 g ( i + 1 ) ,
R ( i + 1 ) + = f i + 1 R ( i + 1 ) f i + 1 R ( i + 1 ) .
n 2 ( x ) = n 0 2 [ 1 ( x / ξ ) 2 ] ,
ϕ = l / ξ ,
s 2 = λ ξ / n 0 .
π ϕ ( x / s ) 2 ,
ϕ = ( λ / s 2 ) ,
p out ( x ) = h ( x , x ) p in ( x ) d x , h ( x , x ) = C exp [ i π ( α x 2 2 β x x + γ x 2 ) ] ,
α s 2 2 = cot ϕ = γ s 1 2 ,
β s 1 s 2 = csc ϕ
s 1 4 = ( β 2 γ / α γ 2 ) 1 ,
s 2 4 = ( β 2 α / γ α 2 ) 1 ,
tan ϕ = ± ( β 2 α / γ 1 ) 1 / 2 ,
[ A B C D ] [ γ / β 1 / β β + α γ / β α / β ] = [ α / β 1 / β β α γ / β γ / β ] 1 ,
[ Ā B ¯ C ¯ D ¯ ] [ γ ¯ / β ¯ 1 / β ¯ β ¯ + α ¯ γ ¯ / β ¯ α ¯ / β ¯ ] = [ A s 1 / s 2 B / s 1 s 2 C s 1 s 2 D s 2 / s 1 ] .
[ cos ϕ sin ϕ sin ϕ cos ϕ ] .
s 1 4 = B 2 ( A / D A 2 ) 1 ,
s 2 4 = B 2 ( D / A D 2 ) 1 ,
cos ϕ = ± ( A D ) 1 / 2 ,
[ A B C D ] = [ 1 0 K 2 1 ] [ A B C D ] [ 1 0 K 1 1 ] ,

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