Abstract

An extension of the Fresnel transform to first-order optical systems that can be represented by an ABCD matrix is analyzed. We present and discuss a definition of the generalized transform, which is recognized to belong to the class of linear canonical transforms. A general mathematical characterization is performed by listing a number of meaningful theorems that hold for this operation and can be exploited for simplyfying the analysis of optical systems. The relevance to physics of this transform and of the theorems is stressed. Finally, a comprehensive number of possible decompositions of the generalized transform in terms of elementary optical transforms is discussed to obtain further insight into this operation.

© 1997 Optical Society of America

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References

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  1. F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994).
  2. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. XX, p. 777.
  3. A. Yariv, Optical Electronics (Saunders College, Fort Worth, Tex., 1991), Chap. 2, p. 51.
  4. A. Papoulis, “Pulse compression, fiber communication, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13 (1994).
    [CrossRef]
  5. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1174 (1970).
    [CrossRef]
  6. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  7. T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
    [CrossRef]
  8. R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 3, p. 49.
  9. P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electron. 1, 37–44 (1969).
    [CrossRef]
  10. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9, p. 381.
  11. M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation lossless system,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  12. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  13. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
    [CrossRef] [PubMed]
  14. D. F. V. James, G. S. Agarwal, “Generalized Radon transform for tomographic measurements of short pulses,” J. Opt. Soc. Am. B 12, 704–708 (1995).
    [CrossRef]
  15. D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
    [CrossRef]
  16. W. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Chap. 12, p. 390.
  17. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986), Chap. 6, p. 104.
  18. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
    [CrossRef]
  19. H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 48, 397–399 (1980).
    [CrossRef]
  20. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  21. G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
    [CrossRef]

1996 (1)

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

1995 (3)

1994 (3)

1982 (1)

1981 (1)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

1980 (1)

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

1979 (1)

1970 (1)

1969 (1)

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electron. 1, 37–44 (1969).
[CrossRef]

1966 (1)

Abe, S.

Agarwal, G. S.

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

D. F. V. James, G. S. Agarwal, “Generalized Radon transform for tomographic measurements of short pulses,” J. Opt. Soc. Am. B 12, 704–708 (1995).
[CrossRef]

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

Agulló-López, F.

T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

Alieva, T.

T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

Arsenault, H. H.

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

Bastiaans, M. J.

Baues, P.

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electron. 1, 37–44 (1969).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986), Chap. 6, p. 104.

Collins, S. A.

Feynman, R. P.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 3, p. 49.

Flannery, B. P.

W. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Chap. 12, p. 390.

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994).

Hibbs, A. R.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 3, p. 49.

James, D. F. V.

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

D. F. V. James, G. S. Agarwal, “Generalized Radon transform for tomographic measurements of short pulses,” J. Opt. Soc. Am. B 12, 704–708 (1995).
[CrossRef]

Kogelnik, H.

Li, T.

Mendlovic, D.

Nazarathy, M.

Ozaktas, H. M.

Papoulis, A.

Press, W.

W. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Chap. 12, p. 390.

Shamir, J.

Sheridan, J. T.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. XX, p. 777.

Simon, R.

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

Teukolsky, S. A.

W. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Chap. 12, p. 390.

Vetterling, W. T.

W. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Chap. 12, p. 390.

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9, p. 381.

Yariv, A.

A. Yariv, Optical Electronics (Saunders College, Fort Worth, Tex., 1991), Chap. 2, p. 51.

Am. J. Phys. (1)

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

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

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

Opt. Commun. (4)

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

T. Alieva, F. Agulló-López, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Opt. Lett. (1)

Opto-Electron. (1)

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electron. 1, 37–44 (1969).
[CrossRef]

Other (7)

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9, p. 381.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 3, p. 49.

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. XX, p. 777.

A. Yariv, Optical Electronics (Saunders College, Fort Worth, Tex., 1991), Chap. 2, p. 51.

W. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Chap. 12, p. 390.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986), Chap. 6, p. 104.

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Equations (44)

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Eα[f(ξ)](x)=f^α(x)=-iα-+ f(ξ)exp[iπα(x-ξ)2]dξ,
f(x)=iα-+ f^α(ξ)exp[-iπα(x-ξ)2]dξ.
Vz(x)=-iλz exp(ikz)-+ V0(ξ)expi k2z (x-ξ)2dξ,
Vz(x)=-iλB exp(ikl)-+ V0(ξ)×expi πλB (Aξ2-2xξ+Dx2)dξ,
W=ACBD
EA,B,D[f(ξ)](x)=f^A,B,D(x)=-iλB -+ f(ξ)×expi πλB (Aξ2-2xξ+Dx2)dξ,
W-1=D-C-BA,
f(x)=iλB -+ f^A,B,D(ξ)×exp-i πλB (Dξ2-2xξ+Ax2)dξ,
EA,B,D[f(ξ)](x)=expi πCλA x2A EA/(λB)[f(ξ)](x/A),
g˜(ν)=F{g(x)}(ν)=-+ g(x)exp(-2πiνx)dx
F{Eα[f(ξ)](x)}(ν)=f˜(ν)exp-i πα ν2.
F{f^A,B,D}(ν)=iλA/C exp-i πλAC ν2 *[Af˜(Aν)exp(-iπλABν2)],
F{f^A,B,D}(ν)=Af˜(Aν)exp(-iπλABν2)
EA,B,D[f(ξ)](x)=expi πCλA x2F-1{A×exp(-iπλABν2)F[f(ξ)](Aν)}
EA,B,D[f(ξ)exp(iπβξ2)](x)=EA+βλB,B,D[f(ξ)](x).
A=A+λβB,C=C+λβD.
EA,B,D[f(ξ/M)](x)=EA,B/M2,D[f(ξ)](x/M).
(f^A,B,Dg^A,B,D)(x)=-+ f^A,B,D(ξ+x)g^A,B,D*(ξ)dξ=-+ f(ξ+Dx)g*(ξ)×expik2 C(2xξ+Dx2)dξ.
(f^A,B,Dg^A,B,D)(x)=(fg)(Dx).
EA,B,D[(fg)(ξ)](x)=-+ f^A,B,D(Aξ+x)g*(ξ)×exp-ik2 C(Aξ2+2Dxξ)dξ.
EA,B,D[(fg)(ξ)](x)=(f^A,B,Dg)(x).
(f * g)(x)=-+ f(x-ξ)g(ξ)dξ,
EA,B,D[(f * g)(ξ)](x)=-+ f^A,B,D(x-Aξ)g(ξ)×exp-ik2 C(Aξ2-2xξ)dξ.
EA,B,D[(f * g)(ξ)](x)=(f^A,B,D * g)(x)
-+ f^A,B,D(ξ)g^A,B,D*(ξ)dξ=-+ f(ξ)g*(ξ)dξ,
-+|f^A,B,D(x)|2dx=-+|f(x)|2dx,
EA,B,D[f(ξ-x0)](x)=exp-ik2 Cx0(Ax0-2x)×EA,B,D[f(ξ)](x-Ax0),
EA,B,D[f(ξ)exp(ik0ξ)](x)
=expiDk0x-k02k B×EA,B,D[f(ξ)]x-k0k B.
Eαdfdξ(x)=df^αdx.
EA,B,Ddfdξ(x)=Adf^A,B,D(x)dx-i2πCλA xf^A,B,D(x).
f^A,B,D(x)=exp(-iπDx2/λB)f˜xλB.
Eα[ξf(ξ)](x)=xf^α(x)+i2πα df^α(x)dx.
EA,B,D[ξf(ξ)](x)=Dxf^A,B,D(x)+iλB2π df^A,B,D(x)dx.
f(x)=exp-i πAλB x2n=-+expi πλAB4x02 n2×fλ|B|2x0 nsinc2x0λ|B| x-n,
sinc(y)=sin πyπy.
f^A,B,D(x)=expi πDλB x2n=-+exp-i πλDB4x02 n2×f^A,B,Dλ|B|2x0 nsinc2x0λ|B| x-n.
ACBD=1C/A01 A001/A 10B/A1,
ACBD=A001/A 1AC01 10B/A1,
ACBD=1C/A01 10AB1 A001/A,
ACBD=10B/D1 1CD01 1/D00D,
ACBD=10B/D1 1/D00D 1C/D01,
ACBD=1/D00D 10BD1 1C/D01.
ACBD=cos α-sin αsin αcos α.

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