Abstract

The parameters of the transfer matrix describing a first-order optical system that is a cascade of k identical subsystems defined by the transfer matrix M are determined from consideration of the subsystem’s eigenfunctions. A condition for the cascade to be cyclic is derived. Particular examples of cyclic first-order optical systems are presented. Structure and properties of eigenfunctions of cyclic transforms are considered. A method of optical signal encryption by use of cyclic first-order systems is proposed.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Berkeley, Calif., 1966).
  2. M. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  3. T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
    [CrossRef]
  4. T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998).
    [CrossRef]
  5. T. Alieva, M. J. Bastiaans, “Radon–Wigner transform for optical field analysis,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, Kehar Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, 132–135.
  6. D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
    [CrossRef]
  7. D. Stolen, “Operator methods in physical optics,” J. Opt. Soc. Am. 71, 334–341 (1981).
    [CrossRef]
  8. M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  9. C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).
  10. T. Alieva, F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
    [CrossRef]
  11. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  12. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  13. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transform and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  14. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  15. J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
    [CrossRef]
  16. M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991).
    [CrossRef]
  17. G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
    [CrossRef]
  18. A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
    [CrossRef]
  19. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self-Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
    [CrossRef]
  20. T. Alieva, “On the self-fractional Fourier functions,” J. Phys. A 29, L377–L379 (1996).
    [CrossRef]
  21. T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
    [CrossRef]
  22. T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
    [CrossRef]
  23. D. Choudhury, P. N. Puntambekar, A. K. Chakraborty, “Optical synthesis of self-Fourier functions,” Opt. Commun. 119, 279–282 (1995).
    [CrossRef]
  24. B. Javidi, “Securing information with optical technologies,” Phys. Today 50, No. 3, 27–32 (1997).
    [CrossRef]

1998

T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998).
[CrossRef]

1997

T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

B. Javidi, “Securing information with optical technologies,” Phys. Today 50, No. 3, 27–32 (1997).
[CrossRef]

1996

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

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

T. Alieva, “On the self-fractional Fourier functions,” J. Phys. A 29, L377–L379 (1996).
[CrossRef]

1995

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

D. Choudhury, P. N. Puntambekar, A. K. Chakraborty, “Optical synthesis of self-Fourier functions,” Opt. Commun. 119, 279–282 (1995).
[CrossRef]

J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
[CrossRef]

1994

M. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

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

1993

1992

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
[CrossRef]

1991

M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

1982

1981

1980

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

Agarwal, G. S.

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

Agullo-Lopez, F.

T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

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

Alieva, T.

T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998).
[CrossRef]

T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
[CrossRef]

T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

T. Alieva, “On the self-fractional Fourier functions,” J. Phys. A 29, L377–L379 (1996).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

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

T. Alieva, M. J. Bastiaans, “Radon–Wigner transform for optical field analysis,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, Kehar Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, 132–135.

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Barbé, A.

T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
[CrossRef]

T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, “Radon–Wigner transform for optical field analysis,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, Kehar Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, 132–135.

Beck, M.

M. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Caola, M. J.

M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

Chakraborty, A. K.

D. Choudhury, P. N. Puntambekar, A. K. Chakraborty, “Optical synthesis of self-Fourier functions,” Opt. Commun. 119, 279–282 (1995).
[CrossRef]

Choudhury, D.

D. Choudhury, P. N. Puntambekar, A. K. Chakraborty, “Optical synthesis of self-Fourier functions,” Opt. Commun. 119, 279–282 (1995).
[CrossRef]

Cincotti, G.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

Cohen, N.

Gomez-Reino, C.

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

Gori, F.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

James, D. F. V.

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

Javidi, B.

B. Javidi, “Securing information with optical technologies,” Phys. Today 50, No. 3, 27–32 (1997).
[CrossRef]

Lohmann, A. W.

Luneburg, R. K.

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

McAlister, D. F.

M. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Mendlovic, D.

Namias, V.

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

Nazarathy, M.

Ozaktas, H. M.

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

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

Puntambekar, P. N.

D. Choudhury, P. N. Puntambekar, A. K. Chakraborty, “Optical synthesis of self-Fourier functions,” Opt. Commun. 119, 279–282 (1995).
[CrossRef]

Raymer, M.

M. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Santarsiero, M.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

Shamir, J.

Stolen, D.

IEEE Trans. Signal Process.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Int. J. Optoelectron.

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

J. Inst. Math. Appl.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

T. Alieva, “On the self-fractional Fourier functions,” J. Phys. A 29, L377–L379 (1996).
[CrossRef]

T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

Opt. Commun.

T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
[CrossRef]

D. Choudhury, P. N. Puntambekar, A. K. Chakraborty, “Optical synthesis of self-Fourier functions,” Opt. Commun. 119, 279–282 (1995).
[CrossRef]

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

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

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

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

Phys. Rev. Lett.

M. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Phys. Today

B. Javidi, “Securing information with optical technologies,” Phys. Today 50, No. 3, 27–32 (1997).
[CrossRef]

Other

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

T. Alieva, M. J. Bastiaans, “Radon–Wigner transform for optical field analysis,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, Kehar Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, 132–135.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (55)

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

fo(u)=RM[fi(x)](u)=-fi(x)KM(x, u)d x,
KM(x, u)
=(1/iB)exp[iπ(Ax2+Du2-2xu)/B]B0A exp(iπCu2/A)δ(x-Au)B=0
M=ABCD,
RM2RM1=RM3,
Mk=ξkM-μkI,
μk=λ1λ2ξk-1,
ξk=λ1k-1+λ1k-2λ2++λ1λ2k-2+λ2k-1,
RM[ fM(x)](u)=afM(u),
Φn(x)=(π2nλn!)-1/2×exp[-12(1+iβ)(x/λ)2]Hn(x/λ)
θ=arccos[12(A+D)],
λ2=2B[4-(A+D)2]-1/2,
β=(A-D)[4-(A+D)2]-1/2.
RM[Φn(x)](u)=exp[-i(n+12)θ]Φn(u)
A=cos θ+β sin θ,
B=λ2 sin θ,
C=-[(β2+1)/λ2]sin θ,
D=cos θ-β sin θ.
Φn(x)=(π2nλn!)-1/2 exp[-12(x/λ)2]Hn(x/λ).
2π(N+ϕ)=-(n+12)θ,
Φn(x)=[π2n exp(iπ/4)n!]-1/2×exp(12ix2)Hn[x exp(-iπ/4)].
Φ0(x)=[π exp(iπ/4)]-1/2 exp(12ix2)
A(k)=cos kθ+β sin kθ,
B(k)=λ2 sin kθ,
C(k)=-[(β2+1)/λ2]sin kθ,
D(k)=cos kθ-β sin kθ.
A(k)=cos kθ+12(A-D)sin kθ/sin θ,
B(k)=B sin kθ/sin θ,
C(k)=C sin kθ/sin θ,
D(k)=cos kθ-12(A-D)sin kθ/sin θ,
θ=2πm/k,
Mk=I=1001.
12(A+D)=cos(2πm/k)
Ak=cos(2π/k)+β sin(2π/k),
Bk=λ2 sin(2π/k),
Ck=-[(β2+1)/λ2]sin(2π/k),
Dk=cos(2π/k)-β sin(2π/k).
Mj=Mkmj=Mkl+Nk=Mkl=Ml/m,
M=AB-(A2+1)/B-A.
KI1/4(x, u)=(1/iB)exp{iπ[A(x2-u2)-2xu]/B}.
M=1-z2/fz1+z2-z1z2/f-1/f1-z1/f,
I1/4=(z1-z2)/2 f-1/f(z12+z22)/2 f-(z1-z2)/2 f.
I1/4=-12 f-1/f1.
Mk=01-10,
N+ϕ=-(n+12)/k,
n=L+kl,
fkL(u)=l=0gL+klΦL+kl(u),
g(u)=n=0gnΦn(u).
g(u)=L=0k-1l=0gL+klΦL+kl(u)=L=0k-1fkL(u),
fkL(u)=1k l=0k-1 expi2π(L+12)lkRMl[g(x)](u).
g(u)fk0(u)fkL(u)fkk-1(u)a0aLak-1a0 fk0(u)aL fkL(u)ak-1 fkk-1(u)
G(u).
G(u)a0 fk0(u)aL fkL(u)ak-1 fkk-1(u)a0-1aL-1ak-1fk0(u)fkL(u)fkk-1(u)
 g(u);
G(u)=[ g(u)(a0+a1)+g(-u)(a0-a1)]/2,

Metrics