Abstract

We study the problem of paraxial propagation in two grade-index media by using invariant techniques that allow a continuous solution of the problem. By using the well-known fact that this problem is analogous to the time-dependent harmonic oscillator in quantum mechanics, known methods there may be imported producing on the one hand a solution to the propagation problem and on the other hand a realization of a quantum-mechanical invariant.

© 2009 Optical Society of America

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References

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  1. S. Chávez-Cerda, J. R. Moya-Cessa, and H. M. Moya-Cessa, J. Opt. Soc. Am. B 24, 404 (2007) and references therein.
    [CrossRef]
  2. R. Mar-Sarao and H. Moya-Cessa, Opt. Lett. 33, 1966 (2008).
    [CrossRef] [PubMed]
  3. C. Gomez-Reino, M. Victoria-Pérez, C. Bao, and M. T. Flores-Arias, Laser Photonics Rev. 2, 203 (2008).
    [CrossRef]
  4. C. Gomez-Reino, M. Victoria-Pérez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).
  5. B. A. Malomed, Soliton Management in Periodic Systems (Springer Verlag, 2006).
  6. H. R. Lewis, Jr., Phys. Rev. Lett. 18, 510 (1967).
    [CrossRef]
  7. H. S. Eisenberg and Y. Silberberg, Phys. Rev. Lett. 85, 1863 (2000).
    [CrossRef] [PubMed]
  8. J. A. Arnaud, Appl. Opt. 8, 1909 (1969).
    [CrossRef] [PubMed]
  9. A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford U. Press, 2007).
  10. M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A 36, 2069 (2003).
    [CrossRef]
  11. H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
    [CrossRef]
  12. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).
    [CrossRef]
  13. R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).
    [CrossRef]

2008 (2)

C. Gomez-Reino, M. Victoria-Pérez, C. Bao, and M. T. Flores-Arias, Laser Photonics Rev. 2, 203 (2008).
[CrossRef]

R. Mar-Sarao and H. Moya-Cessa, Opt. Lett. 33, 1966 (2008).
[CrossRef] [PubMed]

2007 (1)

2003 (2)

M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A 36, 2069 (2003).
[CrossRef]

H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
[CrossRef]

2000 (1)

H. S. Eisenberg and Y. Silberberg, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

1969 (1)

1967 (1)

H. R. Lewis, Jr., Phys. Rev. Lett. 18, 510 (1967).
[CrossRef]

1948 (1)

R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).
[CrossRef]

Arnaud, J. A.

Bao, C.

C. Gomez-Reino, M. Victoria-Pérez, C. Bao, and M. T. Flores-Arias, Laser Photonics Rev. 2, 203 (2008).
[CrossRef]

C. Gomez-Reino, M. Victoria-Pérez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).

Chávez-Cerda, S.

Eisenberg, H. S.

H. S. Eisenberg and Y. Silberberg, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Fernández-Guasti, M.

H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
[CrossRef]

M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A 36, 2069 (2003).
[CrossRef]

Feynman, R. P.

R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).
[CrossRef]

Flores-Arias, M. T.

C. Gomez-Reino, M. Victoria-Pérez, C. Bao, and M. T. Flores-Arias, Laser Photonics Rev. 2, 203 (2008).
[CrossRef]

Gomez-Reino, C.

C. Gomez-Reino, M. Victoria-Pérez, C. Bao, and M. T. Flores-Arias, Laser Photonics Rev. 2, 203 (2008).
[CrossRef]

C. Gomez-Reino, M. Victoria-Pérez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).

Lewis, H. R.

H. R. Lewis, Jr., Phys. Rev. Lett. 18, 510 (1967).
[CrossRef]

Malomed, B. A.

B. A. Malomed, Soliton Management in Periodic Systems (Springer Verlag, 2006).

Mar-Sarao, R.

Moya-Cessa, H.

R. Mar-Sarao and H. Moya-Cessa, Opt. Lett. 33, 1966 (2008).
[CrossRef] [PubMed]

H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
[CrossRef]

M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A 36, 2069 (2003).
[CrossRef]

Moya-Cessa, H. M.

Moya-Cessa, J. R.

Schleich, W. P.

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).
[CrossRef]

Silberberg, Y.

H. S. Eisenberg and Y. Silberberg, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Victoria-Pérez, M.

C. Gomez-Reino, M. Victoria-Pérez, C. Bao, and M. T. Flores-Arias, Laser Photonics Rev. 2, 203 (2008).
[CrossRef]

C. Gomez-Reino, M. Victoria-Pérez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).

Yariv, A.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford U. Press, 2007).

Yeh, P.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford U. Press, 2007).

Appl. Opt. (1)

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

J. Phys. A (1)

M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A 36, 2069 (2003).
[CrossRef]

Laser Photonics Rev. (1)

C. Gomez-Reino, M. Victoria-Pérez, C. Bao, and M. T. Flores-Arias, Laser Photonics Rev. 2, 203 (2008).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

H. R. Lewis, Jr., Phys. Rev. Lett. 18, 510 (1967).
[CrossRef]

H. S. Eisenberg and Y. Silberberg, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).
[CrossRef]

Other (4)

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford U. Press, 2007).

C. Gomez-Reino, M. Victoria-Pérez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).

B. A. Malomed, Soliton Management in Periodic Systems (Springer Verlag, 2006).

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Arbitrary field created at z = 0 and propagated through a first GRIN medium, z L (left side), and a second GRIN medium, z > L (right side).

Fig. 2
Fig. 2

Example of a step function for the functions ν ( z ) , μ ( z ) , or β ( z ) .

Fig. 3
Fig. 3

Solution of the Ermakov equation for the function of Fig. 2.

Equations (15)

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

2 i k 0 E z = 2 E + k 2 ( x , y , z ) E ,
k 2 ( x , y , z ) = { k 1 2 = β 1 2 + ν 1 2 2 x 2 + μ 1 2 2 y 2 , 0 < z < L k 2 2 = β 2 2 + ν 2 2 2 x 2 + μ 2 2 2 y 2 z L } ,
i E z = ( p x 2 + ν 2 ( z ) x 2 2 p y 2 + μ 2 ( z ) y 2 2 + β ( z ) ) E ,
α ( z ) = { α 1 z < L α 2 z L } ,
E = e i z β ( z ) d z ϵ ,
i ϵ z = ( p x 2 + ν 2 ( z ) x 2 2 p y 2 + μ 2 ( z ) y 2 2 ) ϵ .
H ̂ ( t ) = 1 2 ( p ̂ q 2 + Ω 2 ( t ) q ̂ 2 ) ,
I ̂ = 1 2 [ ( q ̂ ρ q ) 2 + ( ρ q p ̂ ρ ̇ q q ̂ ) 2 ] ,
ρ ̈ q + Ω 2 ( t ) ρ q = ρ q 3 .
T q = S q D q , q = x , y ,
S q = e i ( q p q + p q q ) ln ρ q 2 ,
D q = exp { i q 2 2 ρ q d ρ q d z } .
ϵ ( x , y , z ) = T x e i z d z ρ x 2 ( z ) ( N x + 1 2 ) ϵ 1 ( x ) T y e i z d z ρ y 2 ( z ) ( N y + 1 2 ) ϵ 2 ( y ) ,
u n ( q ) = 1 2 n n ! H n ( q ) e q 2 2 ,
ϵ j ( q ) = n = 0 C n ( j ) u n ( q ) ,

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