Abstract

We analyze theoretically the dispersion of linearly polarized light propagating in a uniaxial anisotropic medium where multibeam interference is present. Explicit expressions of the group-delay dispersion for transmitting waves are derived for the simplest situation, and the effect of dispersion on pulse broadening is analyzed for a few selected cases. Our results reveal that at normal incidence and in the situation where the optic axis is parallel to the surface of birefringent plate (in the xy plane), the dispersion of the refracted wave decreases with the extent of birefringence. In particular, the dispersion for the electric field parallel to the polarization direction of the incident light changes with the rotation angle between the optic axis and the polarization direction of the incident field, whereas the dispersion for the refracted field whose direction is vertical to the polarization of incident light is independent of this angle. For oblique incidence, dispersion varies substantially for different incident angles. In the situation where the optic axis is in the xz plane at either normal or oblique incidence, the dispersion increases in a periodically oscillating manner as a function of the relative thickness of the birefringent plate.

© 2005 Optical Society of America

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  8. J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from stratified anisotropic media,” IEEE Trans. Antennas Propag. 39, 35–39 (1991).
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  12. A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
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  13. M. Ware, W. E. Dibble, S. A. Glasgow, J. Peatross, “Energy flow in angularly dispersive optical systems,” J. Opt. Soc. Am. B 18, 839–845 (2001).
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  14. G. F. Torres del Castillo, C. J. Pérez Ballinas, “Total reflection of light propagation from an isotropic medium to an anisotropic medium,” Opt. Lett. 26, 1251–1252 (2001).
    [CrossRef]
  15. A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 20, 333–341 (2003).
  16. L. Dettwiller, “Shape evolution of a light pulse in a linear birefringent medium,” J. Opt. Soc. Am. A 21, 288–297 (2004).
    [CrossRef]
  17. Y. Jia, Y. Zhao, X. Zhu, “Dispersion characteristics of F–P etalons in devices of optical communications,” J. Optoelectron. Laser 15, 156–159 (2004) (in Chinese).
  18. G. P. Agrawal, Nonlinear Fiber Optics and Applications of Nonlinear Fiber Optics, in Chinese: translated from English into Chinese by D. F. Jia, Z. H. Yu, B. Tan, Z. Y. Hu (Publishing House of Electronics Industry, Beijing, 2002), Chap. 3, pp. 42–62.

2004

L. Dettwiller, “Shape evolution of a light pulse in a linear birefringent medium,” J. Opt. Soc. Am. A 21, 288–297 (2004).
[CrossRef]

Y. Jia, Y. Zhao, X. Zhu, “Dispersion characteristics of F–P etalons in devices of optical communications,” J. Optoelectron. Laser 15, 156–159 (2004) (in Chinese).

2003

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 20, 333–341 (2003).

A. V. Zvyagin, E. D. J. Smith, D. D. Sampson, “Delay and dispersion characteristics of a frequency-domain optical delay line for scanning interferometry,” J. Opt. Soc. Am. A 20, 333–341 (2003).
[CrossRef]

2001

1993

1992

W. Zhang, “General ray-tracing formulas for crystal,” Appl. Opt. 31, 7328–7331 (1992).
[CrossRef] [PubMed]

M. Deschamps, P. Chevée, “Reflection and refraction of a heterogeneous plane wave by a solid layer,” Wave Motion 15, 61–75 (1992).
[CrossRef]

1991

J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from stratified anisotropic media,” IEEE Trans. Antennas Propag. 39, 35–39 (1991).
[CrossRef]

1986

1983

1979

1977

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “Response of Fabry–Perot interferometers to amplitude-modulated light beams,” Opt. Acta 24, 1217–1236 (1977).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics and Applications of Nonlinear Fiber Optics, in Chinese: translated from English into Chinese by D. F. Jia, Z. H. Yu, B. Tan, Z. Y. Hu (Publishing House of Electronics Industry, Beijing, 2002), Chap. 3, pp. 42–62.

Carroll, J. B.

Cesini, G.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “Response of Fabry–Perot interferometers to amplitude-modulated light beams,” Opt. Acta 24, 1217–1236 (1977).
[CrossRef]

Chevée, P.

M. Deschamps, P. Chevée, “Reflection and refraction of a heterogeneous plane wave by a solid layer,” Wave Motion 15, 61–75 (1992).
[CrossRef]

Ciattoni, A.

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 20, 333–341 (2003).

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

Cincotti, G.

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 20, 333–341 (2003).

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

Corones, J.

Deschamps, M.

M. Deschamps, P. Chevée, “Reflection and refraction of a heterogeneous plane wave by a solid layer,” Wave Motion 15, 61–75 (1992).
[CrossRef]

O. Poncelet, M. Deschamps, “Reflection and refractionof an inhomogeneous plane wave on a fluid/anisotropic solid interface,” in Proceedings of the IEEE Ultrasonics Symposium (IEEE Press, Piscataway, N.J., 1994), pp. 753–756.

Dettwiller, L.

Dibble, W. E.

Feit, M. D.

Fleck, J. A.

Glasgow, S. A.

Guattari, G.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “Response of Fabry–Perot interferometers to amplitude-modulated light beams,” Opt. Acta 24, 1217–1236 (1977).
[CrossRef]

Huber, D.

Jia, Y.

Y. Jia, Y. Zhao, X. Zhu, “Dispersion characteristics of F–P etalons in devices of optical communications,” J. Optoelectron. Laser 15, 156–159 (2004) (in Chinese).

Lucarini, G.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “Response of Fabry–Perot interferometers to amplitude-modulated light beams,” Opt. Acta 24, 1217–1236 (1977).
[CrossRef]

Palma, C.

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 20, 333–341 (2003).

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “Response of Fabry–Perot interferometers to amplitude-modulated light beams,” Opt. Acta 24, 1217–1236 (1977).
[CrossRef]

Peatross, J.

Pérez Ballinas, C. J.

Poncelet, O.

O. Poncelet, M. Deschamps, “Reflection and refractionof an inhomogeneous plane wave on a fluid/anisotropic solid interface,” in Proceedings of the IEEE Ultrasonics Symposium (IEEE Press, Piscataway, N.J., 1994), pp. 753–756.

Sampson, D. D.

Smith, E. D. J.

Stewart, R.

Titchener, J. B.

J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from stratified anisotropic media,” IEEE Trans. Antennas Propag. 39, 35–39 (1991).
[CrossRef]

Torres del Castillo, G. F.

Ware, M.

Willis, J. R.

J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from stratified anisotropic media,” IEEE Trans. Antennas Propag. 39, 35–39 (1991).
[CrossRef]

Yeh, P.

Yu, J.

Yuan, S.

Zhang, W.

Zhao, Y.

Y. Jia, Y. Zhao, X. Zhu, “Dispersion characteristics of F–P etalons in devices of optical communications,” J. Optoelectron. Laser 15, 156–159 (2004) (in Chinese).

Zhu, X.

Y. Jia, Y. Zhao, X. Zhu, “Dispersion characteristics of F–P etalons in devices of optical communications,” J. Optoelectron. Laser 15, 156–159 (2004) (in Chinese).

Zvyagin, A. V.

Appl. Opt.

IEEE Trans. Antennas Propag.

J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from stratified anisotropic media,” IEEE Trans. Antennas Propag. 39, 35–39 (1991).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Optoelectron. Laser

Y. Jia, Y. Zhao, X. Zhu, “Dispersion characteristics of F–P etalons in devices of optical communications,” J. Optoelectron. Laser 15, 156–159 (2004) (in Chinese).

Opt. Acta

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “Response of Fabry–Perot interferometers to amplitude-modulated light beams,” Opt. Acta 24, 1217–1236 (1977).
[CrossRef]

Opt. Commun.

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 20, 333–341 (2003).

Opt. Lett.

Wave Motion

M. Deschamps, P. Chevée, “Reflection and refraction of a heterogeneous plane wave by a solid layer,” Wave Motion 15, 61–75 (1992).
[CrossRef]

Other

O. Poncelet, M. Deschamps, “Reflection and refractionof an inhomogeneous plane wave on a fluid/anisotropic solid interface,” in Proceedings of the IEEE Ultrasonics Symposium (IEEE Press, Piscataway, N.J., 1994), pp. 753–756.

G. P. Agrawal, Nonlinear Fiber Optics and Applications of Nonlinear Fiber Optics, in Chinese: translated from English into Chinese by D. F. Jia, Z. H. Yu, B. Tan, Z. Y. Hu (Publishing House of Electronics Industry, Beijing, 2002), Chap. 3, pp. 42–62.

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Figures (10)

Fig. 1
Fig. 1

Three-layered structure in our study with layer II representing a birefringent plate sandwiched between two infinite isotropic media.

Fig. 2
Fig. 2

(a) Dispersion for the x component of a propagating wave as a function of the relative thickness L = 2 π l / λ , where l is the thickness of birefringent plate and λ is the wavelength of the incident light. The results are obtained for the optic axis in the x y plane and are shown for three different rotation angles at normal incidence. (b) Same as (a) except for weak birefringence. (c) Dispersion for the x component for L = 0.08 . (d) Pulse broadening (top) caused by the x-component dispersion (bottom) for λ 0 = 1600   nm and T o = 40   fs .

Fig. 3
Fig. 3

(a) Dispersion for larger relative thickness ( L 10 ) , with the other parameters being the same as Fig. 2(a); (b) same as (a) except for weak birefringence.

Fig. 4
Fig. 4

(a) Dispersion for larger relative thickness ( L 100 ) than in Fig. 3, with the other parameters the same as Fig. 2(a); (b) same as (a) except for weak birefringence.

Fig. 5
Fig. 5

Dispersion for the y component with n e = 2.903 , n o = 2.901 ; n e = 2.903 , n o = 2.900 ; and n e = 2.903 , n o = 2.861 , calculated under the same conditions as for Fig. 2(a).

Fig. 6
Fig. 6

(a) Same as Fig. 5 except for L 10 (dashed–dotted curve, n e = 2.903 , n o = 2.901 ; solid curve, n e = 2.903 , n o = 1.461 ) ; (b) zoom-in plot of the two large dispersion peaks associated with L = 8.68 and L = 13.03 shown in (a).

Fig. 7
Fig. 7

Same as Fig. 6(a) except for L 100 .

Fig. 8
Fig. 8

Dispersion calculated for the optic axis in the x z plane and with a 60° angle to the x axis at three different incident angles.

Fig. 9
Fig. 9

X-component dispersion calculated for the optic axis in the x y plane and with a 30° angle to the x axis at two different incident angles.

Fig. 10
Fig. 10

X-component dispersion calculated for two different orientation angles of the optic axis within x y z coordinates at normal incidence.

Equations (29)

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˜ = A n e 2 0 0 0 n o 2 0 0 0 n o 2 A - 1 ,
A = cos   ψ   cos   ϕ - cos   θ   sin   ϕ   sin   ψ - sin   ψ   cos   ϕ - cos   θ   sin   ϕ   cos   ψ sin   θ   sin   ϕ cos   ψ   sin   ϕ - cos   θ   cos   ϕ   sin   ψ - sin   ψ   sin   ϕ + cos   θ   cos   ϕ   cos   ψ - sin   θ   cos   ϕ sin   θ   sin   ψ sin   θ   cos   ψ cos   θ .
k × ( k × E ) + ω 2 μ ˜ E = 0 .
t xx = 4 ( cos   ϕ ) 2 B + ( sin   ϕ ) 2 C ,
t xy = 4 1 C - 1 B sin   ϕ   cos   ϕ ,
B = 1 + n e n 1 1 + n 2 n e exp - i   2 π λ   n e l + 1 - n e n 1 1 - n 2 n e exp i   2 π λ   n e l ,
C = 1 + n o n 1 1 + n 2 n o exp - i   2 π λ   n o l + 1 - n o n 1 1 - n 2 n o exp i   2 π λ   n o l ,
ϕ xx = arctan ( g / f ) and ϕ xy = arctan ( q / p ) ,
g = ( a o - b o ) [ a e 2 + b e 2 + 2 a e b e cos ( 2 φ e ) ] × sin ( φ o ) cos 2   ϕ + ( a e - b e ) [ a o 2 + b o 2 + 2 a o b o cos ( 2 φ o ) ] sin ( φ e ) sin 2   ϕ ,
f = ( a o + b o ) [ a e 2 + b e 2 + 2 a e b e cos ( 2 φ e ) ] × cos ( φ o ) cos 2   ϕ + ( a e + b e ) [ a o 2 + b o 2 + 2 a o b o cos ( 2 φ o ) ] cos ( φ e ) sin 2   ϕ ,
p = ( a o - b o ) [ a e 2 + b e 2 + 2 a e b e cos ( 2 φ e ) ] × sin ( φ o ) - ( a e - b e ) [ a o 2 + b o 2 + 2 a o b o cos ( 2 φ o ) ] sin ( φ e ) ,
q = ( a o + b o ) [ a e 2 + b e 2 + 2 a e b e cos ( 2 φ e ) ] cos ( φ o ) - ( a e + b e ) [ a o 2 + b o 2 + 2 a o b o cos ( 2 φ o ) ] cos ( φ e ) ,
a o = 1 + n o n 1 1 + n 2 n o ,
a e = 1 + n e n 1 1 + n 2 n e ,
b o = 1 - n o n 1 1 - n 2 n o ,
b e = 1 - n e n 1 1 - n 2 n e ,
φ o = Ln o , φ e = Ln e ,
τ xx = λ 2 2 c π d ϕ xx d λ and D xx = d τ xx d λ ,
τ xx = - l c f L g - g L f f 2 + g 2 ,
D xx = L 2 2 π l   L τ xx ,
τ xy = - l c p L q - q L p p 2 + q 2 ,
D xy = L 2 2 π l   L τ xy .
t = 2   exp [ iL ( γ 1 + γ 2 ) ] n 1 ( p 2 q 1 - p 1 q 2 ) cos   ρ exp ( iL γ 1 ) ( - n 1 p 1 + q 1 cos   ρ ) ( n 1 p 2 + q 2 cos   ρ ) + exp ( iL γ 2 ) ( n 1 p 1 + q 1 cos   ρ ) ( n 1 p 2 - q 2 cos   ρ ) ,
γ 1 = n 1 sin   ρ [ n 1 ( - n e 2 + n o 2 ) sin   2 θ   sin   ρ + n e n o ( n e 2 sin 2   θ + n o 2 cos 2   θ - n 1 2 sin 2   ρ ) 0.5 ] n e 2 sin 2   θ + n o 2 cos 2   θ ,
γ 2 = n 1 sin   ρ [ n 1 ( - n e 2 + n o 2 ) sin   2 θ   sin   ρ - n e n o ( n e 2 sin 2   θ + n o 2 cos 2   θ - n 1 2 sin 2   ρ ) 0.5 ] n e 2 sin 2   θ + n o 2 cos 2   θ ,
p 1 = p 2 = n e 2 sin 2   θ + n o 2 cos 2   θ - n 1 2 sin 2   ρ ,
p ± = ( - n e 2 + n o 2 ) cos   θ   sin   θ - n 1 sin   ρ [ n 1 ( - n e 2 + n o 2 ) sin   2 θ   sin   ρ ± n e n o ( n e 2 sin 2   θ + n o 2 cos 2   θ - n 1 2 sin 2   ρ ) 0.5 ] / ( n e 2 sin 2   θ + n o 2 cos 2   θ ) ,
q 1 = γ 1 p 1 - p + n 1 sin   ρ ,
q 2 = γ 2 p 2 - p - n 1 sin   ρ .

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