Abstract

The flux of energy given by the Poynting vector Se and the kt-wave vector normal to the geometrical wavefront for the extraordinary ray propagating through uniaxial crystals can be evaluated by using the theory developed by Avendaño-Alejo et al. [J. Opt. Soc. Am. A 19, 1668 (2002) ] and Avendaño-Alejo and Stavroudis [J. Opt. Soc. Am. A 19, 1674 (2002) ]. We give here the equations necessary to evaluate the general dispersion angle Sekt. Additionally we define two new dispersion angles, SeA and ktA, where A is the crystal axis vector. With these new dispersion angles we evaluate the optical path length traversed by the extraordinary ray in a plane-parallel uniaxial plate when the crystal axis lies in the plane of incidence.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.
  2. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.
  3. J. P. Mathieu, Optics Parts 1 and 2 (Pergamon, 1976), Chap. 4, pp. 77-102.
  4. S. Huard, Polarization of Light (Wiley, 1997), Chap. 2.
  5. J. P. Lesso, A. J. Duncan, W. Sibbett, and M. J. Podgett, "Aberrations introduced by a lens made from a birefringent material," Appl. Opt. 39, 592-598 (2000).
    [CrossRef]
  6. M. Avendaño-Alejo and M. Rosete-Aguilar, "Paraxial theory of birefringent lenses," J. Opt. Soc. Am. A 22, 881-891 (2005).
    [CrossRef]
  7. M. C. Simon and K. V. Gottschalk, "Optical path in birefringent media and Fermat's principle," Pure Appl. Opt. 7, 1403-1410 (1998).
    [CrossRef]
  8. S. Prunet, B. Journet, and G. Fortunato, "Exact calculation of the optical path difference and description of a new birefringent interferometer," Opt. Eng. 38, 983-990 (1999).
    [CrossRef]
  9. C. L. Lin and J. J. Wu, "Total reflection of waves propagating from an isotropic medium to an anisotropic medium," Opt. Lett. 23, 22-24 (1998).
    [CrossRef]
  10. Q. T. Liang, "Simple ray tracing formulas for uniaxial optical crystals," Appl. Opt. 29, 1008-1010 (1990).
    [CrossRef] [PubMed]
  11. M. Avendaño-Alejo, O. Stavroudis, and A. R. Boyain, "Huygens's principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface," J. Opt. Soc. Am. A 19, 1668-1673 (2002).
    [CrossRef]
  12. M. Avendaño-Alejo and O. Stavroudis, "Huygens's principle and rays in uniaxial anisotropic media. II. Crystal axis with arbitrary orientation," J. Opt. Soc. Am. A 19, 1674-1679 (2002).
    [CrossRef]
  13. E. Cojocaru, "Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media," Appl. Opt. 36, 302-306 (1997).
    [CrossRef] [PubMed]
  14. M. C. Simon, "Ray tracing formulas for monoaxial optical components," Appl. Opt. 22, 354-360 (1983).
    [CrossRef] [PubMed]
  15. M. Avendaño-Alejo, "Analysis of the refraction of the extraordinary ray in a plane-parallel uniaxial plate with an arbitrary orientation of the optical axis," Opt. Express 13, 2549-2555 (2005).
    [CrossRef] [PubMed]

2005

2002

2000

1999

S. Prunet, B. Journet, and G. Fortunato, "Exact calculation of the optical path difference and description of a new birefringent interferometer," Opt. Eng. 38, 983-990 (1999).
[CrossRef]

1998

C. L. Lin and J. J. Wu, "Total reflection of waves propagating from an isotropic medium to an anisotropic medium," Opt. Lett. 23, 22-24 (1998).
[CrossRef]

M. C. Simon and K. V. Gottschalk, "Optical path in birefringent media and Fermat's principle," Pure Appl. Opt. 7, 1403-1410 (1998).
[CrossRef]

1997

1990

1983

Avendaño-Alejo, M.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

Boyain, A. R.

Cojocaru, E.

Duncan, A. J.

Fortunato, G.

S. Prunet, B. Journet, and G. Fortunato, "Exact calculation of the optical path difference and description of a new birefringent interferometer," Opt. Eng. 38, 983-990 (1999).
[CrossRef]

Gottschalk, K. V.

M. C. Simon and K. V. Gottschalk, "Optical path in birefringent media and Fermat's principle," Pure Appl. Opt. 7, 1403-1410 (1998).
[CrossRef]

Huard, S.

S. Huard, Polarization of Light (Wiley, 1997), Chap. 2.

Journet, B.

S. Prunet, B. Journet, and G. Fortunato, "Exact calculation of the optical path difference and description of a new birefringent interferometer," Opt. Eng. 38, 983-990 (1999).
[CrossRef]

Lesso, J. P.

Liang, Q. T.

Lin, C. L.

Mathieu, J. P.

J. P. Mathieu, Optics Parts 1 and 2 (Pergamon, 1976), Chap. 4, pp. 77-102.

Podgett, M. J.

Prunet, S.

S. Prunet, B. Journet, and G. Fortunato, "Exact calculation of the optical path difference and description of a new birefringent interferometer," Opt. Eng. 38, 983-990 (1999).
[CrossRef]

Rosete-Aguilar, M.

Sibbett, W.

Simon, M. C.

M. C. Simon and K. V. Gottschalk, "Optical path in birefringent media and Fermat's principle," Pure Appl. Opt. 7, 1403-1410 (1998).
[CrossRef]

M. C. Simon, "Ray tracing formulas for monoaxial optical components," Appl. Opt. 22, 354-360 (1983).
[CrossRef] [PubMed]

Stavroudis, O.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

Wu, J. J.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

S. Prunet, B. Journet, and G. Fortunato, "Exact calculation of the optical path difference and description of a new birefringent interferometer," Opt. Eng. 38, 983-990 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Pure Appl. Opt.

M. C. Simon and K. V. Gottschalk, "Optical path in birefringent media and Fermat's principle," Pure Appl. Opt. 7, 1403-1410 (1998).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.

J. P. Mathieu, Optics Parts 1 and 2 (Pergamon, 1976), Chap. 4, pp. 77-102.

S. Huard, Polarization of Light (Wiley, 1997), Chap. 2.

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.


Figures (10)

Fig. 1
Fig. 1

Direction cosines for incident and refracted rays and optical axis in the Y Z plane.

Fig. 2
Fig. 2

Index ellipsoid for a negative uniaxial crystal. Angles for the incident and refracted rays and crystal axis measured with respect to the normal to the refracted surface are θ i , θ o , θ e , ϕ, respectively, and the angle of the normal to the wavefront is θ t .

Fig. 3
Fig. 3

Refracted wavefronts in and out of the uniaxial crystal and optical path length for the extraordinary ray along S e or along k t .

Fig. 4
Fig. 4

Dispersion angles θ a , θ m , θ n , and their respective relationships.

Fig. 5
Fig. 5

OPL e versus angle of incidence when ϕ = arctan ( n e n o ) , for a plane-parallel plate of thickness d = 1 cm .

Fig. 6
Fig. 6

Effective refractive indices for θ i = 0 as a function of crystal axis position.

Fig. 7
Fig. 7

Difference between the effective refractive indices for θ i = 0 as a function of crystal axis position.

Fig. 8
Fig. 8

Effective refractive indices for ϕ = arctan ( n e n o ) as a function of the angle of incidence.

Fig. 9
Fig. 9

Difference between the effective refractive indices for ϕ = arctan ( n e n o ) as a function of the angle of incidence.

Fig. 10
Fig. 10

Optical path difference between the ordinary and extraordinary rays for ϕ = arctan ( n e n o ) as a function of the angle of incidence for a plane-parallel plate of thickness d = 1 cm .

Equations (52)

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

( ξ o η o ζ o ) = [ n i n o 0 0 0 n i n o 0 0 0 1 ζ i 1 ( n i n o ) 2 ( 1 ζ i 2 ) ] ( ξ i η i ζ i ) ,
( ξ e η e ζ e ) = 1 δ e g { [ n o 2 ( n e 2 + ( n o 2 n e 2 ) β 2 ) n o 2 ( n o 2 n e 2 ) α β 0 n o 2 ( n o 2 n e 2 ) α β n o 2 ( n e 2 + ( n o 2 n e 2 ) α 2 ) 0 0 0 Γ Δ ζ o ] ( ξ o η o ζ o ) γ ( n o 2 n e 2 ) Δ [ 1 0 0 0 1 0 0 0 0 ] ( α β γ ) } ,
δ eg 2 = n e 2 { n o 2 Γ 2 ( n o 2 n e 2 ) [ γ Δ + n o 2 ( α ξ o + β η o ) ] 2 } ,
Δ 2 = Γ δ o 2 + n o 2 ( n o 2 n e 2 ) ( α ξ o + β η o ) 2 ,
Γ = n e 2 γ 2 + n o 2 ( α 2 + β 2 ) ,
δ o = n e 2 n o 2 ( ξ o 2 + η o 2 ) ,
( ξ o η o ζ o ) = ( 0 ( n i n o ) η i 1 ( n i n o ) 2 ( 1 ζ i 2 ) )
( ξ e η e ζ e ) = 1 δ eg ( 0 n o 2 n e 2 η o γ ( n o 2 n e 2 ) Δ β Γ Δ ) ,
δ eg 2 = n e 2 [ n o 2 Γ 2 ( n o 2 n e 2 ) ( γ Δ + n o 2 β η o ) 2 ] ,
Δ 2 = Γ δ o 2 + n o 2 ( n o 2 n e 2 ) β 2 η o 2 .
tan θ e = n i n e n o sin θ i + ( n e 2 n o 2 ) cos ϕ sin ϕ n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ( n e 2 cos 2 ϕ + n o 2 sin 2 ϕ ) n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ;
tan θ o = n i sin θ i n o 2 n i 2 sin 2 θ i ,
f e ( x e , y e , z e ) = D x e + E y e + F z e = 0 ,
D = ξ o [ ( n o 2 n e 2 ) ( α ξ o + β η o ) γ Δ ] ,
E = η o [ ( n o 2 n e 2 ) ( α ξ o + β η o ) γ Δ ] ,
F = δ o 2 + ( n o 2 n e 2 ) ( α ξ o + β η o ) 2 ;
k t = f e f e .
tan θ t = k t y k t z = [ n i ( n e 2 n o 2 ) sin ϕ cos ϕ sin θ i + n e n o n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i n e 2 n o 2 n i 2 sin 2 θ i ( n o 2 cos 2 ϕ + n e 2 sin 2 ϕ ) ] n i sin θ i ,
cos θ m = S e k t = ( ξ e , η e , ζ e ) ( k t x , k t y , k t z ) ,
S e = [ 0 , ( n e 2 n o 2 ) β γ γ 2 n e 4 + β 2 n o 4 , ( γ 2 n e 2 + β 2 n o 2 ) γ 2 n e 4 + β 2 n o 4 ] ;
cos θ m = ( γ 2 n e 2 + β 2 n o 2 ) γ 2 n e 4 + β 2 n o 4 ,
tan θ m = β γ ( n e 2 n o 2 ) γ 2 n e 2 + β 2 n o 2 = ( n e 2 n o 2 ) tan ϕ n e 2 + n o 2 tan 2 ϕ .
n t Y 2 n e 2 + n t Z 2 n o 2 = 1 .
( n t Y n t Z ) = [ cos ϕ sin ϕ sin ϕ cos ϕ ] ( n Y n Z ) .
n Y 2 ( sin 2 ϕ n o 2 + cos 2 ϕ n e 2 ) + n Z 2 ( cos 2 ϕ n o 2 + sin 2 ϕ n e 2 ) + 2 n Y n Z sin ϕ cos ϕ ( 1 n o 2 1 n e 2 ) = 1 .
tan θ t = n Y n Z ,
n Y 2 = tan 2 θ t A tan 2 θ t + B + C tan θ t ,
n Z 2 = 1 A tan 2 θ t + B + C tan θ t ,
A = ( sin 2 ϕ n o 2 + cos 2 ϕ n e 2 ) , B = ( cos 2 ϕ n o 2 + sin 2 ϕ n e 2 ) ,
C = 2 sin ϕ cos ϕ ( 1 n o 2 1 n e 2 ) .
n w f ( k t ) = n Y 2 + n Z 2 = n o 2 n e 2 ( 1 + tan 2 θ t ) n o 2 ( cos ϕ tan θ t sin ϕ ) 2 + n e 2 ( sin ϕ tan θ t + cos ϕ ) 2 ,
n w f ( k t ) = n o n e n o 2 sin 2 ( ϕ θ t ) + n e 2 cos 2 ( ϕ θ t ) ,
OPL e = n w f ( k t ) l w f ,
ψ a = 2 π n u λ v ( k ̂ t r ) ω t ,
ψ b = 2 π n u λ v ( k t r ) ω t + Δ ψ ,
Δ ψ = 2 π c λ v u a 1 b 1 ¯ ,
Δ ψ = 2 π c λ v v a 1 b 2 ¯ + 2 π c λ v v i b 2 b 3 ¯ ,
a 1 b 1 ¯ u = a 1 b 2 ¯ v + b 2 b 3 ¯ v i ,
u 2 = u e 2 sin 2 θ a + u o 2 cos 2 θ a ,
1 v 2 = sin 2 θ n u e 2 + cos 2 θ n u o 2 ,
cos θ a = k t A ,
cos θ n = S e A ,
OPL e = a 1 ¯ b 1 ¯ c u ,
OPL e = a 1 ¯ b 2 ¯ c v + b 2 ¯ b 3 ¯ c v i .
n w f = c u = n o n e n o 2 sin 2 θ a + n e 2 cos 2 θ a ,
n p v = c v = n e 2 sin 2 θ n + n o 2 cos 2 θ n ,
OPL e = n p v d cos θ e = d cos θ t [ n w f + n i ( sin θ i cos θ e ) sin θ m ] ,
OPD = d ( n o cos θ o n p v cos θ e + n i [ tan θ e tan θ o ] sin θ i ) .
n w f = n e 2 + n o 2 2 ,
OPL e = n e 2 + n o 2 2 d .
OPL o = n o d .
Δ OPL = d ( n o n c 2 + n o 2 2 ) .

Metrics