Abstract

We theoretically analyze waves propagating between an isotropic nonabsorbing medium and a uniaxial absorbing medium in the general case where the incidence plane does not coincide with the principal section of the uniaxial crystal. Expressions for the reflection and transmission coefficients are derived by a combination of complex ray tracing and the 4×4 matrix method. The work presented here is useful for the design of optical systems that incorporate the use of birefringent absorbing components.

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  7. J. D. Trolinger and R. A. Chipman, "Polarization ray tracing in birefringent media," Opt. Eng. (Bellingham) 30, 461-465 (1991).
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    [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. 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]
  14. W. Shen and Z. Shao, "Dispersion between ordinary ray and extraordinary ray in uhiaxial crystals for any orientation of optical axis," Acta Opt. Sin. 22, 765-768 (2002) (in Chinese).
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    [CrossRef]
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    [CrossRef]
  23. C.-J. Chen, A. Lien, and M. I. Nathan, "4×4 and 2×2 matrix formulations for the optics in stratified and biaxial media," J. Opt. Soc. Am. A 14, 3125-3134 (1997).
    [CrossRef]
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    [CrossRef]
  25. M. A. Dupertuis, M. Proctor, and B. Acklin, "Generalization of complex Snell-Descartes and Fresnel laws," J. Opt. Soc. Am. A 11, 1159-1166 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. R. A. Egorchenkov and Y. A. Kravtsov, "Complex ray tracing algorithms with application to optical problems," J. Opt. Soc. Am. A 18, 650-656 (2001).
    [CrossRef]
  30. M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, June 18-22, 2001.
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  33. E. P. Oscar, "Optical properties, reflectance, and transmittance of anisotropic absorbing crystal plates," Phys. Rev. B 36, 3427-3435 (1987).
    [CrossRef]
  34. F. Bréhat and B. Wyncke, "Reflectivity, transmissivity and optical constants of anisotropic absorbing crystals," J. Phys. D 24, 2055-2066 (1991).
    [CrossRef]
  35. T. Hasegawa, J. Umemura, and T. Takenaka, "Simple refraction law for uniaxial anisotropic absorbing media," Appl. Spectrosc. 47, 338-340 (1993).
    [CrossRef]
  36. H. Greiner, "Power splitting between refracted ordinary and extraordinary waves in uniaxiai crystals with absorption," Optik (Stuttgart) 114, 109-112 (2003).
    [CrossRef]
  37. R. Echarri and M. T. Garea, "Behaviour of the Poynting vector in uniaxial absorbing media," Pure Appl. Opt. 3, 931-941 (1994).
    [CrossRef]
  38. S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, and C. Sáenz, "Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in absorbing isotropic media," J. Opt. Soc. Am. A 21, 1776-1784 (2004).
    [CrossRef]
  39. C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernández, "Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media," J. Mod. Opt. 49, 1553-1566 (2002).
    [CrossRef]
  40. J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernández, and C. Sáenz, "Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media," J. Opt. Soc. Am. A 24, 1767-1775 (2007).
    [CrossRef]

2007 (2)

L. Wu, C. Zhang, and B. Zhao, "Analysis of the lateral displacement and optical path difference in wide-field-of-view polarization interference imaging spectrometer," Opt. Commun. 273, 67-73 (2007).
[CrossRef]

J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernández, and C. Sáenz, "Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media," J. Opt. Soc. Am. A 24, 1767-1775 (2007).
[CrossRef]

2005 (3)

L. Wu, C. Zhang, Y. Yuan, and B. Zhao, "Exact calculation of the lateral displacement and optical path difference of Savart polariscopes," Acta Opt. Sin. 25, 885-890 (2005) (in Chinese).

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]

P. C. Y. Changa, J. G. Walkera, and K. I. Hopcraft, "Ray tracing in absorbing media," J. Quant. Spectrosc. Radiat. Transf. 96, 327-341 (2005).
[CrossRef]

2004 (1)

2003 (1)

H. Greiner, "Power splitting between refracted ordinary and extraordinary waves in uniaxiai crystals with absorption," Optik (Stuttgart) 114, 109-112 (2003).
[CrossRef]

2002 (4)

W. Shen and Z. Shao, "Dispersion between ordinary ray and extraordinary ray in uhiaxial crystals for any orientation of optical axis," Acta Opt. Sin. 22, 765-768 (2002) (in Chinese).

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]

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]

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernández, "Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media," J. Mod. Opt. 49, 1553-1566 (2002).
[CrossRef]

2001 (1)

1998 (1)

1997 (2)

1995 (1)

1994 (3)

1993 (3)

1992 (1)

1991 (4)

T. A. Maldonado and T. K. Gaylord, "Light propagation characteristics for arbitrary wavevector directions in biaxial media by a coordinate-free approach," Appl. Opt. 30, 2465-2480 (1991).
[CrossRef] [PubMed]

J. Lekner, "Reflection and refraction by uniaxial crystal," J. Phys.: Condens. Matter 3, 6122-6133 (1991).
[CrossRef]

J. D. Trolinger and R. A. Chipman, "Polarization ray tracing in birefringent media," Opt. Eng. (Bellingham) 30, 461-465 (1991).
[CrossRef]

F. Bréhat and B. Wyncke, "Reflectivity, transmissivity and optical constants of anisotropic absorbing crystals," J. Phys. D 24, 2055-2066 (1991).
[CrossRef]

1990 (1)

1987 (1)

E. P. Oscar, "Optical properties, reflectance, and transmittance of anisotropic absorbing crystal plates," Phys. Rev. B 36, 3427-3435 (1987).
[CrossRef]

1983 (2)

F. Parmigiani, "Some aspects of the reflection and refraction of an electromagnetic wave at an absorbing surface," Am. J. Phys. 51, 245-247 (1983).
[CrossRef]

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

1982 (1)

1981 (1)

1975 (1)

1972 (1)

1968 (1)

1962 (1)

Acta Opt. Sin. (2)

W. Shen and Z. Shao, "Dispersion between ordinary ray and extraordinary ray in uhiaxial crystals for any orientation of optical axis," Acta Opt. Sin. 22, 765-768 (2002) (in Chinese).

L. Wu, C. Zhang, Y. Yuan, and B. Zhao, "Exact calculation of the lateral displacement and optical path difference of Savart polariscopes," Acta Opt. Sin. 25, 885-890 (2005) (in Chinese).

Am. J. Phys. (1)

F. Parmigiani, "Some aspects of the reflection and refraction of an electromagnetic wave at an absorbing surface," Am. J. Phys. 51, 245-247 (1983).
[CrossRef]

Appl. Opt. (8)

Appl. Spectrosc. (1)

J. Mod. Opt. (1)

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernández, "Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media," J. Mod. Opt. 49, 1553-1566 (2002).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (10)

C. Gu and P. Yeh, "Extended Jones matrix method II," J. Opt. Soc. Am. A 10, 507-513 (1993).
[CrossRef]

S. C. McClain, L. W. Hillman, and R. A. Chipman, "Polarization ray tracing in anisotropic optically active media. I. Algorithms," J. Opt. Soc. Am. A 10, 2371-2382 (1993).
[CrossRef]

G. D. Landry and T. A. Maldonado, "Complete method to determine transmission and reflection characteristics at a planar interface between arbitrarily oriented biaxial media," J. Opt. Soc. Am. A 12, 2048-2060 (1995).
[CrossRef]

C.-J. Chen, A. Lien, and M. I. Nathan, "4×4 and 2×2 matrix formulations for the optics in stratified and biaxial media," J. Opt. Soc. Am. A 14, 3125-3134 (1997).
[CrossRef]

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]

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]

R. A. Egorchenkov and Y. A. Kravtsov, "Complex ray tracing algorithms with application to optical problems," J. Opt. Soc. Am. A 18, 650-656 (2001).
[CrossRef]

M. A. Dupertuis, M. Proctor, and B. Acklin, "Generalization of complex Snell-Descartes and Fresnel laws," J. Opt. Soc. Am. A 11, 1159-1166 (1994).
[CrossRef]

J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernández, and C. Sáenz, "Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media," J. Opt. Soc. Am. A 24, 1767-1775 (2007).
[CrossRef]

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, and C. Sáenz, "Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in absorbing isotropic media," J. Opt. Soc. Am. A 21, 1776-1784 (2004).
[CrossRef]

J. Phys. D (1)

F. Bréhat and B. Wyncke, "Reflectivity, transmissivity and optical constants of anisotropic absorbing crystals," J. Phys. D 24, 2055-2066 (1991).
[CrossRef]

J. Phys.: Condens. Matter (1)

J. Lekner, "Reflection and refraction by uniaxial crystal," J. Phys.: Condens. Matter 3, 6122-6133 (1991).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

P. C. Y. Changa, J. G. Walkera, and K. I. Hopcraft, "Ray tracing in absorbing media," J. Quant. Spectrosc. Radiat. Transf. 96, 327-341 (2005).
[CrossRef]

Opt. Commun. (1)

L. Wu, C. Zhang, and B. Zhao, "Analysis of the lateral displacement and optical path difference in wide-field-of-view polarization interference imaging spectrometer," Opt. Commun. 273, 67-73 (2007).
[CrossRef]

Opt. Eng. (Bellingham) (1)

J. D. Trolinger and R. A. Chipman, "Polarization ray tracing in birefringent media," Opt. Eng. (Bellingham) 30, 461-465 (1991).
[CrossRef]

Opt. Express (1)

Optik (Stuttgart) (1)

H. Greiner, "Power splitting between refracted ordinary and extraordinary waves in uniaxiai crystals with absorption," Optik (Stuttgart) 114, 109-112 (2003).
[CrossRef]

Phys. Rev. B (1)

E. P. Oscar, "Optical properties, reflectance, and transmittance of anisotropic absorbing crystal plates," Phys. Rev. B 36, 3427-3435 (1987).
[CrossRef]

Pure Appl. Opt. (1)

R. Echarri and M. T. Garea, "Behaviour of the Poynting vector in uniaxial absorbing media," Pure Appl. Opt. 3, 931-941 (1994).
[CrossRef]

Other (3)

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, June 18-22, 2001.

Y. A. Kravtsov, G. W. Forbes, and A. A. Asatryan, "Theory and applications of complex rays," in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1999), Vol. 39, pp. 1-62.
[CrossRef]

M. Born and E. Wolf, Principles of Optics7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 840-852.

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Figures (3)

Fig. 1
Fig. 1

Reflection and refraction at the interface between nonabsorbing and absorbing isotropic media.

Fig. 2
Fig. 2

Planes of constant phase and of constant amplitude in the absorbing isotropic medium.

Fig. 3
Fig. 3

Propagation directions of the incident, reflected, and refracted waves. For simplicity, the ordinary wave is not shown in the figure.

Equations (37)

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n i sin θ i = n ̃ sin θ ̃ t ,
D ̃ = n ̃ c s ̃ × H ̃ ,
μ 0 H ̃ = n ̃ c s ̃ × E ̃ ,
D ̃ s ̃ = 0 ,
H ̃ s ̃ = 0 ,
s ̂ pr = ( n 2 s ̃ Re 2 + κ 2 s ̃ Im 2 ) 1 2 ( n s ̃ Re + κ s ̃ Im ) ,
s ̂ at = ( n 2 s ̃ Im 2 + κ 2 s ̃ Re 2 ) 1 2 ( n s ̃ Im κ s ̃ Re ) ,
γ = arccos ( s ̂ pr s ̂ at ) .
D ̃ = ε ̃ E ̃ ,
ε ̃ = [ n ̃ o 2 0 0 0 n ̃ o 2 0 0 0 n ̃ e 2 ] ,
s ̃ = D ̃ × H ̃ D ̃ × H ̃ ,
t ̃ = E ̃ × H ̃ E ̃ × H ̃ .
s ̃ x 2 n ̃ 2 n ̃ o 2 + s ̃ y 2 n ̃ 2 n ̃ o 2 + s ̃ z 2 n ̃ 2 n ̃ e 2 = 0 .
s ̃ e = ( cos ω sin θ ̃ e , sin ω sin θ ̃ e , cos θ ̃ e ) .
n i sin θ i = n ̃ e f f ( θ ̃ ) sin θ ̃ e ,
cos θ ̃ = s ̃ e c ̂ = cos β cos θ ̃ e + cos ω sin β sin θ ̃ e .
n ̃ e f f ( θ ̃ ) = n ̃ o n ̃ e n ̃ e 2 cos 2 θ ̃ + n ̃ o 2 sin 2 θ ̃ .
tan θ ̃ e = A ̃ n i sin i n ̃ o n ̃ e 2 A ̃ [ A ̃ ( n ̃ o 2 n ̃ e 2 ) sin 2 β cos 2 ω ] n i 2 sin 2 i + ( n ̃ o 2 n ̃ e 2 ) n i sin β cos β cos ω sin i
t ̃ e = η ̃ s ̃ e + λ ̃ c ̂ .
t ̃ e x t ̃ e y t ̃ e z cos ω sin θ ̃ e sin ω sin θ ̃ e cos θ ̃ e sin β 0 cos β = 0 .
η ̃ = ± n ̃ o 2 cos ξ ̃ n ̃ e 2 cos θ ̃ ,
λ ̃ = ( 1 n ̃ o 2 n ̃ e 2 ) cos ξ ̃ ,
tan α ̃ = tan ( ξ ̃ θ ̃ ) = ( n ̃ o 2 n ̃ e 2 ) tan θ ̃ n ̃ e 2 + n ̃ o 2 tan 2 θ ̃ .
U ̃ = U ̃ exp [ i ω ( n ̃ e f f c s ̃ r t ) ] exp ( i φ ̃ U ) u ̃ ,
h ̃ o = b ̃ o = s ̃ o × c ̂ s ̃ o × c ̂ ,
e ̃ o = d ̃ o = h ̃ o × s ̃ o ,
h ̃ e = b ̃ e = s ̃ e × c ̂ s ̃ e × c ̂ ,
e ̂ o = h ̂ o × t ̂ o ,
D ̃ = n ̃ e f f 2 E ̃ ,
D ̃ = D ̃ d ̃ .
n ̂ × ( E ̃ ( I ) E ̃ ( II ) ) = 0 ,
n ̂ × ( H ̃ ( I ) H ̃ ( II ) ) = 0 ,
A ̃ X ̃ = B ,
A ̃ = [ E ̃ p r x ̂ E ̃ s r x ̂ E ̃ o x ̂ E ̃ e x ̂ E ̃ p r y ̂ E ̃ s r y ̂ E ̃ o y ̂ E ̃ e y ̂ H ̃ p r x ̂ H ̃ s r x ̂ H ̃ o x ̂ H ̃ e x ̂ H ̃ p r y ̂ H ̃ s r y ̂ H ̃ o y ̂ H ̃ e y ̂ ] ,
X ̃ = ( E ̃ p r , E ̃ s r , E ̃ o , E ̃ e ) T ,
B = ( E p i , E s i , H p i , H s i ) T .
X ̃ = A ̃ 1 B ̃ ,

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