Abstract

Using the electromagnetic wave method and Huygens’ principle, we have studied light refraction and reflection on the boundary between two uniaxial crystals with arbitrary optic axis orientations. We provide a set of direct formulas that calculate the directions of the refracted and reflected extraordinary rays and their associated waves, as well as the ray indices and refractive indices. The formulas use only the initial input variables and do not require the calculation of intermediate quantities. Refraction and reflection are combined in one formula. We also prove that the electromagnetic wave method and Huygens’ principle produce analytically equivalent results for geometrical ray tracing.

© 2018 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Ray refraction in uniaxial crystals by Fermat’s principle

Pengqian Wang
Appl. Opt. 57(18) 4950-4954 (2018)

Reflection formulae for ray tracing in uniaxial anisotropic media using Huygens’s principle

Luis A. Alemán-Castañeda and Martha Rosete-Aguilar
J. Opt. Soc. Am. A 33(11) 2198-2205 (2016)

Simple ray tracing formulas for uniaxial optical crystals

Quan-Ting Liang
Appl. Opt. 29(7) 1008-1010 (1990)

References

  • View by:
  • |
  • |
  • |

  1. M. C. Simon, “Ray tracing in monoaxial crystals; are the exact formulas necessary?” Appl. Opt. 26, 3187–3189 (1987).
    [Crossref]
  2. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
  3. L. D. Landau, L. P. Pitaevskii, and E. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinemann, 1984).
  4. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley-Interscience, 2002).
  5. J. J. Stamnes and G. C. Sherman, “Reflection and refraction of an arbitrary wave at a plane interface separating two uniaxial crystals,” J. Opt. Soc. Am. 67, 683–695 (1977).
    [Crossref]
  6. J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–465 (1991).
    [Crossref]
  7. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
    [Crossref]
  8. M. C. Simon and R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
    [Crossref]
  9. M. C. Simon and R. M. Echarri, “Internal reflection in uniaxial crystals I. Geometrical description,” J. Mod. Opt. 37, 121–129 (1990).
    [Crossref]
  10. M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
    [Crossref]
  11. Q.-T. Liang, “Simple ray tracing formulas for uniaxial optical crystals,” Appl. Opt. 29, 1008–1010 (1990).
    [Crossref]
  12. Z. Zhang and H. Caulfield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. Laser Technol. 28, 549–553 (1996).
    [Crossref]
  13. Y. Wang, L. Liang, H. Xin, and L. Wu, “Complex ray tracing in uniaxial absorbing media,” J. Opt. Soc. Am. A 25, 653–657 (2008).
    [Crossref]
  14. H. Wu, C. Zhang, and X. Bai, “A complete description of polarization and transmission of nonnormal incident rays in a uniaxial birefringent plate with arbitrary optic axis,” Opt. Commun. 283, 4129–4134 (2010).
    [Crossref]
  15. E. Cojocaru, “Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media,” Appl. Opt. 36, 302–306 (1997).
    [Crossref]
  16. E. Cojocaru, “Explicit relations for the extraordinary-ray trajectory at the back of a rotating uniaxial birefringent plate,” Appl. Opt. 36, 8886–8888 (1997).
    [Crossref]
  17. S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
    [Crossref]
  18. B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
    [Crossref]
  19. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. 3, 6121–6133 (1991).
    [Crossref]
  20. H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983).
  21. G. Beyerle and I. S. McDermid, “Ray-tracing formulas for refraction and internal reflection in uniaxial crystals,” Appl. Opt. 37, 7947–7953 (1998).
    [Crossref]
  22. S. C. McClain and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media,” Proc. SPIE 1746, 107–118 (1992).
    [Crossref]
  23. 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]
  24. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
    [Crossref]
  25. M. C. Simon, “Refraction in biaxial crystals: a formula for the indices,” J. Opt. Soc. Am. A 4, 2201–2204 (1987).
    [Crossref]
  26. 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, 2048–2063 (1995).
    [Crossref]
  27. W.-Q. Zhang, “General ray-tracing formulas for crystal,” Appl. Opt. 31, 7328–7331 (1992).
    [Crossref]
  28. C. Huygens, Treatise on Light (Dover, 1962).
  29. O. N. Stavroudis, “Ray-tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187–191 (1962).
    [Crossref]
  30. M. Avendaño-Alejo, O. N. Stavroudis, and A. R. B. Y. Goitia, “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]
  31. M. Avendaño-Alejo and O. N. Stavroudis, “Huygens’s principle and rays in uniaxial anisotropic media. II. Crystal axis orientation arbitrary,” J. Opt. Soc. Am. A 19, 1674–1679 (2002).
    [Crossref]
  32. M. Avendaño-Alejo, I. Moreno, and O. Stavroudis, “Minimum deviation angle in uniaxial prisms,” J. Opt. Soc. Am. A 24, 2431–2437 (2007).
    [Crossref]
  33. M. Avendaño-Alejo and M. Rosete-Aguilar, “Paraxial theory for birefringent lenses,” J. Opt. Soc. Am. A 22, 881–891 (2005).
    [Crossref]
  34. M. Avendaño-Alejo and M. Rosete-Aguilar, “Optical path difference in a plane-parallel uniaxial plate,” J. Opt. Soc. Am. A 23, 926–932 (2006).
    [Crossref]
  35. L. A. Alemán-Castañeda and M. Rosete-Aguilar, “Reflection formulae for ray tracing in uniaxial anisotropic media using Huygens’s principle,” J. Opt. Soc. Am. A 33, 2198–2205 (2016).
    [Crossref]
  36. W. Swindell, “Extraordinary-ray and -wave tracing in uniaxial crystals,” Appl. Opt. 14, 2298–2301 (1975).
    [Crossref]
  37. H. Ren, L. Liu, Z. Song, and D. Liu, “Double refraction and reflection of sequential interfaces in a crystal and application to integration of 1×N optical switch,” J. Mod. Opt. 50, 2231–2242 (2003).
  38. H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529–539 (2005).
    [Crossref]
  39. M. C. Simon and R. M. Echarri, “Internal total reflection in monoaxial crystals,” Appl. Opt. 26, 3878–3883 (1987).
    [Crossref]
  40. M. C. Simon and R. M. Echarri, “Inhibited reflection in uniaxial crystals,” Opt. Lett. 14, 257–259 (1989).
    [Crossref]
  41. M. C. Simon, K. B. Bastida, and K. V. Gottschalk, “Total reflection in a uniaxial crystal-uniaxial crystal interface,” Optik 116, 586–594 (2005).
    [Crossref]
  42. A. A. Muryiand and V. I. Stroganov, “Conditions for bringing the ordinary and extraordinary rays into coincidence in a plane-parallel plate fabricated from an optical uniaxial crystal,” J. Opt. Technol. 71, 283–285 (2004).
    [Crossref]
  43. 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]
  44. M. C. Simon, K. V. Gottschalk, and J. M. Simon, “The coincidence of ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959–974 (2008).
    [Crossref]
  45. R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
    [Crossref]
  46. M. C. Simon and L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
    [Crossref]

2016 (1)

2015 (1)

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

2014 (1)

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

2010 (1)

H. Wu, C. Zhang, and X. Bai, “A complete description of polarization and transmission of nonnormal incident rays in a uniaxial birefringent plate with arbitrary optic axis,” Opt. Commun. 283, 4129–4134 (2010).
[Crossref]

2008 (2)

Y. Wang, L. Liang, H. Xin, and L. Wu, “Complex ray tracing in uniaxial absorbing media,” J. Opt. Soc. Am. A 25, 653–657 (2008).
[Crossref]

M. C. Simon, K. V. Gottschalk, and J. M. Simon, “The coincidence of ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959–974 (2008).
[Crossref]

2007 (2)

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
[Crossref]

M. Avendaño-Alejo, I. Moreno, and O. Stavroudis, “Minimum deviation angle in uniaxial prisms,” J. Opt. Soc. Am. A 24, 2431–2437 (2007).
[Crossref]

2006 (1)

2005 (4)

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529–539 (2005).
[Crossref]

M. Avendaño-Alejo and M. Rosete-Aguilar, “Paraxial theory for birefringent lenses,” J. Opt. Soc. Am. A 22, 881–891 (2005).
[Crossref]

M. C. Simon, K. B. Bastida, and K. V. Gottschalk, “Total reflection in a uniaxial crystal-uniaxial crystal interface,” Optik 116, 586–594 (2005).
[Crossref]

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]

2004 (1)

2003 (1)

H. Ren, L. Liu, Z. Song, and D. Liu, “Double refraction and reflection of sequential interfaces in a crystal and application to integration of 1×N optical switch,” J. Mod. Opt. 50, 2231–2242 (2003).

2002 (2)

1998 (1)

1997 (2)

1996 (1)

Z. Zhang and H. Caulfield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. Laser Technol. 28, 549–553 (1996).
[Crossref]

1995 (2)

1993 (2)

1992 (2)

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

S. C. McClain and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media,” Proc. SPIE 1746, 107–118 (1992).
[Crossref]

1991 (3)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. 3, 6121–6133 (1991).
[Crossref]

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–465 (1991).
[Crossref]

M. C. Simon and L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[Crossref]

1990 (2)

Q.-T. Liang, “Simple ray tracing formulas for uniaxial optical crystals,” Appl. Opt. 29, 1008–1010 (1990).
[Crossref]

M. C. Simon and R. M. Echarri, “Internal reflection in uniaxial crystals I. Geometrical description,” J. Mod. Opt. 37, 121–129 (1990).
[Crossref]

1989 (1)

1987 (3)

1986 (1)

1983 (1)

1977 (1)

1975 (1)

1962 (1)

Alemán-Castañeda, L. A.

Avendaño-Alejo, M.

Bai, X.

H. Wu, C. Zhang, and X. Bai, “A complete description of polarization and transmission of nonnormal incident rays in a uniaxial birefringent plate with arbitrary optic axis,” Opt. Commun. 283, 4129–4134 (2010).
[Crossref]

Bastida, K. B.

M. C. Simon, K. B. Bastida, and K. V. Gottschalk, “Total reflection in a uniaxial crystal-uniaxial crystal interface,” Optik 116, 586–594 (2005).
[Crossref]

Beyerle, G.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Caulfield, H.

Z. Zhang and H. Caulfield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. Laser Technol. 28, 549–553 (1996).
[Crossref]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983).

Chipman, R. A.

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[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]

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
[Crossref]

S. C. McClain and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media,” Proc. SPIE 1746, 107–118 (1992).
[Crossref]

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–465 (1991).
[Crossref]

Choi, H. C.

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

Cojocaru, E.

Echarri, R. M.

Goitia, A. R. B. Y.

Gottschalk, K. V.

M. C. Simon, K. V. Gottschalk, and J. M. Simon, “The coincidence of ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959–974 (2008).
[Crossref]

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
[Crossref]

M. C. Simon, K. B. Bastida, and K. V. Gottschalk, “Total reflection in a uniaxial crystal-uniaxial crystal interface,” Optik 116, 586–594 (2005).
[Crossref]

Hillman, L. W.

Huygens, C.

C. Huygens, Treatise on Light (Dover, 1962).

Kang, B.

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

Kim, B. K.

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

Landau, L. D.

L. D. Landau, L. P. Pitaevskii, and E. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinemann, 1984).

Landry, G. D.

Lee, G. D.

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

Lee, G.-D.

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

Lee, J. H.

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

Lee, S. H.

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

Lekner, J.

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. 3, 6121–6133 (1991).
[Crossref]

Liang, L.

Liang, Q.-T.

Lifshitz, E.

L. D. Landau, L. P. Pitaevskii, and E. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinemann, 1984).

Liu, D.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529–539 (2005).
[Crossref]

H. Ren, L. Liu, Z. Song, and D. Liu, “Double refraction and reflection of sequential interfaces in a crystal and application to integration of 1×N optical switch,” J. Mod. Opt. 50, 2231–2242 (2003).

Liu, L.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529–539 (2005).
[Crossref]

H. Ren, L. Liu, Z. Song, and D. Liu, “Double refraction and reflection of sequential interfaces in a crystal and application to integration of 1×N optical switch,” J. Mod. Opt. 50, 2231–2242 (2003).

Maldonado, T. A.

McClain, S. C.

McDermid, I. S.

Moreno, I.

Mun, B.-J.

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

Muryiand, A. A.

Perez, L. I.

M. C. Simon and L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[Crossref]

Pitaevskii, L. P.

L. D. Landau, L. P. Pitaevskii, and E. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinemann, 1984).

Ren, H.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529–539 (2005).
[Crossref]

H. Ren, L. Liu, Z. Song, and D. Liu, “Double refraction and reflection of sequential interfaces in a crystal and application to integration of 1×N optical switch,” J. Mod. Opt. 50, 2231–2242 (2003).

Rosete-Aguilar, M.

Sherman, G. C.

Simon, J. M.

M. C. Simon, K. V. Gottschalk, and J. M. Simon, “The coincidence of ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959–974 (2008).
[Crossref]

Simon, M. C.

M. C. Simon, K. V. Gottschalk, and J. M. Simon, “The coincidence of ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959–974 (2008).
[Crossref]

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
[Crossref]

M. C. Simon, K. B. Bastida, and K. V. Gottschalk, “Total reflection in a uniaxial crystal-uniaxial crystal interface,” Optik 116, 586–594 (2005).
[Crossref]

M. C. Simon and L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[Crossref]

M. C. Simon and R. M. Echarri, “Internal reflection in uniaxial crystals I. Geometrical description,” J. Mod. Opt. 37, 121–129 (1990).
[Crossref]

M. C. Simon and R. M. Echarri, “Inhibited reflection in uniaxial crystals,” Opt. Lett. 14, 257–259 (1989).
[Crossref]

M. C. Simon, “Refraction in biaxial crystals: a formula for the indices,” J. Opt. Soc. Am. A 4, 2201–2204 (1987).
[Crossref]

M. C. Simon and R. M. Echarri, “Internal total reflection in monoaxial crystals,” Appl. Opt. 26, 3878–3883 (1987).
[Crossref]

M. C. Simon, “Ray tracing in monoaxial crystals; are the exact formulas necessary?” Appl. Opt. 26, 3187–3189 (1987).
[Crossref]

M. C. Simon and R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
[Crossref]

M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
[Crossref]

Song, Z.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529–539 (2005).
[Crossref]

H. Ren, L. Liu, Z. Song, and D. Liu, “Double refraction and reflection of sequential interfaces in a crystal and application to integration of 1×N optical switch,” J. Mod. Opt. 50, 2231–2242 (2003).

Stamnes, J. J.

Stavroudis, O.

Stavroudis, O. N.

Stroganov, V. I.

Swindell, W.

Trolinger, J. D.

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–465 (1991).
[Crossref]

Wang, Y.

Wilson, D. K.

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–465 (1991).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Wu, H.

H. Wu, C. Zhang, and X. Bai, “A complete description of polarization and transmission of nonnormal incident rays in a uniaxial birefringent plate with arbitrary optic axis,” Opt. Commun. 283, 4129–4134 (2010).
[Crossref]

Wu, L.

Xin, H.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley-Interscience, 2002).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley-Interscience, 2002).

Youn, S.-H.

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

Zhang, C.

H. Wu, C. Zhang, and X. Bai, “A complete description of polarization and transmission of nonnormal incident rays in a uniaxial birefringent plate with arbitrary optic axis,” Opt. Commun. 283, 4129–4134 (2010).
[Crossref]

Zhang, W.-Q.

Zhang, Z.

Z. Zhang and H. Caulfield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. Laser Technol. 28, 549–553 (1996).
[Crossref]

Appl. Opt. (10)

J. Mod. Opt. (6)

H. Ren, L. Liu, Z. Song, and D. Liu, “Double refraction and reflection of sequential interfaces in a crystal and application to integration of 1×N optical switch,” J. Mod. Opt. 50, 2231–2242 (2003).

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529–539 (2005).
[Crossref]

M. C. Simon, K. V. Gottschalk, and J. M. Simon, “The coincidence of ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959–974 (2008).
[Crossref]

M. C. Simon and L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[Crossref]

S.-H. Youn, B.-J. Mun, J. H. Lee, B. K. Kim, H. C. Choi, S. H. Lee, B. Kang, and G.-D. Lee, “Multidimensional calculation of ray path in a twisted nematic liquid crystal cell,” J. Mod. Opt. 61, 257–262 (2014).
[Crossref]

M. C. Simon and R. M. Echarri, “Internal reflection in uniaxial crystals I. Geometrical description,” J. Mod. Opt. 37, 121–129 (1990).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (11)

M. Avendaño-Alejo, O. N. Stavroudis, and A. R. B. Y. Goitia, “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. N. Stavroudis, “Huygens’s principle and rays in uniaxial anisotropic media. II. Crystal axis orientation arbitrary,” J. Opt. Soc. Am. A 19, 1674–1679 (2002).
[Crossref]

M. Avendaño-Alejo, I. Moreno, and O. Stavroudis, “Minimum deviation angle in uniaxial prisms,” J. Opt. Soc. Am. A 24, 2431–2437 (2007).
[Crossref]

M. Avendaño-Alejo and M. Rosete-Aguilar, “Paraxial theory for birefringent lenses,” J. Opt. Soc. Am. A 22, 881–891 (2005).
[Crossref]

M. Avendaño-Alejo and M. Rosete-Aguilar, “Optical path difference in a plane-parallel uniaxial plate,” J. Opt. Soc. Am. A 23, 926–932 (2006).
[Crossref]

L. A. Alemán-Castañeda and M. Rosete-Aguilar, “Reflection formulae for ray tracing in uniaxial anisotropic media using Huygens’s principle,” J. Opt. Soc. Am. A 33, 2198–2205 (2016).
[Crossref]

Y. Wang, L. Liang, H. Xin, and L. Wu, “Complex ray tracing in uniaxial absorbing media,” J. Opt. Soc. Am. A 25, 653–657 (2008).
[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]

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
[Crossref]

M. C. Simon, “Refraction in biaxial crystals: a formula for the indices,” J. Opt. Soc. Am. A 4, 2201–2204 (1987).
[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, 2048–2063 (1995).
[Crossref]

J. Opt. Technol. (1)

J. Phys. (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. 3, 6121–6133 (1991).
[Crossref]

Mol. Cryst. Liq. Cryst. (1)

B.-J. Mun, S.-H. Youn, J. H. Lee, B. K. Kim, H. C. Choi, B. Kang, and G. D. Lee, “Calculation of ray path in liquid crystal modes,” Mol. Cryst. Liq. Cryst. 613, 75–81 (2015).
[Crossref]

Opt. Commun. (1)

H. Wu, C. Zhang, and X. Bai, “A complete description of polarization and transmission of nonnormal incident rays in a uniaxial birefringent plate with arbitrary optic axis,” Opt. Commun. 283, 4129–4134 (2010).
[Crossref]

Opt. Eng. (2)

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–465 (1991).
[Crossref]

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

Z. Zhang and H. Caulfield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. Laser Technol. 28, 549–553 (1996).
[Crossref]

Opt. Lett. (1)

Optik (2)

M. C. Simon, K. B. Bastida, and K. V. Gottschalk, “Total reflection in a uniaxial crystal-uniaxial crystal interface,” Optik 116, 586–594 (2005).
[Crossref]

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
[Crossref]

Proc. SPIE (1)

S. C. McClain and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media,” Proc. SPIE 1746, 107–118 (1992).
[Crossref]

Other (5)

C. Huygens, Treatise on Light (Dover, 1962).

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

L. D. Landau, L. P. Pitaevskii, and E. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinemann, 1984).

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley-Interscience, 2002).

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 (2)

Fig. 1.
Fig. 1. Geometrical ray tracing using the electromagnetic wave method.
Fig. 2.
Fig. 2. Geometrical ray tracing using Huygens’ principle.

Tables (1)

Tables Icon

Table 1. Numerical Comparison between Our Direct Formulas and Previous Ray-Tracing Examples

Equations (36)

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

ψ ( r ) = r 2 ( r · c ) 2 n e 2 + ( r · c ) 2 n o 2 = 1 .
n o 2 ( x 2 + y 2 ) + ( n e 2 n o 2 ) ( x 0 x + y 0 y ) 2 n o 2 n e 2 = 0 .
n m t = n e n o 2 + ( n e 2 n o 2 ) x 0 2 n e 2 + ( n o 2 n e 2 ) z 0 2 .
[ n o 2 + ( n e 2 n o 2 ) x 0 2 ] cot 2 θ e + 2 ( n e 2 n o 2 ) x 0 y 0 cot θ e + n o 2 + ( n e 2 n o 2 ) y 0 2 n e 2 n o 2 / ( n i sin θ i ) 2 = 0.
cot θ e = [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] 1 · [ x 0 y 0 ( n o 2 n e 2 ) ± n o d n i sin θ i ] ,
d = n e 2 [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] n i 2 sin 2 θ i [ n e 2 + ( n o 2 n e 2 ) z 0 2 ] .
n = n i sin θ i 1 + cot 2 θ e .
t = n o 2 s + ( n e 2 n o 2 ) ( s · c ) c n o 4 + ( n e 4 n o 4 ) ( s · c ) 2 .
s = n e 2 t + ( n o 2 n e 2 ) ( t · c ) c n e 4 + ( n o 4 n e 4 ) ( t · c ) 2 .
t = ( t x , t y , t z ) = ( u x , u y , u z ) / U ,
u x = n o 2 cot θ e + ( n e 2 n o 2 ) ( x 0 cot θ e + y 0 ) x 0 , u y = n o 2 + ( n e 2 n o 2 ) ( x 0 cot θ e + y 0 ) y 0 , u z = ( n e 2 n o 2 ) ( x 0 cot θ e + y 0 ) z 0 ,
U = n o n e ( n o 2 + n e 2 ) / ( n i sin θ i ) 2 ( 1 + cot 2 θ e ) .
n r = n i sin θ i ( t x cot θ e + t y ) = n o n e n o 2 + n e 2 ( n i sin θ i ) 2 ( 1 + cot 2 θ e ) .
ϕ ( r ) = n e 2 r 2 + ( n o 2 n e 2 ) ( r · c ) 2 = 1 ,
n e 2 ( x 2 + y 2 + z 2 ) + ( n o 2 n e 2 ) ( x 0 x + y 0 y + z 0 z ) 2 = 1 .
ϕ ( r ) = 2 [ n e 2 r + ( n o 2 n e 2 ) ( r · c ) c ] ,
ϕ ( r ) · r = 2 ,
n e 2 z + ( n o 2 n e 2 ) z 0 ( x 0 x + y 0 y + z 0 z ) = 0 .
n e 2 y + ( n o 2 n e 2 ) y 0 ( x 0 x + y 0 y + z 0 z ) = n i sin θ i ,
T = ( T x , T y , T z ) = ( x 1 , y 1 , z 1 ) / r 1 ,
x 1 = ± d n e 2 n o , y 1 = [ n e 2 n o ( n e 2 n o 2 ) n o z 0 2 ] n i sin θ i ± ( n e 2 n o 2 ) x 0 y 0 d n e 2 n o [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] , z 1 = ( n e 2 n o 2 ) z 0 ( n o y 0 n i sin θ i ± x 0 d ) n e 2 n o [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] ,
r 1 = { 1 n e 2 + ( n e 2 n o 2 ) ( n o y 0 n i sin θ i ± x 0 d ) 2 n e 2 n o 2 [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] 2 } 1 2 .
cot Θ e = x 0 y 0 ( n o 2 n e 2 ) ± ( n o / n i sin θ ) d n o 2 + ( n e 2 n o 2 ) x 0 2 ,
N r = { 1 n e 2 + ( n e 2 n o 2 ) ( n o y 0 n i sin θ i ± x 0 d ) 2 n e 2 n o 2 [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] 2 } 1 2 ,
N 2 = N r 2 cos 2 α = N r 2 r 1 2 | ϕ ( r 1 ) | 2 [ r 1 · ϕ ( r 1 ) ] 2 = 1 4 | ϕ ( r 1 ) | 2 .
N 2 = n o 2 + n e 2 n o 2 n e 2 N r 2 .
N = { n e 2 ( n e 2 n o 2 ) ( n o y 0 n i sin θ i ± x 0 d ) 2 [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] 2 } 1 2 ,
x 1 u x = y 1 u y = z 1 u z = n i sin θ i n e 2 n o 2 .
n 2 = n i 2 sin 2 θ i + [ x 0 y 0 ( n o 2 n e 2 ) n i sin θ i ± n o d ] 2 [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] 2 = A × n i 2 sin 2 θ i + B × n i sin θ i + C [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] 2 ,
A = ( n e 2 n o 2 ) [ ( n e 2 n o 2 ) ( x 0 2 + y 0 2 ) x 0 2 + n o 2 ( x 0 2 y 0 2 ) ] , B = ± 2 n o ( n o 2 n e 2 ) x 0 y 0 d , C = n o 2 n e 2 [ n o 2 + ( n e 2 n o 2 ) x 0 2 ] .
N = 1 2 ϕ ( r ) .
N 2 ( N · c ) 2 n e 2 + ( N · c ) 2 n o 2 = 1 .
N · z ^ = 0 ,
N · y ^ = n i sin θ i ,
R = 1 2 ψ ( r ) = 1 n e 2 n + ( 1 n o 2 1 n e 2 ) ( n · c ) c .
n = n e 2 R + ( n o 2 n e 2 ) ( R · c ) c .

Metrics