Abstract

Ray vector traces that can be obtained at the back of a biaxial crystal through crystal rotation are calculated numerically by variation of one of the three Euler angles. Negative and positive lossless biaxial crystals are considered at normal incidence. Interesting features of these traces are evidenced by reference to the circular ring that forms at internal conical refraction.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Mentel, E. Schmidt, T. Mavrudis, “Birefringent filter with arbitrary orientation of the optic axis: an analysis of improved accuracy,” Appl. Opt. 31, 5022–5029 (1992).
    [CrossRef] [PubMed]
  2. P. J. Valle, F. Moreno, “Theoretical study of birefringent filters as intracavity wavelength selectors,” Appl. Opt. 31, 528–535 (1992).
    [CrossRef] [PubMed]
  3. J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
    [CrossRef]
  4. Z. M. Li, B. T. Sullivan, R. R. Parsons, “Use of the 4 × 4 matrix method in the optics of multilayer magnetooptic recording media,” Appl. Opt. 27, 1334–1338 (1988).
    [CrossRef] [PubMed]
  5. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  6. J. Schesser, G. Eichmann, “Propagation of plane waves in biaxially anisotropic media,” J. Opt. Soc. Am. 62, 786–791 (1972).
    [CrossRef]
  7. H. Ito, H. Naito, H. Inaba, “Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992–3998 (1975).
    [CrossRef]
  8. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  9. N. J. Damaskos, A. L. Maffett, P. L. E. Uslenghi, “Reflection and transmission for gyroelectromagnetic biaxial layered media,” J. Opt. Soc. Am. A 2, 454–461 (1985).
    [CrossRef]
  10. D. J. De Smet, “Reflection from an oriented biaxial surface,” Appl. Opt. 26, 995–998 (1987).
    [CrossRef] [PubMed]
  11. M. A. Dreger, J. H. Erkkila, “Improved method for calculating phase-matching criteria in biaxial nonlinear materials,” Opt. Lett. 17, 787–788 (1992).
    [CrossRef] [PubMed]
  12. D. A. Roberts, “Simplified characteristics of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
    [CrossRef]
  13. M. C. Simon, “Refraction in biaxial crystals: a formula for the indices,” J. Opt. Soc. Am. A 4, 2201–2204 (1987).
    [CrossRef]
  14. M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
    [CrossRef]
  15. T. A. Maldonado, 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]
  16. G. D. Landry, 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]
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–690.
  18. G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart and Winston, New York, 1968), Chap. 5, pp. 152–199.
  19. Z. Shao, C. Yi, “Behavior of extraordinary rays in uniaxial crystals,” Appl. Opt. 33, 1209–1212 (1994).
    [CrossRef] [PubMed]
  20. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.

1995

1994

1992

1991

1988

1987

1985

1982

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

1979

1975

H. Ito, H. Naito, H. Inaba, “Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992–3998 (1975).
[CrossRef]

1972

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–690.

Damaskos, N. J.

De Smet, D. J.

Dreger, M. A.

Eichmann, G.

Erkkila, J. H.

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart and Winston, New York, 1968), Chap. 5, pp. 152–199.

Gaylord, T. K.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.

Gottschalk, K. V.

M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
[CrossRef]

Henderson, D. M.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Inaba, H.

H. Ito, H. Naito, H. Inaba, “Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992–3998 (1975).
[CrossRef]

Ito, H.

H. Ito, H. Naito, H. Inaba, “Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992–3998 (1975).
[CrossRef]

Landry, G. D.

Li, Z. M.

Lotspeich, J. F.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Maffett, A. L.

Maldonado, T. A.

Mavrudis, T.

Mentel, J.

Moreno, F.

Naito, H.

H. Ito, H. Naito, H. Inaba, “Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992–3998 (1975).
[CrossRef]

Parsons, R. R.

Roberts, D. A.

D. A. Roberts, “Simplified characteristics of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

Schesser, J.

Schmidt, E.

Shao, Z.

Simon, M. C.

M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
[CrossRef]

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

Stephens, R. R.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Sullivan, B. T.

Uslenghi, P. L. E.

Valle, P. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–690.

Yeh, P.

Yi, C.

Appl. Opt.

IEEE J. Quantum Electron.

D. A. Roberts, “Simplified characteristics of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

IEEE J. Quantum. Electron.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum. Electron. QE-18, 1253–1258 (1982).
[CrossRef]

J. Appl. Phys.

H. Ito, H. Naito, H. Inaba, “Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992–3998 (1975).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Pure Appl. Opt.

M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–690.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart and Winston, New York, 1968), Chap. 5, pp. 152–199.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.

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 (5)

Fig. 1
Fig. 1

Relationship between the (x p , y p , z p ) principal axes and the (x, y, z) laboratory coordinate systems. The transformation between these coordinate systems is specified by three Euler angles (x convention): ϕ, θ, and ψ.

Fig. 2
Fig. 2

Schematic diagram showing the relative orientation of relevant vectors and planes. The wave vector k p is perpendicular to the plane Θ p that contains vectors H ip , D ip (i = 1, 2), and N p± . A translation of k p and Θ p on the distance O 1 O 2 permits the separate representation of vectors with indices 1 and 2. Vectors k p , E ip , D ip , and S ip lie in the planes Π ip (i = 1, 2), which are perpendicular to each other. To show them separately, we have translated each plane Π ip further on the distance O i O i ′, with i = 1, 2.

Fig. 3
Fig. 3

Ray vector traces at the back of the aragonite crystal obtained by varying ψ from 0 to 180° in steps of 1°, at ϕ = 0 (a) for ray 1, with θ from 10° to 80° (θ < Ω p ) in steps of 10° (curves 1–8) and (b) for ray 2, with θ from 10° to 70° in steps of 10° (curves 1–7). Ray vector traces obtained by varying ψ from 0 to 180° in steps of 0.1° (c) for ray 1 and (d) for ray 2, at ϕ = 0 and θ values of 81° and 82°, which are smaller and greater than Ω p = 81.4275°. Ray vector traces obtained by varying ψ from 0 to 180° in steps of 0.1° at ϕ = 0 and θ > Ω p for (e) ray 1 and (f) ray 2, with θ from 84° to 88°. The x i and y i axes (i = 1, 2) coincide with the x and y coordinates axes. On each trace the starting point (at ψ = 0) is marked by a small circle and the point at ψ = 90° is marked by an asterisk. The circular ring that corresponds to the internal conical refraction is represented by a dotted curve.

Fig. 4
Fig. 4

Similar to Fig. 3, ray vector traces at the back of the topaz crystal at ϕ = 0 and θ < Ω p (a) for ray 1 and (b) ray 2, with θ from 5° to 30° in steps of 5°. Ray vector traces at ϕ = 0 for (c) ray 1 and (d) for ray 2, at θ values of 33° and 34°, which are smaller and greater than Ω p = 33.3326°. Ray vector traces at ϕ = 0 for (e) ray 1, with θ from 35° to 50° in steps of 5°, and (f) for ray 2, with θ from 35° to 85° in steps of 10°.

Fig. 5
Fig. 5

Similar to Figs. 3(a) and 3(b), ray vector traces at the back of the aragonite crystal at ϕ = 45° and θ < Ω p for (a) ray 1 and (b) ray 2. Lines AA′ and BB′ are perpendicular to each other and are rotated by 45° with respect to the x and y coordinate axes.

Tables (1)

Tables Icon

Table 1 Unit Electric-Field Vectors in the System of Principal Axes and the Polarization Angles of Normally Incident, Forward-Traveling Waves into the Biaxial Crystal in the Cases of Singularity When the Wave Vector Lies in Either Principal Plane

Equations (21)

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

tan2 Ωp=1/εxp-1/εyp/1/εyp-1/εzp.
Hˆ1p=Nˆp+-Nˆp-/|Nˆp+-Nˆp-|,  Hˆ2p=-Nˆp++Nˆp-/|Nˆp++Nˆp-|
Sˆi=M¯TSˆip,  i=1, 2.
Sˆi=cos αi sin ηi, sin αi sin ηi, cos ηiT.
ηi=-signkˆp·EˆiparcosEˆip · Dˆip,  i=1, 2,
αi=signkˆp·yˆp×Hˆiparcosyˆp·Hˆip,  i=1, 2.
|tan η2|=εzp-εyptan θ/εzp+εyp tan2 θ.
Sˆ1·Sˆ2=cos η1 cos η2,
tan η=-C sin α,
C=εyp/εxp-11-εyp/εzp1/2.
xi=d cos αi tan ηi, yi=d sin αi tan ηi, i=1, 2,
xi, yiTϕ0=cos ϕ-sin ϕsin ϕcos ϕxi, yiTϕ=0,  i=1, 2.
αiϕ0=αiϕ=0+ϕ,  i=1, 2,
rixi2+yi21/2=d|tan ηi|,  i=1, 2.
x=-Cd sin α cos α,  y=-Cd sin2 α,
x2+y2+yCd=0.
θm=arctannzp/nyp,  r2m=d/tan2θm.
δs=dεzp-εyp2sin 4θ/4εypεzp+εzp-εyp2sin22θ.
θm*=arctannzp/nxp,  rjm*=d/tan2θm*,
ψm=arctannyp/nxp,  r1m=d/tan2ψm.
εxpC tan2 θc-εzp-εxptan θc+εzpC=0,

Metrics