When this research was performed, the authors were with the Jet Propulsion Laboratory, Table Mountain Facility, California Institute of Technology, P.O. Box 367, Wrightwood, Calif. 92397.
G. Beyerle (gbeyerle@awi-potsdam.de) is now at Telegraphenberg A43, D-14473 Potsdam, Germany.
Georg Beyerle and I. Stuart McDermid, "Ray-tracing formulas for refraction and internal reflection in uniaxial crystals," Appl. Opt. 37, 7947-7953 (1998)
Formulas for the calculation of the direction cosines of refracted and internally
reflected rays in anisotropic uniaxial crystals are presented. The method is
based on a transformation to a nonorthonormal coordinate system in which the
normal surface associated with the extraordinary ray is of spherical shape. A
numerical example for the case of refraction and internal reflection in calcite
is given.
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Numerical Example of Vectors ŝ and k̂ for a
Ray with Incidence Angle θ(1) on Calcite
(mo = 1.54426,
me = 1.55335)a
θ(1)
(ŝ)x
(ŝ)y
(ŝ)z
(k̂)x
(k̂)y
(k̂)z
30°
0.94551581
0.32554625
0.00441537
0.94628810
0.32332465
0
45°
0.88878265
0.45830648
0.00453605
0.88927917
0.45736480
0
60°
0.82839146
0.56013102
0.00456451
0.82836490
0.56018890
0
The optic axis is taken to be (0.75, 0.5, 0.433), and the surface normal
is (1, 0, 0). The plane of incidence is the xy plane.
The numbers are derived from the formulas given in Simon and
Echarri,2 Liang,3 Zhang,5 and the present study. The results agree within
the computer’s numerical precision of 14 decimals.
Table 2
Numerical Example of Vectors ŝ and k̂ for a
Ray Internally Reflected on a Calcite–Air Boundary
(mo = 1.54426,
me = 1.55335) with Incidence Angle
θ(1)a
θ(1)
(ŝ)x
(ŝ)y
(ŝ)z
(k̂)x
(k̂)y
(k̂)z
30°
-0.86929881
0.49424257
-0.00662259
-0.86740484
0.49758210
-0.00457123
45°
-0.71021345
0.70396561
-0.00541063
-0.70888715
0.70530764
-0.00448758
60°
-0.50190702
0.86491313
-0.00382368
-0.50240012
0.86462554
-0.00409856
The optic axis is taken to be (0.75, 0.5, 0.433), and the surface normal
is (1, 0, 0). The incident ray vectors lie in the xy
plane. The incident wave vectors are (0.86632350, 0.49946224,
-0.00458850), (0.70579723, 0.70839957, -0.00450725), and (0.49670473,
0.86790984, -0.00411413) for incidence angles of 30°, 45°, and 60°,
respectively.
Tables (2)
Table 1
Numerical Example of Vectors ŝ and k̂ for a
Ray with Incidence Angle θ(1) on Calcite
(mo = 1.54426,
me = 1.55335)a
θ(1)
(ŝ)x
(ŝ)y
(ŝ)z
(k̂)x
(k̂)y
(k̂)z
30°
0.94551581
0.32554625
0.00441537
0.94628810
0.32332465
0
45°
0.88878265
0.45830648
0.00453605
0.88927917
0.45736480
0
60°
0.82839146
0.56013102
0.00456451
0.82836490
0.56018890
0
The optic axis is taken to be (0.75, 0.5, 0.433), and the surface normal
is (1, 0, 0). The plane of incidence is the xy plane.
The numbers are derived from the formulas given in Simon and
Echarri,2 Liang,3 Zhang,5 and the present study. The results agree within
the computer’s numerical precision of 14 decimals.
Table 2
Numerical Example of Vectors ŝ and k̂ for a
Ray Internally Reflected on a Calcite–Air Boundary
(mo = 1.54426,
me = 1.55335) with Incidence Angle
θ(1)a
θ(1)
(ŝ)x
(ŝ)y
(ŝ)z
(k̂)x
(k̂)y
(k̂)z
30°
-0.86929881
0.49424257
-0.00662259
-0.86740484
0.49758210
-0.00457123
45°
-0.71021345
0.70396561
-0.00541063
-0.70888715
0.70530764
-0.00448758
60°
-0.50190702
0.86491313
-0.00382368
-0.50240012
0.86462554
-0.00409856
The optic axis is taken to be (0.75, 0.5, 0.433), and the surface normal
is (1, 0, 0). The incident ray vectors lie in the xy
plane. The incident wave vectors are (0.86632350, 0.49946224,
-0.00458850), (0.70579723, 0.70839957, -0.00450725), and (0.49670473,
0.86790984, -0.00411413) for incidence angles of 30°, 45°, and 60°,
respectively.