Abstract

We have experimentally investigated the behavior of extraordinary rays (E rays) in uniaxial crystals for two cases: that in which optical axes are parallel to the surfaces and that in which they are inclined. The E ray always rotates around the ordinary ray (O ray) in the same direction that the crystal rotates around its surface normal. For the case when the axes are parallel to the surface, we discovered that the E ray rotates up to α = 2π as the crystal rotates to ϕ = π. The E ray traces a series of ellipses as the angle of incidence is varied. Snell’s law is valid for the E ray only when the optical axes are perpendicular to the plane of incidence. For the case in which the optical axes are incident, the E ray and the crystal rotate at different speeds except for the case of normal incidence. The speed of rotation increases with the incidence angle. The ray traces a curve known as the Pascal worm, which is described by the equation (x 2 + z 2mx)2 = n 2(x 2 + z 2). When the optical axes coincide with the plane of incidence, the space between the rays in the plane is not related to the angle of incidence.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 14, pp. 676, 685.
  2. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Chap. 24.9, pp. 508–509.
  3. W. H. A. Fincham, M. H. Freeman, Optics (Butterworth, London, 1980), pp. 340–341.
  4. M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986), Chap. 9, p. 601.
  5. Q. T. Liang, “Simple ray tracing formulas for uniaxial optical crystals,” Appl. Opt. 29, 1008–1010 (1990).
    [CrossRef] [PubMed]
  6. M. C. Simon, R. M. Echarry, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
    [CrossRef] [PubMed]
  7. Z. X. Zhang, Acta Phys. Sin. (in Chinese) 29, 1483–1487 (1980).

1990 (1)

1986 (1)

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

1980 (1)

Z. X. Zhang, Acta Phys. Sin. (in Chinese) 29, 1483–1487 (1980).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 14, pp. 676, 685.

Echarry, R. M.

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

Fincham, W. H. A.

W. H. A. Fincham, M. H. Freeman, Optics (Butterworth, London, 1980), pp. 340–341.

Freeman, M. H.

W. H. A. Fincham, M. H. Freeman, Optics (Butterworth, London, 1980), pp. 340–341.

Furtak, T. E.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986), Chap. 9, p. 601.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Chap. 24.9, pp. 508–509.

Klein, M. V.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986), Chap. 9, p. 601.

Liang, Q. T.

Simon, M. C.

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

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Chap. 24.9, pp. 508–509.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 14, pp. 676, 685.

Zhang, Z. X.

Z. X. Zhang, Acta Phys. Sin. (in Chinese) 29, 1483–1487 (1980).

Acta Phys. Sin. (1)

Z. X. Zhang, Acta Phys. Sin. (in Chinese) 29, 1483–1487 (1980).

Appl. Opt. (1)

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

Appl. Opt. (1)

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 14, pp. 676, 685.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Chap. 24.9, pp. 508–509.

W. H. A. Fincham, M. H. Freeman, Optics (Butterworth, London, 1980), pp. 340–341.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986), Chap. 9, p. 601.

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 (3)

Fig. 1
Fig. 1

Diagram showing double refraction in a uniaxial crystal. The figure at the bottom shows the rotation of the XYZ′ system with respect to the XYZ system.

Fig. 2
Fig. 2

Traces of the E ray on the rear surface of the calcite APS at a different incidence. The O ray acts as the origin of the coordinates, the X″ axis is parallel to the X axis. Curves I–V are the traces for θ = 0, 5π/12, π/3, π/4, π/6, respectively. The symbols on the curves denote the rotation of the crystal in steps of ϕ = π/16: ●, curve V; ▲, curve IV; ▼, curve III; +, curve II.

Fig. 3
Fig. 3

Traces of E rays of the calcite AIS (natural cleavage). The solid curves are the experimental results, the dashed curves are the Pascal worm calculated by (x 2 + z 2mx)2 = n 2(x 2 + z 2). The symbols denote the rotation of the crystal in steps of ϕ = π/8. The other factors are the same as in Fig. 2. ●, curve V; ▲, curve IV; ▲, curve III; +, curve II; ○, curve I.

Tables (2)

Tables Icon

Table 1 Axes a and b of the Ellipses in Fig. 2 for Different Angles of Incidence

Tables Icon

Table 2 Data in the Experiments and Comparison between the Data and Theoretical Calculationa

Equations (18)

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

( A cos 2 ϕ + n e 2 sin 2 ϕ ) x 2 + B y 2 + ( A sin 2 ϕ + n e 2 cos 2 ϕ ) z 2 + 2 C cos ϕ x y + 2 C sin ϕ y z + 2 D sin ϕ cos ϕ z x = 1 ,
A = n o 2 cos 2 γ + n e 2 sin 2 γ , B = n o 2 sin 2 γ + n e 2 cos 2 γ , C = ( n e 2 - n o 2 ) sin γ cos γ , D = ( n o 2 - n e 2 ) cos 2 γ .
A x 1 x + B y 1 y + C ( y 1 x + x 1 y ) = 1.
A x 1 + C y 1 - sin θ = 0
y 1 = ( sin θ - A x 1 ) / C .
A x 1 + B y 1 + 2 C x 1 y 1 = 1.
x 1 = ( sin θ ± C K ) / A , y 1 = k ,
k = [ ( A - sin 2 θ ) / ( A B - C 2 ) 1 / 2 ] .
r e = arctan [ ( sin θ + C K ) / ( A K ) ] ,
r e = arctan [ ( sin θ - C K ) / ( A K ) ] ,
Δ = d [ tan ( r e ) - tan ( r e ) ] = 2 d C / A .
r e = arctan ( C / A ) = - r e .
n o 2 x 1 x + n e 2 y 1 y = 1.
x 1 = sin θ / n o 2 , y 1 = [ 1 - ( sin θ / n o ) 2 ] 1 / 2 / n e .
x 1 = sin θ / n e 2 , y 1 = [ 1 - ( sin θ / n e ) 2 ] 1 / 2 / n e .
sin θ / sin ( r e ) = n e ( θ )
n e ( θ ) = [ n o 4 + ( n e 2 - n o 2 ) sin 2 θ ] 1 / 2 / n e .
sin θ / sin ( r e ) = n e .

Metrics