Abstract

Dispersion in a uniaxial prism provides an example of the application of the law of refraction for a uniaxial crystal. Formulas for the minimum deviation angle for the extraordinary ray are given when the crystal axis lies in the plane of incidence. Three particular cases for the crystal axis position are presented and are shown to have a behavior similar to that of an ordinary prism.

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.
  2. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1993), Chap. 4, pp. 69-120.
  3. J. P. Mathieu, Optics (Pergamon, 1975), Parts 1 and 2, Chap. 4, pp. 77-102.
  4. S. Huard, Polarization of Light (Wiley, 1997), Chap. 2.
  5. F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976), Chap. 2, pp. 30-32.
  6. M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986), Chap. 2, pp. 86-90.
  7. M. C. Simon and P. A. Larocca, "Minimum deviation for uniaxial prisms," Appl. Opt. 34, 709-715 (1995).
    [CrossRef] [PubMed]
  8. M. Avendaño-Alejo and O. Stavroudis, "Huygens' principle and rays in uniaxial anisotropic media: II. Crystal axis with arbitrary orientation," J. Opt. Soc. Am. A 19, 1674-1679 (2002).
    [CrossRef]
  9. M. Fraçon, N. Krauzman, J. P. Mathieu, and M. May, Experiments in Physical Optics (Gordon & Breach, 1970), pp. 128-139.
  10. M. Fraçon, Optical Interferometry (Academic, 1966), pp. 137-161.

2002 (1)

1995 (1)

Avendaño-Alejo, M.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

Fraçon, M.

M. Fraçon, N. Krauzman, J. P. Mathieu, and M. May, Experiments in Physical Optics (Gordon & Breach, 1970), pp. 128-139.

M. Fraçon, Optical Interferometry (Academic, 1966), pp. 137-161.

Furtak, T. E.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986), Chap. 2, pp. 86-90.

Huard, S.

S. Huard, Polarization of Light (Wiley, 1997), Chap. 2.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976), Chap. 2, pp. 30-32.

Klein, M. V.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986), Chap. 2, pp. 86-90.

Krauzman, N.

M. Fraçon, N. Krauzman, J. P. Mathieu, and M. May, Experiments in Physical Optics (Gordon & Breach, 1970), pp. 128-139.

Larocca, P. A.

Mathieu, J. P.

M. Fraçon, N. Krauzman, J. P. Mathieu, and M. May, Experiments in Physical Optics (Gordon & Breach, 1970), pp. 128-139.

J. P. Mathieu, Optics (Pergamon, 1975), Parts 1 and 2, Chap. 4, pp. 77-102.

May, M.

M. Fraçon, N. Krauzman, J. P. Mathieu, and M. May, Experiments in Physical Optics (Gordon & Breach, 1970), pp. 128-139.

Simon, M. C.

Stavroudis, O.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976), Chap. 2, pp. 30-32.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1993), Chap. 4, pp. 69-120.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1993), Chap. 4, pp. 69-120.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Other (8)

M. Fraçon, N. Krauzman, J. P. Mathieu, and M. May, Experiments in Physical Optics (Gordon & Breach, 1970), pp. 128-139.

M. Fraçon, Optical Interferometry (Academic, 1966), pp. 137-161.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1993), Chap. 4, pp. 69-120.

J. P. Mathieu, Optics (Pergamon, 1975), Parts 1 and 2, Chap. 4, pp. 77-102.

S. Huard, Polarization of Light (Wiley, 1997), Chap. 2.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976), Chap. 2, pp. 30-32.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986), Chap. 2, pp. 86-90.

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Figures (7)

Fig. 1
Fig. 1

Direction cosines for incident, ordinary, and extraordinary refracted rays and the optical crystal axis in both Y Z and Y Z system coordinates. The fictive ordinary ray also is drawn.

Fig. 2
Fig. 2

Angles involved for obtaining the total deviation angle in a common prism.

Fig. 3
Fig. 3

Thicker lines give the angle of refraction for which we obtain the minimum deviation angle for both ordinary and extraordinary rays as a function of the apex angle. Dashed lines give possible orientations of the crystal axis with respect to the normal to the refracting surface as a function of the apex angle.

Fig. 4
Fig. 4

Angles and direction cosines involved for obtaining the total deviation angle for the extraordinary ray in a uniaxial prism.

Fig. 5
Fig. 5

Minimum deviation angle for the extraordinary ray as a function of the crystal axis position for quartz as a uniaxial crystal with n e = 1.5533 , n o = 1.5442 , and the apex angle given by Ω = 30 ° .

Fig. 6
Fig. 6

Gray curve, total deviation angle for an isotropic prism with index of refraction n o and δ o given by Eq. (7). Dashed curve, total deviation angle for the extraordinary ray when the crystal axis is parallel to the base of the prism, δ e , given by Eq. (18), by use of Eqs. (21, 22) for ϕ = Ω 2 .

Fig. 7
Fig. 7

Solid curve, total deviation angle for an isotropic prism with index of refraction n e and δ e given by Eq. (18) by use of Eqs. (16, 17). Dashed curve, total deviation angle for the extraordinary ray when the crystal axis is perpendicular to the base of the prism δ e given by Eq. (18), by use of Eqs. (21, 22) for ϕ = ( Ω π ) 2 .

Equations (45)

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θ o 1 = arcsin ( n i sin θ i n o ) ,
( ξ e η e ζ e ) = 1 Υ e g [ ( n o 2 [ n e 2 + ( n o 2 n e 2 ) β 1 2 ] n o 2 ( n o 2 n e 2 ) α 1 β 1 0 n o 2 ( n o 2 n e 2 ) α 1 β 1 n o 2 [ n e 2 + ( n o 2 n e 2 ) α 1 2 ] 0 0 0 Γ Δ ζ o ) ( ξ o η o ζ o ) γ 1 ( n o 2 n e 2 ) Δ ( 1 0 0 0 1 0 0 0 0 ) ( α 1 β 1 γ 1 ) ] ,
Υ e g 2 = n e 2 { n o 2 Γ 2 ( n o 2 n e 2 ) [ γ 1 Δ + n o 2 ( α 1 ξ o + β 1 η o ) ] 2 } ,
Δ 2 = Γ [ n e 2 n o 2 ( ξ o 2 + η o 2 ) ] + n o 2 ( n o 2 n e 2 ) ( α 1 ξ o + β 1 η o ) 2 ,
Γ = n e 2 γ 1 2 + n o 2 ( α 1 2 + β 1 2 ) ,
( ξ o f η o f ζ o f ) = 1 n o Γ e [ ( n e 2 + ( n o 2 n e 2 ) α 1 2 ( n o 2 n e 2 ) α 1 β 1 0 ( n o 2 n e 2 ) α 1 β 1 n e 2 + ( n o 2 n e 2 ) β 1 2 0 0 0 Δ e ζ e 1 ) ( ξ e 1 η e 1 ζ e 1 ) + ζ e 1 γ 1 ( n o 2 n e 2 ) ( 1 0 0 0 1 0 0 0 0 ) ( α 1 β 1 γ 1 ) ] ,
Γ e 2 = n e 2 sin 2 ψ + n o 2 cos 2 ψ ,
Δ e 2 = n e 2 ( n o 2 n e 2 ) sin 2 ψ + [ ( n o 2 n e 2 ) γ 1 cos ψ + n e 2 ζ e 1 ] 2 ,
cos ψ = α 1 ξ e 1 + β 1 η e 1 + γ 1 ζ e 1 ,
( ξ e 2 η e 2 ζ e 2 ) = 1 n i ( n o 0 0 0 n o 0 0 0 1 ζ o f n i 2 n o 2 ( 1 ζ o f 2 ) ) ( ξ o f η o f ζ o f ) .
δ o = θ i + θ t 2 Ω , Ω = θ o 1 + θ i 2 ,
θ t 2 ( θ i ) = arcsin [ n o sin ( Ω θ o 1 ) n i ] = arcsin { n o sin [ Ω arcsin ( n i sin θ i n o ) ] n i } .
d δ o d θ i = 1 [ n i cos θ i ( cos Ω + n i sin Ω sin θ i n o 2 n i 2 sin 2 θ i ) n i 2 ( n i cos Ω sin θ i sin Ω n o 2 n i 2 sin 2 θ i ) 2 ] = 0 .
n i 2 cos 2 θ i ( n i sin Ω sin θ i + cos Ω n o 2 n i 2 sin 2 θ i ) 2 = [ n o 2 n i 2 sin 2 θ i n i 2 ( n i cos Ω sin θ i sin Ω n o 2 n i 2 sin 2 θ i ) 2 ] 2 .
( n i 2 n o 2 ) 2 [ n o 4 sin 2 Ω 4 n i 2 n o 2 sin 2 θ i + 4 n i 4 sin 4 θ i ] sin 2 Ω = 0 ,
θ i + = ± arcsin { n o sin [ ( π Ω ) 2 ] n i } ,
θ i = ± arcsin [ n o sin ( Ω 2 ) n i ] .
δ o ( θ i + + ) = arcsin [ n o cos ( Ω 2 ) n i ] arcsin [ n o cos ( 3 Ω 2 ) n i ] Ω ,
δ o ( θ i ) = arcsin [ n o sin ( Ω 2 ) n i ] + arcsin [ n o sin ( 3 Ω 2 ) n i ] Ω ,
δ o ( θ i + ) = Ω ,
δ o ( θ i + ) = arcsin [ n o sin ( Ω 2 ) n i ] + arcsin [ n o sin ( Ω 2 ) n i ] Ω .
δ m o = 2 arcsin [ n o sin ( Ω 2 ) n i ] Ω ,
n o = n i sin [ ( δ m o + Ω ) 2 ] sin ( Ω 2 ) .
θ i = arcsin { n o sin [ Ω arcsin ( n i sin θ i n o ) ] n i } .
n i sin θ i n o = sin [ Ω arcsin ( n i sin θ i n o ) ] ,
n i 2 sin 2 θ i n o 2 sin 2 ( Ω 2 ) = 0 .
θ i m o = θ i + = arcsin [ n o sin ( Ω 2 ) n i ] ,
θ e 1 = arctan ( n i sin θ i n e 2 n i 2 sin 2 θ i ) ,
θ e 2 = arctan [ n e sin ( Ω θ e 1 ) n i 2 n e 2 sin 2 ( Ω θ e 1 ) ] .
δ e = θ i + θ e 2 Ω ,
( n e 4 sin 2 Ω 4 n i 2 n e 2 sin 2 θ i + 4 n i 4 sin 4 θ i ) sin 2 Ω = 0 ,
θ i m e = arcsin [ n e sin ( Ω 2 ) n i ] ,
δ m e = 2 arcsin [ n e sin ( Ω 2 ) n i ] Ω .
tan θ e 1 = n i n e n o sin θ i + ( n e 2 n o 2 ) cos ϕ sin ϕ n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ( n e 2 cos 2 ϕ + n o 2 sin 2 ϕ ) n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ,
tan θ e 2 = n e 2 cos ϕ sin φ + n o 2 cos φ sin ϕ n i 2 ( n o 2 cos 2 φ + n e 2 sin 2 φ ) ( n e 2 cos ϕ sin φ + n o 2 cos φ sin ϕ ) 2 ,
tan ( Ω 2 ) = n i n e n o sin θ i + ( n e 2 n o 2 ) cos ϕ sin ϕ n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ( n e 2 cos 2 ϕ + n o 2 sin 2 ϕ ) n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i .
θ i = arcsin [ { n e 2 cos ϕ sin [ ( Ω 2 ) ϕ ] + n o 2 sin ϕ cos [ ( Ω 2 ) ϕ ] } n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i n e 2 n o 2 cos 2 [ ( Ω 2 ) ] + { n e 2 cos ϕ sin [ ( Ω 2 ) ϕ ] + n o 2 cos [ ( Ω 2 ) ϕ ] sin ϕ } 2 ] .
θ i m e ϕ = θ i = arcsin { n e 2 cos ϕ sin [ ( Ω 2 ) ϕ ] + n o 2 sin ϕ cos [ ( Ω 2 ) ϕ ] n i n o 2 cos 2 [ ( Ω 2 ) ϕ ] + n e 2 sin 2 [ ( Ω 2 ) ϕ ] } ,
δ m e ϕ = 2 arcsin { n e 2 cos ϕ sin [ ( Ω 2 ) ϕ ] + n o 2 sin ϕ cos [ ( Ω 2 ) ϕ ] n i n o 2 cos 2 [ ( Ω 2 ) ϕ ] + n e 2 sin 2 [ ( Ω 2 ) ϕ ] } Ω .
θ i m e = arcsin [ n o sin ( Ω 2 ) n i ] = θ i m o ,
δ m e = δ m o for θ i = θ i m o = θ i m e ,
δ e ( θ i ) δ o ( θ i ) for θ i θ i m o = θ i m e .
θ i m e = arcsin [ n e sin ( Ω 2 ) n i ] = θ i m e ,
δ m e = δ m e for θ i = θ i m e = θ i m e ,
δ e ( θ i ) δ e ( θ i ) for θ i θ i m e = θ i m e ,

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