Abstract

The conventional solutions to the problem of normally incident light transmission through homogeneous, birefringent, nonoptically active, nonabsorbing, crystalline plate are not exact. When it is treated as a boundary value problem in electromagnetic field theory, exact expressions are obtained for the retardation or phase difference and the electric field amplitude ratio. The two solutions differ in some interesting ways that become of substantial importance in the examination of laser light. The nature of the exact solutions is examined in detail and numerical comparisons with the conventional solutions are given for the cases of calcite and quartz, neglecting the optical activity of crystalline quartz. For quartz it is shown that one can obtain a quarter-wave plate by using any one of a number of different crystal thicknesses. The application of wave plates in the investigation of elliptically polarized light is briefly discussed. For small angles of incidence, the effects to be expected for light which is obliquely incident on the plate surface are investigated.

© 1964 Optical Society of America

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References

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  1. For an equivalent analysis see, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 688.
  2. B. I. Stepanov and A. P. Khapalyuk, Opt. i Spectroskopiya 13, 714 (1962)[English transl.: Opt. Spectry. 13, 404 (1962)].
  3. F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
    [Crossref]
  4. Thornton C. Fry, J. Opt. Soc. Am. and Rev. Sci. Instr. 16, 1 (1928).
    [Crossref]
  5. H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
    [Crossref]
  6. S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio (John Wiley & Sons, Inc., New York, 1960), p. 290.
  7. H. Jacobs, F. A. Brand, J. D. Meindl, S. Weitz, R. Benjamin, and D. A. Holmes, Proc. IEEE 51, 581 (1963).
    [Crossref]
  8. For the purposes of numerical computations, we have chosen optical constants for calcite and quartz which correspond to near infrared wavelengths because we are using GaAs injection laser sources (8400 Å) in some of our work. The general concepts advanced in this work, however, are valid for other wavelengths. The numerical values for the optical constants were taken from: Dwight E. Gray, Coordinating Editor, American Institute of Physics Handbook (McGraw-Hill Book Company Inc., New York, 2nd edition, 1963), Calcite, p. 6-18; Crystal Quartz, p. 6-24; Rutile, p. 6-33.
  9. G. N. Ramachandran and S. Ramaseshan in Handbuch der Physik, edited by S. Flügge, (Springer-Verlag, Berlin, 1961), Vol. 25, Chap. 1, p. 76.
  10. Reference 1, p. 24.
  11. A. C. Hall, J. Opt. Soc. Am. 53, 801 (1963).We have observed a typographical error in Eq. (12) of Hall’s work. The corrected forms, in Hall’s notation, arecos2ψ=[+cosδsinδ±cot2γ(KtanΔ−sin2δ)12]/K,sin2ψ=[−sinδcot2γ±cosδ(KtanΔ−sin2δ)12]/K.The geometry used by Hall is equivalent to that used in the present work.
    [Crossref]
  12. D. Bergman, J. Opt. Soc. Am. 52, 1080 (1962).
    [Crossref]
  13. H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964).
    [Crossref]

1964 (1)

1963 (3)

1962 (2)

D. Bergman, J. Opt. Soc. Am. 52, 1080 (1962).
[Crossref]

B. I. Stepanov and A. P. Khapalyuk, Opt. i Spectroskopiya 13, 714 (1962)[English transl.: Opt. Spectry. 13, 404 (1962)].

1940 (1)

F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
[Crossref]

1928 (1)

Thornton C. Fry, J. Opt. Soc. Am. and Rev. Sci. Instr. 16, 1 (1928).
[Crossref]

Benjamin, R.

H. Jacobs, F. A. Brand, J. D. Meindl, S. Weitz, R. Benjamin, and D. A. Holmes, Proc. IEEE 51, 581 (1963).
[Crossref]

Bergman, D.

Born, M.

For an equivalent analysis see, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 688.

Brand, F. A.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

H. Jacobs, F. A. Brand, J. D. Meindl, S. Weitz, R. Benjamin, and D. A. Holmes, Proc. IEEE 51, 581 (1963).
[Crossref]

Fry, Thornton C.

Thornton C. Fry, J. Opt. Soc. Am. and Rev. Sci. Instr. 16, 1 (1928).
[Crossref]

Gabler, F.

F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
[Crossref]

Hall, A. C.

Harris, J.

Hatkin, L.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

Holmes, D. A.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

H. Jacobs, F. A. Brand, J. D. Meindl, S. Weitz, R. Benjamin, and D. A. Holmes, Proc. IEEE 51, 581 (1963).
[Crossref]

Jacobs, H.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

H. Jacobs, F. A. Brand, J. D. Meindl, S. Weitz, R. Benjamin, and D. A. Holmes, Proc. IEEE 51, 581 (1963).
[Crossref]

Khapalyuk, A. P.

B. I. Stepanov and A. P. Khapalyuk, Opt. i Spectroskopiya 13, 714 (1962)[English transl.: Opt. Spectry. 13, 404 (1962)].

Meindl, J. D.

H. Jacobs, F. A. Brand, J. D. Meindl, S. Weitz, R. Benjamin, and D. A. Holmes, Proc. IEEE 51, 581 (1963).
[Crossref]

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan in Handbuch der Physik, edited by S. Flügge, (Springer-Verlag, Berlin, 1961), Vol. 25, Chap. 1, p. 76.

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan in Handbuch der Physik, edited by S. Flügge, (Springer-Verlag, Berlin, 1961), Vol. 25, Chap. 1, p. 76.

Ramo, S.

S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio (John Wiley & Sons, Inc., New York, 1960), p. 290.

Sokob, P.

F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
[Crossref]

Stepanov, B. I.

B. I. Stepanov and A. P. Khapalyuk, Opt. i Spectroskopiya 13, 714 (1962)[English transl.: Opt. Spectry. 13, 404 (1962)].

Weinberger, H.

Weitz, S.

H. Jacobs, F. A. Brand, J. D. Meindl, S. Weitz, R. Benjamin, and D. A. Holmes, Proc. IEEE 51, 581 (1963).
[Crossref]

Whinnery, J. R.

S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio (John Wiley & Sons, Inc., New York, 1960), p. 290.

Wolf, E.

For an equivalent analysis see, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 688.

J. Appl. Phys. (1)

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. and Rev. Sci. Instr. (1)

Thornton C. Fry, J. Opt. Soc. Am. and Rev. Sci. Instr. 16, 1 (1928).
[Crossref]

Opt. i Spectroskopiya (1)

B. I. Stepanov and A. P. Khapalyuk, Opt. i Spectroskopiya 13, 714 (1962)[English transl.: Opt. Spectry. 13, 404 (1962)].

Proc. IEEE (1)

H. Jacobs, F. A. Brand, J. D. Meindl, S. Weitz, R. Benjamin, and D. A. Holmes, Proc. IEEE 51, 581 (1963).
[Crossref]

Z. Physik (1)

F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
[Crossref]

Other (5)

For an equivalent analysis see, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 688.

For the purposes of numerical computations, we have chosen optical constants for calcite and quartz which correspond to near infrared wavelengths because we are using GaAs injection laser sources (8400 Å) in some of our work. The general concepts advanced in this work, however, are valid for other wavelengths. The numerical values for the optical constants were taken from: Dwight E. Gray, Coordinating Editor, American Institute of Physics Handbook (McGraw-Hill Book Company Inc., New York, 2nd edition, 1963), Calcite, p. 6-18; Crystal Quartz, p. 6-24; Rutile, p. 6-33.

G. N. Ramachandran and S. Ramaseshan in Handbuch der Physik, edited by S. Flügge, (Springer-Verlag, Berlin, 1961), Vol. 25, Chap. 1, p. 76.

Reference 1, p. 24.

S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio (John Wiley & Sons, Inc., New York, 1960), p. 290.

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Figures (9)

F. 1
F. 1

Geometry used in this work. The anisotropic plate is assumed to have its principal dielectric axes aligned with the Cartesian coordinate system shown. The x direction is that for a right-hand coordinate system.

F. 2
F. 2

Maximum error angle emax, for an isotropic slab vs index of refraction n.

F. 3
F. 3

Quantity (Δe−Δa) vs d0 for calcite at a wavelength of 8010 Å.

F. 4
F. 4

Shift in the amplitude ratio F ≡ (E0y/E0x)/(Eiy/Eix) vs d0 for calcite at 8010 Å.

F. 5
F. 5

Comparison between Δe (———) and Δa (– – –) for calcite in the vicinity of 90° phase difference.

F. 6
F. 6

Comparison between Δe (———) and Δ0 (– – –) for quartz in vicinity of 90° phase difference. The optical constants used are for 8325 Å.

F. 7
F. 7

Shift in amplitude ratio F ≡ (E0y/E0x)/(Eiy/Eix) vs d0 for quartz at 8325 Å.

F. 8
F. 8

Exact (Δe) and approximate (Δa) values of phase difference for quartz as a function of normalized crystal thickness d0 with angle of incidence i as a parameter. ny = 1.54661, nx = nz = 1.53773.

F. 9
F. 9

Δe and Δa vs angle of incidence for quartz. The numbers arranged in a column on the right correspond to values of d0. The slowly rising curves represent the approximate phase difference Δa. For λ0 = 8325Å, nx = nz = 1.53773, nv = 1.54661.

Equations (39)

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Ɛ i x = E i x e j T ,
Ɛ i y = E i y e j T ,
Ɛ 0 x = E 0 x e j [ T ( β x β ) d ] ,
Ɛ 0 y = E 0 y e j [ T ( β y β 0 ) d ] ,
Δ a = ( β y β x ) d .
Ɛ i = E i e j T , Ɛ 0 = E 0 e j [ T θ + β 0 d ] ,
E 0 / E i = [ cos 2 β d + K 2 sin 2 β d ] 1 2 ,
θ = arc tan [ K tan β d ] ,
K = ( n 2 + 1 ) / 2 n , β = ( 2 π n ) / λ 0 .
e arc tan [ K tan β d ] β d ,
e max = arc tan [ ( K 1 ) / 2 ( K ) 1 2 ] .
e = arc tan [ tan β d + ( K 1 ) tan β d ] arc tan [ tan β d ] ,
e ( K 1 ) tan β d / [ 1 + tan 2 β d ] = 1 2 ( K 1 ) sin 2 β d .
θ β d + 1 2 ( K 1 ) sin 2 β d .
Ɛ 0 x = [ E i x e j ( T θ x + β 0 d ) ] / ( cos 2 β x d + K x 2 sin 2 β x d ) 1 2 ,
Ɛ 0 y = [ E i y e j ( T θ y + β 0 d ) ] / ( cos 2 β y d + K y 2 sin 2 β y d ) 1 2 ,
K k = ( n k 2 + 1 ) / 2 n k , θ k = arc tan [ K k tan β k d ] ; k = x , y .
Δ e = θ y θ x = arc tan ( K y tan β y d K x tan β x d 1 + K y K x tan β y d tan β x d ) ,
E 0 y E 0 x = E i y E i x F E i y E i x ( cos 2 β x d + K x 2 sin 2 β x d cos 2 β y d + K y 2 sin 2 β y d ) 1 2 .
Δ e Δ a = { arc tan [ tan β y d + ( K y 1 ) tan β y d ] arc tan × [ tan β y d ] } { arc tan [ tan β x d + ( K x 1 ) tan β x d ] arc tan [ tan β x d ] } ,
Δ e Δ a 1 2 ( K y 1 ) sin 2 β y d 1 2 ( K x 1 ) sin 2 β x d ,
Δ e Δ a 1 2 ( K y + K x 2 ) cos [ ( β y + β x ) d ] × sin [ ( β y β x ) d ] .
1 2 ( K y + K x 2 ) = [ ( n x + n y ) ( 1 + n x n y ) 4 n x n y ] / 4 n x n y 0.105 rad 6 ° .
F = { [ 1 + ( K x 2 1 ) sin 2 β x d ] / [ 1 + ( K y 2 1 ) sin 2 β x d ] } 1 2 , F 1 1 4 ( K y 2 + K x 2 2 ) sin [ ( β y + β x ) d ] sin [ ( β y β x ) d ] + 1 4 ( K x 2 K y 2 ) { 1 cos [ ( β y + β x ) d ] × cos [ ( β y β x ) d ] } .
sin 2 ϕ = ± sin 2 γ sin δ ,
tan 2 ψ = tan 2 γ cos δ .
2 F tan ν / ( F 2 + tan 2 ν ) = ± sin 2 γ sin δ .
2 F tan ν sin Δ e / ( F 2 + tan 2 ν ) = ± sin 2 γ sin δ .
Ɛ 0 x = E 0 x exp [ j ( T θ x + β 0 d cos i ) ] , Ɛ 0 y = E 0 y exp [ j ( T θ x + β 0 d cos i ) ] ,
β y = ( 2 π / λ 0 ) ( n y 2 sin 2 i ) 1 2 ,
β x = ( 2 π n x ) / ( λ 0 n z ) ( n z 2 sin 2 i ) 1 2 ,
K y = 1 2 { [ cos i / ( n y 2 sin 2 i ) 1 2 ] + [ ( n y 2 sin 2 i ) 1 2 / cos i ] } ,
K x = 1 2 { [ n x n z cos i / ( n z 2 sin 2 i ) 1 2 ] + [ ( n z 2 sin 2 i ) 1 2 / n x n z cos i ] } .
n x d ( 2 m λ 0 / 4 ) = 0 ,
n y d ( 2 m λ 0 / 4 ) = λ 0 / 4 ,
m = n x / [ 2 ( n y n x ) ] ,
d = λ 0 / 4 ( n y n x ) ] .
1 + K x K y tan β x d tan β y d = 0 .
cos2ψ=[+cosδsinδ±cot2γ(KtanΔsin2δ)12]/K,sin2ψ=[sinδcot2γ±cosδ(KtanΔsin2δ)12]/K.