Abstract

The theory conventionally used to describe the operation of Berek and Ehringhaus rotary compensators is based on geometrical optics and, hence, is not exact. When rotary compensators are analyzed within the framework of classical electromagnetic theory, exact solutions for the phase difference and amplitude ratio of the transmitted light can be determined. The approximate and exact solutions differ in some interesting ways which become of substantial importance in the examination of monochromatic plane waves of light. In particular, the discrepancies between exact and approximate solutions become more pronounced at high angles of incidence. Exact theory predicts the possibility of using a high-refractive index isotropic plate for measuring small phase differences at relatively long wavelengths.

© 1964 Optical Society of America

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References

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  1. R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).
    [Crossref]
  2. F. Partovi, J. Opt. Soc. Am. 52, 918 (1962).
    [Crossref]
  3. C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925).
    [Crossref]
  4. H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948).
    [Crossref]
  5. A. C. Hall, J. Opt. Soc. Am. 53, 801 (1963).
    [Crossref]
  6. M. Berek, Mikroskopische Mineralbestimmung mil Hilfe der Universaldrehtischmelhoden (Gebruder Borntrager, Berlin, 1924).
  7. F. Rinne and M. Berek, Anleitung zu Optischen Untersuchungen mit dent Polarizationsmikroskop (Schweizerbart’sche Verlagsbuchhandlung, Stuttgart, 1953).
  8. M. Berek, Zentr. Mineral.1913 pp. 388, 427, 464, 580.
  9. A. Ehringhaus, Z. Krist. 76, 315 (1931);Z. Krist. 98, 394 (1938);Z. Krist. 102, 85 (1939).
  10. Conrad Burri, Z. Angew. Math. Phys. 4, 418 (1953).
    [Crossref]
  11. H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964).
    [Crossref]
  12. D. A. Holmes, J. Opt. Soc. Am. 54, 1115 (1964).
    [Crossref]
  13. J. Gahm, Zeiss Mitt. Fortschr. Tech. Opt. 3, 152 (1964).
  14. A. B. Winterbottom, Kgl. Norske Vindenskab. Selskabs Skrifter 1, 27, 37 (1955).
  15. H. Schopper, Z. Physik 132, 146 (1952).
    [Crossref]
  16. For illustrative numerical examples, we have chosen optical constants for calcite and quartz corresponding to near-infrared wavelengths because we are using gallium arsenide injection laser sources (8400 Å) in some of our work. We assume that no absorption occurs at the wavelengths used. The general concepts advanced in this work, however, clearly apply at other wavelengths. The optical constants were taken from American Institute of Physics Handbook, edited by D. E. Gray (McGraw-Hill Book Company Inc., New York, 1963), 2nd ed., Calcite, pp. 6–18; Crystal Quartz, pp. 6–33.
  17. Some aspects of propagation through an isotropic slab are discussed by M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 60 ff.
  18. D. Bergman, J. Opt. Soc. Am. 52, 1080 (1962).
    [Crossref]
  19. A summary of the optical properties of thallium bromide iodide (KRS-5) is given in Synthetic Optical Crystals (The Harshaw Chemical Company, Cleveland 6, Ohio, 1955), rev. ed., pp. 23–24. We have taken the index of refraction at 10μ as n=2.37274.
  20. See Ref.14, p. 68.
  21. The geometrical optics theory of the Soleil compensator is treated in Ref. 17, p. 691.
  22. See Ref.17, pp. 699–700.

1964 (3)

1963 (1)

1962 (3)

1955 (1)

A. B. Winterbottom, Kgl. Norske Vindenskab. Selskabs Skrifter 1, 27, 37 (1955).

1953 (1)

Conrad Burri, Z. Angew. Math. Phys. 4, 418 (1953).
[Crossref]

1952 (1)

H. Schopper, Z. Physik 132, 146 (1952).
[Crossref]

1948 (1)

1931 (1)

A. Ehringhaus, Z. Krist. 76, 315 (1931);Z. Krist. 98, 394 (1938);Z. Krist. 102, 85 (1939).

1925 (1)

C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925).
[Crossref]

Archer, R. J.

Berek, M.

M. Berek, Mikroskopische Mineralbestimmung mil Hilfe der Universaldrehtischmelhoden (Gebruder Borntrager, Berlin, 1924).

F. Rinne and M. Berek, Anleitung zu Optischen Untersuchungen mit dent Polarizationsmikroskop (Schweizerbart’sche Verlagsbuchhandlung, Stuttgart, 1953).

M. Berek, Zentr. Mineral.1913 pp. 388, 427, 464, 580.

Bergman, D.

Born, M.

Some aspects of propagation through an isotropic slab are discussed by M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 60 ff.

Burri, Conrad

Conrad Burri, Z. Angew. Math. Phys. 4, 418 (1953).
[Crossref]

Ehringhaus, A.

A. Ehringhaus, Z. Krist. 76, 315 (1931);Z. Krist. 98, 394 (1938);Z. Krist. 102, 85 (1939).

Gahm, J.

J. Gahm, Zeiss Mitt. Fortschr. Tech. Opt. 3, 152 (1964).

Hall, A. C.

Harris, J.

Holmes, D. A.

Jerrard, H. G.

Partovi, F.

Rinne, F.

F. Rinne and M. Berek, Anleitung zu Optischen Untersuchungen mit dent Polarizationsmikroskop (Schweizerbart’sche Verlagsbuchhandlung, Stuttgart, 1953).

Schopper, H.

H. Schopper, Z. Physik 132, 146 (1952).
[Crossref]

Skinner, C. A.

C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925).
[Crossref]

Weinberger, H.

Winterbottom, A. B.

A. B. Winterbottom, Kgl. Norske Vindenskab. Selskabs Skrifter 1, 27, 37 (1955).

Wolf, E.

Some aspects of propagation through an isotropic slab are discussed by M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 60 ff.

J. Opt. Soc. Am. (7)

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

C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925).
[Crossref]

Kgl. Norske Vindenskab. Selskabs Skrifter (1)

A. B. Winterbottom, Kgl. Norske Vindenskab. Selskabs Skrifter 1, 27, 37 (1955).

Z. Angew. Math. Phys. (1)

Conrad Burri, Z. Angew. Math. Phys. 4, 418 (1953).
[Crossref]

Z. Krist. (1)

A. Ehringhaus, Z. Krist. 76, 315 (1931);Z. Krist. 98, 394 (1938);Z. Krist. 102, 85 (1939).

Z. Physik (1)

H. Schopper, Z. Physik 132, 146 (1952).
[Crossref]

Zeiss Mitt. Fortschr. Tech. Opt. (1)

J. Gahm, Zeiss Mitt. Fortschr. Tech. Opt. 3, 152 (1964).

Other (9)

For illustrative numerical examples, we have chosen optical constants for calcite and quartz corresponding to near-infrared wavelengths because we are using gallium arsenide injection laser sources (8400 Å) in some of our work. We assume that no absorption occurs at the wavelengths used. The general concepts advanced in this work, however, clearly apply at other wavelengths. The optical constants were taken from American Institute of Physics Handbook, edited by D. E. Gray (McGraw-Hill Book Company Inc., New York, 1963), 2nd ed., Calcite, pp. 6–18; Crystal Quartz, pp. 6–33.

Some aspects of propagation through an isotropic slab are discussed by M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 60 ff.

A summary of the optical properties of thallium bromide iodide (KRS-5) is given in Synthetic Optical Crystals (The Harshaw Chemical Company, Cleveland 6, Ohio, 1955), rev. ed., pp. 23–24. We have taken the index of refraction at 10μ as n=2.37274.

See Ref.14, p. 68.

The geometrical optics theory of the Soleil compensator is treated in Ref. 17, p. 691.

See Ref.17, pp. 699–700.

M. Berek, Mikroskopische Mineralbestimmung mil Hilfe der Universaldrehtischmelhoden (Gebruder Borntrager, Berlin, 1924).

F. Rinne and M. Berek, Anleitung zu Optischen Untersuchungen mit dent Polarizationsmikroskop (Schweizerbart’sche Verlagsbuchhandlung, Stuttgart, 1953).

M. Berek, Zentr. Mineral.1913 pp. 388, 427, 464, 580.

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Figures (8)

F. 1
F. 1

Geometry considered in this work. The xyz coordinate system is assumed to be oriented such that the xy plane is located at the interface of plate 1 and medium I. Media I and II are considered to have unity of refraction. Crystalline plates 1 and 2 are both assumed to have their respective principal axes aligned with the xyz coordinate axes. The input plane wave is incident in the xz plane, with the incident wave normal making an angle i with the z axis. The permeabilities of all media are assumed to be equal to the permeability μ0 of vacuum. Further, we assume that all media are lossless, nonoptically active, and homogeneous.

F. 2
F. 2

The error angle Δe−Δa in degrees (vertical scale) as a function of the angle of incidence i in degrees (upper horizontal scale) for a calcite Berek compensator. The values of Δa are given in degrees (lower horizontal scale) for 2° increments of i. Δe−Δa is plotted for d=120λ0 (——) and d=50λ0(— —). The envelope function is also shown (– – –). The wavelength used is 8010 Å, for which we have ω=1.64869 and =1.48216. [Curve (b) is a continuation of curve (a).]

F. 3
F. 3

The amplitude ratio factor T=tanα0/tanαi (vertical scale) vs i in degrees (horizontal scale) for a calcite Berek compensator with d=120λ0 (——) and d=50λ0(— —). [Curve (b) is a continuation of curve (a).]

F. 4
F. 4

Envelope function ±Φ sinΔa in degrees (vertical scale) vs angle of incidence i in degrees (horizontal scale) for a quartz Ehringhaus compensator with d=1200λ0. The range 0°≦i≦75° covers about 5 1 2 orders of Δa at a wavelength of λ0=8325 Å. We have taken =1.54661 and ω=1.53773. The intervals delineated by the dotted lines are shown in greater detail in Figs. 5 and 6.

F. 5
F. 5

Detail of Fig. 4 for the range 5.0°≦i≦7.5°.Δe and Δa are in degrees (vertical scale) and i is in degrees (horizontal scale). The slowly rising curve is the approximate phase difference Δa. The two dashed (— —) curves correspond to the variation of the envelope function, ±Φ sinΔa.

F. 6
F. 6

Detail of Fig. 4 for the range 47.4°≦i≦47.6°. Δe−Δa is in degrees (vertical scale) and i is in degrees (upper horizontal scale). The envelope function ±Φ sinΔa is virtually a straight line. Δe−Δa is shown for d=1200.0λ0 (——) and d=1200.1λ0 (— —). The values of the approximate phase difference Δa are given in degrees for d=1200.0λ0 (middle horizontal scale) and for d=1200.1λ0 (lower horizontal scale).

F. 7
F. 7

Δmax in degrees (vertical scale) vs angle of incidence in degrees (horizontal scale) for isotropic plate. (1) n=1.25; (2)n=1.50; (3)n=1.75; (4) n=2.25; (5)n=3.00; (6)n=3.75.

F. 8
F. 8

Wave optics value of phase shift Δ in degrees (vertical scale) vs angle of incidence i in degrees (horizontal scale) for transmission through a thallium bromide iodide optical crystal. The index of refraction at λ0=10μ is taken as n=2.37274.

Equations (31)

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E p i / E s i = exp ( j δ i ) tan α i ,
E p 0 / E s 0 = exp ( j δ 0 ) tan α 0 .
exp ( j δ 0 ) tan α 0 = exp [ j ( Δ + δ i ) ] T tan α i .
n x 1 = n x 2 = n y 1 = n y 2 = ω , n z 1 = n z 2 = , d 1 + d 2 = d 0.1 mm ,
n x 1 = n z 1 = n y 2 = n z 2 = ω , n y 1 = n x 2 = , d 1 = d 2 = d 1.0 mm .
Δ a = 2 π d λ 0 [ ( ω 2 sin 2 i ) 1 2 ω ( 2 sin 2 i ) 1 2 ] ,
Δ a = 2 π d λ 0 [ ( 2 sin 2 i ) 1 2 ω ( ω 2 sin 2 i ) 1 2 ] ,
T e j Δ e = [ cos β s 1 d 1 cos β s 2 d 2 K s 12 sin β s 1 d 1 sin β s 2 d 2 + j ( K s 1 sin β s 1 d 1 cos β s 2 d 2 + K s 2 cos β s 1 d 1 sin β s 2 d 2 ) ] × [ cos β p 1 d 1 cos β p 2 d 2 K p 12 sin β p 1 d 1 sin β p 2 d 2 + j ( K p 1 sin β p 1 d 1 cos β p 2 d 2 + K p 2 cos β p 1 d 1 sin β p 2 d 2 ) ] 1 ,
β s k = ( 2 π / λ 0 ) ( n y k 2 sin 2 i ) 1 2 , k = 1 , 2 ,
β p k = ( 2 π n x k / λ 0 n z k ) ( n y k 2 sin 2 i ) 1 2 , k = 1 , 2 ,
2 K p k = n x k n z k cos i ( n z k 2 sin 2 i ) 1 2 + ( n z k 2 sin 2 i ) 1 2 n x k n z k cos i , k = 1 , 2 ,
2 K s k = cos i ( n y k 2 sin 2 i ) 1 2 + ( n y k 2 sin 2 i ) 1 2 cos i , k = 1 , 2 ,
2 K p 12 = n x 2 n z 2 n x 1 n z 1 ( n z 1 2 sin 2 i n z 2 2 sin 2 i ) 1 2 + n x 1 n z 1 n x 2 n z 2 ( n z 2 2 sin 2 i n z 1 2 sin 2 i ) 1 2 ,
2 K s 12 = ( n y 1 2 sin 2 i n y 2 2 sin 2 i ) 1 2 + ( n y 2 2 sin 2 i n y 1 2 sin 2 i ) 1 2 .
T = 1 ,
Δ a = ( β s 1 β p 1 ) d 1 + ( β s 2 β p 2 ) d 2 .
Δ e = arctan ( K s 1 tan β s 1 d 1 + K s 2 tan β s 2 d 2 1 K s 12 tan β s 1 d 1 tan β s 2 d 2 ) arctan ( K p 1 tan β p 1 d 1 + K p 2 tan β p 2 d 2 1 K p 12 tan β p 1 d 1 tan β p 2 d 2 ) .
K s 12 1 , K p 12 1 , | K s 1 K s 2 | K s 1 + K s 2 , | K p 1 K p 2 | K p 1 + K p 2 .
Δ e arctan [ 1 2 ( K s 1 + K s 2 ) tan ( β s 1 d 1 + β s 2 d 2 ) ] arctan [ 1 2 ( K p 1 + K p 2 ) tan ( β p 1 d 1 + β p 2 d 2 ) ] .
Δ e Δ a + Φ cos θ sin Δ a ,
Φ = ( K s 1 + K s 2 + K p 1 + K p 2 4 ) / 4 , θ = ( β s 1 + β p 1 ) d 1 + ( β s 2 + β p 2 ) d 2 .
( K s 1 + K s 2 2 ) / 2 1 , ( K p 1 + K p 2 2 ) / 2 1 .
Δ e = arctan ( K s tan β s d K p tan β p d 1 + K s K p tan β s d tan β p d ) ,
T = [ 1 + ( K s 2 1 ) sin 2 β s d 1 + ( K p 2 1 ) sin 2 β s d ] 1 2 ,
β s = ( 2 π / λ 0 ) ( ω 2 sin 2 i ) 1 2 , β p = ( 2 π ω / λ 0 ) ( 2 sin 2 i ) 1 2 , 2 K s = [ cos i / ( ω 2 sin 2 i ) 1 2 ] + [ ( ω 2 sin 2 i ) 1 2 / cos i ] , 2 K p = [ ω cos i / ( 2 sin 2 i ) 1 2 ] + [ ( 2 sin 2 i ) 1 2 / ω cos i ] .
Δ = arctan [ ( K s K p ) tan β d 1 + K s K p tan 2 β d ] ,
T = ( 1 + K s 2 tan 2 β d 1 + K p 2 tan 2 β d ) 1 2 ,
β d = ( 2 π d / λ 0 ) ( n 2 sin 2 i ) 1 2 , 2 K s = [ cos i / ( n 2 sin 2 i ) 1 2 ] + [ ( n 2 sin 2 i ) 1 2 / cos i ] , 2 K p = [ n 2 cos i / ( n 2 sin 2 i ) 1 2 ] + [ ( n 2 sin 2 i ) 1 2 / n 2 cos i ] .
Δ max = arctan [ ( K s / 4 K p ) 1 2 ( K p / 4 K s ) 1 2 ] .
( b / a ) tan ( Δ / 2 ) ( a / b ) ,
( 4 d / λ 0 ) ( n 2 sin 2 i 2 ) 1 2 m ( 4 d / λ 0 ) ( n 2 sin 2 i 1 ) 1 2 .