Abstract

An analysis is given of Savart plates for arbitrary angles between the optic axis and the plate normal. Conoscopic interference patterns of thin Savart plates cut nearly parallel to the optic axis are shown and the use of such plates combined with diffraction gratings is discussed.

© 1971 Optical Society of America

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References

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  1. cf. Ann. Physik 49, 292 (1840).
  2. A. C. S. van Heel, A. Walther, Opt. Acta 5, 47 (1958).
    [CrossRef]
  3. M. Francon, Optical Interferometry (Academic Press, New York, 1966), p. 137–141.
  4. J. M. Burch, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam1963), Vol. 2, p. 75.
    [CrossRef]
  5. H. de Lang, E. T. Ferguson, G. C. M. Schoenaker, Philips Techn. Rev. 30, 153 (1969).
  6. T. H. Peek, to be published.
  7. T. H. Peek, Opt. Comm. 2, 377 (1971).
    [CrossRef]
  8. R. H. Hammerschlag, Opt. Acta 16, 491 (1969).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 673.
  10. See Ref. 9, Chap. 14.2.3.
  11. M. Born, Optik (Springer Verlag, Berlin, 1933), p. 250.
  12. A. Schuster, An Introduction to the Theory of Optics (Edward Arnold & Co., London, 1925), p. 208.
  13. R. C. Jones et al., J. Opt. Soc. Amer. 31, 488 (1941); J. Opt. Soc. Amer. 32, 486 (1942); J. Opt. Soc. Amer. 37, 107 (1947).
    [CrossRef]

1971 (1)

T. H. Peek, Opt. Comm. 2, 377 (1971).
[CrossRef]

1969 (2)

R. H. Hammerschlag, Opt. Acta 16, 491 (1969).
[CrossRef]

H. de Lang, E. T. Ferguson, G. C. M. Schoenaker, Philips Techn. Rev. 30, 153 (1969).

1958 (1)

A. C. S. van Heel, A. Walther, Opt. Acta 5, 47 (1958).
[CrossRef]

1941 (1)

R. C. Jones et al., J. Opt. Soc. Amer. 31, 488 (1941); J. Opt. Soc. Amer. 32, 486 (1942); J. Opt. Soc. Amer. 37, 107 (1947).
[CrossRef]

1840 (1)

cf. Ann. Physik 49, 292 (1840).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 673.

M. Born, Optik (Springer Verlag, Berlin, 1933), p. 250.

Burch, J. M.

J. M. Burch, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam1963), Vol. 2, p. 75.
[CrossRef]

de Lang, H.

H. de Lang, E. T. Ferguson, G. C. M. Schoenaker, Philips Techn. Rev. 30, 153 (1969).

Ferguson, E. T.

H. de Lang, E. T. Ferguson, G. C. M. Schoenaker, Philips Techn. Rev. 30, 153 (1969).

Francon, M.

M. Francon, Optical Interferometry (Academic Press, New York, 1966), p. 137–141.

Hammerschlag, R. H.

R. H. Hammerschlag, Opt. Acta 16, 491 (1969).
[CrossRef]

Jones, R. C.

R. C. Jones et al., J. Opt. Soc. Amer. 31, 488 (1941); J. Opt. Soc. Amer. 32, 486 (1942); J. Opt. Soc. Amer. 37, 107 (1947).
[CrossRef]

Peek, T. H.

T. H. Peek, Opt. Comm. 2, 377 (1971).
[CrossRef]

T. H. Peek, to be published.

Schoenaker, G. C. M.

H. de Lang, E. T. Ferguson, G. C. M. Schoenaker, Philips Techn. Rev. 30, 153 (1969).

Schuster, A.

A. Schuster, An Introduction to the Theory of Optics (Edward Arnold & Co., London, 1925), p. 208.

van Heel, A. C. S.

A. C. S. van Heel, A. Walther, Opt. Acta 5, 47 (1958).
[CrossRef]

Walther, A.

A. C. S. van Heel, A. Walther, Opt. Acta 5, 47 (1958).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 673.

Ann. Physik (1)

cf. Ann. Physik 49, 292 (1840).

J. Opt. Soc. Amer. (1)

R. C. Jones et al., J. Opt. Soc. Amer. 31, 488 (1941); J. Opt. Soc. Amer. 32, 486 (1942); J. Opt. Soc. Amer. 37, 107 (1947).
[CrossRef]

Opt. Acta (2)

A. C. S. van Heel, A. Walther, Opt. Acta 5, 47 (1958).
[CrossRef]

R. H. Hammerschlag, Opt. Acta 16, 491 (1969).
[CrossRef]

Opt. Comm. (1)

T. H. Peek, Opt. Comm. 2, 377 (1971).
[CrossRef]

Philips Techn. Rev. (1)

H. de Lang, E. T. Ferguson, G. C. M. Schoenaker, Philips Techn. Rev. 30, 153 (1969).

Other (7)

T. H. Peek, to be published.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 673.

See Ref. 9, Chap. 14.2.3.

M. Born, Optik (Springer Verlag, Berlin, 1933), p. 250.

A. Schuster, An Introduction to the Theory of Optics (Edward Arnold & Co., London, 1925), p. 208.

M. Francon, Optical Interferometry (Academic Press, New York, 1966), p. 137–141.

J. M. Burch, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam1963), Vol. 2, p. 75.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Geometry of Savart’s plate. The optic axes 1 and 2 of both components make angles, α, with the plate normal X3; the second plate is rotated about the plate normal over 90°. Also, indicated are the angle of incidence, ϑ, of the incident light and the angle, φ, which the plane of incidence makes with the X1X3-plane.

Fig. 2
Fig. 2

Calculated hyperbolic fringe pattern of Savart’s plate between crossed polarizers. The angular distance of the center of the fringe pattern from the origin of the coordinate system is only dependent on the angle, α, between the optic axis and the plate normal. The angular fringe separation in the direction AB is dependent both on α and the thickness, d, of the plates; 1 and 2 denote the projections of the optic axes of the two components of Savart’s plate on the X1X2-plane.

Fig. 3
Fig. 3

Conoscopic fringe pattern of a twice 2 mm-thick Savart plate cut parallel to the optic axis (α = 90°). The hyperbolic fringes are centered around normal incidence in this case.

Fig. 4
Fig. 4

Conoscopic fringe patterns of (a) twice 2 mm-thick plate cut at 5° to the optic axis (α = 85°) and (b) twice 0.3-mm thick plate with α = 78.5°. The center of the hyperbolic fringe pattern moves outside the point corresponding to normal incidence with decreasing α; the much larger fringe separation (b) is mainly due to the slight thickness used here (d = 0.3 mm).

Fig. 5
Fig. 5

Nearly straight interference lines obtained with a twice 1.18-mm thick plate cut at 45° to the optic axis. The fringe modulation for large angles of incidence can be eliminated by a rotation of the two components with respect to each other, indicating nonorthogonal eigenvectors (see text).

Fig. 6
Fig. 6

Strongly modulated fringe pattern obtained with a twice 1-mm thick plate with α = 30°; Savart fringes are still recognized, but the secondary line pattern disturbs the fringes even for small values of the angle of incidence.

Fig. 7
Fig. 7

Conoscopic fringe pattern of a twice 2-mm thick plate cut at 85° to the optic axis (α = 5°). The modulation due to the secondary line pattern and the effects of the optical activity clearly are visible.

Fig. 8
Fig. 8

Conoscopic fringe pattern of a 2.9-mm thick quartz plate cut perpendicular to the optic axis (α = 0). With two plates a similar fringe pattern is obtained, since the indicatrices of the two plates coincide.

Fig. 9
Fig. 9

Principle of operation of Savart’s plate as an achromatic wave plate in combination with a diffraction grating; as before, 1 and 2 denote the projections of the optic axes of the plate components on the surface of the plates. With a phase retardation of 180° for the first diffracted order, adjacent diffracted orders are orthogonally polarized.

Fig. 10
Fig. 10

(a) Diffracted orders of an amplitude diffraction grating. Due to the broad spectrum of the light source used, the higher orders partly coincide. After polarizing the incident beam parallel to the grating and inserting a Savart plate with 180° phase retardation for the first diffracted order, (b) the orders ± 1 and ± 3 are visible behind a crossed polarizer, and (c) the orders 0, ± 2 with a parallel polarizer.

Equations (23)

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x 1 2 / n o 2 + x 2 2 / n o 2 + x 3 2 / n e 2 = 1 ,
( N r ,             r ) = 1 ,
N = ( 1 / n o 2 0 0 0 1 / n o 2 0 0 0 1 / n e 2 ) ,
r = ( x 1 ,             x 2 ,             x 3 ) .
N = 1 / n o 2 - 2 ( Δ n / n o 3 ) N ;
( N r ,             r ) = r 2 / n o 2 - 2 ( Δ n / n o 3 ) ( N r ,             r ) .
D 1 ( α ) N D 1 ( - α ) = N 1 ,
D 1 ( α ) N D 1 ( - α ) = N 1 ;
N 1 = ( 0 0 0 0 sin 2 α sin α cos α 0 sin α cos α cos 2 α ) .
N ˜ ( 3 ) = D 2 ( - ϑ ) D 3 ( - φ ) N 1 D 3 ( φ ) D 2 ( ϑ ) ,
N ˜ ( 2 ) = ( a c c b ) ,
a = sin 2 ϑ cos 2 α + cos 2 ϑ sin 2 φ sin 2 α + sin ϑ cos ϑ sin φ sin 2 α b = cos 2 φ sin 2 α c = - cos ϑ sin φ cos φ sin 2 α - sin ϑ cos φ sin α cos α .
N ˜ ( 2 ) = ( a c c b ) ,
a = sin 2 ϑ cos 2 α + cos 2 ϑ cos 2 φ sin 2 α - sin ϑ cos ϑ cos φ sin 2 α ; b = sin 2 φ sin 2 α ; c = cos ϑ sin φ cos φ sin 2 α - sin ϑ sin φ sin α cos α .
n ± = [ n o + Δ n ( a + b ) / 2 ] ± Δ n ( a + b ) / 2 , n ± = [ n o + Δ n ( a + b ) / 2 ] ± Δ n ( a + b ) / 2.
M = ( exp ( i ϕ ) 0 0 exp ( - i ϕ ) ) ,
ϕ = π d Δ n ( a + b - a - b ) / ( λ cos ϑ ) ;
2 ϕ = 2 π d Δ n / ( λ cos ϑ ) [ sin 2 α sin 2 ϑ cos 2 φ + 2 sin ϑ cos ϑ sin 2 α cos ( φ - π / 4 ) ] .
u = cos φ tan ϑ             v = sin φ tan ϑ ,
( u + 1 / tan α ) 2 - ( v - 1 / tan α ) 2 = k λ / ( 2 d Δ n sin 2 α ) .
2 ϕ = 2 π d Δ n ( 2 ) 1 2 sin ϑ sin 2 α / ( λ n o ) ,
2 ϕ m = 2 π m d Δ n ( 2 ) 1 2 sin 2 α / ( p n o ) ,
2 ϕ m = m π .

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