Abstract

In a study of the Sénarmont compensator method of flow birefringence measurement, the effects of polychromatism and of error in the quarter-wave plate are investigated by a theoretical treatment of the behavior of the cross of isocline. It is found that with monochromatic light an error in the quarter-wave plate introduces asymmetry into the scissors-type motion of the cross of isocline. The arms of the cross do not close completely. Rather, one arm disappears while the other continues its motion toward the 45° position in the annulus. It is shown by example that, with the proper quarter-wave plate, polychromatic light from a filter can give a result which is almost the same as that given by monochromatic light and a quarter-wave plate without error. A mismatch of the quarter-wave plate and polychromatic light gives essentially the same error in the measurement of birefringence as the error caused by a comparable mismatch between the quarter-wave plate and monochromatic light.

© 1964 Optical Society of America

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References

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  1. Part I, C. A. Hollingsworth and W. T. Granquist, J. Opt. Soc. Am. 52, 562 (1962).
    [Crossref]
  2. This is obtained by use of a standard equation. See, for example, H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948), Eq. (4a).
    [Crossref]
  3. H. T. Jessop, Brit. J. Appl. Phys. 8, 138 (1953).
    [Crossref]
  4. “Kodak Wratten Filters,” 19th edition (Eastman Kodak Company, Rochester, New York, 1957), p. 39.
  5. W. T. Granquist and C. A. Hollingsworth, Trans. Faraday Soc. 59, 2192 (1963).
    [Crossref]
  6. The behavior for θ greater than δ/2 was described incorrectly in Ref. 1.

1963 (1)

W. T. Granquist and C. A. Hollingsworth, Trans. Faraday Soc. 59, 2192 (1963).
[Crossref]

1962 (1)

1953 (1)

H. T. Jessop, Brit. J. Appl. Phys. 8, 138 (1953).
[Crossref]

1948 (1)

Granquist, W. T.

W. T. Granquist and C. A. Hollingsworth, Trans. Faraday Soc. 59, 2192 (1963).
[Crossref]

Part I, C. A. Hollingsworth and W. T. Granquist, J. Opt. Soc. Am. 52, 562 (1962).
[Crossref]

Hollingsworth, C. A.

W. T. Granquist and C. A. Hollingsworth, Trans. Faraday Soc. 59, 2192 (1963).
[Crossref]

Part I, C. A. Hollingsworth and W. T. Granquist, J. Opt. Soc. Am. 52, 562 (1962).
[Crossref]

Jerrard, H. G.

Jessop, H. T.

H. T. Jessop, Brit. J. Appl. Phys. 8, 138 (1953).
[Crossref]

Brit. J. Appl. Phys. (1)

H. T. Jessop, Brit. J. Appl. Phys. 8, 138 (1953).
[Crossref]

J. Opt. Soc. Am. (2)

Trans. Faraday Soc. (1)

W. T. Granquist and C. A. Hollingsworth, Trans. Faraday Soc. 59, 2192 (1963).
[Crossref]

Other (2)

The behavior for θ greater than δ/2 was described incorrectly in Ref. 1.

“Kodak Wratten Filters,” 19th edition (Eastman Kodak Company, Rochester, New York, 1957), p. 39.

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Figures (9)

F. 1
F. 1

P represents the polarizer. A and A′ represent the analyzer in the crossed and an arbitrary position, respectively. S represents the optical axis of the sample at position α in the annulus.

F. 2
F. 2

Positions α of minimum and maximum transmittance as a function of the analyzer angle θ for δ = 20° and three values of β. Solid curves: minima; broken curves: maxima.

F. 3
F. 3

Positions α of minimum and maximum transmittance as a function of the analyzer angle θ for δ = 80° and β = 1.17. Solid curves: minima; broken curves: maxima.

F. 4
F. 4

Positions α of minimum and maximum transmittance for δ = 80° and the polychromatic light from filter No. 53, and for β = 1. Solid curves: minima, with filter; dashed curves: maxima with filter; dot–dash curves: minima, β = 1; dotted curves: maxima, β= 1.

F. 5
F. 5

The percentage transmittance at the minimum and maximum positions as a function of the analyzer angle for δ = 80° and β=1.

F. 6
F. 6

The percentage transmittance at the minimum and maximum positions as a function of the analyzer angle for δ=80° and β=1.17.

F. 7
F. 7

The percentage transmittance as a function of position in the annulus for several angles of the analyzer when δ=20° and β = 1.17.

F. 8
F. 8

The percentage transmittance as a function of position in the annulus for several angles of the analyzer when δ=20°, and for the polychromatic light from filter No. 53.

F. 9
F. 9

The behavior of the cross of isocline as the analyzer is turned. Top figure represents all examples when θ = 0. The other figures represent Examples 1, 2, and 3 when θ is increased as indicated.

Tables (2)

Tables Icon

Table I Percent transmittance of Filter No. 53, λ in mμ (Ref. 4).

Tables Icon

Table II Numerical values of parameters, Examples 1 through 4.

Equations (22)

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β = λ q / λ ,
Δ n ( λ ) = n e ( λ ) n 0 ( λ ) = ( λ / 2 π h ) δ ( λ ) ,
| λ λ q | | λ λ A | ,
β ( λ ) = δ ( λ ) / δ ( λ q ) .
T β = sin 2 θ + sin 2 2 α cos 2 θ sin 2 ( β δ / 2 ) sin 2 α sin 2 θ [ cos 2 α sin 2 ( β δ / 2 ) cos ( β π / 2 ) ± 1 2 sin β δ sin ( β δ / 2 ) ] ,
T = sin 2 θ + J 1 cos 2 θ sin 2 2 α 1 2 J 2 sin 2 θ sin 4 α J 3 sin 2 θ sin 2 α ,
J 1 = λ T 0 ( λ ) sin 2 β δ / 2 , J 2 = λ T 0 ( λ ) sin 2 ( β δ / 2 ) cos ( β π / 2 ) , J 3 = ± 1 2 λ T 0 ( λ ) sin β δ sin ( β π / 2 ) ,
λ T 0 ( λ ) = 1 .
T a = 1 2 π 0 2 π T d α = sin 2 θ + 1 2 J 1 cos 2 θ .
T / α = 0 ,
tan 2 θ = J 1 sin 4 α / ( J 2 cos 4 α + J 3 cos 2 α ) .
2 T / α 2 = 0
tan 2 θ = 2 J 1 cos 4 α / ( 2 J 2 sin 4 α + J 3 sin 2 α ) .
cos 2 α c = ( J 2 / J 3 ) 1 3 ,
tan 2 θ c = 2 J 2 / J 3 [ 2 ( J 2 / J 3 ) 2 3 + 1 ] [ 1 ( J 2 / J 3 ) 2 3 ] 1 2 .
T / θ = 0 ,
tan 2 θ = J 2 sin 4 α + 2 J 3 sin 2 α 1 2 J 1 sin 2 2 α .
tan 2 θ α = 45 ° = 2 J 3 / ( 1 2 J 1 ) .
J 2 cos 4 α + J 3 cos 2 α = 0 ,
cos 2 α = 1 4 { ( J 3 / J 2 ) ± [ ( J 3 / J 2 ) 2 + 8 ] 1 2 } .
cos 2 α J 2 / J 3 ,
α ( π / 4 ) ( J 2 / 2 J 3 ) ,