Abstract

Up to the present time no image-forming depolarizing filter has been described which will work for monochromatic light. This paper describes a simple monochromatic depolarizer. This depolarizer consists of two variable wave plates whose axes are separated by an angle of 45°. The retardation of the first wave plate varies from zero to 2π. The second wave plate varies from zero to 4π. This variation is made to occur in a time which is short compared to the time constant of the apparatus in which the depolarizer is used. Typical components are Z cut plates of ammonium-dihydrogen-phosphate. The retardation of these plates can be made to vary by applying an electric field along the optic axis.

A discussion is also given of the Lyot depolarizer which is effective for light which covers a wide wavelength range. Tolerances are derived for the behavior of the Lyot depolarizer and experimental measurements are shown of its behavior.

© 1951 Optical Society of America

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References

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  1. Lyot, Ann. Observatoire Astron. Phys. de Paris (Meudon)—Tomi i Fasc. 1–2 8, 102 (1928).
  2. F. Perrin, J. Chem. Phys. 10, 415 (1942).
    [Crossref]
  3. H. Mueller, M.I.T. Course (8.26), Spring, 1945; J. Opt. Soc. Am. 38, 661(A) (1948).
  4. B. H. Billings and E. H. Land, J. Opt. Soc. Am. 38, 819–829 (1948).
    [Crossref] [PubMed]
  5. R. Clark Jones, J. Opt. Soc. Am. 31, 488–503 (1941).
    [Crossref]
  6. B. H. Billings, J. Opt. Soc. Am. 39, 797–801 (1949); J. Opt. Soc. Am. 39, 802–808 (1949); R. O’B. Carpenter, J. Opt. Soc. Am. 40, 225–229 (1950).
    [Crossref]
  7. G. D. Gottschall, J. Soc. Motion Picture Engrs. 51, 13 (1948).
  8. Hans Jaffe, Phys. Rev. 73, 95 (1948).
  9. B. Zwicker and P. Scherrer, Helv. Phys. Acta 16, 214 (1943).
  10. B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

1949 (1)

1948 (3)

B. H. Billings and E. H. Land, J. Opt. Soc. Am. 38, 819–829 (1948).
[Crossref] [PubMed]

G. D. Gottschall, J. Soc. Motion Picture Engrs. 51, 13 (1948).

Hans Jaffe, Phys. Rev. 73, 95 (1948).

1945 (1)

H. Mueller, M.I.T. Course (8.26), Spring, 1945; J. Opt. Soc. Am. 38, 661(A) (1948).

1944 (1)

B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

1943 (1)

B. Zwicker and P. Scherrer, Helv. Phys. Acta 16, 214 (1943).

1942 (1)

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[Crossref]

1941 (1)

1928 (1)

Lyot, Ann. Observatoire Astron. Phys. de Paris (Meudon)—Tomi i Fasc. 1–2 8, 102 (1928).

Billings, B. H.

Clark Jones, R.

Gottschall, G. D.

G. D. Gottschall, J. Soc. Motion Picture Engrs. 51, 13 (1948).

Jaffe, Hans

Hans Jaffe, Phys. Rev. 73, 95 (1948).

Land, E. H.

Lyot,

Lyot, Ann. Observatoire Astron. Phys. de Paris (Meudon)—Tomi i Fasc. 1–2 8, 102 (1928).

Mueller, H.

H. Mueller, M.I.T. Course (8.26), Spring, 1945; J. Opt. Soc. Am. 38, 661(A) (1948).

Perrin, F.

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[Crossref]

Scherrer, P.

B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

B. Zwicker and P. Scherrer, Helv. Phys. Acta 16, 214 (1943).

Zwicker, B.

B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

B. Zwicker and P. Scherrer, Helv. Phys. Acta 16, 214 (1943).

Ann. Observatoire Astron. Phys. de Paris (Meudon)—Tomi i Fasc. 1–2 (1)

Lyot, Ann. Observatoire Astron. Phys. de Paris (Meudon)—Tomi i Fasc. 1–2 8, 102 (1928).

Helv. Phys. Acta (2)

B. Zwicker and P. Scherrer, Helv. Phys. Acta 16, 214 (1943).

B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

J. Chem. Phys. (1)

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[Crossref]

J. Opt. Soc. Am. (3)

J. Soc. Motion Picture Engrs. (1)

G. D. Gottschall, J. Soc. Motion Picture Engrs. 51, 13 (1948).

M.I.T. Course (8.26) (1)

H. Mueller, M.I.T. Course (8.26), Spring, 1945; J. Opt. Soc. Am. 38, 661(A) (1948).

Phys. Rev. (1)

Hans Jaffe, Phys. Rev. 73, 95 (1948).

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Figures (4)

Fig. 1
Fig. 1

Pattern observed at 546 mμ with basal section of ADP between crossed circular polarizers. Plate 0.31 cm thick.

Fig. 2
Fig. 2

Pattern observed at 546 mμ with basal section of ADP 0.31 cm thick between crossed circular polarizers. 4500 volts on plate.

Fig. 3
Fig. 3

Special optical system for measuring the efficiency of a white light depolarizer. The source was a tungsten filament. The motor driven linear polarizer was rotated at thirty revolutions per second.

Fig. 4
Fig. 4

Relative transmission measurements for different arrangements of the crystal plate in Fig. 3. In each section the first linear polarizer was slowly rotated by hand through one complete revolution.

Equations (71)

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M ¯ = 1 T 0 T M d t = 0 , C ¯ = 1 T 0 T C d t = 0 , S ¯ = 1 T T S d t = 0 ,
| I M C S | = | 1 0 0 0 | .
D | I 1 M 1 C 1 S 1 | = | I 2 M 2 C 2 S 2 | .
A | I 2 M 2 C 2 S s | = | I 3 M 3 C 3 S 3 | .
I 3 = A 11 I 2 + A 12 M 2 + A 13 C 2 + A 14 S 2 .
M 2 = D 21 I 1 + D 22 M 1 + D 23 C 1 + D 24 S 1 = 0
1 τ 0 τ M 2 d t = I τ τ D 21 d t + = 0.
R 2 ( n δ , β ) R 1 ( δ , 0 ) | I 1 M 1 C 1 S 1 | = | I 2 M 2 C 2 S 2 | .
| 1 0 0 0 0 cos 2 2 β + sin 2 2 β cos n δ [ cos 2 β sin 2 β ( 1 - cos n δ ) ] cos δ + sin 2 β sin n δ sin δ - [ cos 2 β sin 2 β ( 1 - cos n δ ) ] sin δ + sin 2 β sin n δ cos δ 0 sin 2 β cos 2 β ( 1 - cos n δ ) cos δ [ sin 2 2 β + cos 2 2 β cos n δ ] + sin δ [ - cos 2 β sin n δ ] - sin δ ( sin 2 2 β + cos 2 2 β cos n δ ) - cos δ cos 2 β sin n δ 0 - sin 2 β sin n δ cos δ ( cos 2 β sin n δ ) + sin δ cos n δ - sin δ ( cos 2 β sin n δ ) + cos δ cos n δ |
D ¯ 22 = 1 T 0 T ( cos 2 2 β + sin 2 2 β cos n δ ) d t = 0
D ¯ 22 = cos 2 2 β + 1 T 0 T sin 2 2 β cos n δ ( t ) d t = 0.
δ = K t             0 t τ .
D ¯ 22 = 1 2 π 0 2 π cos n δ d δ = 0.
D ¯ 23 = 1 2 π ( n - 1 ) [ sin π ( n - 1 ) ] + 1 2 π ( n + 1 ) [ sin π ( n + 1 ) ] .
Γ = d ( n e - n 0 ) / λ = p V ,
sin 2 Ω = p V / k d
δ = δ 0 sin ω t .
D ¯ 22 = ( ω / 2 π ) 0 2 π / ω cos ( n δ 0 sin ω t ) d t .
D ¯ 22 = ( 1 / 2 π ) 0 2 π cos ( n δ 0 sin ϕ ) d ϕ .
D ¯ 23 = ( 1 / 2 π ) 0 2 π sin ( n δ 0 sin ϕ ) sin ( δ 0 sin ϕ ) d ϕ .
sin n α sin α = 1 2 cos ( n α - α ) - 1 2 cos ( n α + α ) .
D ¯ 23 = 1 2 [ J 0 ( n - 1 ) δ 0 - J 0 ( n + 1 ) δ 0 ] .
D ¯ 33 = J 0 ( δ 0 ) ,
I = 1 + cos 2 θ ( D ¯ 22 cos 2 α + D ¯ 23 sin 2 α ) + sin 2 θ ( D ¯ 33 sin 2 α ) .
J 0 ( δ 0 ) = 0 J 0 ( n δ 0 ) = 0 J 0 ( n - 1 ) δ 0 - J 0 ( n + 1 ) δ 0 = 0.
A | I 2 M 2 C 2 S 2 | = | I 3 M 3 C 3 S 3 | .
I 3 = A 11 I 2 + A 12 M 2 + A 13 C 2 + A 14 S 2 .
I ¯ 3 = ( 1 / Δ λ ) λ 1 λ 2 I 3 d λ = ( 1 / Δ λ ) λ 1 λ 2 A 11 I 2 d λ + ( 1 / Δ λ ) λ 1 λ 2 A 12 M 2 d λ + .
λ 1 λ 2 M 2 d λ = λ 1 λ 2 C 2 d λ + λ 1 λ 2 S 2 d λ = 0
M ¯ = C ¯ = S ¯ = 0.
| I 1 M 1 C 1 S 1 | = I 1 | 1 m 1 c 1 s 1 | ,
λ 1 λ 2 D 21 I 1 d λ = 0 ,             m 1 λ 1 λ 2 D 22 I 1 d λ = 0 , c 1 λ 1 λ 2 D 23 I 1 d λ = 0 ,             s 1 λ 2 λ 2 D 24 I 1 d λ = 0.
R 2 ( n δ , β ) R 1 ( δ , 0 ) | I 1 M 1 C 1 S 1 | = | I 2 M 2 C 2 S 2 | .
D ¯ i j = D i j d λ = 0.
δ = [ 2 π ( n e - n 0 ) d / λ ]
D ¯ i j = ( 1 / Δ λ ) λ 1 λ 2 f ( δ ) d λ = ( δ 1 δ 2 / δ 1 - δ 2 ) δ 1 δ 2 ( f ( δ ) d δ / δ 2 ) .
D ¯ i j = ( δ 1 δ 2 / 2 π ) δ 1 δ 2 ( f ( δ ) / δ 2 ) d δ .
D ¯ 22 = δ 1 δ 2 2 π [ 2 π cos 2 2 β + sin 2 2 β δ 1 δ 2 cos n δ d δ δ 2 ] .
I 1 = ( Q / λ 2 ) .
D ¯ i j = ( I Av / 2 π ) δ 1 δ 2 f ( δ ) d ,
D ¯ 22 = ( I Av / 2 π ) δ 1 δ 2 cos n δ d δ
D 22 = ( 1 / 2 π ) δ 1 δ 1 + 2 π cos n δ d δ = ( 1 / π n ) sin π n cos n ( π + δ ) .
D ¯ 23 = [ 1 / 2 π ( n - 1 ) ] sin π ( n - 1 ) cos ( n - 1 ) ( δ + π ) + [ 1 / 2 π ( n + 1 ) ] sin π ( n + 1 ) cos ( n + 1 ) ( δ + π ) .
( Δ I / I ) = ( I max - I min ) / I Av .
P ( θ ) D ¯ P ( α ) | 1 0 0 0 | = | I M C S | .
I = 1 + cos 2 θ ( D ¯ 22 cos 2 α + D ¯ 23 sin 2 α ) + sin 2 θ ( D ¯ 32 cos 2 α + D ¯ 33 sin 2 α ) .
I = 1 + cos 2 θ [ D ¯ 22 cos 2 α + D ¯ 23 sin 2 α ] .
I / α = - 2 D ¯ 22 sin 2 α + 2 D ¯ 23 cos 2 α = 0.
α 1 = 1 2 tan - 1 ( D ¯ 23 / D ¯ 22 )
Δ I / I Av = 2 [ D ¯ 22 cos 2 α 1 + D ¯ 23 sin 2 α 1 ] ,
α 1 = 1 2 tan - 1 × ( n / n - 1 ) sin π ( n - 1 ) + ( n / n + 1 ) sin π ( n + 1 ) sin π n .
n = n + ,
α 1 = 1 2 tan - 1 ( n / n - 1 ) sin [ π ( n - 1 ) + π ] + ( n / n + 1 ) sin [ π ( n + 1 ) + π ] 2 sin ( π n + π ) .
α = 1 2 tan - 1 ( n 2 / n 2 - 1 ) .
sin 2 α = D ¯ 23 / ( D ¯ 22 2 + D ¯ 23 2 ) 1 2 × cos 2 α = D ¯ 22 / ( D ¯ 22 2 + D ¯ 23 2 ) 1 2 .
Δ I = 2 ( D ¯ 22 2 + D ¯ 23 2 ) 1 2 .
Δ I = ( 2 sin π / π ) [ ( n n 2 - 1 ) 2 + 1 n 2 ] 1 2 .
p ( α ) | I 1 M 1 C 1 S 1 | = | I 2 M 2 C 2 S 2 | .
1 2 | I 1 + M 1 cos 2 α + C 1 sin 2 α I 1 cos 2 α + M 1 cos 2 2 α + C 1 sin 2 α cos 2 α I 1 sin 2 α + M 1 sin 2 α cos 2 α + C 1 sin 2 2 α 0 | .
M 2 = M 1 2 π 0 2 π cos 2 2 α d α = M 1 / 2.
| I 2 M 2 C 2 S 2 | = | I 1 1 2 M 1 1 2 C 1 0 | .
R ( β , δ ) | I 1 M 1 C 1 S 1 | = | I 2 M 2 C 2 S 2 | .
| I 2 M 2 C 2 S 2 | = | I 1 1 2 ( 1 + cos δ ) M 1 1 2 ( 1 + cos δ ) C 1 S 1 cos δ | .
| I 1 1 2 M 1 1 2 C 1 0 | .
p = 100 ( M 2 2 + C 2 2 ) / ( M 1 2 + C 1 2 ) = 25 percent .
| I 1 0 0 - S 1 | .
1 4 { ( 1 - cos θ 1 - cos θ 2 + cos θ 1 cos θ 2 ) cos [ 8 π ( n - 1 ) t / T ] + 1 + cos θ 1 + cos θ 2 + cos θ 1 cos θ 2 + ( 1 - cos θ 1 + cos θ 2 - cos θ 1 cos θ 2 ) cos ( 8 π t / T ) + ( 1 + cos θ 1 - cos θ 2 - cos θ 1 cos θ 2 ) cos ( 8 π n t / T ) + 2 sin θ 1 sin θ 2 [ cos 4 π ( 1 + n ) t T - cos 4 π ( n - 1 ) t T ] } = D 22
D 23 = 1 4 { sin [ 8 π ( n - 1 ) t / T ] ( cos θ 1 + cos θ 2 - 1 ) - sin [ 8 π ( n + 1 ) t / T ] ( cos θ 1 cos θ 2 ) + sin ( 8 π t / T ) ( 1 + cos θ 2 - cos θ 1 - cos θ 1 cos θ 2 ) + sin ( 8 n π t / T ) ( 1 - cos θ 2 + cos θ 1 + cos θ 1 cos θ 2 ) + 2 sin 4 π ( n + 1 ) t T sin θ 1 sin θ 2 + 2 sin 4 π ( n - 1 ) t T sin θ 1 sin θ 2 } ,
D ¯ 22 = ( 1 / T ) 0 T D 22 d t = 1 4 ( 1 + cos δ 1 ) ( 1 + cos δ 2 )
D ¯ 23 = ( 1 / 2 π ) 0 2 π D 23 d β = 0.
D ¯ 33 = 1 T 0 T cos δ d t .