Abstract

Colorimetric analysis of the color of light passed through a birefringent crystal placed between Nicol prisms is treated more generally than hitherto discussed, by adding an achromatic retardation. It is found that, by proper choice of the achromatic retardation, the color change with respect to the change of birefringence becomes very sensitive and we call this “hypersensitive (polarization) color.”

The case when the crystal has the power of rotating the plane of polarization, which can be treated in the same way, is also shown.

Then the color of thin layers of dielectrics on a metallic surface is analyzed. This can be treated as an extension of the cases of polarization color. As it is difficult to give a general solution, some special cases (so called cases of “white metal”) are given and the color of the reflected light is calculated and plotted on CIE chromaticity diagram.

© 1955 Optical Society of America

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References

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  1. Kubota, Ara, and Saito, J. Opt. Soc. Am. 41, 537 (1951).
    [Crossref]
  2. H. Kubota, J. Opt. Soc. Am. 42, 144(L) (1952).
    [Crossref]
  3. H. Kubota, “Tables of the Interference Color,” Rept. Inst. Ind. Sci. Tokyo Univ. 2, No. 6 (1952).
  4. D. L. MacAdam, J. Opt. Soc. Am. 27, 294 (1937).
    [Crossref]
  5. M. Françon, Rev. opt. 31, 67 (1952).
  6. Geiger-Scheel, Handbuch der Physik (Berlin, 1928), G. Szivessy, “Licht als Wellenbewegung,” p. 846.
  7. A. Wüllner, Lehrbuch der Experimentalische Physik (Leipzig, 1899) 5th edition, p. 970.
  8. H. Kubota and T. Ose, J. Phys. Soc. Japan. 7, 470 (1952).
    [Crossref]

1952 (4)

H. Kubota, J. Opt. Soc. Am. 42, 144(L) (1952).
[Crossref]

H. Kubota, “Tables of the Interference Color,” Rept. Inst. Ind. Sci. Tokyo Univ. 2, No. 6 (1952).

M. Françon, Rev. opt. 31, 67 (1952).

H. Kubota and T. Ose, J. Phys. Soc. Japan. 7, 470 (1952).
[Crossref]

1951 (1)

1937 (1)

Ara,

Françon, M.

M. Françon, Rev. opt. 31, 67 (1952).

Geiger-Scheel,

Geiger-Scheel, Handbuch der Physik (Berlin, 1928), G. Szivessy, “Licht als Wellenbewegung,” p. 846.

Kubota,

Kubota, H.

H. Kubota and T. Ose, J. Phys. Soc. Japan. 7, 470 (1952).
[Crossref]

H. Kubota, J. Opt. Soc. Am. 42, 144(L) (1952).
[Crossref]

H. Kubota, “Tables of the Interference Color,” Rept. Inst. Ind. Sci. Tokyo Univ. 2, No. 6 (1952).

MacAdam, D. L.

Ose, T.

H. Kubota and T. Ose, J. Phys. Soc. Japan. 7, 470 (1952).
[Crossref]

Saito,

Wüllner, A.

A. Wüllner, Lehrbuch der Experimentalische Physik (Leipzig, 1899) 5th edition, p. 970.

J. Opt. Soc. Am. (3)

J. Phys. Soc. Japan. (1)

H. Kubota and T. Ose, J. Phys. Soc. Japan. 7, 470 (1952).
[Crossref]

Rept. Inst. Ind. Sci. Tokyo Univ. (1)

H. Kubota, “Tables of the Interference Color,” Rept. Inst. Ind. Sci. Tokyo Univ. 2, No. 6 (1952).

Rev. opt. (1)

M. Françon, Rev. opt. 31, 67 (1952).

Other (2)

Geiger-Scheel, Handbuch der Physik (Berlin, 1928), G. Szivessy, “Licht als Wellenbewegung,” p. 846.

A. Wüllner, Lehrbuch der Experimentalische Physik (Leipzig, 1899) 5th edition, p. 970.

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Figures (11)

Fig. 1
Fig. 1

Rotatory polarization color of quartz, plotted on CIE chromaticity diagram. d: thickness of quartz plate, kept constant, θ: angle between Nicols.

Fig. 2
Fig. 2

Rotatory polarization color of quartz, plotted on CIE chromaticity diagram. d: thickness of quartz, θ: angle between Nicols, kept constant.

Fig. 3
Fig. 3

Constants of the ellipses of Fig. 1. L: length of the major axis, e: the eccentricity, ϕ: the angle between major axis and x axis.

Fig. 4
Fig. 4

Thick lines are the loci of the color of chromatic polarization plotted on CIE chromaticity diagram. (μd) and (μd)′: points showing the same hue F. (μd)+Δ(μd) and (μd)′+Δ(μd)′: points showing the same hue (FF).

Fig. 5
Fig. 5

Sensitivity of the “hypersensitive color.” η: achromatic retardation added to the system; ΔS/Δ(μd): sensitivity; (μd): retardation given by the passage through crystal.

Fig. 6
Fig. 6

Phase change at reflection from metal (Ag) surface. ○: calculated from data in International Critical Tables; —: plot of (Eq. 11), φ: angle of incidence to the surface; n: refractive index of the layer on metallic surface.

Fig. 7
Fig. 7

Interference color of thin layer on metallic surface, plotted on CIE chromaticity diagram. (nd): optical thickness of the layer, b and c are the constants of the dispersion of η [see Eq. (11)].

Fig. 8
Fig. 8

Interference color of thin layer on metallic surface. (Notations same as in Fig. 7.)

Fig. 9
Fig. 9

The term Δ is the difference between the values of (nd) which give maximum sensitivity. Lengths of vertical lines attached to small circles show the range of error of calculation.

Fig. 10
Fig. 10

Interference color of the layer on metallic surface plotted on CIE chromaticity diagram. Thick lines: loci of the interference color, in which dispersion of C (reflectivity of metallic surface) is taken into account. Thin lines: dispersion of C is not taken into account.

Fig. 11
Fig. 11

Interference color of thin layer on metallic surface. S.L.: means single reflection, that is, multiple reflection is not taken into account. M.L.: multiple reflection is taken into account. Dis. of η (or C): dispersion of η (or C) is taken into account.

Equations (40)

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I = cos 2 ( π ( μ d ) / λ + η / 2 ) ,
I = cos 2 ( φ ( λ ) d - θ ) .
X = E I x ¯ d λ = X 0 + X r cos 2 θ + X i sin 2 θ ,
Y = E I y ¯ d λ = Y 0 + Y r cos 2 θ + Y i sin 2 θ , Z = E I z ¯ d λ = Z 0 + Z r cos 2 θ + Z i sin 2 θ .
X 0 = 1 2 E x ¯ d λ ,             X r = 1 2 E x ¯ cos ( 2 φ d ) d λ , X i = 1 2 E x ¯ sin ( 2 φ d ) d λ ,             etc .
| X X r X 0 Y Y r Y 0 Z Z r Z 0 | 2 + | X X i X 0 Y Y i Y 0 Z Z i Z 0 | 2 = | X i X r X Y i Y r Y Z i Z r Z | 2 .
X + Y + Z = S ,             X 0 + Y 0 + Z 0 = S 0 ,             X r + Y r + Z r = S r ,             X i + Y i + Z i = S i ,
| x X r X 0 y Y r Y 0 1 S r S 0 | 2 + | x X i X 0 y Y i Y 0 1 S i S 0 | 2 = | x X r X i y Y r Y i 1 S r S i | 2 .
φ ( λ ) ~ 1 / λ 2
φ ( λ ) = α + β / λ ,
X r = 1 2 E x ¯ cos ( 2 φ d ) d λ = R x · cos 2 α d - I x · sin 2 α d , etc . ,
2 R x = E x ¯ cos ( 2 β d / λ ) d λ , 2 I x = E x ¯ sin ( 2 β d / λ ) d λ , etc .
d / ( 1 + 2 θ / π )
( μ d ) / ( 1 - η / π )
I = cos 2 ( π ( μ d ) / λ 0 + η / 2 ) = 0
π ( μ d ) / λ 0 + η / 2 = ( 2 m + 1 ) π / 2 ,
( μ d ) / ( 1 - η / π ) = λ 0 / 2 = const .
1 / λ = ( 1 + α ( λ ) ) / λ 0 ,
I = cos 2 [ π ( μ d ) ( 1 + α ) / λ 0 + η / 2 ] .
I = sin 2 [ π ( μ d ) α / λ 0 ] α 2 [ π ( μ d ) / λ 0 ] 2 .
d / ( 1 + 2 θ / π ) = π / 2 φ ( λ 0 ) .
d / ( 1 + 2 θ / π )
( μ d ) / ( 1 - η / π ) = ( μ d ) / ( 1 - η / π ) .
[ ( μ d ) + Δ ( μ d ) ] / ( 1 - η / π ) = [ ( μ d ) + Δ ( μ d ) ] / ( 1 - η / π ) .
Δ ( μ d ) / Δ ( μ d ) = ( 1 - η / π ) / ( 1 - η / π ) .
λ 0 = 531 m μ .
θ + 27.51 d = 90 ° ( θ in degree , d in mm ) .
I = B 2 + C 2 + 2 B C cos ( 4 π ( n d ) / λ + η ) .
I = A [ + cos 2 ( 2 π ( n d ) / λ + η / 2 ) ] .
η = η 0 + b ( λ - λ 0 ) + c ( λ - λ 0 ) 2 ,
I = cos 2 ( 2 π ( n d ) / λ + η / 2 ) .
I ~ ( λ ) + C cos 2 ( 2 π ( n d ) / λ + η / 2 ) ,
= ( B - C ) 2 / 4 B ;
X = E x ¯ d λ + E x ¯ C cos 2 ( 2 π ( n d ) / λ + η / 2 ) d λ .
I = B 2 + C 2 + 2 B C cos ( 4 π ( n d ) / λ + η ) 1 + B 2 C 2 + 2 B C cos ( 4 π ( n d ) / λ + η .
I = cos ρ · cos ( θ - φ ) e i ( δ + η ) + sin ρ · cos ( θ - φ ) · sin φ e i δ - cos ρ · sin φ · sin ( θ - φ ) e i η + sin ρ · sin ( θ - φ ) · cos φ .
J = I · I * = A + B cos ( δ + η ) + C cos ( δ - η ) + D cos δ ,
J ~ + cos 2 ( ( σ + η + δ ) / 2 ) ,
tan σ = sin 2 ρ · sin 2 φ · sin 2 η + cos 2 ρ · sin 2 φ · sin η sin 2 ρ · cos 2 φ - sin 2 ρ · sin 2 φ cos 2 η - cos 2 ρ · sin 2 φ · cos η .
ρ = π / 4 ± α ,             φ = π / 2 ± β ,             or             0 ± β ,