Abstract

This paper shows how a single Z-cut plate of a uniaxial crystal can be used as a two-beam interferometer or Fourier-transform spectrometer. As a visual device, it permits easy identification of a set of achromatic fringes that can be seen in any two-beam interferometer when a white-light source is observed through it. These fringes apparently have not been described previously. They are expected when we understand that the white-light fringe pattern is in fact the Fourier transform of the spectral-sensitivity curve of the eye.

© 1975 Optical Society of America

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References

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  1. B. H. Billings, J. Phys. (Paris) 28, Suppl. 3–4, C2-205 (1967).
    [CrossRef]
  2. J. D. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).
  3. F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw–Hill, New York, 1937).
  4. D. L. MacAdam, in American Institute of Physics Handbook, 2nd ed. (McGraw–Hill, New York, 1963), p. 6–140.
  5. A. C. Hardy and F. H. Perrin, Principles of Optics (McGraw–Hill, New York, 1932).
  6. D. L. MacAdam, in American Institute of Physics Handbook, 3rd ed. (McGraw–Hill, New York, 1972), p. 6–183.
  7. H. Kubota, Proc. Jpn. Acad. 36, 418 (1960).

1967 (1)

B. H. Billings, J. Phys. (Paris) 28, Suppl. 3–4, C2-205 (1967).
[CrossRef]

1960 (1)

H. Kubota, Proc. Jpn. Acad. 36, 418 (1960).

Billings, B. H.

B. H. Billings, J. Phys. (Paris) 28, Suppl. 3–4, C2-205 (1967).
[CrossRef]

Hardy, A. C.

A. C. Hardy and F. H. Perrin, Principles of Optics (McGraw–Hill, New York, 1932).

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw–Hill, New York, 1937).

Kubota, H.

H. Kubota, Proc. Jpn. Acad. 36, 418 (1960).

MacAdam, D. L.

D. L. MacAdam, in American Institute of Physics Handbook, 2nd ed. (McGraw–Hill, New York, 1963), p. 6–140.

D. L. MacAdam, in American Institute of Physics Handbook, 3rd ed. (McGraw–Hill, New York, 1972), p. 6–183.

Perrin, F. H.

A. C. Hardy and F. H. Perrin, Principles of Optics (McGraw–Hill, New York, 1932).

Strong, J. D.

J. D. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw–Hill, New York, 1937).

J. Phys. (Paris) (1)

B. H. Billings, J. Phys. (Paris) 28, Suppl. 3–4, C2-205 (1967).
[CrossRef]

Proc. Jpn. Acad. (1)

H. Kubota, Proc. Jpn. Acad. 36, 418 (1960).

Other (5)

J. D. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw–Hill, New York, 1937).

D. L. MacAdam, in American Institute of Physics Handbook, 2nd ed. (McGraw–Hill, New York, 1963), p. 6–140.

A. C. Hardy and F. H. Perrin, Principles of Optics (McGraw–Hill, New York, 1932).

D. L. MacAdam, in American Institute of Physics Handbook, 3rd ed. (McGraw–Hill, New York, 1972), p. 6–183.

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Figures (14)

FIG. 1
FIG. 1

Fourier transform of rectangular approximation to eye-sensitivity (luminosity) curve.

FIG. 2
FIG. 2

Photograph of fringe patterns in film produced by light from an incandescent-tungsten lamp, transmitted through crystal plate.

FIG. 3
FIG. 3

Eye spectral-sensitivity (luminosity) curve multiplied by A(ν)I(ν) cos2πνx with x = 3 μm.

FIG. 4
FIG. 4

Eye spectral-sensitivity (luminosity) curve multiplied by A(ν)I(ν) cos2πνx with x = 975 nm.

FIG. 5
FIG. 5

Calculated Fourier transform of product of luminosity curve, skylight (illuminant D), and crystal-plate transmittance to xmax = 7.5 μm.

FIG. 6
FIG. 6

Calculated Fourier transform of product of luminosity curve, skylight (illuminant D), and crystal-plate transmittance from x = 3 to x = 10.5 μm.

FIG. 7
FIG. 7

Calculated Fourier transform of product of luminosity curve, skylight (illuminant D), and crystal-plate transmittance from x = 7 to x = 14.5 μm.

FIG. 8
FIG. 8

Calculated Fourier transform of product of luminosity curve and tungsten light 2858.7 K (illuminant A) to x = 7.5 μm.

FIG. 9
FIG. 9

Calculated Fourier transform of product of luminosity curve and tungsten light 2858.7 K (illuminant A) from x = 3 to x =10.5 μm.

FIG. 10
FIG. 10

Calculated Fourier transform of product of luminosity curve and tungsten light 2858.7 K (illuminant A) from x = 7 to x = 14.5 μm.

FIG. 11
FIG. 11

Chromaticity plot at 30-nm intervals of fringe pattern seen by the eye in skylight from x = 0 to x = 3 μm.

FIG. 12
FIG. 12

Chromaticity plot at 30-nm intervals of fringe pattern seen by the eye in skylight from x = 3 to x = 6 μm.

FIG. 13
FIG. 13

Chromaticity plot at 30-nm intervals of fringe pattern seen by the eye in skylight from x = 6 to x = 9 μm.

FIG. 14
FIG. 14

Spectral transmittance of red filter.

Equations (32)

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Γ = d λ 2 - ω 2 d 2 ω sin 2 i ,
Γ = x ν ,
G ( x ) = I 0 2 ( 1 + cos 2 π x ν ) ,
G ( x ) = 0 I ( ν ) 2 d ν + 0 I ( ν ) 2 cos 2 π x ν d ν .
F ( x ) = 1 2 0 I ( ν ) A ( ν ) D ( ν ) cos 2 π x ν d ν ,
I ( ν ) A ( v ) D ( ν ) = 0 F ( x ) cos 2 π x ν d x .
F ( x ) = ν 1 ν 2 cos 2 π x ν d ν ,
I ( ν ) A ( ν ) D ( ν ) = 0 , ν < ν 1 I ( ν ) A ( ν ) D ( ν ) = 1 , ν 2 > ν > ν 1 I ( ν ) A ( ν ) D ( ν ) = 0 , ν > ν 2
F ( x ) = ( ν 1 - ν 2 ) [ sin π x ( ν 1 - ν 2 ) π x ( ν 1 - ν 2 ) cos π x ( ν 1 + ν 2 ) ] .
ν 1 = 1.6390 μ m - 1 , ν 2 = 1.9620 μ m - 1 ,
x = 2 N - 1 ν 2 + ν 1 ,
x = 1 ν 2 - ν 1 .
N = 6.6
F ( x ) = ( ν 1 - ν 2 ) [ sin 2 [ π x ( ν 1 - ν 2 ) ] π 2 x 2 ( ν 1 - ν 2 ) 2 cos π x ( ν 1 + ν 2 ) ] .
λ 2 - λ 1 = 2 λ 0 2 N - 1 = 74 nm ,
F ( x ) = ( ν 1 - ν 2 ) [ sin 2 π ( ν 1 - ν 2 ) x 2 π ( ν 1 - ν 2 ) x ] [ 1 1 - 4 ( ν 1 - ν 2 ) 2 x 2 ] × cos π ( ν 1 + ν 2 ) x .
x = 1 ν 2 - ν 1 ,
λ 2 - λ 1 = 74 nm .
F ( x ) = ( ν 2 - ν 1 ) ( π 4 ) 1 / 2 exp [ - π 2 x 2 4 ( ν 2 - ν 1 ) 2 ] cos π ( ν 2 + ν 1 ) x .
0.02 = exp [ - ( π 2 x 2 4 ( ν 2 - ν 1 ) 2 ) ] .
λ 2 - λ 1 = 8 π λ 0 2 N - 1 .
λ 2 - λ 1 = 94.2 nm ,
F ( x ) = 0 D ( ν ) A ( ν ) I ( ν ) ν 2 cos 2 π x ν d ν .
x 0 = X X + Y + Z , y 0 = Y X + Y + Z ,
X = 1 2 0 x ¯ ( ν ) A ( ν ) I ( ν ) ν 2 ( 1 + cos 2 π x ν ) d ν , Z = 1 2 0 z ( ν ) A ( ν ) I ( ν ) ν 2 ( 1 + cos 2 π x ν ) d ν ,
I ( ν ) = S ( ν ) 2 ( 1 + cos 2 π p ν ) ,
F ( x ) = 0 S ( ν ) D ( ν ) 2 ν 2 cos 2 π ν x d ν + 0 S ( ν ) D ( ν ) 2 ν 2 cos 2 π p ν cos 2 π x ν d ν .
F ( x ) = ν 1 - ν 2 2 [ sin π ( ν 1 - ν 2 ) x π ( ν 1 - ν 2 ) x cos π ( ν 1 + ν 2 ) x ] + ν 1 - ν 2 2 [ sin π ( x - p ) ( ν 1 - ν 2 ) π ( x - p ) ( ν 1 - ν 2 ) cos π ( x - p ) ( ν 1 + ν 2 ) ] .
F ( x 1 ) = 0.02 = sin 2 π x 1 ( ν 1 - ν 2 ) π 2 x 1 2 ( ν 1 - ν 2 ) 2 ,
x 1 = 2 N - 1 ν 2 + ν 1
ν 2 + ν 1 = 2 555 = 3.60 μ m - 1 .
ν 1 - ν 2 = 0.0408 μ m - 1 , λ 1 - λ 2 = 12.6 nm .