Abstract

The effect on photometric-setting accuracy of mechanical misalignments and surface errors of the end mirrors of a double-passed Michelson interferometer has been analyzed. Tolerance limits are given for the mirror quality and their misalignments.

© 1968 Optical Society of America

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References

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  1. F. Zernike, J. Opt. Soc. Am. 40, 326 (1950).
    [Crossref]
  2. M. Bottema and F. Zernike, J. Opt. Soc. Am. 41, 870A (1951).
  3. M. Bottema, Physica 24, 519 (1958).
    [Crossref]
  4. M. Bottema, thesis, University of Groningen (1957).
  5. M. Bottema, in Optics in Metrology, P. Mollet, Ed. (Pergamon Press, Inc., New York, 1960), p. 42.
  6. B. Vittoz, Helv. Phys. Act. 26, 400 (1953).
  7. P. Hariharan and D. Sen, J. Sci. Instr. 36, 72 (1959).
    [Crossref]
  8. P. Hariharan and D. Sen, J. Sci. Instr. 36, 70 (1959).
    [Crossref]
  9. P. Hariharan and D. Sen, J. Opt. Soc. Am. 50, 357 (1960).
    [Crossref]
  10. P. Hariharan and D. Sen, J. Opt. Soc. Am. 50, 999 (1960).
    [Crossref]
  11. P. Hariharan and D. Sen, J. Opt. Soc. Am. 50, 1026 (1960).
    [Crossref]
  12. P. Hariharan and D. Sen, J. Opt. Soc. Am. 51, 617 (1961).
    [Crossref]
  13. P. Hariharan and D. Sen, J. Opt. Soc. Am. 51, 1212 (1961).
    [Crossref]
  14. P. Hariharan and D. Sen, J. Sci. Instr. 37, 278 (1960).
    [Crossref]
  15. P. Hariharan and D. Sen, Brit. J. Appl. Phys. 12, 20 (1961).
    [Crossref]
  16. R. E. Kinzly, Appl. Opt. 6, 137 (1967).
    [Crossref] [PubMed]
  17. P. K. Katti and K. Singh, Appl. Opt. 5, 1962 (1966).
    [Crossref] [PubMed]
  18. P. K. Katti and K. Singh, Opt. Acta,  14, 289 (1967).
    [Crossref]
  19. R. M. Hill, Opt. Acta,  10, 141 (1963).
    [Crossref]

1967 (2)

P. K. Katti and K. Singh, Opt. Acta,  14, 289 (1967).
[Crossref]

R. E. Kinzly, Appl. Opt. 6, 137 (1967).
[Crossref] [PubMed]

1966 (1)

1963 (1)

R. M. Hill, Opt. Acta,  10, 141 (1963).
[Crossref]

1961 (3)

1960 (4)

1959 (2)

P. Hariharan and D. Sen, J. Sci. Instr. 36, 72 (1959).
[Crossref]

P. Hariharan and D. Sen, J. Sci. Instr. 36, 70 (1959).
[Crossref]

1958 (1)

M. Bottema, Physica 24, 519 (1958).
[Crossref]

1953 (1)

B. Vittoz, Helv. Phys. Act. 26, 400 (1953).

1951 (1)

M. Bottema and F. Zernike, J. Opt. Soc. Am. 41, 870A (1951).

1950 (1)

Bottema, M.

M. Bottema, Physica 24, 519 (1958).
[Crossref]

M. Bottema and F. Zernike, J. Opt. Soc. Am. 41, 870A (1951).

M. Bottema, thesis, University of Groningen (1957).

M. Bottema, in Optics in Metrology, P. Mollet, Ed. (Pergamon Press, Inc., New York, 1960), p. 42.

Hariharan, P.

Hill, R. M.

R. M. Hill, Opt. Acta,  10, 141 (1963).
[Crossref]

Katti, P. K.

P. K. Katti and K. Singh, Opt. Acta,  14, 289 (1967).
[Crossref]

P. K. Katti and K. Singh, Appl. Opt. 5, 1962 (1966).
[Crossref] [PubMed]

Kinzly, R. E.

Sen, D.

Singh, K.

P. K. Katti and K. Singh, Opt. Acta,  14, 289 (1967).
[Crossref]

P. K. Katti and K. Singh, Appl. Opt. 5, 1962 (1966).
[Crossref] [PubMed]

Vittoz, B.

B. Vittoz, Helv. Phys. Act. 26, 400 (1953).

Zernike, F.

M. Bottema and F. Zernike, J. Opt. Soc. Am. 41, 870A (1951).

F. Zernike, J. Opt. Soc. Am. 40, 326 (1950).
[Crossref]

Appl. Opt. (2)

Brit. J. Appl. Phys. (1)

P. Hariharan and D. Sen, Brit. J. Appl. Phys. 12, 20 (1961).
[Crossref]

Helv. Phys. Act. (1)

B. Vittoz, Helv. Phys. Act. 26, 400 (1953).

J. Opt. Soc. Am. (7)

J. Sci. Instr. (3)

P. Hariharan and D. Sen, J. Sci. Instr. 37, 278 (1960).
[Crossref]

P. Hariharan and D. Sen, J. Sci. Instr. 36, 72 (1959).
[Crossref]

P. Hariharan and D. Sen, J. Sci. Instr. 36, 70 (1959).
[Crossref]

Opt. Acta (2)

P. K. Katti and K. Singh, Opt. Acta,  14, 289 (1967).
[Crossref]

R. M. Hill, Opt. Acta,  10, 141 (1963).
[Crossref]

Physica (1)

M. Bottema, Physica 24, 519 (1958).
[Crossref]

Other (2)

M. Bottema, thesis, University of Groningen (1957).

M. Bottema, in Optics in Metrology, P. Mollet, Ed. (Pergamon Press, Inc., New York, 1960), p. 42.

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Figures (6)

Fig. 1
Fig. 1

Double-passed Michelson interferometer; experimental arrangement used to obtain double-passed fringes of equal inclination. A, mercury arc lamp; G, green filter; M0, beam divider; M1, M2, end mirrors; M3 plane mirror; M4, semireflecting mirror; L, telescope objective; E, eye-piece; N1, N2, polarizers; R, quarter-wave plate; S, air-cell compensator.

Fig. 2
Fig. 2

Paths of the rays in the double-passed Michelson interferometer.

Fig. 3
Fig. 3

Nature of sinusoidal errors on the surface of an interferometer mirror. Δ1 and Δ2 are the amplitudes of error in the Y and Z directions, respectively.

Fig. 4
Fig. 4

Intensity distribution in the fringes for (bottom) b = 2, (center) b = (2n + 1)π/2, and (top) b = (2n + 1)π. Circular mirrors misaligned. Curve 1 (broken), Δmax = 0. Curve 2, Δmax = λ/20, Curve 3, Δmax = λ/10. Curve 4, Δmax = λ/5.

Fig. 5
Fig. 5

Intensity distribution in the fringes for (bottom) b = 2, (center) b = (2n + 1)π/2, and (top) b = (2n + 1)π. Sinusoidal polish errors on the surface of the mirror. Curve 1 (broken), Δ1 = Δ2 = 0. Curve 2, Δ1 = Δ2 = λ/20. Curve 3, Δ1 = Δ2 = λ/10. Curve 4, Δ1 = Δ2 = λ/5.

Fig. 6
Fig. 6

Plot of setting accuracy as a function of surface irregularities and misalignments, (a) Misalignment (circular mirrors), (b) Curvature, misalignment (square mirrors), (c) Random errors, (d) Sinusoidal errors.

Equations (23)

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I ( θ ) = 4 I 0 ( cos a θ + cos b ) 2 ,
a = ( 2 π / λ ) 2 t sin α ; b = ( 2 π / λ ) 2 t cos α ,
E = K S I ( θ ) d S = K I ( θ ) S ,
I ( θ ) = ( 1 / S ) S I ( θ ) d S .
d S = f ( Δ ) d Δ = f ( δ b ) d ( δ b ) ,
I ( θ ) = I ( b + δ b ) d S S d = I ( b + δ b ) f ( δ b ) d ( δ b ) f ( δ b ) d ( δ b ) .
f ( δ b ) = ( β / π ) 1 2 exp ( β δ b 2 ) .
I ( θ ) = exp ( β δ b 2 ) 4 I 0 [ cos a θ + cos ( b + δ b ) ] 2 d ( δ b ) / exp ( β δ b 2 ) d ( δ b ) ,
I ( θ ) = 4 I 0 [ cos 2 a θ + 0.5 { 1 + exp ( 1 / β ) cos 2 b } + 2 exp ( 1 / 4 β ) cos a θ cos b ] .
f ( δ b ) = K , where K is a constant .
I ( θ ) = 4 I 0 δ b m δ b m K [ cos a θ + cos ( b + δ b ) ] 2 d ( δ b ) / δ b m δ b m K d ( δ b ) ,
I ( θ ) = 4 I 0 [ cos 2 a θ + 0.5 + ( cos 2 b / 2 ) · ( sin 2 δ b m / 2 δ b m ) + 2 cos a θ cos b · sin δ b m / δ b m ) ] .
2 ( t 1 + Δ 1 sin θ 1 + Δ 2 sin θ 2 ) ,
I ( θ ) = 4 I 0 [ cos 2 a θ + ( 1 / 4 π 2 ) 0 2 π 0 2 π cos 2 ( 2 π / λ ) 2 ( t + Δ 1 sin θ 1 + Δ 2 sin θ 2 ) d θ 1 d θ 2 + 2 cos a θ · ( 1 / 4 π 2 ) × 0 2 π 0 2 π cos ( 2 π / λ ) ( t 1 + Δ 1 sin θ 1 + Δ 2 sin θ 2 ) d θ 1 d θ 2 ] .
I ( θ ) = 4 I 0 [ cos 2 a θ + 0.5 + J 0 ( 2 δ b 1 ) J 0 ( 2 δ b 2 ) 0.5 cos 2 b + 2 J 0 ( δ b 1 ) J 0 ( δ b 2 ) cos a θ cos b ] ,
I ( θ ) = 4 I 0 [ cos 2 a θ + 0.5 + 0.5 J 0 2 ( 2 δ b ) cos 2 b + 2 J 0 2 ( δ b ) cos a θ cos b ] .
d S = 2 ( r 2 x 2 ) 1 2 d x
I ( θ ) = 4 I 0 r r 2 ( r 2 x 2 ) 1 2 [ cos a θ + cos { b + ( 2 π / λ ) 2 γ x } ] 2 d x / 2 r r ( r 2 x 2 ) 1 2 d x ,
I ( θ ) = 4 I 0 { cos a θ + 0.5 + 0.5 [ 2 J 1 ( 2 δ b ) / 2 δ b ] cos 2 b + 2 [ J 1 ( δ b ) / δ b ] cos a θ cos b } ,
{ 4 exp ( 1 / 4 β ) / [ 1.5 0.5 exp ( 1 / β ) ] } Δ b ;
{ 4 ( sin δ b ) / δ b / [ 1.5 0.5 ( sin 2 δ b ) / 2 δ b ] } Δ b ;
{ 4 J 0 2 ( δ b ) / [ 1.5 0.5 J 0 2 ( 2 δ b ) ] } Δ b ;
{ 4 J 1 ( δ b ) / δ b / [ 1.5 J 1 ( 2 δ b ) / 2 δ b ] } Δ b .