Abstract

A brief review of lateral- and radial-shearing interferometry is presented together with a thorough discussion of the theory of rotational-shearing interferometry. A rotational-shearing interferometer is described based on the Jamin principle that was built by the authors while at the Institute of Optics at the University of Rochester in Rochester, New York. Graphs and photographs of the interference patterns of the lenses and a lightweight mirror that were tested with this interferometer are presented and discussed. Finally, the effect of the size of the source on the degree of coherence of the interference pattern is considered.

© 1966 Optical Society of America

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References

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  1. H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, Oxford, 1950), p. 11.
  2. A. E. Conrady, Applied Optics and Optical Design (Dover Publications, Inc., New York, 1960), p. 615.
  3. W. J. BatesProc. Phys. Soc. (London) B59, 940 (1947).
    [CrossRef]
  4. R. L. Drew, Proc. Phys. Soc. (London) B64, 1005 (1951).
  5. D. S. Brown, Proc. Phys. Soc. (London) B67, 232 (1954).
  6. J. B. Saunders, J. Res. Natl. Bur. Std. 68C, 155 (1964).
  7. M. V. R. K. Murty, Appl. Opt. 3, 531 (1964).
    [CrossRef]
  8. M. V. R. K. Murty, Appl. Opt. 3, 853 (1964).
    [CrossRef]
  9. D. S. Brown, Interferometry (Her Majesty’s Stationery Office, London, 1960), p. 253.
  10. P. Hariharan, D. Sen, Opt. Acta 9, 159 (1962).
    [CrossRef]
  11. C. Candler, Modern Interferometers (Hilger and Watts, Ltd., London, 1951), p. 18.
  12. Ref. 11, p. 485.
  13. M. Born, E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1964), p. 511.

1964 (3)

1962 (1)

P. Hariharan, D. Sen, Opt. Acta 9, 159 (1962).
[CrossRef]

1954 (1)

D. S. Brown, Proc. Phys. Soc. (London) B67, 232 (1954).

1951 (1)

R. L. Drew, Proc. Phys. Soc. (London) B64, 1005 (1951).

1947 (1)

W. J. BatesProc. Phys. Soc. (London) B59, 940 (1947).
[CrossRef]

Bates, W. J.

W. J. BatesProc. Phys. Soc. (London) B59, 940 (1947).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1964), p. 511.

Brown, D. S.

D. S. Brown, Proc. Phys. Soc. (London) B67, 232 (1954).

D. S. Brown, Interferometry (Her Majesty’s Stationery Office, London, 1960), p. 253.

Candler, C.

C. Candler, Modern Interferometers (Hilger and Watts, Ltd., London, 1951), p. 18.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Dover Publications, Inc., New York, 1960), p. 615.

Drew, R. L.

R. L. Drew, Proc. Phys. Soc. (London) B64, 1005 (1951).

Hariharan, P.

P. Hariharan, D. Sen, Opt. Acta 9, 159 (1962).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, Oxford, 1950), p. 11.

Murty, M. V. R. K.

Saunders, J. B.

J. B. Saunders, J. Res. Natl. Bur. Std. 68C, 155 (1964).

Sen, D.

P. Hariharan, D. Sen, Opt. Acta 9, 159 (1962).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1964), p. 511.

Appl. Opt. (2)

J. Res. Natl. Bur. Std. (1)

J. B. Saunders, J. Res. Natl. Bur. Std. 68C, 155 (1964).

Opt. Acta (1)

P. Hariharan, D. Sen, Opt. Acta 9, 159 (1962).
[CrossRef]

Proc. Phys. Soc. (London) (3)

W. J. BatesProc. Phys. Soc. (London) B59, 940 (1947).
[CrossRef]

R. L. Drew, Proc. Phys. Soc. (London) B64, 1005 (1951).

D. S. Brown, Proc. Phys. Soc. (London) B67, 232 (1954).

Other (6)

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, Oxford, 1950), p. 11.

A. E. Conrady, Applied Optics and Optical Design (Dover Publications, Inc., New York, 1960), p. 615.

C. Candler, Modern Interferometers (Hilger and Watts, Ltd., London, 1951), p. 18.

Ref. 11, p. 485.

M. Born, E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1964), p. 511.

D. S. Brown, Interferometry (Her Majesty’s Stationery Office, London, 1960), p. 253.

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Figures (10)

Fig. 1
Fig. 1

Sheared wavefront.

Fig. 2
Fig. 2

Rotationally sheared wavefront.

Fig. 3
Fig. 3

Variation of sin(ϕ/2) and sinϕ with ϕ.

Fig. 4
Fig. 4

Fringe pattern for two wavelengths of coma for a rotational shear of 180°.

Fig. 5
Fig. 5

Fringe pattern for minus two wavelengths of astigmatism for a rotational shear of 90°.

Fig. 6
Fig. 6

Fringe pattern for minus two wavelengths of astigmatism for a rotational shear of 36° for a lightweight mirror with five ribs.

Fig. 7
Fig. 7

Schematic diagram of rotational-shearing interferometer.

Fig. 8
Fig. 8

Rotational-shearing interferometer.

Fig. 9
Fig. 9

Fringe pattern of cylindrical lens.

Fig. 10
Fig. 10

Interchange of bright and dark fringes.

Equations (21)

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W ( x , y + Δ y ) - W ( x , y ) Δ y ( W / y ) = n λ ,
α = δ OPD / δ y = T A / f ,
W ( R , ϕ ) = A R i cos j ϕ ,
W e = A ( R / S e ) i cos j ϕ .
W c = A ( R / S c ) i cos j ϕ .
W e - W c = A cos j ϕ R i [ ( 1 / S e ) i - ( 1 / S c ) i ] .
W ( x , y ) = B ( x 2 + y 2 ) 2 + F y ( x 2 + y 2 ) + C ( x 2 + 3 y 2 ) + a ( x 2 + y 2 ) + g x + h y .
W ( x , y ) = B ( x 2 + y 2 ) 2 + F y ( x 2 + y 2 ) + C 1 ( x 2 - y 2 ) + C 2 ( x 2 + y 2 ) + D y + a ( x 2 + y 2 ) + g x + h y ,
W ( x , y ) = B ( x 2 + y 2 ) 2 + F y ( x 2 + y 2 ) + C 1 ( x 2 - y 2 ) + C 2 ( x 2 + y 2 ) + a ( x 2 + y 2 ) + g x + E y .
W 1 ( x , y ) = B ( x 2 + y 2 ) 2 + F y ( x 2 + y 2 ) + C 1 ( x 2 - y 2 ) + C 2 ( x 2 + y 2 ) + a ( x 2 + y 2 ) + g x + E y .
W 2 ( x , y ) = B ( x 2 + y 2 ) 2 + F y ( x 2 + y 2 ) + C 1 ( x 2 - y 2 ) + C 2 ( x 2 + y 2 ) + a ( x 2 + y 2 ) + g x + E y .
x = r cos ( θ - ϕ / 2 ) , y = r sin ( θ - ϕ / 2 ) , x = r cos ( θ + ϕ / 2 ) , y = r sin ( θ + ϕ / 2 ) .
W 2 ( r , θ + ϕ / 2 ) - W 1 ( r , θ - ϕ / 2 ) = F r 3 [ sin ( θ + ϕ / 2 ) - sin ( θ - ϕ / 2 ) ] + C 1 r 2 [ cos 2 ( θ + ϕ / 2 ) - cos 2 ( θ - ϕ / 2 ) ] + g r [ cos ( θ + ϕ / 2 ) - cos ( θ - ϕ / 2 ) ] + E r [ sin ( θ + ϕ / 2 ) - sin ( θ - ϕ / 2 ) ] = n λ .
F = F / r max 3 ,             g = g / r max , C 1 = C 1 / r max 2 ,             E = E / r max ,
W 2 ( R , θ + ϕ / 2 ) - W 1 ( R , θ - ϕ / 2 ) = 2 F R 3 cos θ sin ϕ / 2 - 2 C 1 R 2 sin 2 θ sin ϕ - 2 g R sin θ sin ϕ / 2 + 2 E R cos θ sin ϕ / 2 = n λ .
- 2 C 1 R 2 sin 2 θ sin ϕ = n λ .
- 2 C 1 R 2 sin m θ sin m ϕ 2 = n λ ,
P P = 2 r sin ϕ / 2 ,
μ = 2 J 1 ( ν ) / ν ,
ν = 2 π ρ P P / λ f ,
ν = ( 4 π ρ / λ f ) r sin ϕ / 2 ,

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