Abstract

A compact in-line radial shearing interferometer using laser as a light source is presented. The interferometer is made out of a cube-type beam splitter so that the two opposite surfaces are generated with different curvatures while the normal to the entrance and exit surfaces are in the same line. The interferometer is simple to make and easy to align. Aberration analysis of the interferometer is also presented. Some applications of the interferometer for testing lenses and infrared optical systems and for accessing the quality of an emerging wave front from the exit slit of a monochromator are suggested.

© 1992 Optical Society of America

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References

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  1. O. Bryngdhal, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. II.
    [CrossRef]
  2. W. H. Steel, “Two-beam interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. V, Chap. III.
    [CrossRef]
  3. K. M. Baird, G. R. Hanes, “Interferometers,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1967), Vol. 4, p. 336.
  4. D. Malacara, “Radial, rotational, and reversal shear interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), p. 149.
  5. P. Hariharan, “The study of optical wavefronts,” in Optical Interferometry, (Academic, Sydney, 1985), Chap. 8, pp. 137–138, 147.
  6. M. V. R. K. Murty, “Interferometry applied to testing to optics,” Bull. Opt. Soc. India 1, 29–37 (1967).
  7. J. D. Briers, “Interferometric testing of optical systems and components: a review,” Opt. Laser Technol. 4, 28–4 (1972).
    [CrossRef]
  8. J. C. Fouéré, D. Malacara, “Generalized shearing interferometry,” Bol. Inst. Tonantzintla, 1, 227–232 (1975).
  9. J. Dyson, “Common path interferometer for testing purposes,” J. Opt. Soc. Am. 47, 386–390 (1957).
    [CrossRef]
  10. D. S. Brown, “Radial shear interferograms,” in Interferometry, N.P.L. Symposium No. 11, National Physical Laboratory, Teddington, UK (Her Majesty’s Stationary Office, London, 1959), p. 253.
  11. P. Hariharan, D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961b).
    [CrossRef]
  12. P. Hariharan, D. Sen, “Interferometric measurements of the abberrations of microscope objectives,” Opt. Acta 9, 159–175 (1962).
    [CrossRef]
  13. D. S. Brown, “Radial shear interferometry,” J. Sci. Instrum. 39, 71–72 (1962).
    [CrossRef]
  14. M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt. 3, 853–857 (1964).
    [CrossRef]
  15. W. H. Steel, “A radial shear interferometer for testing microscope objectives,” J. Sci. Instrum. 42, 102–104 (1965).
    [CrossRef]
  16. S. C. Som, “Theory of a compact radial shearing laser interferometer,” Opt. Acta 17, 107–113 (1970).
    [CrossRef]
  17. W. H. Steel, “A radial-shear interferometer for use with a laser source,” Opt. Acta 17, 721–724 (1970).
    [CrossRef]
  18. O. Bryngdahl, “Reversed-radial shearing interferometry,” J. Opt. Soc. Am. 60, 915–917 (1970).
    [CrossRef]
  19. O. Bryngdahl, “Shearing interferometry with constant radial displacement,” J. Opt. Soc. Am. 61, 169–172 (1971).
    [CrossRef]
  20. M. V. R. K. Murty, R. P. Shukla, “Radial shearing interferometers using a laser source,” Appl. Opt. 12, 2765–2767 (1973).
    [CrossRef] [PubMed]
  21. J. C. Fouéré, D. Malacara, “Holographic radial shear interferometer,” Appl. Opt. 13, 2035–2039 (1974).
    [CrossRef] [PubMed]
  22. J. C. Fouéré, “Holographic interferometers for optical testing,” Opt. Laser Technol. 6, 181–183 (1974).
    [CrossRef]
  23. D. Malacara, “Mathematical interpretation of radial shearing interferometers,” Appl. Opt. 13, 1781–1784 (1974).
    [CrossRef] [PubMed]
  24. M. V. R. K. Murty, R. P. Shukla, A. Cornejo, “Aberration in a radial shearing interferometer using a laser source,” Indian J. Pure Appl. Phys., 13, 384–387 (1975).
  25. W. H. Steel, “A simple radial shear interferometer,” Opt. Commun. 14, 108–109 (1975).
    [CrossRef]
  26. W. Zhou, “Reflecting radial shear interferometer,” Opt. Commun. 49, 83–85 (1984).
    [CrossRef]
  27. Optics Guide 5 (Melles Griot, Hengelder 23, Posbus 272, 6900 AG Zevenaar, The Netherlands), pp. 19.20–19.24.
  28. W. J. Smith, Modern Optical Engineering (Mc-Graw Hill, New York, 1966), p. 84.
  29. M. V. R. K. Murty, “Theory and principles of monochromators, spectrometers and spectographs,” Opt. Eng. 31, 23–39 (1974).
  30. Ref. 28, p. 392.
  31. Ref. 28, p.295.

1984 (1)

W. Zhou, “Reflecting radial shear interferometer,” Opt. Commun. 49, 83–85 (1984).
[CrossRef]

1975 (3)

J. C. Fouéré, D. Malacara, “Generalized shearing interferometry,” Bol. Inst. Tonantzintla, 1, 227–232 (1975).

M. V. R. K. Murty, R. P. Shukla, A. Cornejo, “Aberration in a radial shearing interferometer using a laser source,” Indian J. Pure Appl. Phys., 13, 384–387 (1975).

W. H. Steel, “A simple radial shear interferometer,” Opt. Commun. 14, 108–109 (1975).
[CrossRef]

1974 (4)

J. C. Fouéré, D. Malacara, “Holographic radial shear interferometer,” Appl. Opt. 13, 2035–2039 (1974).
[CrossRef] [PubMed]

J. C. Fouéré, “Holographic interferometers for optical testing,” Opt. Laser Technol. 6, 181–183 (1974).
[CrossRef]

D. Malacara, “Mathematical interpretation of radial shearing interferometers,” Appl. Opt. 13, 1781–1784 (1974).
[CrossRef] [PubMed]

M. V. R. K. Murty, “Theory and principles of monochromators, spectrometers and spectographs,” Opt. Eng. 31, 23–39 (1974).

1973 (1)

1972 (1)

J. D. Briers, “Interferometric testing of optical systems and components: a review,” Opt. Laser Technol. 4, 28–4 (1972).
[CrossRef]

1971 (1)

1970 (3)

S. C. Som, “Theory of a compact radial shearing laser interferometer,” Opt. Acta 17, 107–113 (1970).
[CrossRef]

W. H. Steel, “A radial-shear interferometer for use with a laser source,” Opt. Acta 17, 721–724 (1970).
[CrossRef]

O. Bryngdahl, “Reversed-radial shearing interferometry,” J. Opt. Soc. Am. 60, 915–917 (1970).
[CrossRef]

1967 (1)

M. V. R. K. Murty, “Interferometry applied to testing to optics,” Bull. Opt. Soc. India 1, 29–37 (1967).

1965 (1)

W. H. Steel, “A radial shear interferometer for testing microscope objectives,” J. Sci. Instrum. 42, 102–104 (1965).
[CrossRef]

1964 (1)

1962 (2)

P. Hariharan, D. Sen, “Interferometric measurements of the abberrations of microscope objectives,” Opt. Acta 9, 159–175 (1962).
[CrossRef]

D. S. Brown, “Radial shear interferometry,” J. Sci. Instrum. 39, 71–72 (1962).
[CrossRef]

1961 (1)

P. Hariharan, D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961b).
[CrossRef]

1957 (1)

Baird, K. M.

K. M. Baird, G. R. Hanes, “Interferometers,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1967), Vol. 4, p. 336.

Briers, J. D.

J. D. Briers, “Interferometric testing of optical systems and components: a review,” Opt. Laser Technol. 4, 28–4 (1972).
[CrossRef]

Brown, D. S.

D. S. Brown, “Radial shear interferometry,” J. Sci. Instrum. 39, 71–72 (1962).
[CrossRef]

D. S. Brown, “Radial shear interferograms,” in Interferometry, N.P.L. Symposium No. 11, National Physical Laboratory, Teddington, UK (Her Majesty’s Stationary Office, London, 1959), p. 253.

Bryngdahl, O.

Bryngdhal, O.

O. Bryngdhal, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. II.
[CrossRef]

Cornejo, A.

M. V. R. K. Murty, R. P. Shukla, A. Cornejo, “Aberration in a radial shearing interferometer using a laser source,” Indian J. Pure Appl. Phys., 13, 384–387 (1975).

Dyson, J.

Fouéré, J. C.

J. C. Fouéré, D. Malacara, “Generalized shearing interferometry,” Bol. Inst. Tonantzintla, 1, 227–232 (1975).

J. C. Fouéré, “Holographic interferometers for optical testing,” Opt. Laser Technol. 6, 181–183 (1974).
[CrossRef]

J. C. Fouéré, D. Malacara, “Holographic radial shear interferometer,” Appl. Opt. 13, 2035–2039 (1974).
[CrossRef] [PubMed]

Hanes, G. R.

K. M. Baird, G. R. Hanes, “Interferometers,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1967), Vol. 4, p. 336.

Hariharan, P.

P. Hariharan, D. Sen, “Interferometric measurements of the abberrations of microscope objectives,” Opt. Acta 9, 159–175 (1962).
[CrossRef]

P. Hariharan, D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961b).
[CrossRef]

P. Hariharan, “The study of optical wavefronts,” in Optical Interferometry, (Academic, Sydney, 1985), Chap. 8, pp. 137–138, 147.

Malacara, D.

J. C. Fouéré, D. Malacara, “Generalized shearing interferometry,” Bol. Inst. Tonantzintla, 1, 227–232 (1975).

J. C. Fouéré, D. Malacara, “Holographic radial shear interferometer,” Appl. Opt. 13, 2035–2039 (1974).
[CrossRef] [PubMed]

D. Malacara, “Mathematical interpretation of radial shearing interferometers,” Appl. Opt. 13, 1781–1784 (1974).
[CrossRef] [PubMed]

D. Malacara, “Radial, rotational, and reversal shear interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), p. 149.

Murty, M. V. R. K.

M. V. R. K. Murty, R. P. Shukla, A. Cornejo, “Aberration in a radial shearing interferometer using a laser source,” Indian J. Pure Appl. Phys., 13, 384–387 (1975).

M. V. R. K. Murty, “Theory and principles of monochromators, spectrometers and spectographs,” Opt. Eng. 31, 23–39 (1974).

M. V. R. K. Murty, R. P. Shukla, “Radial shearing interferometers using a laser source,” Appl. Opt. 12, 2765–2767 (1973).
[CrossRef] [PubMed]

M. V. R. K. Murty, “Interferometry applied to testing to optics,” Bull. Opt. Soc. India 1, 29–37 (1967).

M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt. 3, 853–857 (1964).
[CrossRef]

Sen, D.

P. Hariharan, D. Sen, “Interferometric measurements of the abberrations of microscope objectives,” Opt. Acta 9, 159–175 (1962).
[CrossRef]

P. Hariharan, D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961b).
[CrossRef]

Shukla, R. P.

M. V. R. K. Murty, R. P. Shukla, A. Cornejo, “Aberration in a radial shearing interferometer using a laser source,” Indian J. Pure Appl. Phys., 13, 384–387 (1975).

M. V. R. K. Murty, R. P. Shukla, “Radial shearing interferometers using a laser source,” Appl. Opt. 12, 2765–2767 (1973).
[CrossRef] [PubMed]

Smith, W. J.

W. J. Smith, Modern Optical Engineering (Mc-Graw Hill, New York, 1966), p. 84.

Som, S. C.

S. C. Som, “Theory of a compact radial shearing laser interferometer,” Opt. Acta 17, 107–113 (1970).
[CrossRef]

Steel, W. H.

W. H. Steel, “A simple radial shear interferometer,” Opt. Commun. 14, 108–109 (1975).
[CrossRef]

W. H. Steel, “A radial-shear interferometer for use with a laser source,” Opt. Acta 17, 721–724 (1970).
[CrossRef]

W. H. Steel, “A radial shear interferometer for testing microscope objectives,” J. Sci. Instrum. 42, 102–104 (1965).
[CrossRef]

W. H. Steel, “Two-beam interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. V, Chap. III.
[CrossRef]

Zhou, W.

W. Zhou, “Reflecting radial shear interferometer,” Opt. Commun. 49, 83–85 (1984).
[CrossRef]

Appl. Opt. (4)

Bol. Inst. Tonantzintla (1)

J. C. Fouéré, D. Malacara, “Generalized shearing interferometry,” Bol. Inst. Tonantzintla, 1, 227–232 (1975).

Bull. Opt. Soc. India (1)

M. V. R. K. Murty, “Interferometry applied to testing to optics,” Bull. Opt. Soc. India 1, 29–37 (1967).

Indian J. Pure Appl. Phys. (1)

M. V. R. K. Murty, R. P. Shukla, A. Cornejo, “Aberration in a radial shearing interferometer using a laser source,” Indian J. Pure Appl. Phys., 13, 384–387 (1975).

J. Opt. Soc. Am. (3)

J. Sci. Instrum. (3)

P. Hariharan, D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961b).
[CrossRef]

W. H. Steel, “A radial shear interferometer for testing microscope objectives,” J. Sci. Instrum. 42, 102–104 (1965).
[CrossRef]

D. S. Brown, “Radial shear interferometry,” J. Sci. Instrum. 39, 71–72 (1962).
[CrossRef]

Opt. Acta (3)

S. C. Som, “Theory of a compact radial shearing laser interferometer,” Opt. Acta 17, 107–113 (1970).
[CrossRef]

W. H. Steel, “A radial-shear interferometer for use with a laser source,” Opt. Acta 17, 721–724 (1970).
[CrossRef]

P. Hariharan, D. Sen, “Interferometric measurements of the abberrations of microscope objectives,” Opt. Acta 9, 159–175 (1962).
[CrossRef]

Opt. Commun. (2)

W. H. Steel, “A simple radial shear interferometer,” Opt. Commun. 14, 108–109 (1975).
[CrossRef]

W. Zhou, “Reflecting radial shear interferometer,” Opt. Commun. 49, 83–85 (1984).
[CrossRef]

Opt. Eng. (1)

M. V. R. K. Murty, “Theory and principles of monochromators, spectrometers and spectographs,” Opt. Eng. 31, 23–39 (1974).

Opt. Laser Technol. (2)

J. C. Fouéré, “Holographic interferometers for optical testing,” Opt. Laser Technol. 6, 181–183 (1974).
[CrossRef]

J. D. Briers, “Interferometric testing of optical systems and components: a review,” Opt. Laser Technol. 4, 28–4 (1972).
[CrossRef]

Other (10)

D. S. Brown, “Radial shear interferograms,” in Interferometry, N.P.L. Symposium No. 11, National Physical Laboratory, Teddington, UK (Her Majesty’s Stationary Office, London, 1959), p. 253.

O. Bryngdhal, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. II.
[CrossRef]

W. H. Steel, “Two-beam interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. V, Chap. III.
[CrossRef]

K. M. Baird, G. R. Hanes, “Interferometers,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1967), Vol. 4, p. 336.

D. Malacara, “Radial, rotational, and reversal shear interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), p. 149.

P. Hariharan, “The study of optical wavefronts,” in Optical Interferometry, (Academic, Sydney, 1985), Chap. 8, pp. 137–138, 147.

Ref. 28, p. 392.

Ref. 28, p.295.

Optics Guide 5 (Melles Griot, Hengelder 23, Posbus 272, 6900 AG Zevenaar, The Netherlands), pp. 19.20–19.24.

W. J. Smith, Modern Optical Engineering (Mc-Graw Hill, New York, 1966), p. 84.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of an in-line radial shearing interferometer with a laser source of light; the shear is obtained by means of two concave mirrors of unequal focal lengths: r1, radius curvature of the first mirror; r2, radius curvature of the second mirror.

Fig. 2
Fig. 2

Schematic diagram showing the solid version of Fig. 1.

Fig. 3
Fig. 3

Schematic diagram of a radial shearing interferometer; a concave mirror is used to form the magnified image of the point source of light and the plane mirror is used to fold the path for coinciding the image with the point source.

Fig. 4
Fig. 4

Schematic diagram showing the solid version of Fig. 3.

Fig. 5
Fig. 5

Schematic diagram of a radial shearing interferometer; the concave mirror is used to form the demagnified image of the point source of light and the plane mirror is used to fold the light path of the object beam so that the object and image coincide on the beam-splitting surface.

Fig. 6
Fig. 6

Schematic diagram showing the solid version of Fig. 5.

Fig. 7
Fig. 7

Schematic optical arrangement of an in-line radial shearing interferometer suitable for testing infrared Optics. Beam splitter is made of zinc selenide. CO2 laser is used as a light source, and is focused away from the beam-splitting surface to avoid the damage caused by the high power of the laser.

Fig. 8
Fig. 8

Ellipse showing the position of light source S and its image S′, which are the two focii of the ellipse.

Fig. 9
Fig. 9

Typical radial shearing interferogram of a good lens taken into the radial shearing interferometer.

Fig. 10
Fig. 10

Typical radial shearing interferogram of a lens suffering from spherical aberration taken into the radial shearing interferometer: (a) with defocusing and tilt; (b) without defocusing and with tilt.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

S = f 1 f 2 = r 1 r 2 ,
R = a ( a + 2 b ) ( a + b ) ,
S = a ( a + 2 b ) .
S = b + 2 a b .
R = ( 2 a + b ) b a + b ,
OPD = 2 [ y 4 8 r 1 3 + y 4 8 r 2 3 ] ,
OPD = r 1 1024 f # 4 [ 1 + ( r 1 r 2 ) 3 ] ,
f number = f # = f 1 2 y max ,
OPD = n r 1 1024 f # 4 [ 1 + ( r 1 r 2 ) 3 ] ,
1 4 · n r 1 1024 f # 4 [ 1 + ( r 1 r 2 ) 3 ] λ 20 .
f # 4 = 5 n r 1 1024 λ [ 1 + ( r 1 r 2 ) 3 ] .
Δ x = - e 2 y 4 8 R 3 ,
x 2 a 0 2 + y 2 b 0 2 = 1             ( e < 1 ) ,
wave aberration = - n e 2 y 4 4 R 3 .
1 4 · n e 2 y 4 4 R 3 = λ 20 .
f # 4 = 5 n e 2 a 4 64 λ R 3 ,
f # 4 = 5 n e 2 ( 2 a + b ) 4 64 λ R 3 .

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