Abstract

A simple lateral shearing interferometer based on the use of gratings is discussed, which facilitates the display of the spatial complex degree of coherence (CDC). A first version consisting of two identical gratings inclined relative to each other was used to demonstrate the working principle. For this purpose partially coherent light was used which was generated by imaging different object distributions on a rotating scatterer (e.g., double-slit or linear grid patterns). Furthermore, an achromatic lateral shearing interferometer based on the use of three gratings in a series arrangement is discussed and a theoretical proof for the achromatism of this arrangement is given.

© 1984 Optical Society of America

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References

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  1. K. Itoh, Y. Ohtsuka, “Spatial Coherence Measurements through Turbulent Atmosphere Using a Computer Aided Interferometer,” Opt. Commun. 36, 250 (1981).
    [CrossRef]
  2. H. Madjidi-Zolbanine, “Spatial Coherence of a Ruby Laser Beam by Interferometry,” Appl. Opt. 17, 3500 (1978). A. S. Marathay, Elements of Optical Coherence (Wiley, New York, 1982), p. 163.
    [CrossRef] [PubMed]
  3. D. Hariharan, D. Sen, “Radial Shearing Interferometer,” J. Sci. Instrum. 38, 428 (1961); D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
    [CrossRef]
  4. J. C. Fouere, D. Malacara-Hernandez, “Holographic Radial Shear Interferometer,” Appl. Opt. 13, 2035 (1974).
    [CrossRef] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).
  6. W. Martienssen, E. Spiller, “The Coherence of Laser Light,” Am. J. Phys. 32, 919 (1964).
    [CrossRef]
  7. G. Minkwitz, G. Schulz, “Der raeumliche Interferenzstreifenverlauf der Keilinterferenzen,” Opt. Acta 11, 89 (1964).
    [CrossRef]
  8. R. Landwehr, “Lage und Sichtbarkeit von Keilinterferenzen bei instrumenteller Beobachtung,” Opt. Acta 6, 52 (1959).
    [CrossRef]
  9. A. W. Lohmann, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475 (1983); A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial Periodicities in Coherent and Partially Coherent Fields,” Opt. Acta 30, 1259 (1983).
    [CrossRef]
  10. E. N. Leigh, B. J. Chang, “Space-Invariant Holography with Quasi-Coherent Light,” Appl. Opt. 12, 1957 (1973); R. H. Katyl, “Compensating Optical Systems. 1: Broadband Holographic Reconstruction,” Appl. Opt. 11, 1241 (1972); O. Bryngdahl, A. W. Lohmann, “Holography in White Light,” J. Opt. Soc. Am. 60, 281 (1970).
    [CrossRef] [PubMed]
  11. J. Schwider, R. Spolaczyk, K.-E. Elssner, R. Burow, Akt. Z. WPG01J/252 0836 (16.6.83).

1983 (1)

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475 (1983); A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial Periodicities in Coherent and Partially Coherent Fields,” Opt. Acta 30, 1259 (1983).
[CrossRef]

1981 (1)

K. Itoh, Y. Ohtsuka, “Spatial Coherence Measurements through Turbulent Atmosphere Using a Computer Aided Interferometer,” Opt. Commun. 36, 250 (1981).
[CrossRef]

1978 (1)

1974 (1)

1973 (1)

1964 (2)

W. Martienssen, E. Spiller, “The Coherence of Laser Light,” Am. J. Phys. 32, 919 (1964).
[CrossRef]

G. Minkwitz, G. Schulz, “Der raeumliche Interferenzstreifenverlauf der Keilinterferenzen,” Opt. Acta 11, 89 (1964).
[CrossRef]

1961 (1)

D. Hariharan, D. Sen, “Radial Shearing Interferometer,” J. Sci. Instrum. 38, 428 (1961); D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
[CrossRef]

1959 (1)

R. Landwehr, “Lage und Sichtbarkeit von Keilinterferenzen bei instrumenteller Beobachtung,” Opt. Acta 6, 52 (1959).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

Burow, R.

J. Schwider, R. Spolaczyk, K.-E. Elssner, R. Burow, Akt. Z. WPG01J/252 0836 (16.6.83).

Chang, B. J.

Elssner, K.-E.

J. Schwider, R. Spolaczyk, K.-E. Elssner, R. Burow, Akt. Z. WPG01J/252 0836 (16.6.83).

Fouere, J. C.

Hariharan, D.

D. Hariharan, D. Sen, “Radial Shearing Interferometer,” J. Sci. Instrum. 38, 428 (1961); D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
[CrossRef]

Itoh, K.

K. Itoh, Y. Ohtsuka, “Spatial Coherence Measurements through Turbulent Atmosphere Using a Computer Aided Interferometer,” Opt. Commun. 36, 250 (1981).
[CrossRef]

Landwehr, R.

R. Landwehr, “Lage und Sichtbarkeit von Keilinterferenzen bei instrumenteller Beobachtung,” Opt. Acta 6, 52 (1959).
[CrossRef]

Leigh, E. N.

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475 (1983); A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial Periodicities in Coherent and Partially Coherent Fields,” Opt. Acta 30, 1259 (1983).
[CrossRef]

Madjidi-Zolbanine, H.

Malacara-Hernandez, D.

Martienssen, W.

W. Martienssen, E. Spiller, “The Coherence of Laser Light,” Am. J. Phys. 32, 919 (1964).
[CrossRef]

Minkwitz, G.

G. Minkwitz, G. Schulz, “Der raeumliche Interferenzstreifenverlauf der Keilinterferenzen,” Opt. Acta 11, 89 (1964).
[CrossRef]

Ohtsuka, Y.

K. Itoh, Y. Ohtsuka, “Spatial Coherence Measurements through Turbulent Atmosphere Using a Computer Aided Interferometer,” Opt. Commun. 36, 250 (1981).
[CrossRef]

Ojeda-Castaneda, J.

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475 (1983); A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial Periodicities in Coherent and Partially Coherent Fields,” Opt. Acta 30, 1259 (1983).
[CrossRef]

Schulz, G.

G. Minkwitz, G. Schulz, “Der raeumliche Interferenzstreifenverlauf der Keilinterferenzen,” Opt. Acta 11, 89 (1964).
[CrossRef]

Schwider, J.

J. Schwider, R. Spolaczyk, K.-E. Elssner, R. Burow, Akt. Z. WPG01J/252 0836 (16.6.83).

Sen, D.

D. Hariharan, D. Sen, “Radial Shearing Interferometer,” J. Sci. Instrum. 38, 428 (1961); D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
[CrossRef]

Spiller, E.

W. Martienssen, E. Spiller, “The Coherence of Laser Light,” Am. J. Phys. 32, 919 (1964).
[CrossRef]

Spolaczyk, R.

J. Schwider, R. Spolaczyk, K.-E. Elssner, R. Burow, Akt. Z. WPG01J/252 0836 (16.6.83).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

Am. J. Phys. (1)

W. Martienssen, E. Spiller, “The Coherence of Laser Light,” Am. J. Phys. 32, 919 (1964).
[CrossRef]

Appl. Opt. (3)

J. Sci. Instrum. (1)

D. Hariharan, D. Sen, “Radial Shearing Interferometer,” J. Sci. Instrum. 38, 428 (1961); D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
[CrossRef]

Opt. Acta (3)

G. Minkwitz, G. Schulz, “Der raeumliche Interferenzstreifenverlauf der Keilinterferenzen,” Opt. Acta 11, 89 (1964).
[CrossRef]

R. Landwehr, “Lage und Sichtbarkeit von Keilinterferenzen bei instrumenteller Beobachtung,” Opt. Acta 6, 52 (1959).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475 (1983); A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial Periodicities in Coherent and Partially Coherent Fields,” Opt. Acta 30, 1259 (1983).
[CrossRef]

Opt. Commun. (1)

K. Itoh, Y. Ohtsuka, “Spatial Coherence Measurements through Turbulent Atmosphere Using a Computer Aided Interferometer,” Opt. Commun. 36, 250 (1981).
[CrossRef]

Other (2)

J. Schwider, R. Spolaczyk, K.-E. Elssner, R. Burow, Akt. Z. WPG01J/252 0836 (16.6.83).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

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Figures (10)

Fig. 1
Fig. 1

Schematic view of the grating combination forming the continuous lateral shear interferometer: s, shear amount; G1,G2, gratings; α, wedge angle.

Fig. 2
Fig. 2

Scheme of the setup used to demonstrate the working principle of the interferometer.

Fig. 3
Fig. 3

Shear interferogram made with gratings holographically produced on 10E56 plates (note that the fringe distortions are caused by glass carrier imperfections; an ideal fringe pattern would consist of straight and parallel fringes perpendicular to the edge). Above, demonstration of lateral shear by a dentate screen. Below, shear interferogram for coherent illumination (point source).

Fig. 4
Fig. 4

Disk source experiment (diameter increasing from top to bottom). On the left, source disk; middle, shear interferogram (shear zero at the right edge); on the right, speckle image with scatterer at rest. The size of the speckle patch gives a measure for the shear needed to decorrelate the wave fields.

Fig. 5
Fig. 5

Double-slit experiment. [The distance between the two bright slits is varied by axial translation of the imaging microobjective (Fig. 2).] On the left, source distribution; on the right, corresponding shear interference pattern.

Fig. 6
Fig. 6

Spatially periodic light source. [From top to bottom the microobjective has been translated in a direction off the rotating scatterer (Fig. 2). In the first line the diffraction orders of the grating transparency are imaged sharply on the scatterer. The following lines show different stages of defocusing.] On the left, source distribution; on the right, corresponding shear interference pattern.

Fig. 7
Fig. 7

Shear image with fringes parallel to the edge of the grating combination together with an intensity scan by a vidicon: above, grating defocused on the rotating scatterer; below, grating orders sharp on the rotating scatterer.

Fig. 8
Fig. 8

Scheme of an achromatic continuous lateral shear interferometer; α/2, α, diffraction angles of the first order of the gratings.

Fig. 9
Fig. 9

Ray path for the OPD derivation; light rays ➀ and ➁ interfere at point C.

Fig. 10
Fig. 10

Scheme of ray path for two different wavelengths: ray ABC represents the centroid wavelength λg;ray ADE represents a longer wavelength λr.

Equations (24)

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s = x sin 2 α ,
OPD = n A A ¯ - n A A ¯ ,
OPD = n [ x sin α - ( x - s ) tan α ] , = n x sin α ( 1 - cos α ) ,
OPD = n s 1 - cos α sin α .
μ 1 , 2 ( p , q ) = exp ( i ψ ) I ( ξ , η ) exp [ - i k ( ξ p + η q ) ] d ξ d η I ( ξ , η ) d ξ d η ,
μ 1 , 2 = 2 | F 1 ( v ) v | with v = k ρ R ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 ,
μ 1 , 2 = | sinc k q b 2 | | sinc k p a 2 | | cos k p 2 ( a + 2 c ) | ,
μ 1 , 2 = sinc k q b | sinc k p a 2 | | sin N k p d 2 sin k p d 2 | ,
μ 1 , 2 = I 1 + I 2 2 I 1 I 2 V = I 1 + I 2 2 I 1 I 2 I M - I m I M + I m ,
μ 1 , 2 = V ( s ) V ( 0 ) .
g 0 = λ sin α / 2 ,
OPD = B C ¯ - ( A F ¯ + D C ¯ ) ,
B C ¯ = x sin α , D C ¯ = x sin α cos α ,
OPD = x sin α [ 1 - sin α tan α - cos α ] .
OPD r = D E ¯ - ( A F ¯ + D E ¯ ) .
D E ¯ = D E ¯ cos ɛ ,
A F ¯ = D D ¯ tan γ = D E ¯ sin ɛ tan γ .
OPD r = D E ¯ [ 1 - sin ɛ tan γ - cos ɛ ] .
ɛ = 2 γ
ɛ = δ + γ - α 2 .
δ = γ + α 2 .
G 0 : sin γ sin α 2 = λ r λ g ,
G 1 : sin δ = λ r λ g sin α - sin ( γ - α 2 ) .
sin δ = sin γ cos α 2 + cos γ sin α 2 = sin ( γ + α 2 ) ,

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