Abstract

We analyze the structure and properties of the fringe pattern formed by a high-reflectivity Fizeau interferometer. Our analysis is based on the plane-wave representation of the wave field that is generated when an incident wave interacts with two planar mirrors that form a slight wedge. In particular, we show that within the parabolic approximation the field behind the wedge is self-imaging; it is periodic both in the direction of the incoming beam and in the perpendicular direction. Our approach lends itself to a unified and comprehensive treatment of the fringe transformation properties that occur when the system parameters are varied. The differences between the paraxial-optics and actual fringe patterns are discussed, and three-dimensional intensity plots are used to illustrate the main characteristics of the complex interference structures. Optimal detector locations are evaluated, and some concepts in Fizeau interferometry are clarified.

© 1994 Optical Society of America

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References

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  1. P. Langenbeck, “Fizeau interferometer—fringe sharpening,” Appl. Opt. 9, 2053–2058 (1970).
    [CrossRef] [PubMed]
  2. T. T. Kajava, H. M. Lauranto, R. R. E. Salomaa, “Mode structure fluctuations in a pulsed dye laser,” Appl. Opt. 31, 6987–6992 (1992).
    [CrossRef] [PubMed]
  3. J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1980), Sec. 7.6.
  5. S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), Chap. II.
  6. T. A. Hall, “Fizeau interferometer profiles at finite acceptance angles,” J. Phys. E 2, 837–840 (1969).
    [CrossRef]
  7. J. R. Rogers, “Fringe shifts in multiple-beam Fizeau interferometry,” J. Opt. Soc. Am. 72, 638–643 (1982).
    [CrossRef]
  8. Y. H. Meyer, “Fringe shape with an interferential wedge,” J. Opt. Soc. Am. 71, 1255–1263 (1981).
    [CrossRef]
  9. G. C. Sherman, “Introduction to the angular-spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
    [CrossRef]
  10. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 2.
  11. P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Sec. 14.4.
  12. T. T. Kajava, H. M. Lauranto, R. R. E. Salomaa, “Fizeau interferometer in spectral measurements,” J. Opt. Soc. Am. B 10, 1980–1989 (1993).
    [CrossRef]
  13. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
    [CrossRef]
  14. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  15. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 1–108.
    [CrossRef]
  16. B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
    [CrossRef]
  17. N. Barakat, A. S. Farghaly, A. Abd-El-Azim, “Studies on multiple beam interference fringes formed on high order planes of localization: intensity distribution and the fringe shift between successive planes of localization,” Opt. Acta 12, 205–212 (1965).
    [CrossRef]
  18. R. Roychoudhuri, “Multiple-beam interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 6.

1993

1992

1986

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

1982

1981

1970

1969

1967

1965

N. Barakat, A. S. Farghaly, A. Abd-El-Azim, “Studies on multiple beam interference fringes formed on high order planes of localization: intensity distribution and the fringe shift between successive planes of localization,” Opt. Acta 12, 205–212 (1965).
[CrossRef]

1947

J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

Abd-El-Azim, A.

N. Barakat, A. S. Farghaly, A. Abd-El-Azim, “Studies on multiple beam interference fringes formed on high order planes of localization: intensity distribution and the fringe shift between successive planes of localization,” Opt. Acta 12, 205–212 (1965).
[CrossRef]

Barakat, N.

N. Barakat, A. S. Farghaly, A. Abd-El-Azim, “Studies on multiple beam interference fringes formed on high order planes of localization: intensity distribution and the fringe shift between successive planes of localization,” Opt. Acta 12, 205–212 (1965).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1980), Sec. 7.6.

Brossel, J.

J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

Bryngdahl, O.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Eberly, J. H.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Sec. 14.4.

Eschbach, R.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Farghaly, A. S.

N. Barakat, A. S. Farghaly, A. Abd-El-Azim, “Studies on multiple beam interference fringes formed on high order planes of localization: intensity distribution and the fringe shift between successive planes of localization,” Opt. Acta 12, 205–212 (1965).
[CrossRef]

Hall, T. A.

T. A. Hall, “Fizeau interferometer profiles at finite acceptance angles,” J. Phys. E 2, 837–840 (1969).
[CrossRef]

Kajava, T. T.

Langenbeck, P.

Lauranto, H. M.

Meyer, Y. H.

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Sec. 14.4.

Montgomery, W. D.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 2.

Packross, B.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 1–108.
[CrossRef]

Rogers, J. R.

Roychoudhuri, R.

R. Roychoudhuri, “Multiple-beam interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 6.

Salomaa, R. R. E.

Sherman, G. C.

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
[CrossRef]

G. C. Sherman, “Introduction to the angular-spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
[CrossRef]

Tolansky, S.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), Chap. II.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1980), Sec. 7.6.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Phys. E

T. A. Hall, “Fizeau interferometer profiles at finite acceptance angles,” J. Phys. E 2, 837–840 (1969).
[CrossRef]

Opt. Acta

N. Barakat, A. S. Farghaly, A. Abd-El-Azim, “Studies on multiple beam interference fringes formed on high order planes of localization: intensity distribution and the fringe shift between successive planes of localization,” Opt. Acta 12, 205–212 (1965).
[CrossRef]

Opt. Commun.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Proc. Phys. Soc. London

J. Brossel, “Multiple-beam localized fringes. Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1980), Sec. 7.6.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Dover, New York, 1970), Chap. II.

R. Roychoudhuri, “Multiple-beam interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 6.

G. C. Sherman, “Introduction to the angular-spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 2.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Sec. 14.4.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 1–108.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Illustration of the geometry and notation related to the conventional analysis of fringe formation in Fizeau interferometry. Wedge angle α is exaggerated in the figure. The optical path difference between the planar wave fronts Wn and W0 at a point P behind the wedge is NnPN0P.

Fig. 2
Fig. 2

Interferometer geometry and coordinate system used in the present theoretical analysis. The optical field emerging from the device consists of a coherent superposition of plane waves with wave vectors kn. The z axis coincides with the incident-beam wave vector k = k0.

Fig. 3
Fig. 3

Fizeau intensity distributions (in arbitrary units) in the neighborhood of two lattice points l, 0 for three mirror reflectivities R. The wedge angle is α = 2.5 × 10−5, the wavelength is λ = 500 nm, and the y coordinate is shifted as y′ = ylλ/2α. (a) l = 100, R = 0.8; (b) l = 1.2 × 105, R = 0.8; (c) l = 100, R = 0.9; (d) l = 1.2 × 105, R = 0.9; (e) l = 100, R = 0.95; (f) l = 1.2 × 105, R = 0.95.

Fig. 4
Fig. 4

Magnified portion of Fig. 3(f) near the intensity maximum depicts the modulation that occurs in a high-order fringe ridge when the mirror reflectivity is large. Note that the flat plateaus on both sides of the ridge are artificial, caused by the intensity cutoff in the plot.

Fig. 5
Fig. 5

Detailed structure of the Fizeau intensity pattern (in arbitrary units) within the area 0 ≤ y′ ≤ Δy and 0 ≤ z ≤ Δz, where y′ = ylΔy, l = 2 × 104, Δy = 10 mm, and Δz = 400 m. The mirror reflectivity is R = 0.9. To resolve the main details, we show the pattern along the z direction in four consecutive parts (a)–(d) with slight overlap.

Fig. 6
Fig. 6

Optimum curves, i.e., the positions of the actual Fizeau-fringe intensity maxima, for α = 2.5 × 10−5, λ = 500 nm, m = 0, and mirror reflectivities: curve a, R = 0.8; curve b, R = 0.9; curve c, R = 0.95. The coordinate system corresponds to that in Fig. 2.

Equations (21)

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E ( y , z ) = E 0 n = 0 N R n exp { i k [ sin ( 2 n α ) y + cos ( 2 n α ) z ] } ,
sin ( 2 n α ) = 2 n α ( 2 n α ) 3 6 + O ( 2 n α ) 5 ,
cos ( 2 n α ) = 1 ( 2 n α ) 2 2 + ( 2 n α ) 4 24 + O ( 2 n α ) 6 .
E ( y , z ) = E 0 exp ( ikz ) n = 0 N R n exp { i k [ 2 n α y 2 ( n α ) 2 z ] } .
I ( y , z ) = I 0 | n = 0 N R n exp [ i k δ n ( y , z ) ] | 2 ,
E ( y , 0 ) = E 0 1 R exp [ i k ( 2 α y ) ] .
I ( y , 0 ) = I max 1 + F sin 2 ( k α y ) ,
Δ y = λ 2 α .
Δ z = 2 ( Δ y ) 2 λ = λ 2 α 2 .
E ( y , z + Δ z ) = E ( y , z ) exp ( i π / α 2 )
E ( y + Δ y , z + Δ z ) = E ( y , z ) exp ( i π / α 2 ) .
E ( y ± Δ y 2 , z ± Δ z 2 ) = E ( y , z ) exp ( i π / 2 α 2 ) .
E ( 2 l Δ y y , 2 m Δ z z ) = E * ( y , z ) exp ( i 2 π m / α 2 ) ,
Δ δ n ( y , z ) = ( 4 / 3 ) ( n α ) 3 y + ( 2 / 3 ) ( n α ) 4 z .
y 3 λ 8 ( n α ) 3 ,
z 3 λ 4 ( n α ) 4 ,
y Δ y 3 4 n 3 α 2 ,
z Δ z 3 2 n 4 α 2 .
ϕ = n = 0 ( 2 n α ) R n n = 0 R n = 2 α R 1 R .
S l m ( y ) = y l m l Δ y ,
S l m ( z ) = z l m m Δ z .

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