Abstract

The spectrum of the intensity profile of multiple-beam Fizeau interferograms is presented. Knowledge of this spectrum provides valuable information about the characteristics of Fizeau interferograms, allowing one to calculate the phase error when the Fizeau profile is evaluated by means of two-beam phase-stepping algorithms, as is usual for low-reflectivity coefficients.

© 1996 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Interscience, New York, 1992), pp. 501–598.
  2. K. Creath, “Temporal phase measurement method,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.
  3. P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
    [CrossRef] [PubMed]
  4. C. Ai, J. C. Wyant, “Effect of retroreflection on a Fizeau phase-shifting interferometer,” Appl. Opt. 32, 3470–3478 (1993).
    [CrossRef] [PubMed]
  5. G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).
  6. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1994), p. 366
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), p. 285
  8. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]

1993 (1)

1992 (1)

1989 (1)

G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).

1987 (1)

Ai, C.

Böhme, H.

G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).

Bönsch, G.

G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), p. 285

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Interscience, New York, 1992), pp. 501–598.

Creath, K.

K. Creath, “Temporal phase measurement method,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1994), p. 366

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Interscience, New York, 1992), pp. 501–598.

Hariharan, P.

Larkin, K. G.

Oreb, B. F.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1994), p. 366

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), p. 285

Wyant, J. C.

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

Optik (1)

G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).

Other (4)

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1994), p. 366

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), p. 285

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Interscience, New York, 1992), pp. 501–598.

K. Creath, “Temporal phase measurement method,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

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Figures (3)

Fig. 1
Fig. 1

Percentage deviation from a sinusoidal profile Δ ¯ i k as a function of phase k ϕ. Reflectivity coefficient r is fixed (r = 0.2), and reflectivity coefficient r′ is considered a parameter (r′ = 0.2, 0.7, 0.8, and 0.9).

Fig. 2
Fig. 2

Percentage deviation from a sinusoidal profile Δ ¯ i k as a function of rr′ and k ϕ.

Fig. 3
Fig. 3

(a) Phase error Δϕ for a three-bucket TPSA as a function of phase ϕ for different reflectivity coefficients. The three-bucket TPSA can be written as ϕ = tan−1[(2i 2i 1i 3)/(i 3i 1)], where the phase steps are δ k = [(k − 1)/2] π and k = 1,2,3. (b) Same as (a) but for four-bucket and five-bucket TPSA's. These algorithms can be written, respectively, as ϕ = tan−1[(i 4i 2)/(i 1i 3)] and ϕ = tan−1[(2i 2 − 2i 4)/(2i 3i 5i 1)], with δ k = [(k − 1)/2]π and k = 1,2,3,4,5.

Equations (16)

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i k = i ( k ϕ ) = i ( ϕ + δ k ) = A [ 1 B 1 C cos ( ϕ + δ k ) ] ,
B = ( 1 r 2 ) ( 1 r 2 ) 1 + r 2 r 2 , C = 2 r r 1 + r 2 r 2 ,
ϕ k = ϕ + δ k .
i k = A [ a 0 2 + n = 1 a n cos n ( k ϕ ) ] ,
a m = 2 π 0 π [ 1 B 1 C cos ( k ϕ ) ] cos m ( k ϕ ) d ( k ϕ )
0 π cos m α 1 + D cos α d α = π 1 D 2 ( 1 D 2 1 D ) m ,
a 0 = 2 ( r 2 + r 2 2 r 2 r 2 ) 1 r 2 r 2 ,
a n = 2 ( 1 r 2 ) ( 1 r 2 ) 1 r 2 r 2 ( r r ) n , n = 1 , 2 , 3 , .
Δ i k = i k i k ,
i k = A [ a 0 2 + a 1 cos ( k ϕ ) ] .
Δ i k = A { 2 r 2 r 2 ( 1 r 2 ) ( 1 r 2 ) ( 1 r 2 r 2 ) × [ cos 2 ( k ϕ ) r r cos ( k ϕ ) 2 r r cos ( k ϕ ) ( 1 + r 2 r 2 ) ] } .
Δ ¯ i k = r r [ r r cos ( k ϕ ) cos 2 ( k ϕ ) 2 r r cos ( k ϕ ) 1 r 2 r 2 ] .
ϕ ( i 1 , , i N ) = tan 1 k = 1 N α k i k k = 1 N β k i k ,
Δ ϕ = ϕ ( i 1 , , i N ) ϕ ( i 1 , , i N ) .
ϕ ( i 1 , , i N ) = ϕ ( i 1 , , i N ) + k = 1 N { α k j = 1 N β j i j β k j = 1 N α j i j } Δ i k ( j = 1 N α j i j ) 2 + ( j = 1 N β j i j ) 2 ,
Δ ϕ = k = 1 N ( α k cos ϕ β k sin ϕ ) Δ ¯ i k j = 1 N β j cos δ j .

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