Abstract

We present a method of numerical analysis of interferograms for a diffractive-lens-based common-path interferometer recently introduced by Elfström et al. [Opt. Express 14, 3847 (2006)]. Practical aspects such as the effect of higher diffraction orders upon the interferograms are considered. We show that this method can be used to solve the phase function of the inspected lens. In addition, we show that by using this method it is possible to estimate the focal length and imaging properties of the inspected lens.

© 2008 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992).
  2. J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift, point-diffraction interferometer,” Proc. SPIE 5531, 264-272 (2004).
    [CrossRef]
  3. H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321-5324 (1997).
    [CrossRef] [PubMed]
  4. S. Wolfling, E. Lanzmann, N. Ben-Yosef, and Y. Arieli, “Wavefront reconstruction by spatial-phase-shift imaging interferometry,” Appl. Opt. 45, 2586-2596 (2006).
    [CrossRef] [PubMed]
  5. M. B. North-Morris, J. VanDelden, and J. C. Wyant, “Phase-shifting birefringent scatterplate interferometer,” Appl. Opt. 41, 668-677 (2002).
    [CrossRef] [PubMed]
  6. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723-4730 (1995).
    [CrossRef]
  7. M. Takeda, H. Ina, and S. Kobayashi, “Fourier transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160(1982).
    [CrossRef]
  8. H. Elfström, A. Lehmuskero, T. Saastamoinen, M. Kuittinen, and P. Vahimaa, “Common-path interferometer with diffractive lens,” Opt. Express 14, 3847-3852 (2006).
    [CrossRef] [PubMed]
  9. I. Gurov and M. Volkov, “Fringe evaluation and phase unwrapping of complicated fringe patterns by the data-dependent fringe processing method,” IEEE Trans. Instrum. Meas. 55, 1634-1640 (2006).
    [CrossRef]

2006 (3)

2004 (1)

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift, point-diffraction interferometer,” Proc. SPIE 5531, 264-272 (2004).
[CrossRef]

2002 (1)

1997 (1)

1995 (1)

1982 (1)

Arieli, Y.

Ben-Yosef, N.

Brock, N. J.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift, point-diffraction interferometer,” Proc. SPIE 5531, 264-272 (2004).
[CrossRef]

de Groot, P.

Elfström, H.

Gurov, I.

I. Gurov and M. Volkov, “Fringe evaluation and phase unwrapping of complicated fringe patterns by the data-dependent fringe processing method,” IEEE Trans. Instrum. Meas. 55, 1634-1640 (2006).
[CrossRef]

Hayes, J. B.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift, point-diffraction interferometer,” Proc. SPIE 5531, 264-272 (2004).
[CrossRef]

Ina, H.

Kobayashi, S.

Kuittinen, M.

Lanzmann, E.

Lehmuskero, A.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992).

Millerd, J. E.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift, point-diffraction interferometer,” Proc. SPIE 5531, 264-272 (2004).
[CrossRef]

North-Morris, M. B.

Saastamoinen, T.

Schreiber, H.

Schwider, J.

Takeda, M.

Vahimaa, P.

VanDelden, J.

Volkov, M.

I. Gurov and M. Volkov, “Fringe evaluation and phase unwrapping of complicated fringe patterns by the data-dependent fringe processing method,” IEEE Trans. Instrum. Meas. 55, 1634-1640 (2006).
[CrossRef]

Wolfling, S.

Wyant, J. C.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift, point-diffraction interferometer,” Proc. SPIE 5531, 264-272 (2004).
[CrossRef]

M. B. North-Morris, J. VanDelden, and J. C. Wyant, “Phase-shifting birefringent scatterplate interferometer,” Appl. Opt. 41, 668-677 (2002).
[CrossRef] [PubMed]

Appl. Opt. (4)

IEEE Trans. Instrum. Meas. (1)

I. Gurov and M. Volkov, “Fringe evaluation and phase unwrapping of complicated fringe patterns by the data-dependent fringe processing method,” IEEE Trans. Instrum. Meas. 55, 1634-1640 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Proc. SPIE (1)

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift, point-diffraction interferometer,” Proc. SPIE 5531, 264-272 (2004).
[CrossRef]

Other (1)

D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992).

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Figures (8)

Fig. 1
Fig. 1

Schematic view of the setup.

Fig. 2
Fig. 2

Propagation of diffraction orders after the diffractive lens. The diameter of the diffractive lens is 4 mm , and focal length f 1 = 60 mm .

Fig. 3
Fig. 3

Theoretical effect of an interferogram (a) without noise and (b) with noise created by diffraction order 2 . (c), (d) Their cross sections along the positive x axis. The distance between inspected lens and image plane is z = 100 mm in both cases. The amplitudes of the interfering fields used for images (b) and (d) were U 0 = 1 , U 1 , and U 2 = 0.3 × U 0 .

Fig. 4
Fig. 4

Procedure of fringe counting and wavefront computation. Interferogram in (a) ( x , y ) coordinates, (b) polar coordinates and wavefronts in (c) polar and (d) ( x , y ) coordinates.

Fig. 5
Fig. 5

Interferogram of the diffractive lens (a) and computed wavefront of the diffractive lens (b).

Fig. 6
Fig. 6

(a) Original interferogram of the glass lens ( f = 48 mm ), (b) wavefront of the glass lens ( f = 48 mm ) [and diffractive lens k ( r 1 r 2 ) ], (c) wavefront of the glass lens after the extraction of the wavefront of the diffractive lens, and (d) cosine of the wavefront of the glass lens and diffractive lens.

Fig. 7
Fig. 7

(a) Partial enlargement of the original interferogram in Fig. 6a of the cosine of the wavefront in Fig. 6d.

Fig. 8
Fig. 8

Original interferogram of the injection molded lens (a), wavefront of the injection molded lens (b), and the cosine of the wavefront of the injection molded lens and the diffractive lens (c).

Tables (1)

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Table 1 Theoretical Diffraction Efficiencies

Equations (6)

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U 1 = U 01 exp ( i k Δ z ) ,
U 2 = U 02 exp ( i k r ) r ,
I = U 1 + U 2 2 = U 01 2 + ( U 02 r ) 2 + U 01 U 02 r cos [ k ( r Δ z ) ] ,
I = U 1 + U 2 2 = ( U 01 r ) 2 + ( U 02 r ) 2 + U 01 U 02 r 1 r 2 cos [ k ( r 1 r 2 ) ] .
2 π m = k ( r Δ z ) k x 2 + y 2 2 Δ z ,
2 π m = k ( r 1 r 2 ) k [ x 1 2 + y 1 2 2 Δ z 1 x 2 2 + y 2 2 2 Δ z 2 + Δ z 1 Δ z 2 ] .

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