Abstract

We used a white-light Lau phase interferometer to evaluate the focal distances of two lenses. We found that the variation in the experimentally measured value is less than ±1% from the given values. Limitations of the method and error analysis are presented.

© 2002 Optical Society of America

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References

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  1. J. Zimmerman, “A method for measuring the distortion of photographic objectives,” Appl. Opt. 2, 759–760 (1963).
    [CrossRef]
  2. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  3. O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett. 5, 555–557 (1980).
    [CrossRef] [PubMed]
  4. Y. Nakano, K. Murata, “Measurement of phase objects using the Talbot effect and moiré techniques,” Appl. Opt. 23, 2296–2299 (1984).
    [CrossRef]
  5. Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24, 3162–3166 (1985).
    [CrossRef] [PubMed]
  6. L. M. Bernardo, O. D. D. Soares, “Evaluation of the focal distance of a lens by Talbot interferometry,” Appl. Opt. 27, 296–301 (1988).
    [CrossRef] [PubMed]
  7. E. Keren, K. M. Kreske, O. Kafri, “Universal method for determining the focal length of optical systems by moiré deflectometry,” Appl. Opt. 27, 1383–1385 (1988).
    [CrossRef] [PubMed]
  8. C.-W. Chang, D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
    [CrossRef]
  9. D.-C. Su, C. Wen, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–122 (1989).
    [CrossRef]
  10. J. C. Bhattacharya, A. K. Aggarwal, “Measurement of the focal length of a collimating lens using the Talbot effect and the moiré technique,” Appl. Opt. 30, 4479–4480 (1991).
    [CrossRef] [PubMed]
  11. K. V. Shiram, M. P. Kothiyal, R. S. Sirohi, “Use of a non-collimated beam for determining the focal length of a lens by Talbot interferometry,” J. Opt. 22, 61–66 (1993).
  12. S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
    [CrossRef]
  13. M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of focal lengths of a lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
    [CrossRef]
  14. S. De Nicola, P. Ferraro, “Interferometric focal length measurement of power distributed lenses,” Opt. Commun. 32, 79–87 (1999).
  15. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  16. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  17. R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
    [CrossRef]
  18. R. J. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef] [PubMed]
  19. G. J. Swanson, E. N. Leith, “Lau effect and grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  20. H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
    [CrossRef]
  21. K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
    [CrossRef]
  22. K. Patorski, “Incoherent superposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2403 (1986).
    [CrossRef]
  23. S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation sensitivity of Lau fringes: an analysis based on coherence theory,” Opt. Laser Technol. 21, 265–268 (1989).
    [CrossRef]
  24. S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Sensitivity of Lau fringes to grating rotation: theoretical analysis,” Appl. Opt. 29, 125–128 (1990).
    [CrossRef] [PubMed]

1999 (1)

S. De Nicola, P. Ferraro, “Interferometric focal length measurement of power distributed lenses,” Opt. Commun. 32, 79–87 (1999).

1997 (1)

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of focal lengths of a lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

1996 (1)

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

1993 (1)

K. V. Shiram, M. P. Kothiyal, R. S. Sirohi, “Use of a non-collimated beam for determining the focal length of a lens by Talbot interferometry,” J. Opt. 22, 61–66 (1993).

1991 (1)

1990 (1)

1989 (3)

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation sensitivity of Lau fringes: an analysis based on coherence theory,” Opt. Laser Technol. 21, 265–268 (1989).
[CrossRef]

C.-W. Chang, D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

D.-C. Su, C. Wen, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–122 (1989).
[CrossRef]

1988 (2)

1986 (1)

1985 (1)

1984 (1)

1983 (1)

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

1979 (4)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

1963 (1)

Aggarwal, A. K.

Avudainayagam, K. V.

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Sensitivity of Lau fringes to grating rotation: theoretical analysis,” Appl. Opt. 29, 125–128 (1990).
[CrossRef] [PubMed]

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation sensitivity of Lau fringes: an analysis based on coherence theory,” Opt. Laser Technol. 21, 265–268 (1989).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

Bernardo, L. M.

Bhattacharya, J. C.

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Chang, C.-W.

C.-W. Chang, D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

Chitralekha, S.

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Sensitivity of Lau fringes to grating rotation: theoretical analysis,” Appl. Opt. 29, 125–128 (1990).
[CrossRef] [PubMed]

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation sensitivity of Lau fringes: an analysis based on coherence theory,” Opt. Laser Technol. 21, 265–268 (1989).
[CrossRef]

de Angelis, M.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of focal lengths of a lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

De Nicola, S.

S. De Nicola, P. Ferraro, “Interferometric focal length measurement of power distributed lenses,” Opt. Commun. 32, 79–87 (1999).

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of focal lengths of a lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

Ferraro, P.

S. De Nicola, P. Ferraro, “Interferometric focal length measurement of power distributed lenses,” Opt. Commun. 32, 79–87 (1999).

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of focal lengths of a lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

Finizio, A.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of focal lengths of a lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Jahns, J.

H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Kafri, O.

Keren, E.

Kothiyal, M. P.

K. V. Shiram, M. P. Kothiyal, R. S. Sirohi, “Use of a non-collimated beam for determining the focal length of a lens by Talbot interferometry,” J. Opt. 22, 61–66 (1993).

Kreske, K. M.

Leith, E. N.

Lohmann, A. W.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Murata, K.

Nakano, Y.

Pappu, S. V.

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Sensitivity of Lau fringes to grating rotation: theoretical analysis,” Appl. Opt. 29, 125–128 (1990).
[CrossRef] [PubMed]

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation sensitivity of Lau fringes: an analysis based on coherence theory,” Opt. Laser Technol. 21, 265–268 (1989).
[CrossRef]

Patorski, K.

K. Patorski, “Incoherent superposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2403 (1986).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Pierattini, G.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of focal lengths of a lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

Shiram, K. V.

K. V. Shiram, M. P. Kothiyal, R. S. Sirohi, “Use of a non-collimated beam for determining the focal length of a lens by Talbot interferometry,” J. Opt. 22, 61–66 (1993).

Sirohi, R. S.

K. V. Shiram, M. P. Kothiyal, R. S. Sirohi, “Use of a non-collimated beam for determining the focal length of a lens by Talbot interferometry,” J. Opt. 22, 61–66 (1993).

Soares, O. D. D.

Su, D.-C.

C.-W. Chang, D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

D.-C. Su, C. Wen, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–122 (1989).
[CrossRef]

Sudol, R.

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Sudol, R. J.

Swanson, G. J.

Thompson, B. J.

R. J. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef] [PubMed]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Wen, C.

D.-C. Su, C. Wen, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–122 (1989).
[CrossRef]

Zimmerman, J.

Appl. Opt. (9)

J. Opt. (1)

K. V. Shiram, M. P. Kothiyal, R. S. Sirohi, “Use of a non-collimated beam for determining the focal length of a lens by Talbot interferometry,” J. Opt. 22, 61–66 (1993).

J. Opt. Soc. Am. (1)

Opt. Acta (1)

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Opt. Commun. (9)

H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

C.-W. Chang, D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

D.-C. Su, C. Wen, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–122 (1989).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of focal lengths of a lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

S. De Nicola, P. Ferraro, “Interferometric focal length measurement of power distributed lenses,” Opt. Commun. 32, 79–87 (1999).

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Opt. Laser Technol. (1)

S. Chitralekha, K. V. Avudainayagam, S. V. Pappu, “Rotation sensitivity of Lau fringes: an analysis based on coherence theory,” Opt. Laser Technol. 21, 265–268 (1989).
[CrossRef]

Opt. Lett. (1)

Other (1)

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

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Figures (3)

Fig. 1
Fig. 1

Schematic of the experimental setup for measurement of focal lengths of lenses by Lau phase interferometry.

Fig. 2
Fig. 2

Geometry representing the transformation of distribution U 1 and U 3 by a lens of focal length f. Illumination is provided by a plane wave from the left.

Fig. 3
Fig. 3

Moiré patterns at the (x 3, y 3) plane behind grating G3 for values of (a) d 1 = 218 mm and d 2 = 300 mm ≅ 2f, with no moiré fringes visible; (b) d 1 = 300 mm and d 2 = 218 mm, with moiré fringes visible; (c) d 1 = 246 mm and d 2 = 420 mm ≅ 2f, with no moiré fringes visible; (d) d 1 = 322 mm and d 2 = 344 mm, with moiré fringes visible.

Tables (2)

Tables Icon

Table 1 Experimental Values of f Calculated from Eq. (19) with a Combination of Measured Values of F 2′, F 2″, and ε*,a

Tables Icon

Table 2 Experimental Values of f Calculated from Eq. (19) with a Combination of Measured Values of F 2′, F 2″, and ε*,a

Equations (23)

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

Z=±md2/λ,
t1x1, y1=1+cos ω1x1,
U1x1, y1=1+cos ω1x1.
U2x2, y2=expikLiλLexpik x22+y222L× U1x1, y1expik x12+y122L×exp-ik x1x2+y1y2Ldx1dx2,
U2x2, y2=expiγ1+exp-iϕcos ω1x2,
ϕ=2mπ m=0, ±1, ±2, ,
Δ=2Λ12λ,
U3x3, y3=iωλd1d2expiδx32+y32  U1x1, y1×expiηx12+y12exp-i 2πωλd1d2×x1x3+y1y3dx1dy1,
δ=k2d2l-ωd2,
η=k2d1l-ωd1,
1ω=1d1+1d2-1f.
U3x3, y3=C expiγωx2+y2×1+exp-iθcos ω1x3,
C=ωω-d1d2d1,
γω=kω-d1d1-ωd1-ω2d2d1-ω,
θ=2πΔd12d1-ω,
ω1=ω1fd2-f.
t3=t=1+cos ω1x3
F=ω1-ω1ω1.
F=F1=d2f-d20<d2<fF2=2f-d2d2-fd2>f>0F3=-d2d2+ff<0, d2>0.
f=d21+F22+F2.
f=1+F21+F2F2-F2 ε,
θ=2πΔd12d1-ω.
Δff=ΔF2F2-F21+F21+F2+1+Δεε.

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