Abstract

In this Letter, the transfer characteristics of rectangular periodic phase objects are studied. It turns out that there are significant differences compared to amplitude objects. The imaging of an amplitude object can be understood as a linear process, whereas phase objects behave nonlinearly. It is shown that under certain conditions the correct shape of a rectangular phase grating can be obtained by an interference microscope as long as the first order diffraction component passes the optical imaging system. This result is in a good agreement with experimental observations and computer simulation results.

© 2012 Optical Society of America

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References

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  1. B. S. Lee and C. T. Strand, Appl. Opt. 29, 3784 (1990).
    [CrossRef]
  2. G. S. Kino and S. S. C. Chim, Appl. Opt. 29, 3775 (1990).
    [CrossRef]
  3. F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, Meas. Sci. Technol. 19, 015303 (2008).
    [CrossRef]
  4. T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic Press, 1996).
  5. C. J. R. Sheppard and K. G. Larkin, Appl. Opt. 34, 4731 (1995).
    [CrossRef]
  6. P. de Groot and X. Colonna de Lega, in Proceedings of Fringe 2005 (Springer, 2005), p. 30.
  7. S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge University, 1995).
  8. J. W. Goodman, Introduction to Fourier Optics (Roberts & Co., 2005).
  9. A. Harasaki and J. C. Wyant, Appl. Opt. 39, 2101 (2000).
    [CrossRef]
  10. W. Hillmann, U. Brand, and M. Krystek, Measurement 19, 95 (1996).
    [CrossRef]
  11. W. Xie, P. Lehmann, and J. Niehues, “Lateral resolution and transfer characteristics of vertical scanning white light interferometers,” Appl. Opt. (to be published).

2008 (1)

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

2000 (1)

1996 (1)

W. Hillmann, U. Brand, and M. Krystek, Measurement 19, 95 (1996).
[CrossRef]

1995 (1)

1990 (2)

Brand, U.

W. Hillmann, U. Brand, and M. Krystek, Measurement 19, 95 (1996).
[CrossRef]

Chim, S. S. C.

Colonna de Lega, X.

P. de Groot and X. Colonna de Lega, in Proceedings of Fringe 2005 (Springer, 2005), p. 30.

Corle, T. R.

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic Press, 1996).

Coupland, J. M.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

de Groot, P.

P. de Groot and X. Colonna de Lega, in Proceedings of Fringe 2005 (Springer, 2005), p. 30.

Gao, F.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Co., 2005).

Harasaki, A.

Hillmann, W.

W. Hillmann, U. Brand, and M. Krystek, Measurement 19, 95 (1996).
[CrossRef]

Kino, G. S.

G. S. Kino and S. S. C. Chim, Appl. Opt. 29, 3775 (1990).
[CrossRef]

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic Press, 1996).

Krystek, M.

W. Hillmann, U. Brand, and M. Krystek, Measurement 19, 95 (1996).
[CrossRef]

Larkin, K. G.

Leach, R. K.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Lee, B. S.

Lehmann, P.

W. Xie, P. Lehmann, and J. Niehues, “Lateral resolution and transfer characteristics of vertical scanning white light interferometers,” Appl. Opt. (to be published).

Lipson, H.

S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge University, 1995).

Lipson, S. G.

S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge University, 1995).

Niehues, J.

W. Xie, P. Lehmann, and J. Niehues, “Lateral resolution and transfer characteristics of vertical scanning white light interferometers,” Appl. Opt. (to be published).

Petzing, J.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Sheppard, C. J. R.

Strand, C. T.

Tannhauser, D. S.

S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge University, 1995).

Wyant, J. C.

Xie, W.

W. Xie, P. Lehmann, and J. Niehues, “Lateral resolution and transfer characteristics of vertical scanning white light interferometers,” Appl. Opt. (to be published).

Appl. Opt. (4)

Meas. Sci. Technol. (1)

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Measurement (1)

W. Hillmann, U. Brand, and M. Krystek, Measurement 19, 95 (1996).
[CrossRef]

Other (5)

W. Xie, P. Lehmann, and J. Niehues, “Lateral resolution and transfer characteristics of vertical scanning white light interferometers,” Appl. Opt. (to be published).

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic Press, 1996).

P. de Groot and X. Colonna de Lega, in Proceedings of Fringe 2005 (Springer, 2005), p. 30.

S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge University, 1995).

J. W. Goodman, Introduction to Fourier Optics (Roberts & Co., 2005).

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Figures (4)

Fig. 1.
Fig. 1.

Rectangular input profiles (dashed lines) and reconstructed profiles resulting from Eq. (10): (a) assuming 2kzh0=2π/3, (b) 2kzh0=π/2, and (c) 2kzh0=π/3.

Fig. 2.
Fig. 2.

Rectangular profile of a 2 µm pitch silicon standard of 200 nm PV amplitude measured by a Mirau interferometer (50×/0.5) using white-light LED illumination: (a) result of phase evaluation at λ=600nm, (b) result of phase evaluation at λ=450nm.

Fig. 3.
Fig. 3.

Simulated 180° phase shifted SWLI signals for two different height levels and the corresponding averaged signal.

Fig. 4.
Fig. 4.

Calculated phase differences of two signals for different nominal phase shifts corresponding to different height levels, and phase difference of the averaged signal compared to signal 1.

Equations (11)

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r(x,y)=exp{iφ(x,y)}=exp{i2kzh(x,y)},
kz=kcosθ2π/λeff.
I(x,y,z)=I0+ΔI|γ(zh(x,y))|cos{2kzzφ(x,y)φ0},
|γ(zh(x,y))|=exp{4(zh(x,y))2/lc2}
I(x,y)=I0+ΔIsin{2kzh(x,y)}I0+ΔI2kzh(x,y).
h(x,y)=h0rect(ωxx)=h0cos(ωxx)/|cos(ωxx)|.
I(x,y)I0ΔI2kzh(x,y)4π2kzh0cos(ωxx).
r(x,y)=exp{i2kzh0rect(ωxx)}=cos{2kzh0}+isin{2kzh0}rect(ωxx).
r(x,y)cos{2kzh0}+i4πsin{2kzh0}cos(ωxx).
2kzh(x,y)arctan{4tan(2kzh0)cos(ωxx)/π}=rect(ωxx)arctan{4tan(2kzh0)|cos(ωxx)|/π}.
2kzh(x,y)arctan{4πtan(π2)cos(ωxx)}π2rect(ωxx).

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