Abstract

Chromatic confocal spectral interferomertry (CCSI) is a novel scheme for topography measurements that combines the techniques of spectral interferometry and chromatic confocal microscopy. This hybrid method allows for white-light interferometric detection with a high NA in a single-shot manner. To the best of our knowledge, CCSI is the first interferometric method that utilizes a confocally filtered and chromatically dispersed focus for detection and simultaneously allows for retrieval of the depth position of reflecting or scattering objects utilizing the phase (modulation frequency) of the interferometric signals acquired. With the chromatically dispersed focus, the depth range of the sensor is decoupled from the NA of the microscope objective.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2006 (2)

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, "One-shot line-profiling white-light interferometer with spatial phase shift for measuring rough surfaces," in Proc. SPIE 6188, 14 (2006).
[CrossRef]

E. Papastathopoulos, K. Körner, and W. Osten, "Chromatically dispersed interferometry with wavelet analysis," Opt. Lett. 31, 589-591 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (2)

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

Y. Yasuno, S. Makita, T. Endo, G. Aoki, H. Sumimura, M. Itoh, and T. Yatagai, "One-shot-phase-shifting Fourier domain optical coherence tomography by reference wavefront tilting," Opt. Express 12, 6184-6191 (2004).
[CrossRef] [PubMed]

2003 (1)

P. Hlubina, "Dispersive white-light spectral two-beam interference under general measurement conditions," in Proc. SPIE 5259, 281-288 (2003).

2002 (2)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

C. Bosbach, F. Depiereux, T. Pfeifer, and B. Michelt, "Fiber-optic interferometer for absolute distance measurements with high measuring frequency," in Proc. SPIE 4900, 408-415 (2002).

2000 (1)

1997 (1)

1996 (3)

M. Bail, G. Häusler, J. M. Herrmann, M. W. Lindner, and R. Ringler, "Optical coherence tomography with the "spectral radar": fast optical analysis in volume scatterers by short coherence interferometry," in Proc. SPIE 2925, 298-303 (1996).
[CrossRef]

J. Calatroni, A. L. Guerrero, C. Sainz, and R. Escalona, "Spectrally-resolved white-light interferometry as a profilometry tool," Opt. Laser Technol. 28, 485-489 (1996).
[CrossRef]

P. Sandoz, G. Tribillon, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701 (1996).
[CrossRef]

1995 (2)

U. Schnell, E. Zimmermenn, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

C. J. R. Sheppard and K. G. Larkin, "Effect of numerical aperture on interference fringe spacing," Appl. Opt. 34, 4731-4733 (1995).
[CrossRef] [PubMed]

1994 (2)

1992 (1)

M. A. Browne, O. Akinyemi, and A. Boyde, "Confocal surface profiling utilizing chromatic aberration," Scanning 14, 145-153 (1992).
[CrossRef]

1990 (1)

1987 (1)

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," in Proc. SPIE 775, 233-247 (1987).

1984 (1)

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, "Focus-wavelength encoded optical profilometer," Opt. Commun. 49, 229-233 (1984).
[CrossRef]

Akinyemi, O.

M. A. Browne, O. Akinyemi, and A. Boyde, "Confocal surface profiling utilizing chromatic aberration," Scanning 14, 145-153 (1992).
[CrossRef]

Aoki, G.

Bail, M.

M. Bail, G. Häusler, J. M. Herrmann, M. W. Lindner, and R. Ringler, "Optical coherence tomography with the "spectral radar": fast optical analysis in volume scatterers by short coherence interferometry," in Proc. SPIE 2925, 298-303 (1996).
[CrossRef]

Bajraszewski, T.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Banyay, M.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, "One-shot line-profiling white-light interferometer with spatial phase shift for measuring rough surfaces," in Proc. SPIE 6188, 14 (2006).
[CrossRef]

Bosbach, C.

C. Bosbach, F. Depiereux, T. Pfeifer, and B. Michelt, "Fiber-optic interferometer for absolute distance measurements with high measuring frequency," in Proc. SPIE 4900, 408-415 (2002).

Boyde, A.

M. A. Browne, O. Akinyemi, and A. Boyde, "Confocal surface profiling utilizing chromatic aberration," Scanning 14, 145-153 (1992).
[CrossRef]

Browne, M. A.

M. A. Browne, O. Akinyemi, and A. Boyde, "Confocal surface profiling utilizing chromatic aberration," Scanning 14, 145-153 (1992).
[CrossRef]

Calatroni, J.

J. Calatroni, A. L. Guerrero, C. Sainz, and R. Escalona, "Spectrally-resolved white-light interferometry as a profilometry tool," Opt. Laser Technol. 28, 485-489 (1996).
[CrossRef]

Cha, S. D.

Chim, S. S. C.

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," in Proc. SPIE 775, 233-247 (1987).

Dändliker, R.

U. Schnell, E. Zimmermenn, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," in Proc. SPIE 775, 233-247 (1987).

Depiereux, F.

C. Bosbach, F. Depiereux, T. Pfeifer, and B. Michelt, "Fiber-optic interferometer for absolute distance measurements with high measuring frequency," in Proc. SPIE 4900, 408-415 (2002).

Dodson, S. L.

Endo, T.

Escalona, R.

J. Calatroni, A. L. Guerrero, C. Sainz, and R. Escalona, "Spectrally-resolved white-light interferometry as a profilometry tool," Opt. Laser Technol. 28, 485-489 (1996).
[CrossRef]

Fainman, Y.

Fercher, A. F.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Guerrero, A. L.

J. Calatroni, A. L. Guerrero, C. Sainz, and R. Escalona, "Spectrally-resolved white-light interferometry as a profilometry tool," Opt. Laser Technol. 28, 485-489 (1996).
[CrossRef]

Häusler, G.

P. Pavlicek and G. Häusler, "White-light interferometer with dispersion: an accurate fiber-optic sensor for the measurement of distance," Appl. Opt. 44, 2978-2983 (2005).
[CrossRef] [PubMed]

M. Bail, G. Häusler, J. M. Herrmann, M. W. Lindner, and R. Ringler, "Optical coherence tomography with the "spectral radar": fast optical analysis in volume scatterers by short coherence interferometry," in Proc. SPIE 2925, 298-303 (1996).
[CrossRef]

Hering, M.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, "One-shot line-profiling white-light interferometer with spatial phase shift for measuring rough surfaces," in Proc. SPIE 6188, 14 (2006).
[CrossRef]

Herrmann, J. M.

M. Bail, G. Häusler, J. M. Herrmann, M. W. Lindner, and R. Ringler, "Optical coherence tomography with the "spectral radar": fast optical analysis in volume scatterers by short coherence interferometry," in Proc. SPIE 2925, 298-303 (1996).
[CrossRef]

Herrmann, S.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, "One-shot line-profiling white-light interferometer with spatial phase shift for measuring rough surfaces," in Proc. SPIE 6188, 14 (2006).
[CrossRef]

Hlubina, P.

P. Hlubina, "Dispersive white-light spectral two-beam interference under general measurement conditions," in Proc. SPIE 5259, 281-288 (2003).

Itoh, M.

Jähne, B.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, "One-shot line-profiling white-light interferometer with spatial phase shift for measuring rough surfaces," in Proc. SPIE 6188, 14 (2006).
[CrossRef]

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," in Proc. SPIE 775, 233-247 (1987).

Kino, G. S.

Körner, K.

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, "One-shot line-profiling white-light interferometer with spatial phase shift for measuring rough surfaces," in Proc. SPIE 6188, 14 (2006).
[CrossRef]

E. Papastathopoulos, K. Körner, and W. Osten, "Chromatically dispersed interferometry with wavelet analysis," Opt. Lett. 31, 589-591 (2006).
[CrossRef] [PubMed]

Kowalczyk, A.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Larkin, K. G.

Last, A.

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

Lehman, P.

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

Lehmann, P.

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

Leitgeb, R.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Lin, P. C.

Lindner, M. W.

M. Bail, G. Häusler, J. M. Herrmann, M. W. Lindner, and R. Ringler, "Optical coherence tomography with the "spectral radar": fast optical analysis in volume scatterers by short coherence interferometry," in Proc. SPIE 2925, 298-303 (1996).
[CrossRef]

Lücke, P.

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

Makita, S.

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," in Proc. SPIE 775, 233-247 (1987).

Michelt, B.

C. Bosbach, F. Depiereux, T. Pfeifer, and B. Michelt, "Fiber-optic interferometer for absolute distance measurements with high measuring frequency," in Proc. SPIE 4900, 408-415 (2002).

Mohr, J.

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

Molesini, G.

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, "Focus-wavelength encoded optical profilometer," Opt. Commun. 49, 229-233 (1984).
[CrossRef]

Osten, W.

E. Papastathopoulos, K. Körner, and W. Osten, "Chromatically dispersed interferometry with wavelet analysis," Opt. Lett. 31, 589-591 (2006).
[CrossRef] [PubMed]

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

Papastathopoulos, E.

Pavlicek, P.

Pedrini, G.

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, "Focus-wavelength encoded optical profilometer," Opt. Commun. 49, 229-233 (1984).
[CrossRef]

Perrin, H.

P. Sandoz, G. Tribillon, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701 (1996).
[CrossRef]

Pfeifer, T.

C. Bosbach, F. Depiereux, T. Pfeifer, and B. Michelt, "Fiber-optic interferometer for absolute distance measurements with high measuring frequency," in Proc. SPIE 4900, 408-415 (2002).

Poggi, P.

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, "Focus-wavelength encoded optical profilometer," Opt. Commun. 49, 229-233 (1984).
[CrossRef]

Pruss, C.

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

Quercioli, F.

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, "Focus-wavelength encoded optical profilometer," Opt. Commun. 49, 229-233 (1984).
[CrossRef]

Ringler, R.

M. Bail, G. Häusler, J. M. Herrmann, M. W. Lindner, and R. Ringler, "Optical coherence tomography with the "spectral radar": fast optical analysis in volume scatterers by short coherence interferometry," in Proc. SPIE 2925, 298-303 (1996).
[CrossRef]

Ruprecht, A.

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

Sainz, C.

J. Calatroni, A. L. Guerrero, C. Sainz, and R. Escalona, "Spectrally-resolved white-light interferometry as a profilometry tool," Opt. Laser Technol. 28, 485-489 (1996).
[CrossRef]

Sandoz, P.

P. Sandoz, G. Tribillon, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701 (1996).
[CrossRef]

Schnell, U.

U. Schnell, E. Zimmermenn, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

Schwider, J.

Sheppard, C. J. R.

Sumimura, H.

Sun, P. C.

Tiziani, H.

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

Tiziani, H. J.

Tribillon, G.

P. Sandoz, G. Tribillon, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701 (1996).
[CrossRef]

Uhde, H. M.

Wojtkowski, M.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Yasuno, Y.

Yatagai, T.

Zhou, L.

Zhu, L. J.

Zimmermenn, E.

U. Schnell, E. Zimmermenn, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

Appl. Opt. (6)

J. Biomed. Opt. (1)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

P. Sandoz, G. Tribillon, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701 (1996).
[CrossRef]

Opt. Commun. (1)

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, "Focus-wavelength encoded optical profilometer," Opt. Commun. 49, 229-233 (1984).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

J. Calatroni, A. L. Guerrero, C. Sainz, and R. Escalona, "Spectrally-resolved white-light interferometry as a profilometry tool," Opt. Laser Technol. 28, 485-489 (1996).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (3)

A. Ruprecht, C. Pruss, H. Tiziani, W. Osten, P. Lücke, A. Last, J. Mohr, and P. Lehman, "Confocal micro-optical distance sensor: principle and design," Proc. SPIE 5856, 128-135 (2005).
[CrossRef]

M. Bail, G. Häusler, J. M. Herrmann, M. W. Lindner, and R. Ringler, "Optical coherence tomography with the "spectral radar": fast optical analysis in volume scatterers by short coherence interferometry," in Proc. SPIE 2925, 298-303 (1996).
[CrossRef]

M. Hering, S. Herrmann, M. Banyay, K. Körner, and B. Jähne, "One-shot line-profiling white-light interferometer with spatial phase shift for measuring rough surfaces," in Proc. SPIE 6188, 14 (2006).
[CrossRef]

Pure Appl. Opt. (1)

U. Schnell, E. Zimmermenn, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

Scanning (1)

M. A. Browne, O. Akinyemi, and A. Boyde, "Confocal surface profiling utilizing chromatic aberration," Scanning 14, 145-153 (1992).
[CrossRef]

Other (4)

P. Hlubina, "Dispersive white-light spectral two-beam interference under general measurement conditions," in Proc. SPIE 5259, 281-288 (2003).

C. Bosbach, F. Depiereux, T. Pfeifer, and B. Michelt, "Fiber-optic interferometer for absolute distance measurements with high measuring frequency," in Proc. SPIE 4900, 408-415 (2002).

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," in Proc. SPIE 775, 233-247 (1987).

P. Lücke, A. Last, J. Mohr, A. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, "Confocal micro-optical distance sensor for precision metrology," in Proc. SPIE 5459, 180-184 (2004).

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic of the white-light Linnik-type interferometer with a chromatically dispersed focus, an achromatic reference, and detection in the optical frequency domain, utilizing a grating spectrometer. The entrance plane of the spectrometer is optically conjugated to the detection focus. The spectral components that are sharply focused upon the sample are most effectively coupled into the spectrometer after confocal filtering and interfere with the reference field. (b) Photograph of the experimental setup.

Fig. 2
Fig. 2

(a) Experimentally measured interference spectrum involving a 0.8 NA microscope objective and a 50 μm confocal pinhole. (b) The individual spectra of the reference and object fields are depicted (upper and middle curves, respectively). The interference wavelet of the lower curve results from high-pass filtering the measured signal in (a). (c) Simulated CCSI signal following the numerical evaluation of the integral in Eq. (2).

Fig. 3
Fig. 3

(a) Experimentally measured and high-pass filtered interference wavelets for various axial positions of the reflecting object. (b) Numerically simulated and high-pass filtered interference wavelets corresponding to the experimentally retrieved axial positions of the reflecting object.

Fig. 4
Fig. 4

Second-order interference wavelets experimentally measured after high-pass filtering for various axial positions of the object. The appearance of this interference allows for an expansion of the axial range of the sensor over 40 μm .

Fig. 5
Fig. 5

(a) Experimental investigation of the effect of NA on the CCSI signals. The interference wavelets above (high-pass filtered) were measured for various diameters of the aperture stop of the illumination field and a constant object z position. (b) Theoretical investigation of the effect of NA on the CCSI signals. The shift of the wavelet's modulation envelope, experimentally observed previously, is reproduced here and can be appointed to the nonzero gradient of the spectral profile of the white-light source employed.

Fig. 6
Fig. 6

(a) Experimentally acquired interference wavelet involving CCSI measurement on a mechanically face-ground machined surface with a referenced roughness R a = 2.5 μm . (b) Schematic representation of the definition of the roughness constant R a . (c) Photograph of the investigated sample.

Fig. 7
Fig. 7

Experimentally acquired interference wavelet involving a laser-processed Wolfram plate. This object is courtesy of TRUMPF. It exhibits a step structure that was measured by monitoring the modulation frequency of the interference signals from the sampled areas A and B.

Equations (9)

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I ( z , k , θ ) = A o 2 + A R 2 + 2 A o A R cos ( 2 k z cos θ + ϕ ) ,
I ( z , k ) = S ( k ) 0 π / 2 I ( z , k , θ ) U ( θ ) cos θ sin θ d θ .
2 k z cos θ = Piston + Defocus ,
Piston = 2 k z ,
Defocus = 2 k z ( 1 cos θ ) .
Δ f ( k ) = A ( k k o ) ,
z = z Δ f ( k ) .
Defocus = 2 k z ( 1 cos θ ) .
A o A o sin ( k NA 2 [ z Δ f ( k ) ] / 2 ) k NA 2 [ z Δ f ( k ) ] / 2 .

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