Abstract

An achromatic phase shifter with a rotating half-wave plate (HWP) used at the input and output of the low coherence interference microscopy is presented. This novel achromatic phase-shifter configuration is first investigated, and then its performances are compared with those of traditional phase shifters theoretically by means of Jones matrices. It is evident that the achromatism and the variation of the amplitude ratio of the proposed achromatic phase-shifter configuration is much better than other traditional phase-shifter configurations, and it can provide a phase shift of eight times the rotation angle of the HWP, which is the largest magnification achieved by far. A low coherence interference microscopy system based on the proposed achromatic phase-shifter configuration is also established to confirm the eight times relation between phase-shift and rotation angle of the HWP experimentally. At last, the three- dimensional profile and the groove depth of a step height calibration standard are obtained by using the traditional four-step algorithm to illustrate the capability and the accuracy of the low coherence interference microscopy system based on the proposed achromatic phase-shifter configuration.

© 2011 Optical Society of America

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2010 (2)

2009 (1)

2008 (1)

2007 (1)

Y. J. Zhu, Z. Luan, Q. G. Yang, D. S. Li, W. Lu, and L. R. Liu, “Improved reliability-guided phase unwrapping algorithm based on the fringe modulation and second-order phase difference,” Optik (Jena) 118, 175–180 (2007).
[CrossRef]

2005 (2)

2004 (2)

2002 (2)

P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” Appl. Opt. 41, 4571–4578 (2002).
[CrossRef] [PubMed]

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37, 631–641 (2002).
[CrossRef]

2001 (2)

L. Xue and X. Y. Su, “Phase-unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring-profilometry method,” Appl. Opt. 40, 1207–1215 (2001).
[CrossRef]

X. Y. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

2000 (2)

A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 39, 2107–2115 (2000).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[CrossRef]

1999 (2)

E. J. Galvez and C. D. Holmes, “Geometric phase of optical rotators,” J. Opt. Soc. Am. A 16, 1981–1985 (1999).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Phase shifting by a rotating polarizer in white-light interferometry for surface profiling,” J. Mod. Opt. 46, 993–1001 (1999).

1998 (1)

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249–254 (1998).
[CrossRef]

1997 (2)

N. Baba and K. Shibayama, “Geometric phase observation with dispersed fringes,” Opt. Rev. 4, 593–595 (1997).
[CrossRef]

P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

1996 (3)

1994 (3)

P. Hariharan and M. Roy, “White-light phase-stepping interferometry for surface profiling,” J. Mod. Opt. 41, 2197–2201 (1994).
[CrossRef]

P. Hariharan, K. G. Larkin, and M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

P. Hariharan and P. E. Ciddor, “An achromatic phase-shifter operating on the geometric phase,” Opt. Commun. 110, 113–117 (1994).
[CrossRef]

1990 (1)

1988 (1)

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

1956 (1)

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).

1954 (1)

Baba, N.

N. Baba and K. Shibayama, “Geometric phase observation with dispersed fringes,” Opt. Rev. 4, 593–595 (1997).
[CrossRef]

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

Bhandari, R.

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 30–32.

Bruning, J. H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara, ed. (Wiley2007), pp. 559–560.

Chen, W. J.

X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261(2004).
[CrossRef]

Cherel, L.

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37, 631–641 (2002).
[CrossRef]

Chim, Stanley S. C.

Ciddor, P. E.

P. Hariharan and P. E. Ciddor, “An achromatic phase-shifter operating on the geometric phase,” Opt. Commun. 110, 113–117 (1994).
[CrossRef]

Cox, G.

de Groot, P.

de Lega, X. C.

Devillers, R.

P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

Galvez, E. J.

Gao, W.

Harasaki, A.

Hariharan, P.

M. Roy, J. Schmit, and P. Hariharan, “White-light interference microscopy: minimization of spurious diffraction effects by geometric phase-shifting,” Opt. Express 17, 4495–4499 (2009).
[CrossRef] [PubMed]

M. Roy, G. Cox, and P. Hariharan, “Low-coherence interference microscopy with an improved switchable achromatic phase-shifter,” Opt. Express 13, 9125–9130 (2005).
[CrossRef] [PubMed]

M. Roy, C. J. R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-modulator,” Opt. Express 12, 2512–2516 (2004).
[CrossRef] [PubMed]

P. Hariharan, “Achromatic phase-shifting for white-light interferometry,” Appl. Opt. 35, 6823–6824 (1996).
[CrossRef] [PubMed]

P. Hariharan, “Achromatic and apochromatic halfwave and quarterwave retarders,” Opt. Eng. 35, 3335–3337 (1996).
[CrossRef]

P. Hariharan and P. E. Ciddor, “An achromatic phase-shifter operating on the geometric phase,” Opt. Commun. 110, 113–117 (1994).
[CrossRef]

P. Hariharan, K. G. Larkin, and M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

P. Hariharan and M. Roy, “White-light phase-stepping interferometry for surface profiling,” J. Mod. Opt. 41, 2197–2201 (1994).
[CrossRef]

Hayasaka, Y.

Helen, S. S.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Phase shifting by a rotating polarizer in white-light interferometry for surface profiling,” J. Mod. Opt. 46, 993–1001 (1999).

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249–254 (1998).
[CrossRef]

Holmes, C. D.

Jerrard, H. G.

Kemao, Q.

Kino, G. S.

Kothiyal, M. P.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Phase shifting by a rotating polarizer in white-light interferometry for surface profiling,” J. Mod. Opt. 46, 993–1001 (1999).

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249–254 (1998).
[CrossRef]

Kramer, J.

Larkin, K. G.

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
[CrossRef]

P. Hariharan, K. G. Larkin, and M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

Li, D. S.

Y. J. Zhu, Z. Luan, Q. G. Yang, D. S. Li, W. Lu, and L. R. Liu, “Improved reliability-guided phase unwrapping algorithm based on the fringe modulation and second-order phase difference,” Optik (Jena) 118, 175–180 (2007).
[CrossRef]

Liu, L. R.

Y. J. Zhu, Z. Luan, Q. G. Yang, D. S. Li, W. Lu, and L. R. Liu, “Improved reliability-guided phase unwrapping algorithm based on the fringe modulation and second-order phase difference,” Optik (Jena) 118, 175–180 (2007).
[CrossRef]

Lu, W.

Y. J. Zhu, Z. Luan, Q. G. Yang, D. S. Li, W. Lu, and L. R. Liu, “Improved reliability-guided phase unwrapping algorithm based on the fringe modulation and second-order phase difference,” Optik (Jena) 118, 175–180 (2007).
[CrossRef]

Luan, Z.

Y. J. Zhu, Z. Luan, Q. G. Yang, D. S. Li, W. Lu, and L. R. Liu, “Improved reliability-guided phase unwrapping algorithm based on the fringe modulation and second-order phase difference,” Optik (Jena) 118, 175–180 (2007).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).

Plata, A.

P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

Roy, M.

M. Roy, J. Schmit, and P. Hariharan, “White-light interference microscopy: minimization of spurious diffraction effects by geometric phase-shifting,” Opt. Express 17, 4495–4499 (2009).
[CrossRef] [PubMed]

M. Roy, G. Cox, and P. Hariharan, “Low-coherence interference microscopy with an improved switchable achromatic phase-shifter,” Opt. Express 13, 9125–9130 (2005).
[CrossRef] [PubMed]

M. Roy, C. J. R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-modulator,” Opt. Express 12, 2512–2516 (2004).
[CrossRef] [PubMed]

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37, 631–641 (2002).
[CrossRef]

P. Hariharan and M. Roy, “White-light phase-stepping interferometry for surface profiling,” J. Mod. Opt. 41, 2197–2201 (1994).
[CrossRef]

P. Hariharan, K. G. Larkin, and M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

C. J. R. Sheppared and M. Roy, “Low-coherence interference microscopy,” in Optical Imaging and Microscopy: Techniques and Advanced Systems, P.Török and F.-J.Kao, eds. (Springer2003), p. 336.

Samuel, J.

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

Sandoz, P.

P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

Sato, M.

Y. Watanabe, Y. Hayasaka, M. Sato, and N. Tanno, “Full-field optical coherence tomography by achromatic phase shifting with a rotating polarizer,” Appl. Opt. 44, 1387–1392 (2005).
[CrossRef] [PubMed]

Y. Watanabe and M. Sato, “High-speed high-resolution full-field optical coherence tomography using achromatic phase shifting by a rotating polarizer,” in Pacific Rim Conference on Lasers and Electro-Optics (2005), Vol.  30, pp. 985–986.
[CrossRef]

Schmit, J.

Schouten, H. F.

Schreiber, H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara, ed. (Wiley2007), pp. 559–560.

Sheppard, C. J. R.

M. Roy, C. J. R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-modulator,” Opt. Express 12, 2512–2516 (2004).
[CrossRef] [PubMed]

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37, 631–641 (2002).
[CrossRef]

Sheppared, C. J. R.

C. J. R. Sheppared and M. Roy, “Low-coherence interference microscopy,” in Optical Imaging and Microscopy: Techniques and Advanced Systems, P.Török and F.-J.Kao, eds. (Springer2003), p. 336.

Shibayama, K.

N. Baba and K. Shibayama, “Geometric phase observation with dispersed fringes,” Opt. Rev. 4, 593–595 (1997).
[CrossRef]

Sirohi, R. S.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Phase shifting by a rotating polarizer in white-light interferometry for surface profiling,” J. Mod. Opt. 46, 993–1001 (1999).

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249–254 (1998).
[CrossRef]

Su, X. Y.

X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261(2004).
[CrossRef]

L. Xue and X. Y. Su, “Phase-unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring-profilometry method,” Appl. Opt. 40, 1207–1215 (2001).
[CrossRef]

X. Y. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

Svahn, P.

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37, 631–641 (2002).
[CrossRef]

Tanno, N.

Turzhitsky, M.

Ubachs, W.

van Dijk, T.

Visser, T. D.

Wang, H.

Watanabe, Y.

Y. Watanabe, Y. Hayasaka, M. Sato, and N. Tanno, “Full-field optical coherence tomography by achromatic phase shifting with a rotating polarizer,” Appl. Opt. 44, 1387–1392 (2005).
[CrossRef] [PubMed]

Y. Watanabe and M. Sato, “High-speed high-resolution full-field optical coherence tomography using achromatic phase shifting by a rotating polarizer,” in Pacific Rim Conference on Lasers and Electro-Optics (2005), Vol.  30, pp. 985–986.
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 30–32.

Wyant, J. C.

Xue, L.

L. Xue and X. Y. Su, “Phase-unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring-profilometry method,” Appl. Opt. 40, 1207–1215 (2001).
[CrossRef]

X. Y. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

Yang, Q. G.

Y. J. Zhu, Z. Luan, Q. G. Yang, D. S. Li, W. Lu, and L. R. Liu, “Improved reliability-guided phase unwrapping algorithm based on the fringe modulation and second-order phase difference,” Optik (Jena) 118, 175–180 (2007).
[CrossRef]

Zhu, Y. J.

Y. J. Zhu, Z. Luan, Q. G. Yang, D. S. Li, W. Lu, and L. R. Liu, “Improved reliability-guided phase unwrapping algorithm based on the fringe modulation and second-order phase difference,” Optik (Jena) 118, 175–180 (2007).
[CrossRef]

Appl. Opt. (7)

J. Mod. Opt. (6)

P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. 44, 519–534 (1997).
[CrossRef]

P. Hariharan, K. G. Larkin, and M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Phase shifting by a rotating polarizer in white-light interferometry for surface profiling,” J. Mod. Opt. 46, 993–1001 (1999).

P. Hariharan and M. Roy, “White-light phase-stepping interferometry for surface profiling,” J. Mod. Opt. 41, 2197–2201 (1994).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “White-light interferometry with polarization phase-shifter at the input of the interferometer,” J. Mod. Opt. 47, 1137–1145 (2000).
[CrossRef]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

P. Hariharan and P. E. Ciddor, “An achromatic phase-shifter operating on the geometric phase,” Opt. Commun. 110, 113–117 (1994).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249–254 (1998).
[CrossRef]

Opt. Eng. (2)

P. Hariharan, “Achromatic and apochromatic halfwave and quarterwave retarders,” Opt. Eng. 35, 3335–3337 (1996).
[CrossRef]

X. Y. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

Opt. Express (4)

Opt. Lasers Eng. (2)

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37, 631–641 (2002).
[CrossRef]

X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261(2004).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

N. Baba and K. Shibayama, “Geometric phase observation with dispersed fringes,” Opt. Rev. 4, 593–595 (1997).
[CrossRef]

Optik (Jena) (1)

Y. J. Zhu, Z. Luan, Q. G. Yang, D. S. Li, W. Lu, and L. R. Liu, “Improved reliability-guided phase unwrapping algorithm based on the fringe modulation and second-order phase difference,” Optik (Jena) 118, 175–180 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

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[CrossRef] [PubMed]

Proc. Ind. Acad. Sci. A (1)

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Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 30–32.

Y. Watanabe and M. Sato, “High-speed high-resolution full-field optical coherence tomography using achromatic phase shifting by a rotating polarizer,” in Pacific Rim Conference on Lasers and Electro-Optics (2005), Vol.  30, pp. 985–986.
[CrossRef]

C. J. R. Sheppared and M. Roy, “Low-coherence interference microscopy,” in Optical Imaging and Microscopy: Techniques and Advanced Systems, P.Török and F.-J.Kao, eds. (Springer2003), p. 336.

“The CVI Melles Griot technical guide,” 2 (CVI Melles Griot Inc., 2009), pp. 58, 104.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara, ed. (Wiley2007), pp. 559–560.

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Figures (8)

Fig. 1
Fig. 1

(a) Traditional rotating HWP phase shifter configuration for the output end of the interference systems. QWP, quarter-wave plate; HWP, half-wave plate; P, polarizer; D, detector; Input, linearly polarized light at 0 ° or 90 ° (s- and p-polarized light reflected by the beam splitter from the reference arm and the test arm). (b) Poincaré sphere representation of the operation of the phase shifter. N, right-circular polarization; S, left-circular polarization; A1 and A2, two linear polarizations.

Fig. 2
Fig. 2

A simple optical system (a) generating both s- and p-polarized beams and (b) combining s- and p-polarized beams to a common optical path. PBS, polarized beam splitter; RAP, right-angle prism.

Fig. 3
Fig. 3

Schematic of the proposed rotating HWP phase shifter configuration both at the input end (black) and output end (blue) of our system. Input, linearly polarized light at 0 ° and 90 ° (s- and p-polarized light); QWP, quarter-wave plate; HWP, half-wave plate; PBS, polarized beam splitter; MO, microscope objective; RM, reference mirror; S, sample; P, polarizer; SS, the simple system shown in Fig. 2.

Fig. 4
Fig. 4

Phase-shifting errors of (a) proposed rotating HWP phase shifter, (b) PZT phase shifter, (c) traditional rotating HWP achromatic phase shifter, and (d) rotating polarizer achromatic phase shifter for phase shifts 180 ° , 90 ° , 0 ° , 90 ° , and 180 ° as a function of wavelength. The designed wavelength is at 550 nm .

Fig. 5
Fig. 5

Amplitude ratios of two orthogonally polarized components of (a) proposed rotating HWP phase shifter, (b) traditional rotating HWP achromatic phase shifter, and (c) rotating polarizer achromatic phase shifter for phase shifts 180 ° , 90 ° , 0 ° , 90 ° , and 180 ° as a function of wavelength. The designed wavelength is at 550 nm .

Fig. 6
Fig. 6

Schematic of the low coherence interference microscopy system based on the proposed rotating HWP phase-shifter configuration at both the input and output ends. HLS, halogen light source; LG, light guide; CL, condenser lens; BS, broadband beam splitter ( 450 650 nm ); PBS, broadband polarized beam splitter ( 450 650 nm ); RAP, right-angle prism; QWP, zero- order quarter-wave plate (made of crystal quartz and designed wavelength of 532 nm ); HWP, zero-order half-wave plate (made of crystal quartz and designed wavelength of 532 nm ); MO, microscope objective; S, sample; RM, reference mirror; P, polarizer; TL, tube lens.

Fig. 7
Fig. 7

(a) One of the phase-shifting interference fringe patterns. (b) Intensity profiles at the 400th column of the five captured phase-shifting interference fringe patterns. (c) First and the fifth intensity profiles. The intensities are normalized to 1.

Fig. 8
Fig. 8

(a) Step height calibration standard with three grooves from Taylor Hobson Ltd. (b) 3D profile of the three grooves of the step height calibration standard obtained by Talysurf CCI 3D Non-Contact Surface profiler from Taylor Hobson Ltd. The dimensions of the test surface are 825.5 μm × 825.5 μm . (c) Height plot of the groove depth. The average groove depth was measured to be 4.837 μm . (d) 3D profile of the middle groove of the step height calibration standard obtained by our system. The dimensions of the test surface are 530 μm × 400 μm . (e) Height plot of the middle groove depth after leveling. M1 (yellow thin solid line) is the overall average value of the surface height for a cross section. Line 1 and line 3 (yellow thick solid lines) indicate the pixels that have surface height values larger than M1. Line 2 (yellow thick solid line) indicates the pixels that have surface height values smaller than M1. The groove depth was measured to be 4.788 μm after averaging 50 groove depths.

Tables (1)

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Table 1 Dispersion Constants of Crystal Quartz for Ordinary and Extraordinary Rays ([24], p. 104)

Equations (20)

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H QWP ( 45 ° ) R HWP ( 45 ° + θ ° ) L HWP ( 45 ° θ ° ) R QWP ( 45 ° ) H ,
V QWP ( 45 ° ) L HWP ( 45 ° + θ ° ) R HWP ( 45 ° θ ° ) L QWP ( 45 ° ) V ,
H in = ( 1 0 ) , V in = ( 0 1 ) .
WP ( ϕ , θ ) = [ cos ( ϕ / 2 ) i · sin ( ϕ / 2 ) cos ( 2 θ ) i · sin ( ϕ / 2 ) sin ( 2 θ ) i · sin ( ϕ / 2 ) sin ( 2 θ ) cos ( ϕ / 2 ) + i · sin ( ϕ / 2 ) cos ( 2 θ ) ] .
H out = QWP ( ϕ Q , 45 ° ) · HWP ( ϕ H , 45 ° θ ° ) · HWP ( ϕ H , 45 ° + θ ° ) · QWP ( ϕ Q , 45 ° ) · H in ,
V out = QWP ( ϕ Q , 45 ° ) · HWP ( ϕ H , 45 ° θ ° ) · HWP ( ϕ H , 45 ° + θ ° ) · QWP ( ϕ Q , 45 ° ) · V in .
H out = ( exp ( 4 θ · i ) 0 ) , V out = ( 0 exp ( 4 θ · i ) ) .
ϕ s = 2 π n s ( λ ) t / λ ,
ϕ f = 2 π n f ( λ ) t / λ ,
ϕ = | ϕ s ϕ f | = 2 π | n s ( λ ) n f ( λ ) | t / λ .
n 2 = A 0 + A 1 λ 2 + A 2 λ 2 + A 3 λ 4 + A 4 λ 6 + A 5 λ 8 .
θ = 4 π · Δ h / λ ¯ ,
H out = QWP ( ϕ Q , 45 ° ) · HWP ( ϕ H , 45 ° + θ ° ) · QWP ( ϕ Q , 45 ° ) · H in ,
V out = QWP ( ϕ Q , 45 ° ) · HWP ( ϕ H , 45 ° + θ ° ) · QWP ( ϕ Q , 45 ° ) · V in .
P ( θ ) = [ cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ ] ,
H out = P ( θ ) · QWP ( ϕ Q , 75 ° ) · HWP ( ϕ H , 15 ° ) · H in ,
V out = P ( θ ) · QWP ( ϕ Q , 75 ° ) · HWP ( ϕ H , 15 ° ) · V in .
γ ( x , y ) = 2 [ ( I 4 I 2 ) 2 + ( I 1 I 3 ) 2 ] 1 / 2 I 1 + I 2 + I 3 + I 4 ,
ϕ ( x , y ) = tan 1 [ I 4 I 2 I 1 I 3 ] .
z ( x , y ) = frame number + ( ϕ + 2 k π ) λ ¯ 4 π .

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