Abstract

We present the phase-locked loop (PLL)-based metrology concept using lensed fibers for on-machine surface topography measurement. The shape of a single-mode fiber at the endface was designed using an ABCD matrix method, and two designed lensed fibers—the ball type and the tapered type—were fabricated, and the performance was evaluated, respectively. As a result, the interferometric fringe was not found in the case of the ball lensed fiber, but the machined surface could be measured by utilization of autofocusing and intensity methods. On the other hand, a very clear Fizeau interferometric fringe was observed in the case of the tapered lensed fiber. Its performance was compared with the results of the capacitance sensor and a commercially available white-light interferometer. We confirmed that PLL-based surface profile measurement using the tapered and ball lensed fibers can be applied for on-machine surface topography measurement applications.

© 2011 Optical Society of America

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References

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  1. J. P. Choi, B. K. Min, and S. J. Lee, “Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system,” J. Mater. Process. Technol. 155–156, 2056–2064 (2004).
    [CrossRef]
  2. C. Lee, T. Kuriyagawa, D.-K. Woo, and S.-K. Lee, “Optimizing the fabrication process of a high-efficiency blazed grating through diamond scribing and molding,” J. Micromech. Microeng. 20, 055028 (2010).
    [CrossRef]
  3. J.-Y. Joo, D.-K. Woo, S. S. Park, and S.-K. Lee, “Design and fabrication of a fingerprint imager with compact LED illumination compact imaging optics,” Opt. Express 18, 18932–18944 (2010).
    [CrossRef] [PubMed]
  4. D. J. Whitehouse, Handbook of Surface and Nanometrology, 1st ed. (Taylor & Francis, 2002).
    [CrossRef]
  5. J. Jin, Y.-J. Kim, K. Yunseok, and S.-W. Kim, “Absolute distance measurement using the optical comb of a femtosecond pulse laser,” Int. J. Precis. Eng. Manuf. 8, 22–26 (2007).
  6. S. Kumar Debnath, Y. Joonho, and S.-W. Kim, “Determination of film thickness and surface profile using reflectometry and spectrally resolved phase shifting interferometry,” Int. J. Precis. Eng. Manuf. 10, 5–10 (2009).
    [CrossRef]
  7. K. Zhang, C. Butler, Q. Yang, and Y. Lu, “A fiber optic sensor for the measurement of surface roughness and displacement using artificial neural networks,” IEEE Trans. Instrum. Meas. 46, 899–902 (1997).
    [CrossRef]
  8. Y. Yang, K. Yamazaki, H. Aoyama, and S. Matsumiya, “Fiber optic surface topography measurement sensor and its design study,” Precis. Eng. 24, 32–40 (2000).
    [CrossRef]
  9. A. Majumdar and H. Huang, “Compact optical fiber whitelight interferometric distance sensor for arbitrary small distance measurement,” Appl. Opt. 48, 3702–3708 (2009).
    [CrossRef] [PubMed]
  10. W. L. Emkey and C. A. Jack, “Analysis and evaluation of graded-index fiber-lenses,” J. Lightwave Technol. 5, 1156–1164 (1987).
    [CrossRef]
  11. H. Yoda and K. Shiraishi, “A new scheme of a lensed fiber employing a wedge-shaped graded-index fiber tip for the coupling between high-power laser diodes and single-mode fibers,” J. Lightwave Technol. 19, 1910–1917 (2001).
    [CrossRef]
  12. N. Axelrod, A. Lewis, N. B. Yosef, R. Dekhter, G. Fish, and A. Krol, “Small-focus integral fiber lenses: modeling with the segmented beam-propagation method and near-field characterization,” Appl. Opt. 44, 1270–1282 (2005).
    [CrossRef] [PubMed]
  13. G. J. Kong, J. Kim, H. Y. Choi, J. E. Im, B. H. Park, U. C. Paek, and B. H. Lee, “Lensed photonic crystal fiber obtained by use of an arc discharge,” Opt. Lett. 31, 894–896 (2006).
    [CrossRef] [PubMed]
  14. D. Rugar, H. J. Mamin, and P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
    [CrossRef]
  15. D. P. Hand, T. A. Carolan, J. S. Barton, and J. D. C. Jones, “Profile measurement of optically rough surfaces by fiber-optic interferometry,” Opt. Lett. 18, 1361–1363 (1993).
    [CrossRef] [PubMed]
  16. D. Donlagic and E. Cibula, “All-fiber high-sensitivity pressure sensor with SiO2 diaphragm,” Opt. Lett. 30, 2071–2073(2005).
    [CrossRef] [PubMed]
  17. E. Li, G.-D. Peng, and X. Ding, “High spatial resolution fiber-optic Fizeau interferometric strain sensor based on an in-fiber spherical microcavity,” Appl. Phys. Lett. 92, 101117(2008).
    [CrossRef]
  18. O. Wallner, W. R. Leeb, and P. J. Winzer, “Minimum length of a single mode fiber spatial filter,” J. Opt. Soc. Am. 19, 2445–2448 (2002).
    [CrossRef]
  19. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22, 3406–3410 (1983).
    [CrossRef] [PubMed]
  20. E. Li, “Characterization of a fiber lens,” Opt. Lett. 31, 169–171 (2006).
    [CrossRef] [PubMed]
  21. C. Youk and D. Y. Kim, “A simple reflection-type two-dimensional refractive index profile measurement technique for optical waveguides,” Opt. Commun. 262, 206–210 (2006).
    [CrossRef]
  22. F. Barone, R. E. Rosa, L. D. Fiore, F. Fusco, A. Grado, L. Milano, and G. Russo, “Real-time digital control of optical interferometers by the mechanical-modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
    [CrossRef] [PubMed]
  23. “White light interferometer microscope,” www.Zygo.com.
  24. G. Udupa, M. Singaperumal, R. S. Sirohi, and M. P. Kothiya, “Characterization of surface topography by confocal microscopy: 1. principles and the measurement system,” Meas. Sci. Technol. 11, 305–314 (2000).
    [CrossRef]

2010 (2)

C. Lee, T. Kuriyagawa, D.-K. Woo, and S.-K. Lee, “Optimizing the fabrication process of a high-efficiency blazed grating through diamond scribing and molding,” J. Micromech. Microeng. 20, 055028 (2010).
[CrossRef]

J.-Y. Joo, D.-K. Woo, S. S. Park, and S.-K. Lee, “Design and fabrication of a fingerprint imager with compact LED illumination compact imaging optics,” Opt. Express 18, 18932–18944 (2010).
[CrossRef] [PubMed]

2009 (2)

S. Kumar Debnath, Y. Joonho, and S.-W. Kim, “Determination of film thickness and surface profile using reflectometry and spectrally resolved phase shifting interferometry,” Int. J. Precis. Eng. Manuf. 10, 5–10 (2009).
[CrossRef]

A. Majumdar and H. Huang, “Compact optical fiber whitelight interferometric distance sensor for arbitrary small distance measurement,” Appl. Opt. 48, 3702–3708 (2009).
[CrossRef] [PubMed]

2008 (1)

E. Li, G.-D. Peng, and X. Ding, “High spatial resolution fiber-optic Fizeau interferometric strain sensor based on an in-fiber spherical microcavity,” Appl. Phys. Lett. 92, 101117(2008).
[CrossRef]

2007 (1)

J. Jin, Y.-J. Kim, K. Yunseok, and S.-W. Kim, “Absolute distance measurement using the optical comb of a femtosecond pulse laser,” Int. J. Precis. Eng. Manuf. 8, 22–26 (2007).

2006 (3)

2005 (2)

2004 (1)

J. P. Choi, B. K. Min, and S. J. Lee, “Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system,” J. Mater. Process. Technol. 155–156, 2056–2064 (2004).
[CrossRef]

2002 (1)

O. Wallner, W. R. Leeb, and P. J. Winzer, “Minimum length of a single mode fiber spatial filter,” J. Opt. Soc. Am. 19, 2445–2448 (2002).
[CrossRef]

2001 (1)

2000 (2)

Y. Yang, K. Yamazaki, H. Aoyama, and S. Matsumiya, “Fiber optic surface topography measurement sensor and its design study,” Precis. Eng. 24, 32–40 (2000).
[CrossRef]

G. Udupa, M. Singaperumal, R. S. Sirohi, and M. P. Kothiya, “Characterization of surface topography by confocal microscopy: 1. principles and the measurement system,” Meas. Sci. Technol. 11, 305–314 (2000).
[CrossRef]

1997 (1)

K. Zhang, C. Butler, Q. Yang, and Y. Lu, “A fiber optic sensor for the measurement of surface roughness and displacement using artificial neural networks,” IEEE Trans. Instrum. Meas. 46, 899–902 (1997).
[CrossRef]

1994 (1)

1993 (1)

1989 (1)

D. Rugar, H. J. Mamin, and P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

1987 (1)

W. L. Emkey and C. A. Jack, “Analysis and evaluation of graded-index fiber-lenses,” J. Lightwave Technol. 5, 1156–1164 (1987).
[CrossRef]

1983 (1)

Aoyama, H.

Y. Yang, K. Yamazaki, H. Aoyama, and S. Matsumiya, “Fiber optic surface topography measurement sensor and its design study,” Precis. Eng. 24, 32–40 (2000).
[CrossRef]

Axelrod, N.

Barone, F.

Barton, J. S.

Butler, C.

K. Zhang, C. Butler, Q. Yang, and Y. Lu, “A fiber optic sensor for the measurement of surface roughness and displacement using artificial neural networks,” IEEE Trans. Instrum. Meas. 46, 899–902 (1997).
[CrossRef]

Carolan, T. A.

Choi, H. Y.

Choi, J. P.

J. P. Choi, B. K. Min, and S. J. Lee, “Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system,” J. Mater. Process. Technol. 155–156, 2056–2064 (2004).
[CrossRef]

Cibula, E.

Debnath, S. Kumar

S. Kumar Debnath, Y. Joonho, and S.-W. Kim, “Determination of film thickness and surface profile using reflectometry and spectrally resolved phase shifting interferometry,” Int. J. Precis. Eng. Manuf. 10, 5–10 (2009).
[CrossRef]

Dekhter, R.

Ding, X.

E. Li, G.-D. Peng, and X. Ding, “High spatial resolution fiber-optic Fizeau interferometric strain sensor based on an in-fiber spherical microcavity,” Appl. Phys. Lett. 92, 101117(2008).
[CrossRef]

Donlagic, D.

Emkey, W. L.

W. L. Emkey and C. A. Jack, “Analysis and evaluation of graded-index fiber-lenses,” J. Lightwave Technol. 5, 1156–1164 (1987).
[CrossRef]

Fiore, L. D.

Fish, G.

Fusco, F.

Garetz, B. A.

Grado, A.

Guethner, P.

D. Rugar, H. J. Mamin, and P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

Hand, D. P.

Huang, H.

Im, J. E.

Jack, C. A.

W. L. Emkey and C. A. Jack, “Analysis and evaluation of graded-index fiber-lenses,” J. Lightwave Technol. 5, 1156–1164 (1987).
[CrossRef]

Jin, J.

J. Jin, Y.-J. Kim, K. Yunseok, and S.-W. Kim, “Absolute distance measurement using the optical comb of a femtosecond pulse laser,” Int. J. Precis. Eng. Manuf. 8, 22–26 (2007).

Jones, J. D. C.

Joo, J.-Y.

Joonho, Y.

S. Kumar Debnath, Y. Joonho, and S.-W. Kim, “Determination of film thickness and surface profile using reflectometry and spectrally resolved phase shifting interferometry,” Int. J. Precis. Eng. Manuf. 10, 5–10 (2009).
[CrossRef]

Khosrofian, J. M.

Kim, D. Y.

C. Youk and D. Y. Kim, “A simple reflection-type two-dimensional refractive index profile measurement technique for optical waveguides,” Opt. Commun. 262, 206–210 (2006).
[CrossRef]

Kim, J.

Kim, S.-W.

S. Kumar Debnath, Y. Joonho, and S.-W. Kim, “Determination of film thickness and surface profile using reflectometry and spectrally resolved phase shifting interferometry,” Int. J. Precis. Eng. Manuf. 10, 5–10 (2009).
[CrossRef]

J. Jin, Y.-J. Kim, K. Yunseok, and S.-W. Kim, “Absolute distance measurement using the optical comb of a femtosecond pulse laser,” Int. J. Precis. Eng. Manuf. 8, 22–26 (2007).

Kim, Y.-J.

J. Jin, Y.-J. Kim, K. Yunseok, and S.-W. Kim, “Absolute distance measurement using the optical comb of a femtosecond pulse laser,” Int. J. Precis. Eng. Manuf. 8, 22–26 (2007).

Kong, G. J.

Kothiya, M. P.

G. Udupa, M. Singaperumal, R. S. Sirohi, and M. P. Kothiya, “Characterization of surface topography by confocal microscopy: 1. principles and the measurement system,” Meas. Sci. Technol. 11, 305–314 (2000).
[CrossRef]

Krol, A.

Kuriyagawa, T.

C. Lee, T. Kuriyagawa, D.-K. Woo, and S.-K. Lee, “Optimizing the fabrication process of a high-efficiency blazed grating through diamond scribing and molding,” J. Micromech. Microeng. 20, 055028 (2010).
[CrossRef]

Lee, B. H.

Lee, C.

C. Lee, T. Kuriyagawa, D.-K. Woo, and S.-K. Lee, “Optimizing the fabrication process of a high-efficiency blazed grating through diamond scribing and molding,” J. Micromech. Microeng. 20, 055028 (2010).
[CrossRef]

Lee, S. J.

J. P. Choi, B. K. Min, and S. J. Lee, “Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system,” J. Mater. Process. Technol. 155–156, 2056–2064 (2004).
[CrossRef]

Lee, S.-K.

C. Lee, T. Kuriyagawa, D.-K. Woo, and S.-K. Lee, “Optimizing the fabrication process of a high-efficiency blazed grating through diamond scribing and molding,” J. Micromech. Microeng. 20, 055028 (2010).
[CrossRef]

J.-Y. Joo, D.-K. Woo, S. S. Park, and S.-K. Lee, “Design and fabrication of a fingerprint imager with compact LED illumination compact imaging optics,” Opt. Express 18, 18932–18944 (2010).
[CrossRef] [PubMed]

Leeb, W. R.

O. Wallner, W. R. Leeb, and P. J. Winzer, “Minimum length of a single mode fiber spatial filter,” J. Opt. Soc. Am. 19, 2445–2448 (2002).
[CrossRef]

Lewis, A.

Li, E.

E. Li, G.-D. Peng, and X. Ding, “High spatial resolution fiber-optic Fizeau interferometric strain sensor based on an in-fiber spherical microcavity,” Appl. Phys. Lett. 92, 101117(2008).
[CrossRef]

E. Li, “Characterization of a fiber lens,” Opt. Lett. 31, 169–171 (2006).
[CrossRef] [PubMed]

Lu, Y.

K. Zhang, C. Butler, Q. Yang, and Y. Lu, “A fiber optic sensor for the measurement of surface roughness and displacement using artificial neural networks,” IEEE Trans. Instrum. Meas. 46, 899–902 (1997).
[CrossRef]

Majumdar, A.

Mamin, H. J.

D. Rugar, H. J. Mamin, and P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

Matsumiya, S.

Y. Yang, K. Yamazaki, H. Aoyama, and S. Matsumiya, “Fiber optic surface topography measurement sensor and its design study,” Precis. Eng. 24, 32–40 (2000).
[CrossRef]

Milano, L.

Min, B. K.

J. P. Choi, B. K. Min, and S. J. Lee, “Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system,” J. Mater. Process. Technol. 155–156, 2056–2064 (2004).
[CrossRef]

Paek, U. C.

Park, B. H.

Park, S. S.

Peng, G.-D.

E. Li, G.-D. Peng, and X. Ding, “High spatial resolution fiber-optic Fizeau interferometric strain sensor based on an in-fiber spherical microcavity,” Appl. Phys. Lett. 92, 101117(2008).
[CrossRef]

Rosa, R. E.

Rugar, D.

D. Rugar, H. J. Mamin, and P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

Russo, G.

Shiraishi, K.

Singaperumal, M.

G. Udupa, M. Singaperumal, R. S. Sirohi, and M. P. Kothiya, “Characterization of surface topography by confocal microscopy: 1. principles and the measurement system,” Meas. Sci. Technol. 11, 305–314 (2000).
[CrossRef]

Sirohi, R. S.

G. Udupa, M. Singaperumal, R. S. Sirohi, and M. P. Kothiya, “Characterization of surface topography by confocal microscopy: 1. principles and the measurement system,” Meas. Sci. Technol. 11, 305–314 (2000).
[CrossRef]

Udupa, G.

G. Udupa, M. Singaperumal, R. S. Sirohi, and M. P. Kothiya, “Characterization of surface topography by confocal microscopy: 1. principles and the measurement system,” Meas. Sci. Technol. 11, 305–314 (2000).
[CrossRef]

Wallner, O.

O. Wallner, W. R. Leeb, and P. J. Winzer, “Minimum length of a single mode fiber spatial filter,” J. Opt. Soc. Am. 19, 2445–2448 (2002).
[CrossRef]

Whitehouse, D. J.

D. J. Whitehouse, Handbook of Surface and Nanometrology, 1st ed. (Taylor & Francis, 2002).
[CrossRef]

Winzer, P. J.

O. Wallner, W. R. Leeb, and P. J. Winzer, “Minimum length of a single mode fiber spatial filter,” J. Opt. Soc. Am. 19, 2445–2448 (2002).
[CrossRef]

Woo, D.-K.

C. Lee, T. Kuriyagawa, D.-K. Woo, and S.-K. Lee, “Optimizing the fabrication process of a high-efficiency blazed grating through diamond scribing and molding,” J. Micromech. Microeng. 20, 055028 (2010).
[CrossRef]

J.-Y. Joo, D.-K. Woo, S. S. Park, and S.-K. Lee, “Design and fabrication of a fingerprint imager with compact LED illumination compact imaging optics,” Opt. Express 18, 18932–18944 (2010).
[CrossRef] [PubMed]

Yamazaki, K.

Y. Yang, K. Yamazaki, H. Aoyama, and S. Matsumiya, “Fiber optic surface topography measurement sensor and its design study,” Precis. Eng. 24, 32–40 (2000).
[CrossRef]

Yang, Q.

K. Zhang, C. Butler, Q. Yang, and Y. Lu, “A fiber optic sensor for the measurement of surface roughness and displacement using artificial neural networks,” IEEE Trans. Instrum. Meas. 46, 899–902 (1997).
[CrossRef]

Yang, Y.

Y. Yang, K. Yamazaki, H. Aoyama, and S. Matsumiya, “Fiber optic surface topography measurement sensor and its design study,” Precis. Eng. 24, 32–40 (2000).
[CrossRef]

Yoda, H.

Yosef, N. B.

Youk, C.

C. Youk and D. Y. Kim, “A simple reflection-type two-dimensional refractive index profile measurement technique for optical waveguides,” Opt. Commun. 262, 206–210 (2006).
[CrossRef]

Yunseok, K.

J. Jin, Y.-J. Kim, K. Yunseok, and S.-W. Kim, “Absolute distance measurement using the optical comb of a femtosecond pulse laser,” Int. J. Precis. Eng. Manuf. 8, 22–26 (2007).

Zhang, K.

K. Zhang, C. Butler, Q. Yang, and Y. Lu, “A fiber optic sensor for the measurement of surface roughness and displacement using artificial neural networks,” IEEE Trans. Instrum. Meas. 46, 899–902 (1997).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

D. Rugar, H. J. Mamin, and P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

E. Li, G.-D. Peng, and X. Ding, “High spatial resolution fiber-optic Fizeau interferometric strain sensor based on an in-fiber spherical microcavity,” Appl. Phys. Lett. 92, 101117(2008).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

K. Zhang, C. Butler, Q. Yang, and Y. Lu, “A fiber optic sensor for the measurement of surface roughness and displacement using artificial neural networks,” IEEE Trans. Instrum. Meas. 46, 899–902 (1997).
[CrossRef]

Int. J. Precis. Eng. Manuf. (2)

J. Jin, Y.-J. Kim, K. Yunseok, and S.-W. Kim, “Absolute distance measurement using the optical comb of a femtosecond pulse laser,” Int. J. Precis. Eng. Manuf. 8, 22–26 (2007).

S. Kumar Debnath, Y. Joonho, and S.-W. Kim, “Determination of film thickness and surface profile using reflectometry and spectrally resolved phase shifting interferometry,” Int. J. Precis. Eng. Manuf. 10, 5–10 (2009).
[CrossRef]

J. Lightwave Technol. (2)

J. Mater. Process. Technol. (1)

J. P. Choi, B. K. Min, and S. J. Lee, “Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system,” J. Mater. Process. Technol. 155–156, 2056–2064 (2004).
[CrossRef]

J. Micromech. Microeng. (1)

C. Lee, T. Kuriyagawa, D.-K. Woo, and S.-K. Lee, “Optimizing the fabrication process of a high-efficiency blazed grating through diamond scribing and molding,” J. Micromech. Microeng. 20, 055028 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

O. Wallner, W. R. Leeb, and P. J. Winzer, “Minimum length of a single mode fiber spatial filter,” J. Opt. Soc. Am. 19, 2445–2448 (2002).
[CrossRef]

Meas. Sci. Technol. (1)

G. Udupa, M. Singaperumal, R. S. Sirohi, and M. P. Kothiya, “Characterization of surface topography by confocal microscopy: 1. principles and the measurement system,” Meas. Sci. Technol. 11, 305–314 (2000).
[CrossRef]

Opt. Commun. (1)

C. Youk and D. Y. Kim, “A simple reflection-type two-dimensional refractive index profile measurement technique for optical waveguides,” Opt. Commun. 262, 206–210 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Precis. Eng. (1)

Y. Yang, K. Yamazaki, H. Aoyama, and S. Matsumiya, “Fiber optic surface topography measurement sensor and its design study,” Precis. Eng. 24, 32–40 (2000).
[CrossRef]

Other (2)

D. J. Whitehouse, Handbook of Surface and Nanometrology, 1st ed. (Taylor & Francis, 2002).
[CrossRef]

“White light interferometer microscope,” www.Zygo.com.

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Figures (10)

Fig. 1
Fig. 1

Overall schematic of the lensed fiber with a propagation of a Gaussian beam through a lens and Fizeau interference phenomenon: (a) ball lensed fiber and (b) tapered lensed fiber.

Fig. 2
Fig. 2

Beam spot profile measured by the knife-edge method: tapered fiber, 5.63 and ball lensed fiber, 2.7 μm .

Fig. 3
Fig. 3

Beam profile and microscope photograph (inset) of (a) tapered lensed fiber and (b) ball lensed fiber.

Fig. 4
Fig. 4

Axial response curve of the ball lensed fiber.

Fig. 5
Fig. 5

Fizeau interferometry interference fringe for the tapered lensed fiber.

Fig. 6
Fig. 6

Configuration of the experiment: (a) experimental setup for the PLL-based interferometry and (b) block diagram of the digital PLL.

Fig. 7
Fig. 7

Results of the displacement measurement for calibration: (a) 60 and (b)  20 nm .

Fig. 8
Fig. 8

Results of the machined surface measurement: (a) Zygo and (b) proposed PLL-based interferometry.

Fig. 9
Fig. 9

The axial response curve and its derivative signal in the lock-in amplifier.

Fig. 10
Fig. 10

Results of the machined surface measurement: (a) ball lensed fiber and (b) objective lens.

Equations (6)

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

[ A B C D ] = M 34 M 23 M 12 , M 12 = [ 1 L c 0 1 ] , M 23 = [ 1 0 n 2 n 1 n 2 R n 1 n 2 ] , M 34 = [ 1 L f 0 1 ] ,
ω 01 = ω 0 [ ( n 1 n 2 ) A 2 + a 2 B 2 A D B C ] 1 / 2 .
V m ( t ) = a cos ( ω c t + θ ) .
S ( t ) = S 1 + S 0 cos [ z 0 cos ω c t + α ( z ) ] ,
f ( z 0 + A cos ( ω c t ) ) = f ( z 0 ) + A cos ( ω c t + θ 2 ) f ( z 0 ) + A 2 cos 2 ( ω c t + θ 2 ) 2 ! f ( z 0 ) + ,
f ( z = z 0 ) A 2 2 f ( z 0 ) cos ( θ 1 θ 2 ) ,

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