Abstract

Digital holography (DH) is a promising method for non-contact surface topography because the reconstructed phase image can visualize the nanometer unevenness in a sample. However, the axial range of this method is limited to the range of the optical wavelength due to the phase wrapping ambiguity. Although the use of two different wavelengths of light and the resulting synthetic wavelength, i.e., synthetic wavelength DH, can expand the axial range up to several hundreds of millimeters, its axial precision does not reach sub-micrometer. In this article, we constructed a tunable external cavity laser diode phase-locked to an optical frequency comb, namely, an optical-comb-referenced frequency synthesizer, enabling us to generate multiple synthetic wavelengths within the range of 32 µm to 1.20 m. A multiple cascade link of the phase images among an optical wavelength ( = 1.520 µm) and 5 different synthetic wavelengths ( = 32.39 µm, 99.98 µm, 400.0 µm, 1003 µm, and 4021 µm) enables the shape measurement of a reflective millimeter-sized stepped surface with the axial resolution of 34 nm. The axial dynamic range, defined as the ratio of the axial range ( = 2.0 mm) to the axial resolution ( = 34 nm), achieves 5.9 × 105, which is larger than that of previous synthetic wavelength DH. Such a wide axial dynamic range capability will further expand the application field of DH for large objects with meter dimensions.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (2)

2016 (2)

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
[Crossref] [PubMed]

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

2014 (1)

2011 (1)

2010 (1)

K. M. Molony, B. M. Hennelly, D. P. Kelly, and T. J. Naughton, “Reconstruction algorithms applied to in-line Gabor digital holographic microscopy,” Opt. Commun. 283(6), 903–909 (2010).
[Crossref]

2009 (1)

2008 (3)

2007 (2)

J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15(12), 7231–7242 (2007).
[Crossref] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref] [PubMed]

2006 (2)

M. K. Kim, L. Yu, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A, Pure Appl. Opt. 8(7), S518–S523 (2006).
[Crossref]

D. Parshall and M. K. Kim, “Digital holographic microscopy with dual-wavelength phase unwrapping,” Appl. Opt. 45(3), 451–459 (2006).
[Crossref] [PubMed]

2005 (3)

2003 (2)

2001 (1)

V. Kebbel, H.-J. Hartmann, and W. P. O. Jüptner, “Application of digital holographic microscopy for inspection of micro-optical components,” Proc. SPIE 4398, 189–199 (2001).
[Crossref]

2000 (1)

1999 (1)

1997 (1)

Arai, K.

Araki, T.

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref] [PubMed]

Aspert, N.

Y. Emery, E. Cuche, F. Marquet, N. Aspert, P. Marquet, J. Kuhn, M. Botkine, T. Colomb, F. Montfort, F. Charriere, and C. Depeursinge, “Digital holography microscopy (DHM): fast and robust systems for industrial inspection with interferometer resolution,” Proc. SPIE 5856, 930–938 (2005).
[Crossref]

Asundi, A.

Bevilacqua, F.

Bingham, P. R.

Botkine, M.

Y. Emery, E. Cuche, F. Marquet, N. Aspert, P. Marquet, J. Kuhn, M. Botkine, T. Colomb, F. Montfort, F. Charriere, and C. Depeursinge, “Digital holography microscopy (DHM): fast and robust systems for industrial inspection with interferometer resolution,” Proc. SPIE 5856, 930–938 (2005).
[Crossref]

Charriere, F.

Y. Emery, E. Cuche, F. Marquet, N. Aspert, P. Marquet, J. Kuhn, M. Botkine, T. Colomb, F. Montfort, F. Charriere, and C. Depeursinge, “Digital holography microscopy (DHM): fast and robust systems for industrial inspection with interferometer resolution,” Proc. SPIE 5856, 930–938 (2005).
[Crossref]

Charrière, F.

Chun, B. J.

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
[Crossref] [PubMed]

Colomb, T.

Cuche, E.

Dakoff, A.

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref] [PubMed]

Depeursinge, C.

Emery, Y.

Gass, J.

Hartmann, H.-J.

V. Kebbel, H.-J. Hartmann, and W. P. O. Jüptner, “Application of digital holographic microscopy for inspection of micro-optical components,” Proc. SPIE 4398, 189–199 (2001).
[Crossref]

Hayashi, K.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

Hennelly, B. M.

K. M. Molony, B. M. Hennelly, D. P. Kelly, and T. J. Naughton, “Reconstruction algorithms applied to in-line Gabor digital holographic microscopy,” Opt. Commun. 283(6), 903–909 (2010).
[Crossref]

Hindle, F.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

Holzwarth, R.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref] [PubMed]

Hosako, I.

Hsieh, Y.-D.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

Hyun, S.

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
[Crossref] [PubMed]

Inaba, H.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

T. Yasui, H. Takahashi, K. Kawamoto, Y. Iwamoto, K. Arai, T. Araki, H. Inaba, and K. Minoshima, “Widely and continuously tunable terahertz synthesizer traceable to a microwave frequency standard,” Opt. Express 19(5), 4428–4437 (2011).
[Crossref] [PubMed]

H. Takahashi, Y. Nakajima, H. Inaba, and K. Minoshima, “Ultra-broad absolute-frequency tunable light source locked to a fiber-based frequency comb,” in Conference on Lasers and Electro-Optics (2009), paper CTuK4.
[Crossref]

Iwamoto, Y.

Iwata, T.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

Jang, Y.-S.

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
[Crossref] [PubMed]

Javidi, B.

Jiang, Z.

Jiao, J.

Jüptner, W. P. O.

V. Kebbel, H.-J. Hartmann, and W. P. O. Jüptner, “Application of digital holographic microscopy for inspection of micro-optical components,” Proc. SPIE 4398, 189–199 (2001).
[Crossref]

Kang, H. J.

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
[Crossref] [PubMed]

Kaván, F.

Kawamoto, K.

Kawanishi, T.

Kebbel, V.

V. Kebbel, H.-J. Hartmann, and W. P. O. Jüptner, “Application of digital holographic microscopy for inspection of micro-optical components,” Proc. SPIE 4398, 189–199 (2001).
[Crossref]

Kelly, D. P.

K. M. Molony, B. M. Hennelly, D. P. Kelly, and T. J. Naughton, “Reconstruction algorithms applied to in-line Gabor digital holographic microscopy,” Opt. Commun. 283(6), 903–909 (2010).
[Crossref]

Kemper, B.

Khmaladze, A.

Kim, M.

Kim, M. K.

Kim, S.-W.

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
[Crossref] [PubMed]

Kim, Y.-J.

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
[Crossref] [PubMed]

Kimura, H.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

Kippenberg, T. J.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref] [PubMed]

Kuhn, J.

Y. Emery, E. Cuche, F. Marquet, N. Aspert, P. Marquet, J. Kuhn, M. Botkine, T. Colomb, F. Montfort, F. Charriere, and C. Depeursinge, “Digital holography microscopy (DHM): fast and robust systems for industrial inspection with interferometer resolution,” Proc. SPIE 5856, 930–938 (2005).
[Crossref]

Kühn, J.

Leaird, D. E.

Lédl, V.

Li, H.

Li, Y.

Lo, C. M.

Magistretti, P. J.

Mann, C. J.

C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16(13), 9753–9764 (2008).
[Crossref] [PubMed]

M. K. Kim, L. Yu, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A, Pure Appl. Opt. 8(7), S518–S523 (2006).
[Crossref]

Marquet, F.

Y. Emery, E. Cuche, F. Marquet, N. Aspert, P. Marquet, J. Kuhn, M. Botkine, T. Colomb, F. Montfort, F. Charriere, and C. Depeursinge, “Digital holography microscopy (DHM): fast and robust systems for industrial inspection with interferometer resolution,” Proc. SPIE 5856, 930–938 (2005).
[Crossref]

Marquet, P.

Matoušek, O.

Minamikawa, T.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

Minoshima, K.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

T. Yasui, H. Takahashi, K. Kawamoto, Y. Iwamoto, K. Arai, T. Araki, H. Inaba, and K. Minoshima, “Widely and continuously tunable terahertz synthesizer traceable to a microwave frequency standard,” Opt. Express 19(5), 4428–4437 (2011).
[Crossref] [PubMed]

H. Takahashi, Y. Nakajima, H. Inaba, and K. Minoshima, “Ultra-broad absolute-frequency tunable light source locked to a fiber-based frequency comb,” in Conference on Lasers and Electro-Optics (2009), paper CTuK4.
[Crossref]

Mizutani, Y.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

Mokrý, P.

Molony, K. M.

K. M. Molony, B. M. Hennelly, D. P. Kelly, and T. J. Naughton, “Reconstruction algorithms applied to in-line Gabor digital holographic microscopy,” Opt. Commun. 283(6), 903–909 (2010).
[Crossref]

Montfort, F.

J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15(12), 7231–7242 (2007).
[Crossref] [PubMed]

Y. Emery, E. Cuche, F. Marquet, N. Aspert, P. Marquet, J. Kuhn, M. Botkine, T. Colomb, F. Montfort, F. Charriere, and C. Depeursinge, “Digital holography microscopy (DHM): fast and robust systems for industrial inspection with interferometer resolution,” Proc. SPIE 5856, 930–938 (2005).
[Crossref]

Morohashi, I.

Nakajima, Y.

H. Takahashi, Y. Nakajima, H. Inaba, and K. Minoshima, “Ultra-broad absolute-frequency tunable light source locked to a fiber-based frequency comb,” in Conference on Lasers and Electro-Optics (2009), paper CTuK4.
[Crossref]

Naughton, T. J.

K. M. Molony, B. M. Hennelly, D. P. Kelly, and T. J. Naughton, “Reconstruction algorithms applied to in-line Gabor digital holographic microscopy,” Opt. Commun. 283(6), 903–909 (2010).
[Crossref]

Pan, F.

Paquit, V. C.

Parshall, D.

Psota, P.

Qu, W.

Rappaz, B.

Sakamoto, T.

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref] [PubMed]

Schofield, M. A.

Seo, D. S.

Sotobayashi, H.

Tajahuerce, E.

Takahashi, H.

T. Yasui, H. Takahashi, K. Kawamoto, Y. Iwamoto, K. Arai, T. Araki, H. Inaba, and K. Minoshima, “Widely and continuously tunable terahertz synthesizer traceable to a microwave frequency standard,” Opt. Express 19(5), 4428–4437 (2011).
[Crossref] [PubMed]

H. Takahashi, Y. Nakajima, H. Inaba, and K. Minoshima, “Ultra-broad absolute-frequency tunable light source locked to a fiber-based frequency comb,” in Conference on Lasers and Electro-Optics (2009), paper CTuK4.
[Crossref]

Tian, A.

Tobin, K. W.

von Bally, G.

Wang, G.

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
[Crossref] [PubMed]

Wang, Z.

Wei, H.

Weiner, A. M.

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref] [PubMed]

Wu, X.

Xiao, W.

Yamaguchi, I.

Yamamoto, H.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

Yang, F.

Yang, H.

Yasui, T.

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

T. Yasui, H. Takahashi, K. Kawamoto, Y. Iwamoto, K. Arai, T. Araki, H. Inaba, and K. Minoshima, “Widely and continuously tunable terahertz synthesizer traceable to a microwave frequency standard,” Opt. Express 19(5), 4428–4437 (2011).
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Yu, L.

M. K. Kim, L. Yu, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A, Pure Appl. Opt. 8(7), S518–S523 (2006).
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Appl. Opt. (6)

J. Infrared Millim. Terahertz Waves (1)

Y.-D. Hsieh, H. Kimura, K. Hayashi, T. Minamikawa, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, K. Minoshima, F. Hindle, and T. Yasui, “Terahertz frequency-domain spectroscopy of low-pressure acetonitrile gas by a photomixing terahertz synthesizer referenced to dual optical frequency combs,” J. Infrared Millim. Terahertz Waves 37(9), 903–915 (2016).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

M. K. Kim, L. Yu, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A, Pure Appl. Opt. 8(7), S518–S523 (2006).
[Crossref]

Nature (1)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref] [PubMed]

Opt. Commun. (1)

K. M. Molony, B. M. Hennelly, D. P. Kelly, and T. J. Naughton, “Reconstruction algorithms applied to in-line Gabor digital holographic microscopy,” Opt. Commun. 283(6), 903–909 (2010).
[Crossref]

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B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. 25(9), 610–612 (2000).
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J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003).
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M. A. Schofield and Y. Zhu, “Fast phase unwrapping algorithm for interferometric applications,” Opt. Lett. 28(14), 1194–1196 (2003).
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Proc. SPIE (2)

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Sci. Rep. (1)

Y.-S. Jang, G. Wang, S. Hyun, H. J. Kang, B. J. Chun, Y.-J. Kim, and S.-W. Kim, “Comb-referenced laser distance interferometer for industrial nanotechnology,” Sci. Rep. 6(1), 31770 (2016).
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H. Takahashi, Y. Nakajima, H. Inaba, and K. Minoshima, “Ultra-broad absolute-frequency tunable light source locked to a fiber-based frequency comb,” in Conference on Lasers and Electro-Optics (2009), paper CTuK4.
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Figures (8)

Fig. 1
Fig. 1 Optical frequency comb and optical-comb-referenced frequency synthesizer.
Fig. 2
Fig. 2 Principle of operation in MSW-DH.
Fig. 3
Fig. 3 Experimental setup of the OFS. OFC: optical frequency comb, ECLD: external cavity laser diode, Rb-FS rubidium frequency standard, PI-SC: proportional-integral servo controller, Hs: 1/2 waveplate, Qs:1/4 waveplate, PBS1 and PBS2: polarization beam splitter, PD: photodetector. (b) Experimental setup of the MSW-DH. OFS: optical-comb-referenced frequency synthesizer, OA-PM1 and OA-PM2: off-axis parabolic mirror, BS: beam splitter, and M: mirror. Inset in Fig. 3(b) shows a schematic drawing and a photograph of a sample with a stepped surface.
Fig. 4
Fig. 4 Performance of the OFS. (a) Frequency instability of a rubidium frequency standard. (b) Frequency fluctuation of fceo, frep, fbeat, and mfrep in the OFS. (c) Relation of the wavelength difference between two wavelength lights and the synthetic wavelength in the OFS.
Fig. 5
Fig. 5 Spatial phase noise of (a) λ ( = 1.520302 µm), (b) Λ1 ( = 32.38644 µm), (c) Λ2 ( = 99.97909 µm), (d) Λ3 ( = 400.0234), (e) Λ4 ( = 1,002.524 µm), and (f) Λ5 ( = 4021.204 µm). Dependence of (g) spatial phase noise and (h) unevenness precision on wavelength.
Fig. 6
Fig. 6 Temporal phase noise of (a) λ ( = 1.520302 µm), (b) Λ1 ( = 32.38644 µm), (c) Λ2 ( = 99.97909 µm), (d) Λ3 ( = 400.0234), (e) Λ4 ( = 1,002.524 µm), and (f) Λ5 ( = 4021.204 µm). Dependence of (g) temporal phase noise and (h) height uncertainty on wavelength.
Fig. 7
Fig. 7 Surface topography of a stepped surface sample. Spatial distributions of relative height for (a) an upper surface and (b) a lower surface with respect to the number of cascade links (CLs). The image size is 3 mm by 3 mm. (c) Improvement of the precision in the step height measurement with respect to the number of cascade links. (d) 3D profile of a 1-mm-step sample determined by the full cascaded link of a single optical wavelength and 5 synthetic wavelengths.
Fig. 8
Fig. 8 Comparison of lateral resolution between DH and ranging interferometry. (a) Interference image of a 1-mm-stepped surface at λ ( = 1.520302 µm). (b) Phase image calculated by the ASM-based phase retrieval calculation in DH (z = 134.2 mm). (c) Phase image calculated by the Fourier transform method in the interferometry. Comparison of edge profile between (d) the ASM-based phase retrieval calculation and (e) the Fourier transform method.

Equations (9)

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v m = f ceo +m f rep ,
v ofs = v m + f beat = f ceo +m f rep + f beat ,
Λ= λ 1 λ 2 | λ 2 λ 1 | ,
h(x,y)= ϕ λ 1 (x,y) 4π λ 1 + n λ 1 (x,y) 2 λ 1 = ϕ λ 2 (x,y) 4π λ 2 + n λ 2 (x,y) 2 λ 2 = ϕ Λ (x,y) 4π Λ+ n Λ (x,y) 2 Λ= ϕ Λ (x,y) 4π Λ,
h(x,y)= ϕ λ (x,y) 4π λ+ n λ (x,y) 2 λ= ϕ Λ 1 (x,y) 4π Λ 1 + n Λ 1 (x,y) 2 Λ 1 = ϕ Λ 2 (x,y) 4π Λ 2 + n Λ 2 (x,y) 2 Λ 2 = = ϕ Λ n1 (x,y) 4π Λ n1 + n Λ n1 (x,y) 2 Λ n1 = ϕ Λ n (x,y) 4π Λ n ,
n Λ i (x,y)=INT[ h Λ i+1 (x,y) Λ i /2 ϕ Λ i (x,y) 2π ].
A 0 ( k x , k y )=F[ E 0 ( x 0 , y 0 ) ] = E 0 ( x 0 , y 0 )exp [ i( k x x 0 + k y y 0 ) ]d x 0 d y 0 .
A( k x , k y ;z)= A 0 ( k x , k y )exp[ iz k 2 k x 2 k y 2 ].
E(x,y;z)= F 1 [ A( k x , k y ;z) ] = F 1 [ F[ E 0 ( x 0 , y 0 ) ]exp[ iz k 2 k x 2 k y 2 ] ].

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