Abstract

Swept-wavelength reflectometry is an absolute distance measurement technique with significant sensitivity and detector bandwidth advantages over normal pulsed, time-of-flight methods. Although several tunable laser sources exist, many exhibit short coherence lengths or require mechanical tuning components. Semiconductor distributed feedback laser diodes (DFBs) are advantageous as a swept source because they exhibit a narrow instantaneous linewidth and can be frequency-swept simply via a single injection current. Here, we present a novel bandwidth generation technique that uses a compact, monolithic, 12-element DFB array to create an effectively continuous, gap-free sweep. Each DFB is sequentially swept over 3.5 nm at 1,600 THz/s using a shaped current pulse, ensuring spectral overlap between each element. After combining the self-heterodyned return signatures, the transform-limited resolution of the 43.6 nm sweep is demonstrated to be ~27.4 μm in air with a precision of 0.18 μm at a distance of 1.4 m.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (4)

2014 (2)

2013 (4)

2012 (1)

2011 (3)

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, and K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[Crossref]

K. Iiyama, S. I. Matsui, T. Kobayashi, and T. Maruyama, “High-resolution FMCW reflectometry using a single-mode vertical-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 23(11), 703–705 (2011).
[Crossref]

Z. W. Barber, F. R. Giorgetta, P. A. Roos, I. Coddington, J. R. Dahl, R. R. Reibel, N. Greenfield, and N. R. Newbury, “Characterization of an actively linearized ultrabroadband chirped laser with a fiber-laser optical frequency comb,” Opt. Lett. 36(7), 1152–1154 (2011).
[Crossref] [PubMed]

2010 (4)

2009 (3)

2005 (5)

2004 (1)

2003 (1)

2002 (2)

A. V. Tolmachev, C. D. Masselon, G. A. Anderson, H. R. Udseth, and R. D. Smith, “Frequency shifts due to the interference of resolved peaks in magnitude-mode Fourier-transform ion cyclotron resonance mass spectra,” J. Am. Soc. Mass Spectrom. 13(4), 387–401 (2002).
[Crossref] [PubMed]

W. C. Swann and S. L. Gilbert, “Pressure-induced shift and broadening of 1560-1630 nm carbon monoxide wavelength-calibration lines,” J. Opt. Soc. Am. B 19(10), 2461–2467 (2002).
[Crossref]

2001 (1)

M. C. Amann, T. Bosch, M. Lescure, R. Myllyla, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40(1), 10–19 (2001).
[Crossref]

2000 (1)

1998 (1)

A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photonics Technol. Lett. 10(12), 1709–1711 (1998).
[Crossref]

1996 (1)

K. Iiyama, L. T. Wang, and K. I. Hayashi, “Linearizing optical frequency-sweep of a laser diode for FMCW reflectometry,” J. Lightwave Technol. 14(2), 173–178 (1996).
[Crossref]

1994 (1)

A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
[Crossref]

Ahn, T. J.

Amann, M. C.

M. C. Amann, T. Bosch, M. Lescure, R. Myllyla, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40(1), 10–19 (2001).
[Crossref]

Anderson, G. A.

A. V. Tolmachev, C. D. Masselon, G. A. Anderson, H. R. Udseth, and R. D. Smith, “Frequency shifts due to the interference of resolved peaks in magnitude-mode Fourier-transform ion cyclotron resonance mass spectra,” J. Am. Soc. Mass Spectrom. 13(4), 387–401 (2002).
[Crossref] [PubMed]

Andonovic, I.

A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photonics Technol. Lett. 10(12), 1709–1711 (1998).
[Crossref]

Arbel, D.

Babbitt, W. R.

Barber, Z. W.

Baumann, E.

Behroozpour, B.

B. Behroozpour, N. Quack, P. Sandborn, S. Gerke, W. Yang, C. Chang-Hasnain, M.C. Wu, and B.E. Boser, “Method for increasing the operating distance of MEMS LIDAR beyond Brownian noise limitation,” in CLEO: Appl. and Tech. (2014), paper AW3H–2.

Berg, T.

Bernacil, M. A.

S. O’Connor, M. A. Bernacil, A. DeKelaita, B. Maher, and D. Derickson, “100 kHz axial scan rate swept-wavelength OCT using sampled grating distributed Bragg reflector lasers,” Proc. SPIE 7168, 716825 (2009).
[Crossref]

Bosch, T.

M. C. Amann, T. Bosch, M. Lescure, R. Myllyla, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40(1), 10–19 (2001).
[Crossref]

Boser, B.E.

B. Behroozpour, N. Quack, P. Sandborn, S. Gerke, W. Yang, C. Chang-Hasnain, M.C. Wu, and B.E. Boser, “Method for increasing the operating distance of MEMS LIDAR beyond Brownian noise limitation,” in CLEO: Appl. and Tech. (2014), paper AW3H–2.

Burgner, C. B.

Cable, A. E.

Chang-Hasnain, C.

B. Behroozpour, N. Quack, P. Sandborn, S. Gerke, W. Yang, C. Chang-Hasnain, M.C. Wu, and B.E. Boser, “Method for increasing the operating distance of MEMS LIDAR beyond Brownian noise limitation,” in CLEO: Appl. and Tech. (2014), paper AW3H–2.

Chen, F.

Chen, H.

Choi, D. H.

Choi, W. J.

Choma, M.

Choma, M. A.

M. A. Choma, K. Hsu, and J. A. Izatt, “Swept source optical coherence tomography using an all-fiber 1300-nm ring laser source,” J. Biomed. Opt. 10(4), 044009 (2005).
[Crossref] [PubMed]

Coddington, I.

Cornwell, W. D.

A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photonics Technol. Lett. 10(12), 1709–1711 (1998).
[Crossref]

Dahl, J. R.

DeKelaita, A.

S. O’Connor, M. A. Bernacil, A. DeKelaita, B. Maher, and D. Derickson, “100 kHz axial scan rate swept-wavelength OCT using sampled grating distributed Bragg reflector lasers,” Proc. SPIE 7168, 716825 (2009).
[Crossref]

Delfyett, P. J.

Derickson, D.

S. O’Connor, M. A. Bernacil, A. DeKelaita, B. Maher, and D. Derickson, “100 kHz axial scan rate swept-wavelength OCT using sampled grating distributed Bragg reflector lasers,” Proc. SPIE 7168, 716825 (2009).
[Crossref]

Deschênes, J. D.

Dieckmann, A.

A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
[Crossref]

Ding, Z.

Dong, Y.

Donlagic, D.

Du, Y.

Duker, J.

Eyal, A.

Froggatt, M.

Fujimoto, J.

Fujimoto, J. G.

Gan, Y.

Gerke, S.

B. Behroozpour, N. Quack, P. Sandborn, S. Gerke, W. Yang, C. Chang-Hasnain, M.C. Wu, and B.E. Boser, “Method for increasing the operating distance of MEMS LIDAR beyond Brownian noise limitation,” in CLEO: Appl. and Tech. (2014), paper AW3H–2.

Gifford, D.

Gilbert, S. L.

Giorgetta, F. R.

Gong, H.

Greenfield, N.

Grulkowski, I.

Harris, M.

Hayashi, K. I.

K. Iiyama, L. T. Wang, and K. I. Hayashi, “Linearizing optical frequency-sweep of a laser diode for FMCW reflectometry,” J. Lightwave Technol. 14(2), 173–178 (1996).
[Crossref]

Heim, P. J.

B. Potsaid, V. Jayaraman, J. G. Fujimoto, J. Jiang, P. J. Heim, and A. E. Cable, “MEMS tunable VCSEL light source for ultrahigh speed 60kHz-1MHz axial scan rate and long range centimeter class OCT imaging,” in SPIE BiOS (2012), paper 82130M.

Hsu, K.

Hu, T.

Hu, W.

Huber, R.

Igawa, H.

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, and K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[Crossref]

Iiyama, K.

K. Iiyama, S. I. Matsui, T. Kobayashi, and T. Maruyama, “High-resolution FMCW reflectometry using a single-mode vertical-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 23(11), 703–705 (2011).
[Crossref]

K. Iiyama, L. T. Wang, and K. I. Hayashi, “Linearizing optical frequency-sweep of a laser diode for FMCW reflectometry,” J. Lightwave Technol. 14(2), 173–178 (1996).
[Crossref]

Ishii, H.

H. Ishii, K. Kasaya, and H. Oohashi, “Narrow spectral linewidth operation (< 160 kHz) in widely tunable distributed feedback laser array,” Electron. Lett. 46(10), 1 (2010).
[Crossref]

Izatt, J.

Izatt, J. A.

M. A. Choma, K. Hsu, and J. A. Izatt, “Swept source optical coherence tomography using an all-fiber 1300-nm ring laser source,” J. Biomed. Opt. 10(4), 044009 (2005).
[Crossref] [PubMed]

Jayaraman, V.

Jiang, J.

John, D. D.

Kageyama, K.

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, and K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[Crossref]

Karlsson, C. J.

Kasaya, K.

H. Ishii, K. Kasaya, and H. Oohashi, “Narrow spectral linewidth operation (< 160 kHz) in widely tunable distributed feedback laser array,” Electron. Lett. 46(10), 1 (2010).
[Crossref]

Kaylor, B.

Kim, D. Y.

Ko, T.

Kobayashi, T.

K. Iiyama, S. I. Matsui, T. Kobayashi, and T. Maruyama, “High-resolution FMCW reflectometry using a single-mode vertical-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 23(11), 703–705 (2011).
[Crossref]

Kowalczyk, A.

Lee, B. K.

Lee, J. Y.

Lescure, M.

M. C. Amann, T. Bosch, M. Lescure, R. Myllyla, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40(1), 10–19 (2001).
[Crossref]

Letalick, D.

Leyva, V.

Liu, B.

Liu, G.

Liu, J. J.

Liu, K.

Liu, T.

Liu, Z.

Lu, C.

Maher, B.

S. O’Connor, M. A. Bernacil, A. DeKelaita, B. Maher, and D. Derickson, “100 kHz axial scan rate swept-wavelength OCT using sampled grating distributed Bragg reflector lasers,” Proc. SPIE 7168, 716825 (2009).
[Crossref]

Mandridis, D.

Maruyama, T.

K. Iiyama, S. I. Matsui, T. Kobayashi, and T. Maruyama, “High-resolution FMCW reflectometry using a single-mode vertical-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 23(11), 703–705 (2011).
[Crossref]

Masselon, C. D.

A. V. Tolmachev, C. D. Masselon, G. A. Anderson, H. R. Udseth, and R. D. Smith, “Frequency shifts due to the interference of resolved peaks in magnitude-mode Fourier-transform ion cyclotron resonance mass spectra,” J. Am. Soc. Mass Spectrom. 13(4), 387–401 (2002).
[Crossref] [PubMed]

Mateo, A. B.

Matsui, S. I.

K. Iiyama, S. I. Matsui, T. Kobayashi, and T. Maruyama, “High-resolution FMCW reflectometry using a single-mode vertical-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 23(11), 703–705 (2011).
[Crossref]

Meng, Z.

Murayama, H.

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, and K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[Crossref]

Myllyla, R.

M. C. Amann, T. Bosch, M. Lescure, R. Myllyla, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40(1), 10–19 (2001).
[Crossref]

Newbury, N. R.

Nguyen, D.

Njegovec, M.

O’Connor, S.

S. O’Connor, M. A. Bernacil, A. DeKelaita, B. Maher, and D. Derickson, “100 kHz axial scan rate swept-wavelength OCT using sampled grating distributed Bragg reflector lasers,” Proc. SPIE 7168, 716825 (2009).
[Crossref]

Ohbayashi, K.

Olsson, F. Å.

Omichi, K.

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, and K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[Crossref]

Oohashi, H.

H. Ishii, K. Kasaya, and H. Oohashi, “Narrow spectral linewidth operation (< 160 kHz) in widely tunable distributed feedback laser array,” Electron. Lett. 46(10), 1 (2010).
[Crossref]

Ozdur, I.

Ozharar, S.

Piracha, M. U.

Potsaid, B.

Qin, J.

Quack, N.

B. Behroozpour, N. Quack, P. Sandborn, S. Gerke, W. Yang, C. Chang-Hasnain, M.C. Wu, and B.E. Boser, “Method for increasing the operating distance of MEMS LIDAR beyond Brownian noise limitation,” in CLEO: Appl. and Tech. (2014), paper AW3H–2.

Rakuljic, G.

Reibel, R. R.

Rioux, M.

M. C. Amann, T. Bosch, M. Lescure, R. Myllyla, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40(1), 10–19 (2001).
[Crossref]

Robertson, M. E.

Roos, P. A.

Sagiv, O. Y.

Sandborn, P.

B. Behroozpour, N. Quack, P. Sandborn, S. Gerke, W. Yang, C. Chang-Hasnain, M.C. Wu, and B.E. Boser, “Method for increasing the operating distance of MEMS LIDAR beyond Brownian noise limitation,” in CLEO: Appl. and Tech. (2014), paper AW3H–2.

Sarunic, M.

Satyan, N.

Shalom, H.

A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photonics Technol. Lett. 10(12), 1709–1711 (1998).
[Crossref]

Smith, R. D.

A. V. Tolmachev, C. D. Masselon, G. A. Anderson, H. R. Udseth, and R. D. Smith, “Frequency shifts due to the interference of resolved peaks in magnitude-mode Fourier-transform ion cyclotron resonance mass spectra,” J. Am. Soc. Mass Spectrom. 13(4), 387–401 (2002).
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Tong, Y.

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A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photonics Technol. Lett. 10(12), 1709–1711 (1998).
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D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, and K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
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D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, and K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
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A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photonics Technol. Lett. 10(12), 1709–1711 (1998).
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K. Iiyama, S. I. Matsui, T. Kobayashi, and T. Maruyama, “High-resolution FMCW reflectometry using a single-mode vertical-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 23(11), 703–705 (2011).
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A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photonics Technol. Lett. 10(12), 1709–1711 (1998).
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J. Am. Soc. Mass Spectrom. (1)

A. V. Tolmachev, C. D. Masselon, G. A. Anderson, H. R. Udseth, and R. D. Smith, “Frequency shifts due to the interference of resolved peaks in magnitude-mode Fourier-transform ion cyclotron resonance mass spectra,” J. Am. Soc. Mass Spectrom. 13(4), 387–401 (2002).
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M. A. Choma, K. Hsu, and J. A. Izatt, “Swept source optical coherence tomography using an all-fiber 1300-nm ring laser source,” J. Biomed. Opt. 10(4), 044009 (2005).
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Proc. SPIE (1)

S. O’Connor, M. A. Bernacil, A. DeKelaita, B. Maher, and D. Derickson, “100 kHz axial scan rate swept-wavelength OCT using sampled grating distributed Bragg reflector lasers,” Proc. SPIE 7168, 716825 (2009).
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D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, and K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
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B. Potsaid, V. Jayaraman, J. G. Fujimoto, J. Jiang, P. J. Heim, and A. E. Cable, “MEMS tunable VCSEL light source for ultrahigh speed 60kHz-1MHz axial scan rate and long range centimeter class OCT imaging,” in SPIE BiOS (2012), paper 82130M.

B. Behroozpour, N. Quack, P. Sandborn, S. Gerke, W. Yang, C. Chang-Hasnain, M.C. Wu, and B.E. Boser, “Method for increasing the operating distance of MEMS LIDAR beyond Brownian noise limitation,” in CLEO: Appl. and Tech. (2014), paper AW3H–2.

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Figures (11)

Fig. 1
Fig. 1 Two adjacent regions containing part of the beat signal.
Fig. 2
Fig. 2 (a) Individual laser element frequency sweeps with time, LO (solid line) and delayed return reflection (dashed line). Tn indicates the nth sweep period and τD is the time delay to the target. (b) Combined total bandwidth after resampling at equal optical frequencies and stitching.
Fig. 3
Fig. 3 (a) Combined photodiode response regions with a single target and perfect registration at the combination point (vertical line), (b) Correct Fourier transform after combining, (c) Two regions with a shift error 1 sample point in the second section. (d) FFT resulting in a corrupt main lobe due to the shift error.
Fig. 4
Fig. 4 Experiment Layout. The dotted line indicates components housed inside the butterfly package. The custom printed circuit-board (PCB) drives each DFB element. The light blue lines indicate PM fiber. A Mach-Zehnder reference arm (MZI) and a HCN gas cell aids in wavelength calibration. The Fresnel back reflection from a wedged window acts as the LO, SOA: Semiconductor optical amplifier, A/D: Analog-to-digital data acquisition, TEC: Thermoelectric cooler, BPF: Bandpass filters for DC blocking and anti-aliasing, PD: Photodecter receiver.
Fig. 5
Fig. 5 (a) Current ramp applied to each DFB and (b) resulting response of the DFB module’s 50 GHz etalon wavelength reference from a single element sweep. The shaded section is removed when combining with the previous DFB signal. (c) Custom PCB showing the butterfly package containing the DFB array, current driver, and transistor network. (d) Measurement of linearity for all 12 elements over the 5.5 THz sweep, plotted at the MZI zero-cross points. The dotted line is the approximate standard deviation.
Fig. 6
Fig. 6 (a) Combined signal using all 12 DFB elements. The HCN absorption peaks (red) and a signal from a 50 μm fused silica etalon (blue) are shown. The vertical lines indicate the combination points between DFB regions. The last peak in each DFB region is listed above the HCN trace. (b) Two adjacent measurement signal regions (blue) registered to the absorption peaks (red) of the HCN gas cell. The peak-to-peak distance in zero-crossing-space is labeled as Δζ. The vertical dotted line is the combination point. (c) 2σ uncertainty of the measured gap values (blue bars). The mean MZI FSR (dotted line) and half-FSR (dashed line) are shown for comparison. (d) Optical spectrum analyzer trace averaged over all 12 elements.
Fig. 7
Fig. 7 (a) Air-fiber phase mismatch of the MZI beat signal in terms of the zero-crossings, referenced to the start of the sweep. The contributions of each Taylor series term are shown. For a 1524.9131 nm wavelength start (ng ≈1.4681), the profile follows: (6.37365e-10)m2 + (4.13137e-16)m3. (b) Single reflector FFT peak before (dotted line) and after (solid line) dispersion compensation using all DFB elements. (c) Dispersion compensated signature from a 100 μm fused silica etalon placed at 1 m. The two main lobes correspond to the front and back glass surfaces.
Fig. 8
Fig. 8 Signal processing steps. MZI ZC: The Mach-Zehnder interferometer zero-crossings are used as the resampling reference (black arrow). The HCN peaks (red arrows) provide a known wavelength reference for registering the measurement segments (blue arrows).
Fig. 9
Fig. 9 (a) Incremental improvement in surface resolution of a 1 mm glass slide using 6 elements. All plots are normalized per trace. The measured physical thickness accounting for the refractive index is denoted by Δz. The thickness measured using a micrometer was 1.02( ± 0.01) mm. (b) Thickness measurement of a 100 μm fused silica etalon using up to 12 elements. The thickness reported by the manufacturer was 100.7 μm.
Fig. 10
Fig. 10 (a) Improvement in surface resolution of a 50 μm fused silica etalon using 12 elements. The manufacturer-provided thickness was listed as 47.6 μm. (b) Incremental improvement in surface resolution of a 25 μm fused silica etalon. The actual thickness was not provided.
Fig. 11
Fig. 11 (a) Minimum free-space resolution calculated using the main lobe FWHM of the processed signature from a mirror placed at 1.4 m. The mean of 30 trials is shown. (b) Deviation from the 12 DFB mean range value (dotted line) for elements 6-12. (Nominal range = 143.9173 cm, MZI L = 2.06571 m) The error bars indicate the 1σ standard deviation from the mean. The final 12-element standard deviation was ~180 nm.

Equations (13)

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ε(t)= E 0 exp[ j( πα t 2 +2π ν 0 t+ ϕ e (t)+ ϕ s (t)+ ϕ 0 ) ],
I(t)= | ε LO ( t τ LO )+ ε R ( t τ R ) | 2 .
v(t)=Acos[ απ τ D 2 +2πα τ D t+2π ν 0 τ D +Δ ϕ e +Δ ϕ s ],
v B (t)=v(t)rect( t t 0 T/2 T ),
V B (f)=F[ v(t) ]F[ rect( t t 0 T/2 T ) ],
v tot (t)=v(t)[ rect( t t c + B 1 /2α B 1 /α )+rect( t t c B 2 /2α B 2 /α ) ],
V tot (f)= πBA α exp( jθ )sinc[ πB α ( f f D ) ] ×{ exp[ j π α ( f f D )( 2 ν c B ) ]+exp[ j π α ( f f D )( 2 ν c +B ) ] },
V tot (f)= 2πBA α sinc[ 2πB α ( f f D ) ]exp{ j[ θ 2π ν c α ( f f D ) ] }.
v tot (t)=v(t)[ rect( t t c + B 1 /2α B 1 /α ) ]+v(tΔ t err )[ rect( t t c B 2 /2α B 2 /α ) ].
V tot (f)= πBA α exp( jθ )sinc[ πB α ( f f D ) ] ×{ exp[ j π α ( f f D )( 2 ν c B ) ]+exp[ j π α ( f f D )( 2 ν c +B )j θ err ] }.
γ i (m) Δ ν G,i Δ ζ i c ML n g,i ,
β(m) β 0 +2πm γ 0 β 1 + 1 2 (2πm γ 0 ) 2 β 2 + 1 6 (2πm γ 0 ) 3 β 3 ,
v tot [m']= v tot [ m+ (2π γ 0 ) β 2 2 β 1 m 2 + (2π γ 0 ) 2 β 3 6 β 1 m 3 ].

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