Abstract

We analyze the physics behind the newest generation of rapidly wavelength tunable sources for optical coherence tomography (OCT), retaining a single longitudinal cavity mode during operation without repeated build up of lasing. In this context, we theoretically investigate the currently existing concepts of rapidly wavelength-swept lasers based on tuning of the cavity length or refractive index, leading to an altered optical path length inside the resonator. Specifically, we consider vertical-cavity surface-emitting lasers (VCSELs) with microelectromechanical system (MEMS) mirrors as well as Fourier domain mode-locked (FDML) and Vernier-tuned distributed Bragg reflector (VT-DBR) lasers. Based on heuristic arguments and exact analytical solutions of Maxwell’s equations for a fundamental laser resonator model, we show that adiabatic wavelength tuning is achieved, i.e., hopping between cavity modes associated with a repeated build up of lasing is avoided, and the photon number is conserved. As a consequence, no fundamental limit exists for the wavelength tuning speed, in principle enabling wide-range wavelength sweeps at arbitrary tuning speeds with narrow instantaneous linewidth.

© 2015 Optical Society of America

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2014 (3)

2013 (8)

I. Grulkowski, J. J. Liu, B. Potsaid, V. Jayaraman, J. Jiang, J. G. Fujimoto, and A. E. Cable, “High-precision, high-accuracy ultralong-range swept-source optical coherence tomography using vertical cavity surface emitting laser light source,” Opt. Lett. 38, 673–675 (2013).
[Crossref] [PubMed]

T. Klein, R. André, W. Wieser, T. Pfeiffer, and R. Huber, “Joint aperture detection for speckle reduction and increased collection efficiency in ophthalmic MHz OCT,” Biomed. Opt. Express 4, 619–634 (2013).
[Crossref] [PubMed]

T. Wang, W. Wieser, G. Springeling, R. Beurskens, C. T. Lancee, T. Pfeiffer, A. F. van der Steen, R. Huber, and G. v. Soest, “Intravascular optical coherence tomography imaging at 3200 frames per second,” Opt. Lett. 38, 1715–1717 (2013).
[Crossref] [PubMed]

T.-H. Tsai, B. Potsaid, Y. K. Tao, V. Jayaraman, J. Jiang, P. J. Heim, M. F. Kraus, C. Zhou, J. Hornegger, H. Mashimo, and et al., “Ultrahigh speed endoscopic optical coherence tomography using micromotor imaging catheter and VCSEL technology,” Biomed. Opt. Express 4, 1119–1132 (2013).
[Crossref] [PubMed]

O. O. Ahsen, Y. K. Tao, B. M. Potsaid, Y. Sheikine, J. Jiang, I. Grulkowski, T.-H. Tsai, V. Jayaraman, M. F. Kraus, J. L. Connolly, and et al., “Swept source optical coherence microscopy using a 1310 nm VCSEL light source,” Opt. Express 21, 18021–18033 (2013).
[Crossref] [PubMed]

S. Slepneva, B. Kelleher, B. OShaughnessy, S. P. Hegarty, A. G. Vladimirov, and G. Huyet, “Dynamics of Fourier domain mode-locked lasers,” Opt. Express 21, 19240–19251 (2013).
[Crossref] [PubMed]

T. Klein, W. Wieser, L. Reznicek, A. Neubauer, A. Kampik, and R. Huber, “Multi-MHz retinal OCT,” Biomed. Opt. Express 4, 1890–1908 (2013).
[Crossref] [PubMed]

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave Fourier domain mode-locked lasers,” Nat. Commun. 4, 1848 (2013).
[Crossref] [PubMed]

2012 (6)

2011 (4)

2010 (2)

2009 (4)

Y. Nakazaki and S. Yamashita, “Fast and wide tuning range wavelength-swept fiber laser based on dispersion tuning and its application to dynamic FBG sensing,” Opt. Express 17, 8310–8318 (2009).
[Crossref] [PubMed]

P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17, 18116–18135 (2009).
[Crossref] [PubMed]

J. A. Zeitler and L. F. Gladden, “In-vitro tomography and non-destructive imaging at depth of pharmaceutical solid dosage forms,” Eur. J. Pharm. Biopharm. 71, 2–22 (2009).
[Crossref]

T. Yano, H. Saitou, N. Kanbara, R. Noda, S.-i. Tezuka, N. Fujimura, M. Ooyama, T. Watanabe, T. Hirata, and N. Nishiyama, “Wavelength modulation over 500 kHz of micromechanically tunable InP-based VCSELs with Si-MEMS technology,” IEEE J. Sel. Top. Quantum Electron. 15, 528–534 (2009).
[Crossref]

2008 (2)

2007 (5)

2006 (6)

2005 (5)

2004 (1)

M. F. Yanik and S. Fan, “Time reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004).
[Crossref] [PubMed]

2003 (3)

S.-H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, “High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter,” Opt. Lett. 28, 1981–1983 (2003).
[Crossref] [PubMed]

D. Stifter, P. Burgholzer, O. Höglinger, E. Götzinger, and C. K. Hitzenberger, “Polarisation-sensitive optical coherence tomography for material characterisation and strain-field mapping,” Appl. Phys. A Mater. Sci. 76, 947–951 (2003).
[Crossref]

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90, 203904 (2003).
[Crossref]

1996 (1)

H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y. Yoshikuni, “Quasicontinuous wavelength tuning in super-structure-grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 32, 433–441 (1996).
[Crossref]

1995 (1)

H. Johnston and S. Sarkar, “Moving mirrors and time-varying dielectrics,” Phys. Rev. A 51, 4109–4115 (1995).
[Crossref] [PubMed]

1958 (1)

F. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
[Crossref]

Abundis-Patiño, J. H.

Adler, D. C.

Agrawal, G. P.

Ahsen, O. O.

Akiba, M.

An, X.

Andre, R.

C. Blatter, T. Klein, B. Grajciar, T. Schmoll, W. Wieser, R. Andre, R. Huber, and R. A. Leitgeb, “Ultrahigh-speed non-invasive widefield angiography,” J. Biomed. Opt. 17, 0705051 (2012).
[Crossref]

André, R.

Atia, W.

M. Kuznetsov, W. Atia, B. Johnson, and D. Flanders, “Compact ultrafast reflective Fabry-Perot tunable lasers for OCT imaging applications,” in Proc. SPIE 7554, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIV p. 75541F (2010).
[Crossref]

Barry, S.

Bartula, R. J.

Baumann, B.

Beurskens, R.

Biedermann, B.

Biedermann, B. R.

Blatter, C.

C. Blatter, T. Klein, B. Grajciar, T. Schmoll, W. Wieser, R. Andre, R. Huber, and R. A. Leitgeb, “Ultrahigh-speed non-invasive widefield angiography,” J. Biomed. Opt. 17, 0705051 (2012).
[Crossref]

Bonesi, M.

Boschert, P.

M. Bonesi, M. Minneman, J. Ensher, B. Zabihian, H. Sattmann, P. Boschert, E. Hoover, R. Leitgeb, M. Crawford, and W. Drexler, “Akinetic all-semiconductor programmable swept-source at 1550 nm and 1310 nm with centimeters coherence length,” Opt. Express 22, 2632–2655 (2014).
[Crossref] [PubMed]

J. Ensher, P. Boschert, K. Featherston, J. Huber, M. Crawford, M. Minneman, C. Chiccone, and D. Derickson, “Long coherence length and linear sweep without an external optical k-clock in a monolithic semiconductor laser for inexpensive optical coherence tomography,” in Proc. SPIE 8213, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVI p. 82130T (2012).
[Crossref]

Boudoux, C.

Bouma, B. E.

Burgholzer, P.

D. Stifter, P. Burgholzer, O. Höglinger, E. Götzinger, and C. K. Hitzenberger, “Polarisation-sensitive optical coherence tomography for material characterisation and strain-field mapping,” Appl. Phys. A Mater. Sci. 76, 947–951 (2003).
[Crossref]

Cable, A.

V. Jayaraman, J. Jiang, H. Li, P. Heim, G. Cole, B. Potsaid, J. G. Fujimoto, and A. Cable, “OCT imaging up to 760kHz axial scan rate using single-mode 1310nm MEMs-tunable VCSELs with >100nm tuning range,” in CLEO: Science and Innovations p. PDPB2 (2011).

Cable, A. E.

I. Grulkowski, J. J. Liu, B. Potsaid, V. Jayaraman, J. Jiang, J. G. Fujimoto, and A. E. Cable, “High-precision, high-accuracy ultralong-range swept-source optical coherence tomography using vertical cavity surface emitting laser light source,” Opt. Lett. 38, 673–675 (2013).
[Crossref] [PubMed]

I. Grulkowski, J. J. Liu, B. Potsaid, V. Jayaraman, C. D. Lu, J. Jiang, A. E. Cable, J. S. Duker, and J. G. Fujimoto, “Retinal, anterior segment and full eye imaging using ultrahigh speed swept source OCT with vertical-cavity surface emitting lasers,” Biomed. Opt. Express 3, 2733–2751 (2012).
[Crossref] [PubMed]

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D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nature Photon. 1, 709–716 (2007).
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M. Bonesi, M. Minneman, J. Ensher, B. Zabihian, H. Sattmann, P. Boschert, E. Hoover, R. Leitgeb, M. Crawford, and W. Drexler, “Akinetic all-semiconductor programmable swept-source at 1550 nm and 1310 nm with centimeters coherence length,” Opt. Express 22, 2632–2655 (2014).
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B. Potsaid, B. Baumann, D. Huang, S. Barry, A. E. Cable, J. S. Schuman, J. S. Duker, and J. G. Fujimoto, “Ultrahigh speed 1050nm swept source / Fourier domain OCT retinal and anterior segment imaging at 100,000 to 400,000 axial scans per second,” Opt. Express 18, 20029–20048 (2010).
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D. C. Adler, J. Stenger, I. Gorczynska, H. Lie, T. Hensick, R. Spronk, S. Wolohojian, N. Khandekar, J. Y. Jiang, S. Barry, A. E. Cable, R. Huber, and J. G. Fujimoto, “Comparison of three-dimensional optical coherence tomography and high resolution photography for art conservation studies,” Opt. Express 15, 15972–15986 (2007).
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L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
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D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nature Photon. 1, 709–716 (2007).
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L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
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R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
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R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13, 3513–3528 (2005).
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B. Potsaid, V. Jayaraman, J. G. Fujimoto, J. Jiang, P. J. S. 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 Proc. SPIE 8213, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVI p. 82130M (2012).
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J. Ensher, P. Boschert, K. Featherston, J. Huber, M. Crawford, M. Minneman, C. Chiccone, and D. Derickson, “Long coherence length and linear sweep without an external optical k-clock in a monolithic semiconductor laser for inexpensive optical coherence tomography,” in Proc. SPIE 8213, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVI p. 82130T (2012).
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C. Blatter, T. Klein, B. Grajciar, T. Schmoll, W. Wieser, R. Andre, R. Huber, and R. A. Leitgeb, “Ultrahigh-speed non-invasive widefield angiography,” J. Biomed. Opt. 17, 0705051 (2012).
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T. Klein, W. Wieser, C. M. Eigenwillig, B. R. Biedermann, and R. Huber, “Megahertz OCT for ultrawide-field retinal imaging with a 1050nm Fourier domain mode-locked laser,” Opt. Express 19, 3044–3062 (2011).
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D. C. Adler, J. Stenger, I. Gorczynska, H. Lie, T. Hensick, R. Spronk, S. Wolohojian, N. Khandekar, J. Y. Jiang, S. Barry, A. E. Cable, R. Huber, and J. G. Fujimoto, “Comparison of three-dimensional optical coherence tomography and high resolution photography for art conservation studies,” Opt. Express 15, 15972–15986 (2007).
[Crossref] [PubMed]

L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
[Crossref] [PubMed]

L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
[Crossref]

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nature Photon. 1, 709–716 (2007).
[Crossref]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13, 3513–3528 (2005).
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S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “Time-Encoded Raman: Fiber-based, hyperspectral, broadband stimulated Raman microscopy,” e-print arXiv:1405.4181 [physics.optics] (2014).

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Jang, I.-K.

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I. Grulkowski, J. J. Liu, B. Potsaid, V. Jayaraman, C. D. Lu, J. Jiang, A. E. Cable, J. S. Duker, and J. G. Fujimoto, “Retinal, anterior segment and full eye imaging using ultrahigh speed swept source OCT with vertical-cavity surface emitting lasers,” Biomed. Opt. Express 3, 2733–2751 (2012).
[Crossref] [PubMed]

V. Jayaraman, J. Jiang, H. Li, P. Heim, G. Cole, B. Potsaid, J. G. Fujimoto, and A. Cable, “OCT imaging up to 760kHz axial scan rate using single-mode 1310nm MEMs-tunable VCSELs with >100nm tuning range,” in CLEO: Science and Innovations p. PDPB2 (2011).

B. Potsaid, V. Jayaraman, J. G. Fujimoto, J. Jiang, P. J. S. 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 Proc. SPIE 8213, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVI p. 82130M (2012).
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Jeong, M. Y.

Jiang, J.

I. Grulkowski, J. J. Liu, B. Potsaid, V. Jayaraman, J. Jiang, J. G. Fujimoto, and A. E. Cable, “High-precision, high-accuracy ultralong-range swept-source optical coherence tomography using vertical cavity surface emitting laser light source,” Opt. Lett. 38, 673–675 (2013).
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Biomed. Opt. Express (6)

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[Crossref] [PubMed]

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IEEE J. Quantum Electron. (1)

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S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
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D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nature Photon. 1, 709–716 (2007).
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C. M. Eigenwillig, B. R. Biedermann, G. Palte, and R. Huber, “K-space linear Fourier domain mode locked laser and applications for optical coherence tomography,” Opt. Express 16, 8916–8937 (2008).
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K. Goda, A. Fard, O. Malik, G. Fu, A. Quach, and B. Jalali, “High-throughput optical coherence tomography at 800 nm,” Opt. Express 20, 19612–19617 (2012).
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S. Todor, B. Biedermann, W. Wieser, R. Huber, and C. Jirauschek, “Instantaneous lineshape analysis of Fourier domain mode-locked lasers,” Opt. Express 19, 8802–8807 (2011).
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S. Slepneva, B. Kelleher, B. OShaughnessy, S. P. Hegarty, A. G. Vladimirov, and G. Huyet, “Dynamics of Fourier domain mode-locked lasers,” Opt. Express 21, 19240–19251 (2013).
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D. C. Adler, W. Wieser, F. Trepanier, J. M. Schmitt, and R. A. Huber, “Extended coherence length Fourier domain mode locked lasers at 1310 nm,” Opt. Express 19, 20930–20939 (2011).
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R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13, 3513–3528 (2005).
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W. Wieser, B. R. Biedermann, T. Klein, C. M. Eigenwillig, and R. Huber, “Multi-megahertz OCT: High quality 3D imaging at 20 million A-scans and 4.5 GVoxels per second,” Opt. Express 18, 14685–14704 (2010).
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L. A. Kranendonk, R. J. Bartula, and S. T. Sanders, “Modeless operation of a wavelength-agile laser by high-speed cavity length changes,” Opt. Express 13, 1498–1507 (2005).
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L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
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D. Yelin, W. M. White, J. T. Motz, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Spectral-domain spectrally-encoded endoscopy,” Opt. Express 15, 2432–2444 (2007).
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R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
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S. Slepneva, B. OShaughnessy, B. Kelleher, S. Hegarty, A. Vladimirov, H.-C. Lyu, K. Karnowski, M. Wojtkowski, and G. Huyet, “Dynamics of a short cavity swept source OCT laser,” Opt. Express 22, 18177–18185 (2014).
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E. J. Jung, C.-S. Kim, M. Y. Jeong, M. K. Kim, M. Y. Jeon, W. Jung, and Z. Chen, “Characterization of FBG sensor interrogation based on a FDML wavelength swept laser,” Opt. Express 16, 16552–16560 (2008).
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Y. Nakazaki and S. Yamashita, “Fast and wide tuning range wavelength-swept fiber laser based on dispersion tuning and its application to dynamic FBG sensing,” Opt. Express 17, 8310–8318 (2009).
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T. Klein, W. Wieser, C. M. Eigenwillig, B. R. Biedermann, and R. Huber, “Megahertz OCT for ultrawide-field retinal imaging with a 1050nm Fourier domain mode-locked laser,” Opt. Express 19, 3044–3062 (2011).
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Figures (3)

Fig. 1
Fig. 1 Schematic representation of various methods for wavelength tuning by changing the intracavity optical path length: (a) Resonator with moving end mirror; (b) resonator with time dependent refractive index; (c) grating-based wavelength tuning mechanism.
Fig. 2
Fig. 2 Model parameters for idealized laser resonator with wavelength tuning by (a) moving the end mirror and (b) changing the intracavity refractive index. The optical resonator field is described by a forward and a backward propagating component, denoted by E x + and E x , respectively.
Fig. 3
Fig. 3 (a) Ideal lossless FDML laser model, consisting of an optical fiber for light storage and a tunable bandpass filter. (b) Fabry-Pérot bandpass filter.

Equations (41)

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δ W = F δ L = W δ L L .
W = V ε 0 n 2 E ^ 2 ,
F ( x ) = k 0 x + k 1 x 2 ,
k = d F ( x 0 ) d x = k 0 2 k 1 x 0 .
n = ( 1 + χ N e V ε 0 x ^ 2 E ^ ) 1 / 2 = ( 1 + χ + N e 2 V ε 0 1 k ω 0 2 m e ) ,
δ n = k 1 V ε 0 N e 2 n ( n 2 1 χ ) 2 δ x .
N ω 0 2 π L δ x 0 2 π / ω 0 0 L F [ x 0 + x ^ sin ( k z ) cos ( ω 0 t ) ] d z d t = N ( k 0 x 0 + k 1 x 0 2 + 1 4 k 1 x ^ 2 ) δ x .
δ W = 1 4 N k 1 x ^ 2 δ x = W δ n n ,
t H y = μ 0 1 z E x ,
t D x = z H y ,
D x ( z , t ) = ε 0 n 2 ( t ) E x ( z , t ) .
n 2 ( t ) t 2 D x = c 2 z 2 D x ,
E x = E x + ( t z / c ) + E x ( t + z / c ) .
E x + ( t z / c ) = r 1 E x [ t ( z + 2 L 0 ) / c ] .
E x ( t + z / c ) = γ r 2 E x + [ γ ( t + z / c ) ] .
E x + ( τ ) = γ r 1 r 2 E x + [ γ ( τ 2 L 0 / c ) ] .
E x + ( τ ) = E ^ x exp ( i 0 τ ω ( τ ) d τ ) = E ^ x exp [ i ω 0 b 1 ln ( 1 + b τ ) ] ,
b = c 1 γ 2 L 0 γ = c v ( c v ) L 0 , ω 0 = b ( 2 π m ϕ 1 ϕ 2 ln γ + i ) ,
E x = { E x + ( t z / c ) + E x ( t + z / c ) } = | E ^ x | { cos [ δ ln ( 1 + b t b z / c ) + ϕ ] 1 + b t b z / c + cos [ δ ln ( γ 1 + b t + b z / c ) ϕ 1 + ϕ ] γ 1 + b t + b z / c } ,
H y = | E ^ x | c μ 0 { cos [ δ ln ( 1 + b t b z / c ) + ϕ ] 1 + b t b z / c cos [ δ ln ( γ 1 + b t + b z / c ) ϕ 1 + ϕ ] γ 1 + b t + b z / c } .
E x , m + ( τ ) = | r 1 r 2 | m E x , 0 + ( τ ) .
E x , t + ( τ ) = t 2 m = 0 E x , m + ( τ ) = E x , i + ( τ ) t 1 t 2 m = 0 | r 1 r 2 | m = E x , i + ( τ ) t 1 t 2 1 | r 1 r 2 | .
E x = m = 0 | r 1 r 2 | m { E x , 0 + ( t z / c ) + γ r 2 E x , 0 + [ γ ( t + z / c ) ] } = 1 1 | r 1 r 2 | { E x , 0 + ( t z / c ) + γ r 2 E x , 0 + [ γ ( t + z / c ) ] } = | E ^ x | { cos [ δ ln ( 1 + b t b z / c ϕ ) ] 1 + b t b z / c + | r 2 | cos [ δ ln ( γ 1 + b t + b z / c ) + ϕ ϕ 1 ] γ 1 + b t + b z / c } ,
H y = | E ^ x | c μ 0 { cos [ δ ln ( 1 + b t b z / c ) + ϕ ] 1 + b t b z / c | r 2 | cos [ δ ln ( γ 1 + b t + b z / c ) + ϕ ϕ 1 ] γ 1 + b t + b z / c } .
n 0 2 b 2 4 c 2 F = z 2 F n 0 2 c 2 τ 2 F .
k = n 0 c ω 0 ( 1 + 1 4 b 2 ω 0 2 ) 1 / 2 ,
E x ± ( z , t ) = E ^ x ± ( 1 + b t ) 3 / 2 exp [ ± i k z i ω 0 b 1 ln ( 1 + b t ) ] .
E x + ( z , t ) = r 1 E x ( z 2 L 0 , t ) , E x ( z , t ) = r 2 E x + ( z , t ) ,
E x + ( z , t ) = r 1 r 2 E x + ( z + 2 L 0 , t ) .
k = ( 2 π m ϕ 1 ϕ 2 ) / ( 2 L 0 )
ω 0 = [ c 2 n 0 2 ( 2 π m ϕ 1 ϕ 2 2 L 0 ) 2 1 4 b 2 ] 1 / 2 ,
E x = { E x + ( z , t ) + E x ( z , t ) } = 2 | E ^ x | ( 1 + b t ) 3 / 2 cos ( k z ϕ 2 2 ) cos [ ω 0 b ln ( 1 + b t ) + ϕ + ϕ 2 2 ] ,
H y = 2 ε 0 n 0 2 | E ^ x | k ( 1 + b t ) 1 / 2 sin ( k z ϕ 2 2 ) × { ω 0 sin [ ω 0 b ln ( 1 + b t ) + ϕ + ϕ 2 2 ] + b 2 cos [ ω 0 b ln ( 1 + b t ) + ϕ + ϕ 2 2 ] } .
λ ( t ) = 2 π c / ω ( t ) = 2 π c ( 1 + b t ) / | { ω 0 } | .
λ m = 2 L op ( m ϕ 1 + ϕ 2 2 π ) 1 ,
λ ( t ) = ln ( c + v c v ) ( c v v L 0 + c t ) ( m ϕ 1 + ϕ 2 2 π ) 1 .
λ ( t ) = ( 1 + b t ) [ λ m 0 2 ( b 4 π c ) 2 ] 1 / 2 ,
W = A 2 L 0 z 2 ( ε E x 2 + μ 0 H y 2 ) d z .
W = ε 0 n 0 2 | E ^ x | 2 1 + | r 2 | 2 2 A L 0 1 + b t ,
v t , max c Δ λ ln G net 20 L R .
Δ λ = 2 ln 2 π λ 2 L c ,

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