Abstract

We present a high-speed wavelength-swept fiber laser based on a dispersion tuning method using a reflective semiconductor optical amplifier (RSOA) and a chirped fiber Bragg grating (CFBG). By using these devices, the cavity length can be shortened drastically. The short cavity improves the laser performance at high sweep rates over 200kHz. We achieve a sweep range of 60nm and an output power of 8.4mW at 100kHz sweep. We applied the dispersion-tuned fiber laser to the swept-source OCT system and successfully obtained OCT images of an adhesive tape at up to 250kHz sweep rate.

© 2013 OSA

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References

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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    [Crossref] [PubMed]
  4. R. Huber, M. Wojtkowski, K. Taira, J. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13(9), 3513–3528 (2005).
    [Crossref] [PubMed]
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    [Crossref]
  10. 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(8), 3225–3237 (2006).
    [Crossref] [PubMed]
  11. M. Y. Jeon, J. Zhang, Q. Wang, and Z. Chen, “High-speed and wide bandwidth Fourier domain mode-locked wavelength swept laser with multiple SOAs,” Opt. Express 16(4), 2547–2554 (2008).
    [Crossref] [PubMed]
  12. S. Marschall, T. Klein, W. Wieser, B. R. Biedermann, K. Hsu, K. P. Hansen, B. Sumpf, K. H. Hasler, G. Erbert, O. B. Jensen, C. Pedersen, R. Huber, and P. E. Andersen, “Fourier domain mode-locked swept source at 1050 nm based on a tapered amplifier,” Opt. Express 18(15), 15820–15831 (2010).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  19. B. Burgoyne and A. Villeneuve, “Programmable lasers: design and applications,” Proc. SPIE 7580, 758002, 758002-15 (2010).
    [Crossref]
  20. G. Lamouche, S. Vergnole, Y. Kim, B. Burgoyne, and A. Villeneuve, “Tailoring wavelength sweep for SS-OCT with a programmable picosecond laser,” Proc. SPIE 7889, 78891L, 78891L-6 (2011).
    [Crossref]
  21. Y. Kim, B. Burgoyne, N. Godbout, A. Villeneuve, G. Lamouche, and S. Vergnole, “Picosecond programmable laser sweeping over 50 mega-wavelengths per second,” Proc. SPIE 7914, 79140Y, 79140Y-8 (2011).
    [Crossref]
  22. S. Yamashita and Y. Takubo, “Fast wavelength-swept dispersion-tuned fiber laser over 500kHz using a wideband chirped fiber Bragg grating,” Proc. SPIE 7753, 77537W, 77537W-4 (2011).
    [Crossref]
  23. H. Don Lee, J. H. Lee, M. Y. Jeong, and C. S. Kim, “Characterization of wavelength-swept active mode locking fiber laser based on reflective semiconductor optical amplifier,” Opt. Express 19(15), 14586–14593 (2011).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  25. C. Chong, T. Suzuki, A. Morosawa, and T. Sakai, “Spectral narrowing effect by quasi-phase continuous tuning in high-speed wavelength-swept light source,” Opt. Express 16(25), 21105–21118 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2012 (2)

Y. Takubo and S. Yamashita, “In vivo OCT imaging using wavelength-swept fiber laser based on dispersion tuning,” IEEE Photon. Technol. Lett. 24(12), 979–981 (2012).
[Crossref]

A. Takada, M. Fujino, and S. Nagano, “Dispersion dependence of linewidth in actively mode-locked ring lasers,” Opt. Express 20(4), 4753–4762 (2012).
[Crossref] [PubMed]

2011 (6)

G. Lamouche, S. Vergnole, Y. Kim, B. Burgoyne, and A. Villeneuve, “Tailoring wavelength sweep for SS-OCT with a programmable picosecond laser,” Proc. SPIE 7889, 78891L, 78891L-6 (2011).
[Crossref]

Y. Kim, B. Burgoyne, N. Godbout, A. Villeneuve, G. Lamouche, and S. Vergnole, “Picosecond programmable laser sweeping over 50 mega-wavelengths per second,” Proc. SPIE 7914, 79140Y, 79140Y-8 (2011).
[Crossref]

S. Yamashita and Y. Takubo, “Fast wavelength-swept dispersion-tuned fiber laser over 500kHz using a wideband chirped fiber Bragg grating,” Proc. SPIE 7753, 77537W, 77537W-4 (2011).
[Crossref]

H. Don Lee, J. H. Lee, M. Y. Jeong, and C. S. Kim, “Characterization of wavelength-swept active mode locking fiber laser based on reflective semiconductor optical amplifier,” Opt. Express 19(15), 14586–14593 (2011).
[Crossref] [PubMed]

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(4), 3044–3062 (2011).
[Crossref]

M. P. Minneman, J. Ensher, M. Crawford, and D. Derickson, “All-semiconductor high-speed akinetic swept-source for OCT,” Proc. SPIE 8311, 831116, 831116-10 (2011).
[Crossref]

2010 (6)

2009 (2)

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(10), 8310–8318 (2009).
[Crossref] [PubMed]

S. Yamashita, Y. Nakazaki, R. Konishi, and O. Kusakari, “Wide and fast wavelength-swept fiber laser based on dispersion tuning for dynamic sensing (invited),” Journal of Sensors 2009, A Special Issue on Fiber and Integrated Waveguide-Based Optical Sensors, 572835 (2009).

2008 (2)

2006 (2)

2005 (2)

1997 (2)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Andersen, P. E.

Asano, M.

S. Yamashita and M. Asano, “Wide and fast wavelength-tunable mode-locked fiber laser based on dispersion tuning,” Opt. Express 14(20), 9399–9306 (2006).
[Crossref] [PubMed]

Atia, W.

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

Barry, S.

Baumann, B.

Biedermann, B. R.

Bouma, B. E.

Burgoyne, B.

G. Lamouche, S. Vergnole, Y. Kim, B. Burgoyne, and A. Villeneuve, “Tailoring wavelength sweep for SS-OCT with a programmable picosecond laser,” Proc. SPIE 7889, 78891L, 78891L-6 (2011).
[Crossref]

Y. Kim, B. Burgoyne, N. Godbout, A. Villeneuve, G. Lamouche, and S. Vergnole, “Picosecond programmable laser sweeping over 50 mega-wavelengths per second,” Proc. SPIE 7914, 79140Y, 79140Y-8 (2011).
[Crossref]

B. Burgoyne and A. Villeneuve, “Programmable lasers: design and applications,” Proc. SPIE 7580, 758002, 758002-15 (2010).
[Crossref]

Cable, A. E.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Chen, Z.

Chinn, S. R.

Chong, C.

Crawford, M.

M. P. Minneman, J. Ensher, M. Crawford, and D. Derickson, “All-semiconductor high-speed akinetic swept-source for OCT,” Proc. SPIE 8311, 831116, 831116-10 (2011).
[Crossref]

Derickson, D.

M. P. Minneman, J. Ensher, M. Crawford, and D. Derickson, “All-semiconductor high-speed akinetic swept-source for OCT,” Proc. SPIE 8311, 831116, 831116-10 (2011).
[Crossref]

Don Lee, H.

Duker, J. S.

Eigenwillig, C. M.

Ensher, J.

M. P. Minneman, J. Ensher, M. Crawford, and D. Derickson, “All-semiconductor high-speed akinetic swept-source for OCT,” Proc. SPIE 8311, 831116, 831116-10 (2011).
[Crossref]

Erbert, G.

Fercher, A. F.

Flanders, D.

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

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Fujimoto, J.

Fujimoto, J. G.

Fujino, M.

Godbout, N.

Y. Kim, B. Burgoyne, N. Godbout, A. Villeneuve, G. Lamouche, and S. Vergnole, “Picosecond programmable laser sweeping over 50 mega-wavelengths per second,” Proc. SPIE 7914, 79140Y, 79140Y-8 (2011).
[Crossref]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hansen, K. P.

Hasler, K. H.

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hitzenberger, C. K.

Hsu, K.

Huang, D.

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(19), 20029–20048 (2010).
[Crossref] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Huber, R.

Jensen, O. B.

Jeon, M. Y.

Jeong, M. Y.

Johnson, B.

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

Kim, C. S.

Kim, Y.

G. Lamouche, S. Vergnole, Y. Kim, B. Burgoyne, and A. Villeneuve, “Tailoring wavelength sweep for SS-OCT with a programmable picosecond laser,” Proc. SPIE 7889, 78891L, 78891L-6 (2011).
[Crossref]

Y. Kim, B. Burgoyne, N. Godbout, A. Villeneuve, G. Lamouche, and S. Vergnole, “Picosecond programmable laser sweeping over 50 mega-wavelengths per second,” Proc. SPIE 7914, 79140Y, 79140Y-8 (2011).
[Crossref]

Klein, T.

Konishi, R.

S. Yamashita, Y. Nakazaki, R. Konishi, and O. Kusakari, “Wide and fast wavelength-swept fiber laser based on dispersion tuning for dynamic sensing (invited),” Journal of Sensors 2009, A Special Issue on Fiber and Integrated Waveguide-Based Optical Sensors, 572835 (2009).

Kulhavy, M.

Kusakari, O.

S. Yamashita, Y. Nakazaki, R. Konishi, and O. Kusakari, “Wide and fast wavelength-swept fiber laser based on dispersion tuning for dynamic sensing (invited),” Journal of Sensors 2009, A Special Issue on Fiber and Integrated Waveguide-Based Optical Sensors, 572835 (2009).

Kuznetsov, M.

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

Lamouche, G.

G. Lamouche, S. Vergnole, Y. Kim, B. Burgoyne, and A. Villeneuve, “Tailoring wavelength sweep for SS-OCT with a programmable picosecond laser,” Proc. SPIE 7889, 78891L, 78891L-6 (2011).
[Crossref]

Y. Kim, B. Burgoyne, N. Godbout, A. Villeneuve, G. Lamouche, and S. Vergnole, “Picosecond programmable laser sweeping over 50 mega-wavelengths per second,” Proc. SPIE 7914, 79140Y, 79140Y-8 (2011).
[Crossref]

Lee, J. H.

Lexer, F.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Marschall, S.

Minneman, M. P.

M. P. Minneman, J. Ensher, M. Crawford, and D. Derickson, “All-semiconductor high-speed akinetic swept-source for OCT,” Proc. SPIE 8311, 831116, 831116-10 (2011).
[Crossref]

Morosawa, A.

Nagano, S.

Nakazaki, Y.

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(10), 8310–8318 (2009).
[Crossref] [PubMed]

S. Yamashita, Y. Nakazaki, R. Konishi, and O. Kusakari, “Wide and fast wavelength-swept fiber laser based on dispersion tuning for dynamic sensing (invited),” Journal of Sensors 2009, A Special Issue on Fiber and Integrated Waveguide-Based Optical Sensors, 572835 (2009).

Oh, W. Y.

Oh, W.-Y.

Pedersen, C.

Potsaid, B.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Sakai, T.

Schuman, J. S.

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(19), 20029–20048 (2010).
[Crossref] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Shishkov, M.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Sumpf, B.

Suzuki, T.

Swanson, E. A.

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, “Optical coherence tomography using a frequency-tunable optical source,” Opt. Lett. 22(5), 340–342 (1997).
[Crossref] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Taira, K.

Takada, A.

Takubo, Y.

Y. Takubo and S. Yamashita, “In vivo OCT imaging using wavelength-swept fiber laser based on dispersion tuning,” IEEE Photon. Technol. Lett. 24(12), 979–981 (2012).
[Crossref]

S. Yamashita and Y. Takubo, “Fast wavelength-swept dispersion-tuned fiber laser over 500kHz using a wideband chirped fiber Bragg grating,” Proc. SPIE 7753, 77537W, 77537W-4 (2011).
[Crossref]

Tearney, G. J.

Vakoc, B. J.

Vergnole, S.

Y. Kim, B. Burgoyne, N. Godbout, A. Villeneuve, G. Lamouche, and S. Vergnole, “Picosecond programmable laser sweeping over 50 mega-wavelengths per second,” Proc. SPIE 7914, 79140Y, 79140Y-8 (2011).
[Crossref]

G. Lamouche, S. Vergnole, Y. Kim, B. Burgoyne, and A. Villeneuve, “Tailoring wavelength sweep for SS-OCT with a programmable picosecond laser,” Proc. SPIE 7889, 78891L, 78891L-6 (2011).
[Crossref]

Villeneuve, A.

G. Lamouche, S. Vergnole, Y. Kim, B. Burgoyne, and A. Villeneuve, “Tailoring wavelength sweep for SS-OCT with a programmable picosecond laser,” Proc. SPIE 7889, 78891L, 78891L-6 (2011).
[Crossref]

Y. Kim, B. Burgoyne, N. Godbout, A. Villeneuve, G. Lamouche, and S. Vergnole, “Picosecond programmable laser sweeping over 50 mega-wavelengths per second,” Proc. SPIE 7914, 79140Y, 79140Y-8 (2011).
[Crossref]

B. Burgoyne and A. Villeneuve, “Programmable lasers: design and applications,” Proc. SPIE 7580, 758002, 758002-15 (2010).
[Crossref]

Wang, Q.

Wieser, W.

Wojtkowski, M.

Yamashita, S.

Y. Takubo and S. Yamashita, “In vivo OCT imaging using wavelength-swept fiber laser based on dispersion tuning,” IEEE Photon. Technol. Lett. 24(12), 979–981 (2012).
[Crossref]

S. Yamashita and Y. Takubo, “Fast wavelength-swept dispersion-tuned fiber laser over 500kHz using a wideband chirped fiber Bragg grating,” Proc. SPIE 7753, 77537W, 77537W-4 (2011).
[Crossref]

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(10), 8310–8318 (2009).
[Crossref] [PubMed]

S. Yamashita, Y. Nakazaki, R. Konishi, and O. Kusakari, “Wide and fast wavelength-swept fiber laser based on dispersion tuning for dynamic sensing (invited),” Journal of Sensors 2009, A Special Issue on Fiber and Integrated Waveguide-Based Optical Sensors, 572835 (2009).

S. Yamashita and M. Asano, “Wide and fast wavelength-tunable mode-locked fiber laser based on dispersion tuning,” Opt. Express 14(20), 9399–9306 (2006).
[Crossref] [PubMed]

Yun, S. H.

Zhang, J.

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (1)

Y. Takubo and S. Yamashita, “In vivo OCT imaging using wavelength-swept fiber laser based on dispersion tuning,” IEEE Photon. Technol. Lett. 24(12), 979–981 (2012).
[Crossref]

Opt. Express (12)

R. Huber, M. Wojtkowski, K. Taira, J. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13(9), 3513–3528 (2005).
[Crossref] [PubMed]

S. Yamashita and M. Asano, “Wide and fast wavelength-tunable mode-locked fiber laser based on dispersion tuning,” Opt. Express 14(20), 9399–9306 (2006).
[Crossref] [PubMed]

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(8), 3225–3237 (2006).
[Crossref] [PubMed]

M. Y. Jeon, J. Zhang, Q. Wang, and Z. Chen, “High-speed and wide bandwidth Fourier domain mode-locked wavelength swept laser with multiple SOAs,” Opt. Express 16(4), 2547–2554 (2008).
[Crossref] [PubMed]

C. Chong, T. Suzuki, A. Morosawa, and T. Sakai, “Spectral narrowing effect by quasi-phase continuous tuning in high-speed wavelength-swept light source,” Opt. Express 16(25), 21105–21118 (2008).
[Crossref] [PubMed]

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(10), 8310–8318 (2009).
[Crossref] [PubMed]

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(14), 14685–14704 (2010).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

(a) Configuration of dispersion-tuned fiber laser. (b) Concept of dispersion tuning.

Fig. 2
Fig. 2

Setups of dispersion-tuned fiber lasers. (a) setup-A (b) setup-B (c) setup-C.

Fig. 3
Fig. 3

Static characteristics. (a)-(c) The lasing spectra of setup-A, B, C. (d)-(f) The relation of the lasing wavelength and the mode-locking frequency in setup-A, B, C.

Fig. 4
Fig. 4

Dynamic characteristics. (a)-(c) The peak-hold spectra of setup-A, B, C. (d)-(f) The temporal waveform of setup-A, B, C.

Fig. 5
Fig. 5

Setup of the OCT system.

Fig. 6
Fig. 6

Point spread function (a)-(c) at 1kHz ((a) setup-A (b) setup-B (c) setup-C). (d)-(f) at 50kHz ((d) setup-A (e) setup-B (f) setup-C).

Fig. 7
Fig. 7

OCT images of an adhesive tape (1mm × 10mm).

Tables (1)

Tables Icon

Table 1 Summary of Performances of Three Lasers

Equations (5)

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

F= c nL ,
Δλ= n 0 cD f m0 Δ f m = n 0 L c D total f m0 Δ f m,
Δ λ max = n 0 c| D | f m0 F 0 = 1 | D |L f m0 = 1 | D total | f m0 ,
δz= 2ln2 π λ 0 2 nΔλ ,
l c = 2ln2 π λ 0 2 δλ ,

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