Abstract

A portable method of group velocity measurement is proposed based on a self-seeded gain-switched laser. The calculated group velocity in the optical fiber is obtained by measuring the round-trip frequencies of the gain-switched laser diode with different external cavities, and only a 2 m long fiber is needed. The measurement can be accomplished without oscilloscope or optical spectrum analyzer. The error associated with this test is within 0.65%, which is limited by the jitter of the voltage-controlled oscillator. Its spectrum resolution is 0.1 nm.

© 2012 Optical Society of America

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References

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  1. N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, “Direct measurement of superluminal group velocity and of signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
    [CrossRef]
  2. M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol. 7, 1217–1224 (1989).
    [CrossRef]
  3. W. Mohammed, J. Meier, M. Galle, L. Qian, J. S. Aitchison, and P. W. E. Smith, “Linear and quadratic dispersion characterization of millimeter-length fibers and waveguides using common-path interferometry,” Opt. Lett. 32, 3312–3314 (2007).
    [CrossRef]
  4. J. Y. Lee and D. Y. Kim, “Versatile chromatic dispersion measurement of a single mode fiber using spectral white light interferometry,” Opt. Express 14, 11608–11615 (2006).
    [CrossRef]
  5. M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
    [CrossRef]
  6. D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
    [CrossRef]
  7. S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
    [CrossRef]
  8. Y. Liu and K. S. Chiang, “Wavelength switching of picosecond pulses generated from a self-seeded Fabry–Perot laser diode with a tilted fiber Bragg grating formed in a graded-index multimode fiber,” Appl. Opt. 50, 829–834 (2011).
    [CrossRef]
  9. Y. C. Lee and C. Shu, “Wavelength-tunable nearly transform-limited pulses generated by self-injection seeding of a laser diode at an arbitrary repetition rate,” IEEE Photon. Technol. Lett. 9, 590–592 (1997).
    [CrossRef]
  10. S. Yam and C. Shu, “Fast wavelength-tunable multichannel switching using a self-injection seeding scheme,” IEEE J. Quantum Electron. 35, 228–233 (1999).
    [CrossRef]
  11. K. A. Ahmed, B. J. Eggleton, H. Liu, P. A. Krug, and F. Ouellette, “Simultaneous mode selection and pulse compression of gain-switched pulses from a Fabry–Perot laser using a 40 mm chirped optical fiber grating,” IEEE Photon. Technol. Lett. 7, 158–160 (1995).
    [CrossRef]
  12. J. G. Burnett and J. D. C. Jones, “Cutting optical fibers to equal lengths for broadband stellar interferometry,” Appl. Opt. 31, 2977–2978 (1992).
    [CrossRef]
  13. Corning HI 1060 FLEX and RC HI 1060 FLEX, http://www.corning.com/WorkArea/downloadasset.aspx?id=14719 .
  14. Corning RC 1300 and RC 1550, http://www.corning.com/WorkArea/downloadasset.aspx?id=14801 .
  15. Corning SMF-28e optical fiber with NexCor technology, http://www.princetel.com/datasheets/SMF28e.pdf .
  16. K. T. Vu, A. Malinowski, M. A. F. Roelens, and D. J. Richardson, “Detailed comparison of injection-seeded and self-seeded performance of a 1060 nm gain-switched Fabry–Pérot laser diode,” IEEE J. Quantum Electron. 44, 645–651 (2008).
    [CrossRef]

2011 (1)

2008 (1)

K. T. Vu, A. Malinowski, M. A. F. Roelens, and D. J. Richardson, “Detailed comparison of injection-seeded and self-seeded performance of a 1060 nm gain-switched Fabry–Pérot laser diode,” IEEE J. Quantum Electron. 44, 645–651 (2008).
[CrossRef]

2007 (1)

2006 (1)

2004 (1)

N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, “Direct measurement of superluminal group velocity and of signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef]

2000 (1)

S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
[CrossRef]

1999 (1)

S. Yam and C. Shu, “Fast wavelength-tunable multichannel switching using a self-injection seeding scheme,” IEEE J. Quantum Electron. 35, 228–233 (1999).
[CrossRef]

1997 (1)

Y. C. Lee and C. Shu, “Wavelength-tunable nearly transform-limited pulses generated by self-injection seeding of a laser diode at an arbitrary repetition rate,” IEEE Photon. Technol. Lett. 9, 590–592 (1997).
[CrossRef]

1996 (1)

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

1995 (1)

K. A. Ahmed, B. J. Eggleton, H. Liu, P. A. Krug, and F. Ouellette, “Simultaneous mode selection and pulse compression of gain-switched pulses from a Fabry–Perot laser using a 40 mm chirped optical fiber grating,” IEEE Photon. Technol. Lett. 7, 158–160 (1995).
[CrossRef]

1992 (2)

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

J. G. Burnett and J. D. C. Jones, “Cutting optical fibers to equal lengths for broadband stellar interferometry,” Appl. Opt. 31, 2977–2978 (1992).
[CrossRef]

1989 (1)

M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol. 7, 1217–1224 (1989).
[CrossRef]

Ahmed, K. A.

K. A. Ahmed, B. J. Eggleton, H. Liu, P. A. Krug, and F. Ouellette, “Simultaneous mode selection and pulse compression of gain-switched pulses from a Fabry–Perot laser using a 40 mm chirped optical fiber grating,” IEEE Photon. Technol. Lett. 7, 158–160 (1995).
[CrossRef]

Aitchison, J. S.

Bennion, I.

S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
[CrossRef]

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

Bimberg, D.

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

Bouadma, N.

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

Brunner, N.

N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, “Direct measurement of superluminal group velocity and of signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef]

Burnett, J. G.

Cavelier, M.

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

Chiang, K. S.

Y. Liu and K. S. Chiang, “Wavelength switching of picosecond pulses generated from a self-seeded Fabry–Perot laser diode with a tilted fiber Bragg grating formed in a graded-index multimode fiber,” Appl. Opt. 50, 829–834 (2011).
[CrossRef]

S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
[CrossRef]

Chusseau, L.

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

Eggleton, B. J.

K. A. Ahmed, B. J. Eggleton, H. Liu, P. A. Krug, and F. Ouellette, “Simultaneous mode selection and pulse compression of gain-switched pulses from a Fabry–Perot laser using a 40 mm chirped optical fiber grating,” IEEE Photon. Technol. Lett. 7, 158–160 (1995).
[CrossRef]

Galle, M.

Gambling, W. A.

S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
[CrossRef]

Gisin, N.

N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, “Direct measurement of superluminal group velocity and of signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef]

Horiguchi, T.

M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol. 7, 1217–1224 (1989).
[CrossRef]

Huhse, D.

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

Jones, J. D. C.

Kazmierski, C.

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

Kim, D. Y.

Krug, P. A.

K. A. Ahmed, B. J. Eggleton, H. Liu, P. A. Krug, and F. Ouellette, “Simultaneous mode selection and pulse compression of gain-switched pulses from a Fabry–Perot laser using a 40 mm chirped optical fiber grating,” IEEE Photon. Technol. Lett. 7, 158–160 (1995).
[CrossRef]

Lee, J. Y.

Lee, Y. C.

Y. C. Lee and C. Shu, “Wavelength-tunable nearly transform-limited pulses generated by self-injection seeding of a laser diode at an arbitrary repetition rate,” IEEE Photon. Technol. Lett. 9, 590–592 (1997).
[CrossRef]

Legre, M.

N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, “Direct measurement of superluminal group velocity and of signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef]

Li, S.

S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
[CrossRef]

Liu, H.

K. A. Ahmed, B. J. Eggleton, H. Liu, P. A. Krug, and F. Ouellette, “Simultaneous mode selection and pulse compression of gain-switched pulses from a Fabry–Perot laser using a 40 mm chirped optical fiber grating,” IEEE Photon. Technol. Lett. 7, 158–160 (1995).
[CrossRef]

Liu, Y.

Y. Liu and K. S. Chiang, “Wavelength switching of picosecond pulses generated from a self-seeded Fabry–Perot laser diode with a tilted fiber Bragg grating formed in a graded-index multimode fiber,” Appl. Opt. 50, 829–834 (2011).
[CrossRef]

S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
[CrossRef]

Lourtioz, J.-M.

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

Malinowski, A.

K. T. Vu, A. Malinowski, M. A. F. Roelens, and D. J. Richardson, “Detailed comparison of injection-seeded and self-seeded performance of a 1060 nm gain-switched Fabry–Pérot laser diode,” IEEE J. Quantum Electron. 44, 645–651 (2008).
[CrossRef]

Meier, J.

Mohammed, W.

Ouellette, F.

K. A. Ahmed, B. J. Eggleton, H. Liu, P. A. Krug, and F. Ouellette, “Simultaneous mode selection and pulse compression of gain-switched pulses from a Fabry–Perot laser using a 40 mm chirped optical fiber grating,” IEEE Photon. Technol. Lett. 7, 158–160 (1995).
[CrossRef]

Qian, L.

Richardson, D. J.

K. T. Vu, A. Malinowski, M. A. F. Roelens, and D. J. Richardson, “Detailed comparison of injection-seeded and self-seeded performance of a 1060 nm gain-switched Fabry–Pérot laser diode,” IEEE J. Quantum Electron. 44, 645–651 (2008).
[CrossRef]

Roelens, M. A. F.

K. T. Vu, A. Malinowski, M. A. F. Roelens, and D. J. Richardson, “Detailed comparison of injection-seeded and self-seeded performance of a 1060 nm gain-switched Fabry–Pérot laser diode,” IEEE J. Quantum Electron. 44, 645–651 (2008).
[CrossRef]

Scarani, V.

N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, “Direct measurement of superluminal group velocity and of signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef]

Schell, M.

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

Shu, C.

S. Yam and C. Shu, “Fast wavelength-tunable multichannel switching using a self-injection seeding scheme,” IEEE J. Quantum Electron. 35, 228–233 (1999).
[CrossRef]

Y. C. Lee and C. Shu, “Wavelength-tunable nearly transform-limited pulses generated by self-injection seeding of a laser diode at an arbitrary repetition rate,” IEEE Photon. Technol. Lett. 9, 590–592 (1997).
[CrossRef]

Smith, P. W. E.

Stelmakh, N.

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

Tateda, M.

M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol. 7, 1217–1224 (1989).
[CrossRef]

Utz, W.

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

Vu, K. T.

K. T. Vu, A. Malinowski, M. A. F. Roelens, and D. J. Richardson, “Detailed comparison of injection-seeded and self-seeded performance of a 1060 nm gain-switched Fabry–Pérot laser diode,” IEEE J. Quantum Electron. 44, 645–651 (2008).
[CrossRef]

Wegmuller, M.

N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, “Direct measurement of superluminal group velocity and of signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef]

Williams, J. A. R.

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

Xie, J. M.

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

Yam, S.

S. Yam and C. Shu, “Fast wavelength-tunable multichannel switching using a self-injection seeding scheme,” IEEE J. Quantum Electron. 35, 228–233 (1999).
[CrossRef]

Zhang, L.

S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
[CrossRef]

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. Huhse, M. Schell, W. Utz, D. Bimberg, J. A. R. Williams, L. Zhang, and I. Bennion, “Fast wavelength switching of semiconductor laser pulses by self-seeding,” Appl. Phys. Lett. 69, 2018–2020 (1996).
[CrossRef]

Electron. Lett. (1)

M. Cavelier, N. Stelmakh, J. M. Xie, L. Chusseau, J.-M. Lourtioz, C. Kazmierski, and N. Bouadma, “Picosecond (<2.5  ps) wavelength-tunable (−20  nm) semiconductor laser pulses with repetition rates up to 12 GHz,” Electron. Lett. 28, 224–226 (1992).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. T. Vu, A. Malinowski, M. A. F. Roelens, and D. J. Richardson, “Detailed comparison of injection-seeded and self-seeded performance of a 1060 nm gain-switched Fabry–Pérot laser diode,” IEEE J. Quantum Electron. 44, 645–651 (2008).
[CrossRef]

S. Yam and C. Shu, “Fast wavelength-tunable multichannel switching using a self-injection seeding scheme,” IEEE J. Quantum Electron. 35, 228–233 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

K. A. Ahmed, B. J. Eggleton, H. Liu, P. A. Krug, and F. Ouellette, “Simultaneous mode selection and pulse compression of gain-switched pulses from a Fabry–Perot laser using a 40 mm chirped optical fiber grating,” IEEE Photon. Technol. Lett. 7, 158–160 (1995).
[CrossRef]

S. Li, K. S. Chiang, W. A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “Self-seeding of Fabry–Perot laser diode for generating wavelength-tunable chirp-compensated single-mode pulses with high-sidemode suppression ratio,” IEEE Photon. Technol. Lett. 12, 1441–1443 (2000).
[CrossRef]

Y. C. Lee and C. Shu, “Wavelength-tunable nearly transform-limited pulses generated by self-injection seeding of a laser diode at an arbitrary repetition rate,” IEEE Photon. Technol. Lett. 9, 590–592 (1997).
[CrossRef]

J. Lightwave Technol. (1)

M. Tateda and T. Horiguchi, “Advances in optical time-domain reflectometry,” J. Lightwave Technol. 7, 1217–1224 (1989).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, “Direct measurement of superluminal group velocity and of signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef]

Other (3)

Corning HI 1060 FLEX and RC HI 1060 FLEX, http://www.corning.com/WorkArea/downloadasset.aspx?id=14719 .

Corning RC 1300 and RC 1550, http://www.corning.com/WorkArea/downloadasset.aspx?id=14801 .

Corning SMF-28e optical fiber with NexCor technology, http://www.princetel.com/datasheets/SMF28e.pdf .

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Figures (4)

Fig. 1.
Fig. 1.

Experimental setup. OSC, real-time oscilloscope; OSA, optical spectrum analyzer; CP, fiber coupler; PC, polarization controller; OC, optic circulator.

Fig. 2.
Fig. 2.

Output spectra of FP LD: (a) the repetition frequency and the round-trip frequency are not matched, (b) single mode operation case. The central wavelength is measured as 1550.55 nm.

Fig. 3.
Fig. 3.

Dependence of relative error on the length of L D ( m = 0.15 MHz ). The relative error indicates Δ ( v g ) / v g , and the relative length of L D indicates L D / L S 2 .

Fig. 4.
Fig. 4.

Simplified experimental setup. FC, frequency counter; OPM, optical powermeter; CP, fiber coupler; PC, polarization controller; OC, optic circulator.

Tables (1)

Tables Icon

Table 1. Group Velocity Test

Equations (13)

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

L S = v g f 1 n ,
L S = v g f 2 ( n + 1 ) ,
Δ f = f 2 f 1 = v g L S .
L S 2 = L S L D .
Δ f = v g L S 2 = v g L S L D .
v g = L D Δ f Δ f Δ f Δ f .
v g c = L D c Δ f c Δ f = L D L L L L = L D Δ L L .
c / Δ f = L L ,
δ f i = 1 T 2 Δ T = f i 2 τ ( i = 1 , 2 5 ) ,
δ f = 1 n i n | δ f i | 2 , ( n = 5 ) .
Δ ( v g ) = L D 2 Δ f 4 m 2 ( Δ f Δ f ) 4 + L D 2 Δ f 4 o 2 ( Δ f Δ f ) 4 + Δ f 2 Δ f 2 ( Δ L D ) 2 ( Δ f Δ f ) 2 ,
Δ ( v g ) = m 2 L S 2 4 + ( L S 2 + L D ) 4 L D 2 + v g 2 Δ L D 2 L D 2 = m 2 L S 2 4 + ( L S 2 + L D ) 4 L D 2 + 2.25 × 10 6 × v g 2 .
Δ ( v g ) = m 2 L S 2 4 + ( L S 2 + L D ) 4 L D 2 .

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