Abstract

Abstract: This paper investigates the effect of modal interference on the performance of hollow-core photonic bandgap fiber (HC-PBF) gas sensors. By optimizing mode launch, using proper length of sensing HC-PBF, and applying proper wavelength modulation in combination with lock-in detection, as well as appropriate digital signal processing, an estimated lower detection limit of less than 1 part-per-million by volume (ppmv) acetylene is achieved.

© 2014 Optical Society of America

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2013 (3)

2012 (2)

2011 (4)

K. S. Lee, Y. K. Lee, and S. H. Jang, “A novel grating modulation technique for photonic bandgap fiber gas sensors,” IEEE Photon. Technol. Lett. 23(10), 624–626 (2011).
[Crossref]

H. Lehmann, H. Bartelt, R. Willsch, R. Amézcua-Correa, and J. C. Knight, “In-line gas sensor based on a photonic bandgap fiber with laser-drilled lateral microchannels,” IEEE Sens. J. 11(11), 2926–2931 (2011).
[Crossref]

J. K. Lyngsø, C. Jakobsen, H. R. Simonsen, and J. Broeng, “Single-mode 7-cell core hollow core photonic crystal fiber with increased bandwidth,” Proc. SPIE 7753, 77533Q (2011).
[Crossref]

G. Stewart, W. Johnstone, J. R. P. Bain, K. Puxton, and K. Duffin, “Recovery of absolute gas absorption line shapes using tunable diode laser spectroscopy with wavelength modulation-part 1: Theoretical analysis,” J. Lightwave Technol. 29(6), 811–821 (2011).

2009 (3)

2008 (1)

2007 (3)

2005 (1)

Y. L. Hoo, W. Jin, H. L. Ho, J. Ju, and D. N. Wang, “Gas diffusion measurement using hollow-core photonic bandgap fiber,” Sens. Actuators B Chem. 105(2), 183–186 (2005).
[Crossref]

2004 (1)

2003 (1)

1999 (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

1998 (1)

1994 (2)

1992 (1)

1989 (1)

1982 (1)

D. T. Cassidy and J. Reid, “Harmonic detection with tunable diode lasers—two-tone modulation,” Appl. Phys. B 29(4), 279–285 (1982).
[Crossref]

Aghaie, K. Z.

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Amézcua-Correa, R.

H. Lehmann, H. Bartelt, R. Willsch, R. Amézcua-Correa, and J. C. Knight, “In-line gas sensor based on a photonic bandgap fiber with laser-drilled lateral microchannels,” IEEE Sens. J. 11(11), 2926–2931 (2011).
[Crossref]

Araújo, F. M.

F. Magalhães, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Methane detection system based on wavelength modulation spectroscopy and hollow-core fibres,” in Proceedings of IEEE Sensors Conference (IEEE, 2008), pp. 1277–1280.
[Crossref]

Bain, J. R. P.

Barabadi, B.

Bartelt, H.

H. Lehmann, H. Bartelt, R. Willsch, R. Amézcua-Correa, and J. C. Knight, “In-line gas sensor based on a photonic bandgap fiber with laser-drilled lateral microchannels,” IEEE Sens. J. 11(11), 2926–2931 (2011).
[Crossref]

Birks, T. A.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Broeng, J.

J. K. Lyngsø, C. Jakobsen, H. R. Simonsen, and J. Broeng, “Single-mode 7-cell core hollow core photonic crystal fiber with increased bandwidth,” Proc. SPIE 7753, 77533Q (2011).
[Crossref]

Cao, Y. C.

W. Jin, H. L. Ho, Y. C. Cao, J. Ju, and L. F. Qi, “Gas detection with micro- and nano-engineered optical fibers,” Opt. Fiber Technol. 19(6), 741–759 (2013).
[Crossref]

Carlisle, C. B.

Carr, L. W.

Carvalho, J. P.

F. Magalhães, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Methane detection system based on wavelength modulation spectroscopy and hollow-core fibres,” in Proceedings of IEEE Sensors Conference (IEEE, 2008), pp. 1277–1280.
[Crossref]

Cassidy, D. T.

D. T. Cassidy and J. Reid, “Harmonic detection with tunable diode lasers—two-tone modulation,” Appl. Phys. B 29(4), 279–285 (1982).
[Crossref]

Conde, O. M.

Cooper, D. E.

Creedon, K. J.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Cubillas, A. M.

Demokan, M. S.

Desantolo, A.

Digonnet, M. J. F.

DiMarcello, F.

DiMarcello, F. V.

Duffin, K.

Dulashko, Y.

Fan, S.

Ferreira, L. A.

F. Magalhães, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Methane detection system based on wavelength modulation spectroscopy and hollow-core fibres,” in Proceedings of IEEE Sensors Conference (IEEE, 2008), pp. 1277–1280.
[Crossref]

Fini, J. M.

Gayraud, N.

J. P. Parry, B. C. Griffiths, N. Gayraud, E. D. McNaghten, A. M. Parkes, W. N. MacPherson, and D. P. Hand, “Towards practical gas sensing with micro-structured fibres,” Meas. Sci. Technol. 20(7), 075301 (2009).
[Crossref]

Griffiths, B. C.

J. P. Parry, B. C. Griffiths, N. Gayraud, E. D. McNaghten, A. M. Parkes, W. N. MacPherson, and D. P. Hand, “Towards practical gas sensing with micro-structured fibres,” Meas. Sci. Technol. 20(7), 075301 (2009).
[Crossref]

Hand, D. P.

J. P. Parry, B. C. Griffiths, N. Gayraud, E. D. McNaghten, A. M. Parkes, W. N. MacPherson, and D. P. Hand, “Towards practical gas sensing with micro-structured fibres,” Meas. Sci. Technol. 20(7), 075301 (2009).
[Crossref]

Hansen, T. P.

Hanson, R. K.

Hassan, M.

Ho, H. L.

W. Jin, H. L. Ho, Y. C. Cao, J. Ju, and L. F. Qi, “Gas detection with micro- and nano-engineered optical fibers,” Opt. Fiber Technol. 19(6), 741–759 (2013).
[Crossref]

Y. L. Hoo, W. Jin, H. L. Ho, J. Ju, and D. N. Wang, “Gas diffusion measurement using hollow-core photonic bandgap fiber,” Sens. Actuators B Chem. 105(2), 183–186 (2005).
[Crossref]

Y. L. Hoo, W. Jin, C. Shi, H. L. Ho, D. N. Wang, and S. C. Ruan, “Design and modeling of a photonic crystal fiber gas sensor,” Appl. Opt. 42(18), 3509–3515 (2003).
[Crossref] [PubMed]

Hoo, Y. L.

Y. L. Hoo, W. Jin, H. L. Ho, J. Ju, and D. N. Wang, “Gas diffusion measurement using hollow-core photonic bandgap fiber,” Sens. Actuators B Chem. 105(2), 183–186 (2005).
[Crossref]

Y. L. Hoo, W. Jin, C. Shi, H. L. Ho, D. N. Wang, and S. C. Ruan, “Design and modeling of a photonic crystal fiber gas sensor,” Appl. Opt. 42(18), 3509–3515 (2003).
[Crossref] [PubMed]

Jakobsen, C.

J. K. Lyngsø, C. Jakobsen, H. R. Simonsen, and J. Broeng, “Single-mode 7-cell core hollow core photonic crystal fiber with increased bandwidth,” Proc. SPIE 7753, 77533Q (2011).
[Crossref]

Jang, S. H.

K. S. Lee, Y. K. Lee, and S. H. Jang, “A novel grating modulation technique for photonic bandgap fiber gas sensors,” IEEE Photon. Technol. Lett. 23(10), 624–626 (2011).
[Crossref]

Jeffries, J. B.

Jin, W.

Johnstone, W.

Ju, J.

W. Jin, H. L. Ho, Y. C. Cao, J. Ju, and L. F. Qi, “Gas detection with micro- and nano-engineered optical fibers,” Opt. Fiber Technol. 19(6), 741–759 (2013).
[Crossref]

Y. L. Hoo, W. Jin, H. L. Ho, J. Ju, and D. N. Wang, “Gas diffusion measurement using hollow-core photonic bandgap fiber,” Sens. Actuators B Chem. 105(2), 183–186 (2005).
[Crossref]

Knight, J. C.

H. Lehmann, H. Bartelt, R. Willsch, R. Amézcua-Correa, and J. C. Knight, “In-line gas sensor based on a photonic bandgap fiber with laser-drilled lateral microchannels,” IEEE Sens. J. 11(11), 2926–2931 (2011).
[Crossref]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Koch, K. W.

Lazaro, J. M.

Lee, K. S.

K. S. Lee, Y. K. Lee, and S. H. Jang, “A novel grating modulation technique for photonic bandgap fiber gas sensors,” IEEE Photon. Technol. Lett. 23(10), 624–626 (2011).
[Crossref]

Lee, Y. K.

K. S. Lee, Y. K. Lee, and S. H. Jang, “A novel grating modulation technique for photonic bandgap fiber gas sensors,” IEEE Photon. Technol. Lett. 23(10), 624–626 (2011).
[Crossref]

Lehmann, H.

H. Lehmann, H. Bartelt, R. Willsch, R. Amézcua-Correa, and J. C. Knight, “In-line gas sensor based on a photonic bandgap fiber with laser-drilled lateral microchannels,” IEEE Sens. J. 11(11), 2926–2931 (2011).
[Crossref]

Li, M.

Li, X.

X. Li, J. Liang, S. Lin, Y. Zimin, Y. Zhang, and T. Ueda, “NIR spectrum analysis of natural gas based on hollow-core photonic bandgap fiber,” IEEE Sens. J. 12(7), 2362–2367 (2012).
[Crossref]

Liang, J.

X. Li, J. Liang, S. Lin, Y. Zimin, Y. Zhang, and T. Ueda, “NIR spectrum analysis of natural gas based on hollow-core photonic bandgap fiber,” IEEE Sens. J. 12(7), 2362–2367 (2012).
[Crossref]

Lin, S.

X. Li, J. Liang, S. Lin, Y. Zimin, Y. Zhang, and T. Ueda, “NIR spectrum analysis of natural gas based on hollow-core photonic bandgap fiber,” IEEE Sens. J. 12(7), 2362–2367 (2012).
[Crossref]

Lopez-Higuera, J. M.

Ludvigsen, H.

Lyngsø, J. K.

J. K. Lyngsø, C. Jakobsen, H. R. Simonsen, and J. Broeng, “Single-mode 7-cell core hollow core photonic crystal fiber with increased bandwidth,” Proc. SPIE 7753, 77533Q (2011).
[Crossref]

MacPherson, W. N.

J. P. Parry, B. C. Griffiths, N. Gayraud, E. D. McNaghten, A. M. Parkes, W. N. MacPherson, and D. P. Hand, “Towards practical gas sensing with micro-structured fibres,” Meas. Sci. Technol. 20(7), 075301 (2009).
[Crossref]

Magalhães, F.

F. Magalhães, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Methane detection system based on wavelength modulation spectroscopy and hollow-core fibres,” in Proceedings of IEEE Sensors Conference (IEEE, 2008), pp. 1277–1280.
[Crossref]

Mangan, B.

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Martinelli, R. U.

McNaghten, E. D.

J. P. Parry, B. C. Griffiths, N. Gayraud, E. D. McNaghten, A. M. Parkes, W. N. MacPherson, and D. P. Hand, “Towards practical gas sensing with micro-structured fibres,” Meas. Sci. Technol. 20(7), 075301 (2009).
[Crossref]

Mencaglia, A.

Meng, L.

Menna, R. J.

Monberg, E.

Monberg, E. M.

Nicholson, J. W.

Ortega, A.

Ortiz, R.

Parkes, A. M.

J. P. Parry, B. C. Griffiths, N. Gayraud, E. D. McNaghten, A. M. Parkes, W. N. MacPherson, and D. P. Hand, “Towards practical gas sensing with micro-structured fibres,” Meas. Sci. Technol. 20(7), 075301 (2009).
[Crossref]

Parry, J. P.

J. P. Parry, B. C. Griffiths, N. Gayraud, E. D. McNaghten, A. M. Parkes, W. N. MacPherson, and D. P. Hand, “Towards practical gas sensing with micro-structured fibres,” Meas. Sci. Technol. 20(7), 075301 (2009).
[Crossref]

Petersen, J. C.

Petrovich, M. N.

Philp, W.

Poletti, F.

M. N. Petrovich, F. Poletti, A. Van Brakel, and D. J. Richardson, “Robustly single mode hollow core photonic bandgap fiber,” Opt. Express 16(6), 4337–4346 (2008).
[Crossref] [PubMed]

M. N. Petrovich, F. Poletti, and D. J. Richardson, “Analysis of modal interference in photonic bandgap fibres,” in International Conference on Transparent Optical Networks (ICTON, 2010), pp. 1–4.
[Crossref]

Preier, H.

Puxton, K.

Qi, L. F.

W. Jin, H. L. Ho, Y. C. Cao, J. Ju, and L. F. Qi, “Gas detection with micro- and nano-engineered optical fibers,” Opt. Fiber Technol. 19(6), 741–759 (2013).
[Crossref]

Reid, J.

D. T. Cassidy and J. Reid, “Harmonic detection with tunable diode lasers—two-tone modulation,” Appl. Phys. B 29(4), 279–285 (1982).
[Crossref]

Richardson, D. J.

M. N. Petrovich, F. Poletti, A. Van Brakel, and D. J. Richardson, “Robustly single mode hollow core photonic bandgap fiber,” Opt. Express 16(6), 4337–4346 (2008).
[Crossref] [PubMed]

M. N. Petrovich, F. Poletti, and D. J. Richardson, “Analysis of modal interference in photonic bandgap fibres,” in International Conference on Transparent Optical Networks (ICTON, 2010), pp. 1–4.
[Crossref]

Rieker, G. B.

Riris, H.

Ritari, T.

Roberts, P. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Ruan, S. C.

Russell, P. St. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Santos, J. L.

F. Magalhães, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Methane detection system based on wavelength modulation spectroscopy and hollow-core fibres,” in Proceedings of IEEE Sensors Conference (IEEE, 2008), pp. 1277–1280.
[Crossref]

Shi, C.

Silva-Lopez, M.

Simonsen, H. R.

J. K. Lyngsø, C. Jakobsen, H. R. Simonsen, and J. Broeng, “Single-mode 7-cell core hollow core photonic crystal fiber with increased bandwidth,” Proc. SPIE 7753, 77533Q (2011).
[Crossref]

T. Ritari, J. Tuominen, H. Ludvigsen, J. C. Petersen, T. Sørensen, T. P. Hansen, and H. R. Simonsen, “Gas sensing using air-guiding photonic bandgap fibers,” Opt. Express 12(17), 4080–4087 (2004).
[Crossref] [PubMed]

Sørensen, T.

Stewart, G.

Sun, H. C.

Tuominen, J.

Ueda, T.

X. Li, J. Liang, S. Lin, Y. Zimin, Y. Zhang, and T. Ueda, “NIR spectrum analysis of natural gas based on hollow-core photonic bandgap fiber,” IEEE Sens. J. 12(7), 2362–2367 (2012).
[Crossref]

Van Brakel, A.

Wang, D. N.

Y. L. Hoo, W. Jin, H. L. Ho, J. Ju, and D. N. Wang, “Gas diffusion measurement using hollow-core photonic bandgap fiber,” Sens. Actuators B Chem. 105(2), 183–186 (2005).
[Crossref]

Y. L. Hoo, W. Jin, C. Shi, H. L. Ho, D. N. Wang, and S. C. Ruan, “Design and modeling of a photonic crystal fiber gas sensor,” Appl. Opt. 42(18), 3509–3515 (2003).
[Crossref] [PubMed]

Wang, Y.

Warren, R. E.

West, J. A.

Whittaker, E. A.

Willsch, R.

H. Lehmann, H. Bartelt, R. Willsch, R. Amézcua-Correa, and J. C. Knight, “In-line gas sensor based on a photonic bandgap fiber with laser-drilled lateral microchannels,” IEEE Sens. J. 11(11), 2926–2931 (2011).
[Crossref]

Windeler, R. S.

Wynne, R. M.

Xiao, L.

Zhang, Y.

X. Li, J. Liang, S. Lin, Y. Zimin, Y. Zhang, and T. Ueda, “NIR spectrum analysis of natural gas based on hollow-core photonic bandgap fiber,” IEEE Sens. J. 12(7), 2362–2367 (2012).
[Crossref]

Zhao, C.-L.

Zimin, Y.

X. Li, J. Liang, S. Lin, Y. Zimin, Y. Zhang, and T. Ueda, “NIR spectrum analysis of natural gas based on hollow-core photonic bandgap fiber,” IEEE Sens. J. 12(7), 2362–2367 (2012).
[Crossref]

Appl. Opt. (5)

Appl. Phys. B (1)

D. T. Cassidy and J. Reid, “Harmonic detection with tunable diode lasers—two-tone modulation,” Appl. Phys. B 29(4), 279–285 (1982).
[Crossref]

IEEE Photon. Technol. Lett. (1)

K. S. Lee, Y. K. Lee, and S. H. Jang, “A novel grating modulation technique for photonic bandgap fiber gas sensors,” IEEE Photon. Technol. Lett. 23(10), 624–626 (2011).
[Crossref]

IEEE Sens. J. (2)

H. Lehmann, H. Bartelt, R. Willsch, R. Amézcua-Correa, and J. C. Knight, “In-line gas sensor based on a photonic bandgap fiber with laser-drilled lateral microchannels,” IEEE Sens. J. 11(11), 2926–2931 (2011).
[Crossref]

X. Li, J. Liang, S. Lin, Y. Zimin, Y. Zhang, and T. Ueda, “NIR spectrum analysis of natural gas based on hollow-core photonic bandgap fiber,” IEEE Sens. J. 12(7), 2362–2367 (2012).
[Crossref]

J. Lightwave Technol. (6)

Meas. Sci. Technol. (1)

J. P. Parry, B. C. Griffiths, N. Gayraud, E. D. McNaghten, A. M. Parkes, W. N. MacPherson, and D. P. Hand, “Towards practical gas sensing with micro-structured fibres,” Meas. Sci. Technol. 20(7), 075301 (2009).
[Crossref]

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Y. L. Hoo, W. Jin, H. L. Ho, J. Ju, and D. N. Wang, “Gas diffusion measurement using hollow-core photonic bandgap fiber,” Sens. Actuators B Chem. 105(2), 183–186 (2005).
[Crossref]

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A. Brakel, C. J. Misas, T. T. Ng, P. Petropoulos, J. P. Dakin, C. Grivas, M. N. Petrovich, and D. J. Richardson, “Cavity ring-down in a photonic bandgap fiber gas cell,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2008), paper CThEE4.

F. Magalhães, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Methane detection system based on wavelength modulation spectroscopy and hollow-core fibres,” in Proceedings of IEEE Sensors Conference (IEEE, 2008), pp. 1277–1280.
[Crossref]

M. N. Petrovich, F. Poletti, and D. J. Richardson, “Analysis of modal interference in photonic bandgap fibres,” in International Conference on Transparent Optical Networks (ICTON, 2010), pp. 1–4.
[Crossref]

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Press, 2007), Chap. 2.

A. W. Synder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), Chap. 20.

Y. Cao, “Quartz-enhanced photoacoustic spectroscopy and its association with fiber-optic devices,” Ph.D. Thesis, The Hong Kong Polytechnic University (2012).

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Figures (14)

Fig. 1
Fig. 1 (a) Setup for transmission spectrum measurement, (b) the scanning electron microscopy (SEM) image of HC-1550-02 fiber’s cross-section.
Fig. 2
Fig. 2 (a) Transmission spectrums of HC-PBF samples with different lengths, and (b-e) Fourier transforms of the spectrums in (a).
Fig. 3
Fig. 3 Output mode intensity profile of 35 m HC-1550-02 fiber with (a) optimal launch, (b) 3 μm offset launch, (c) 6 μm offset launch. Output mode intensity profile of 110 cm HC-1550-02 fiber with (d) optimal launch, (e) 3 μm offset launch, (f) 6 μm offset launch.
Fig. 4
Fig. 4 (a) Geometry model of HC-1550-02 fiber used for calculating mode contents, (b) and (c) are the mode fields of HE11 modes, (d)-(g) are the mode fields of quasi-TM01, odd HE21, even HE21 and quasi-TE01 modes respectively, (h)-(k) are the mode fields of odd HE31, odd EH11, even EH11, and even HE31 modes, (l) and (m) are the mode fields of odd HE12 and even HE12 modes.
Fig. 5
Fig. 5 A HC-PBF sample with its input end mechanical spliced to a SMF with a gap
Fig. 6
Fig. 6 (a) Normalized transmission spectrums of 13 m HC-1550-02 fiber with different gap sizes, and (b) Fourier transforms of the transmission spectrums in (a). Details of the mode beatings in (b) are shown in (c) for LP11 and in (d) for LP21.
Fig. 7
Fig. 7 Mode excitation from SMF to HC-1550-02 fiber.
Fig. 8
Fig. 8 Mode launch efficiency of (a) HE11 mode, (b) TE01 mode, (c) odd HE21 mode, (d) even HE21 mode, (e) even HE31 mode. The black and red lines correspond to lateral offsets of 2 and 3 μm, respectively.
Fig. 9
Fig. 9 Experimental setup for gas detection with WMS. DFB: distributed-feedback laser, PD: photodetector, DAQ: data acquisition.
Fig. 10
Fig. 10 Second-harmonic outputs (signals) for modulation voltages of (a) 1 V and (b) 1.9 V with wavenumber scanning across the absorption line. The black curves are the original lock-in outputs and the red curves are the results after digital filtering. (Wavenumber (ν) = 1/λ)
Fig. 11
Fig. 11 Second-harmonic output (standard deviation of fluctuating noise when the wavelength was scanned in the wavelength range away from the absorption peak) as function of modulation voltage.
Fig. 12
Fig. 12 Variation of second-harmonic output with time for the modulation voltage of 1.9 V. The laser wavelength was not scanned by tuned to the peak of the absorption line.
Fig. 13
Fig. 13 Model for analyzing modal interference.
Fig. 14
Fig. 14 Black line: the second-harmonic output (noise) calculated from the standard deviation of the fluctuating noise when the wavelength was scanned in the wavelength range away from the absorption peak, and red line: the second-order Bessel function of the first kind as function of δωΔτ . The two curves are not in the same vertical scale.

Tables (1)

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Table 1 Calculated phase, group and differential group RI, and measured differential group RI at 1530 nm

Equations (17)

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n effg (λ)= n effp (λ)λ·d n effp (λ)/dλ
E( x,y,0 )= E 0 exp[ ( x 2 + y 2 )/ a 2 ]
E( x,y,L )= E 0 e ikL 1i λL π a 2 exp( i k 2R ( x 2 + y 2 ) )exp( x 2 + y 2 ω 2 )
E t ( x,y ) 2 n air n air + n HC E(x,y,L)
H t ( x,y ) 2 n HC n air + n HC H(x,y,L)
E t ( x,y )E(x,y,L)
H t ( x,y )H(x,y,L)
E t ( x,y )= j a j e tj (x,y) + E tr (x,y)
H t ( x,y )= j a j h tj (x,y) + H tr (x,y) ,
a j = A E t × h tj * · z dA A e tj × h tj * · z dA = A e tj * × H t · z dA A e tj * × h tj · z dA
I output = E output 2
E output = k 1 I 0 exp( r 1 αCL)cos(ω(t τ 1 ))+ k 2 I 0 exp( r 2 αCL)cos(ω(t τ 2 ))
I output k 1 I 0 ( 1+2 k 1 k 2 cosωΔτ )
ω=ϖ+δωsin( ω m t)
I MI k 1 I 0 =2 k 1 k 2 cos{ ϖΔτ+[ δωΔτ ]sin( ω m t) }
cos{ ϖΔτ+[ δωΔτ ]sin( ω m t) } = J 0 (δωΔτ)cos(ϖΔτ) n=1 A n sin(n ω m t+ θ n )
I MI,2 | J 2 (δωΔτ)cos(ϖΔτ) |

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