Abstract

The optical low-coherent interferometric technology for long-period grating (LPG) Mach–Zehnder interferometers is described. By including the coupling and recoupling behaviors of a LPG pair, a numerical model is developed to analyze the output reflectogram of the system. The effects of the grating interval, grating length, grating strength, and light source on the output reflectogram have been comprehensively discussed, which reveals that the low-coherence reflectometry offers the capability of interrogating the multiplexed sensors based on LPG pairs. A comparison of the calculated and experimental results is presented, and an excellent agreement between the simulation and the measurement is shown.

© 2006 Optical Society of America

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  1. B. L. Danielson and C. D. Whittenberg, "Guided-wave reflectometery with micrometer resolution," Appl. Opt. 26, 2836-2842 (1987).
    [CrossRef] [PubMed]
  2. S. Keren and M. Horowitz, "Interrogation of fiber gratings by use of low-coherence spectral interferometry of noiselike pulses," Opt. Lett. 26, 328-330 (2001).
    [CrossRef]
  3. S. D. Dyer, K. B. Rochford, and A. H. Rose, "Fast and accurate low-coherence interferometric measurements of fiber Bragg grating dispersion and reflectance," Opt. Express 5, 262-266 (1999).
    [CrossRef] [PubMed]
  4. S. D. Dyer and K. B. Rochford, "Low-coherence interferometric measurements of the dispersion of multiple fiber Bragg gratings," IEEE Photon. Technol. Lett. 13, 230-232 (2001).
    [CrossRef]
  5. Y. Gottesman, E. V. K. Rao, and D. G. Rabus, "New methodology to evaluate the performance of ring resonators using optical low-coherence reflectometry," J. Lightwave Technol. 22, 1566-1572 (2004).
    [CrossRef]
  6. C. Palavicini, Y. Jaouen, G. Debarge, E. Kerrinckx, Y. Quiquempois, M. Douay, C. Lepers, A. F. Obaton, and G. Melin, "Phase-sensitive optical low-coherence reflectometry technique applied to the characterization of photonic crystal fiber properties," Opt. Lett. 30, 361-363 (2005).
    [CrossRef] [PubMed]
  7. Z. G. Guan, A. P. Zhang, M. Jiang, and S. He, "Low-coherence interrogation scheme for the multiplexed sensors based on long-period grating Mach-Zehnder interferometers, IEEE Photon. Technol. Lett. 18, 832-834 (2006).
    [CrossRef]
  8. V. Bhatia and A. M. Vengsarkar, "Optical fiber long-period grating sensors," Opt. Lett. 21, 692-694 (1996).
    [CrossRef] [PubMed]
  9. H. J. Patrick, A. D. Kersey, F. Bucholtz, K. J. Ewing, J. B. Judkins, and A. M. Vengsarkar, "Chemical sensor based on long-period fibre grating response to index of refraction," in Conference on Lasers and Electro-Optics, Vol. 11 of OSA Technical Digests (Optical Society of America, 1997), pp. 420-421.
  10. C. C. Ye, S. W. James, and R. P. Tatam, "Simultaneous temperature and bending sensing with long-period fiber gratings," Opt. Lett. 25, 1007-1009 (2000).
    [CrossRef]
  11. P. L. Swart, "Long-period grating Michelson refractometric sensor," Meas. Sci. Technol. 15, 1576-1580 (2004).
    [CrossRef]
  12. A. P. Zhang, L. Y. Shao, J. F. Ding, and S. He, "Sandwiched long-period gratings for simultaneous measurement of refractive index and temperature," IEEE Photon. Technol. Lett. 17, 2397-2399 (2005).
    [CrossRef]
  13. Y. G. Han, B. H. Lee, W. T. Han, U. C. Paek, and Y. Chung, "Fibre-optic sensing applications of a pair of long-period fibre gratings," Meas. Sci. Technol. 12, 778-781 (2001).
    [CrossRef]
  14. T. Allsop, R. Reeves, D. J. Webb, I. Bennion, and R. Neal, "A high sensitivity refractometer based upon a long period grating Mach-Zehnder interferometer," Rev. Sci. Instrum. 73, 1702-1705 (2002).
    [CrossRef]
  15. J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, "Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor," IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
    [CrossRef]
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    [CrossRef]
  17. A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
    [CrossRef]

2006 (1)

Z. G. Guan, A. P. Zhang, M. Jiang, and S. He, "Low-coherence interrogation scheme for the multiplexed sensors based on long-period grating Mach-Zehnder interferometers, IEEE Photon. Technol. Lett. 18, 832-834 (2006).
[CrossRef]

2005 (4)

A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
[CrossRef]

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, "Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor," IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

A. P. Zhang, L. Y. Shao, J. F. Ding, and S. He, "Sandwiched long-period gratings for simultaneous measurement of refractive index and temperature," IEEE Photon. Technol. Lett. 17, 2397-2399 (2005).
[CrossRef]

C. Palavicini, Y. Jaouen, G. Debarge, E. Kerrinckx, Y. Quiquempois, M. Douay, C. Lepers, A. F. Obaton, and G. Melin, "Phase-sensitive optical low-coherence reflectometry technique applied to the characterization of photonic crystal fiber properties," Opt. Lett. 30, 361-363 (2005).
[CrossRef] [PubMed]

2004 (2)

2002 (1)

T. Allsop, R. Reeves, D. J. Webb, I. Bennion, and R. Neal, "A high sensitivity refractometer based upon a long period grating Mach-Zehnder interferometer," Rev. Sci. Instrum. 73, 1702-1705 (2002).
[CrossRef]

2001 (3)

S. D. Dyer and K. B. Rochford, "Low-coherence interferometric measurements of the dispersion of multiple fiber Bragg gratings," IEEE Photon. Technol. Lett. 13, 230-232 (2001).
[CrossRef]

Y. G. Han, B. H. Lee, W. T. Han, U. C. Paek, and Y. Chung, "Fibre-optic sensing applications of a pair of long-period fibre gratings," Meas. Sci. Technol. 12, 778-781 (2001).
[CrossRef]

S. Keren and M. Horowitz, "Interrogation of fiber gratings by use of low-coherence spectral interferometry of noiselike pulses," Opt. Lett. 26, 328-330 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

1996 (1)

1987 (1)

Allsop, T.

T. Allsop, R. Reeves, D. J. Webb, I. Bennion, and R. Neal, "A high sensitivity refractometer based upon a long period grating Mach-Zehnder interferometer," Rev. Sci. Instrum. 73, 1702-1705 (2002).
[CrossRef]

Bennion, I.

T. Allsop, R. Reeves, D. J. Webb, I. Bennion, and R. Neal, "A high sensitivity refractometer based upon a long period grating Mach-Zehnder interferometer," Rev. Sci. Instrum. 73, 1702-1705 (2002).
[CrossRef]

Bhatia, V.

Bucholtz, F.

H. J. Patrick, A. D. Kersey, F. Bucholtz, K. J. Ewing, J. B. Judkins, and A. M. Vengsarkar, "Chemical sensor based on long-period fibre grating response to index of refraction," in Conference on Lasers and Electro-Optics, Vol. 11 of OSA Technical Digests (Optical Society of America, 1997), pp. 420-421.

Chen, X. W.

A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
[CrossRef]

Chung, Y.

Y. G. Han, B. H. Lee, W. T. Han, U. C. Paek, and Y. Chung, "Fibre-optic sensing applications of a pair of long-period fibre gratings," Meas. Sci. Technol. 12, 778-781 (2001).
[CrossRef]

Danielson, B. L.

Debarge, G.

Ding, J. F.

A. P. Zhang, L. Y. Shao, J. F. Ding, and S. He, "Sandwiched long-period gratings for simultaneous measurement of refractive index and temperature," IEEE Photon. Technol. Lett. 17, 2397-2399 (2005).
[CrossRef]

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, "Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor," IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Douay, M.

Duan, G. H.

Dyer, S. D.

S. D. Dyer and K. B. Rochford, "Low-coherence interferometric measurements of the dispersion of multiple fiber Bragg gratings," IEEE Photon. Technol. Lett. 13, 230-232 (2001).
[CrossRef]

S. D. Dyer, K. B. Rochford, and A. H. Rose, "Fast and accurate low-coherence interferometric measurements of fiber Bragg grating dispersion and reflectance," Opt. Express 5, 262-266 (1999).
[CrossRef] [PubMed]

Ewing, K. J.

H. J. Patrick, A. D. Kersey, F. Bucholtz, K. J. Ewing, J. B. Judkins, and A. M. Vengsarkar, "Chemical sensor based on long-period fibre grating response to index of refraction," in Conference on Lasers and Electro-Optics, Vol. 11 of OSA Technical Digests (Optical Society of America, 1997), pp. 420-421.

Gallion, P.

Gottesman, Y.

Guan, Z. G.

Z. G. Guan, A. P. Zhang, M. Jiang, and S. He, "Low-coherence interrogation scheme for the multiplexed sensors based on long-period grating Mach-Zehnder interferometers, IEEE Photon. Technol. Lett. 18, 832-834 (2006).
[CrossRef]

A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
[CrossRef]

Han, W. T.

Y. G. Han, B. H. Lee, W. T. Han, U. C. Paek, and Y. Chung, "Fibre-optic sensing applications of a pair of long-period fibre gratings," Meas. Sci. Technol. 12, 778-781 (2001).
[CrossRef]

Han, Y. G.

Y. G. Han, B. H. Lee, W. T. Han, U. C. Paek, and Y. Chung, "Fibre-optic sensing applications of a pair of long-period fibre gratings," Meas. Sci. Technol. 12, 778-781 (2001).
[CrossRef]

He, S.

Z. G. Guan, A. P. Zhang, M. Jiang, and S. He, "Low-coherence interrogation scheme for the multiplexed sensors based on long-period grating Mach-Zehnder interferometers, IEEE Photon. Technol. Lett. 18, 832-834 (2006).
[CrossRef]

A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
[CrossRef]

A. P. Zhang, L. Y. Shao, J. F. Ding, and S. He, "Sandwiched long-period gratings for simultaneous measurement of refractive index and temperature," IEEE Photon. Technol. Lett. 17, 2397-2399 (2005).
[CrossRef]

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, "Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor," IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Horowitz, M.

James, S. W.

Jaouen, Y.

Jiang, M.

Z. G. Guan, A. P. Zhang, M. Jiang, and S. He, "Low-coherence interrogation scheme for the multiplexed sensors based on long-period grating Mach-Zehnder interferometers, IEEE Photon. Technol. Lett. 18, 832-834 (2006).
[CrossRef]

Judkins, J. B.

H. J. Patrick, A. D. Kersey, F. Bucholtz, K. J. Ewing, J. B. Judkins, and A. M. Vengsarkar, "Chemical sensor based on long-period fibre grating response to index of refraction," in Conference on Lasers and Electro-Optics, Vol. 11 of OSA Technical Digests (Optical Society of America, 1997), pp. 420-421.

Keren, S.

Kerrinckx, E.

Kersey, A. D.

H. J. Patrick, A. D. Kersey, F. Bucholtz, K. J. Ewing, J. B. Judkins, and A. M. Vengsarkar, "Chemical sensor based on long-period fibre grating response to index of refraction," in Conference on Lasers and Electro-Optics, Vol. 11 of OSA Technical Digests (Optical Society of America, 1997), pp. 420-421.

Lee, B. H.

Y. G. Han, B. H. Lee, W. T. Han, U. C. Paek, and Y. Chung, "Fibre-optic sensing applications of a pair of long-period fibre gratings," Meas. Sci. Technol. 12, 778-781 (2001).
[CrossRef]

Lepers, C.

Melin, G.

Neal, R.

T. Allsop, R. Reeves, D. J. Webb, I. Bennion, and R. Neal, "A high sensitivity refractometer based upon a long period grating Mach-Zehnder interferometer," Rev. Sci. Instrum. 73, 1702-1705 (2002).
[CrossRef]

Obaton, A. F.

Paek, U. C.

Y. G. Han, B. H. Lee, W. T. Han, U. C. Paek, and Y. Chung, "Fibre-optic sensing applications of a pair of long-period fibre gratings," Meas. Sci. Technol. 12, 778-781 (2001).
[CrossRef]

Palavicini, C.

Patrick, H. J.

H. J. Patrick, A. D. Kersey, F. Bucholtz, K. J. Ewing, J. B. Judkins, and A. M. Vengsarkar, "Chemical sensor based on long-period fibre grating response to index of refraction," in Conference on Lasers and Electro-Optics, Vol. 11 of OSA Technical Digests (Optical Society of America, 1997), pp. 420-421.

Quiquempois, Y.

Rabus, D. G.

Rao, E. V. K.

Reeves, R.

T. Allsop, R. Reeves, D. J. Webb, I. Bennion, and R. Neal, "A high sensitivity refractometer based upon a long period grating Mach-Zehnder interferometer," Rev. Sci. Instrum. 73, 1702-1705 (2002).
[CrossRef]

Rochford, K. B.

S. D. Dyer and K. B. Rochford, "Low-coherence interferometric measurements of the dispersion of multiple fiber Bragg gratings," IEEE Photon. Technol. Lett. 13, 230-232 (2001).
[CrossRef]

S. D. Dyer, K. B. Rochford, and A. H. Rose, "Fast and accurate low-coherence interferometric measurements of fiber Bragg grating dispersion and reflectance," Opt. Express 5, 262-266 (1999).
[CrossRef] [PubMed]

Rose, A. H.

Shao, L. Y.

A. P. Zhang, L. Y. Shao, J. F. Ding, and S. He, "Sandwiched long-period gratings for simultaneous measurement of refractive index and temperature," IEEE Photon. Technol. Lett. 17, 2397-2399 (2005).
[CrossRef]

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, "Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor," IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Swart, P. L.

P. L. Swart, "Long-period grating Michelson refractometric sensor," Meas. Sci. Technol. 15, 1576-1580 (2004).
[CrossRef]

Tam, H. Y.

A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
[CrossRef]

Tatam, R. P.

Vengsarkar, A. M.

V. Bhatia and A. M. Vengsarkar, "Optical fiber long-period grating sensors," Opt. Lett. 21, 692-694 (1996).
[CrossRef] [PubMed]

H. J. Patrick, A. D. Kersey, F. Bucholtz, K. J. Ewing, J. B. Judkins, and A. M. Vengsarkar, "Chemical sensor based on long-period fibre grating response to index of refraction," in Conference on Lasers and Electro-Optics, Vol. 11 of OSA Technical Digests (Optical Society of America, 1997), pp. 420-421.

Webb, D. J.

T. Allsop, R. Reeves, D. J. Webb, I. Bennion, and R. Neal, "A high sensitivity refractometer based upon a long period grating Mach-Zehnder interferometer," Rev. Sci. Instrum. 73, 1702-1705 (2002).
[CrossRef]

Whittenberg, C. D.

Wiedmann, U.

Yan, J. H.

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, "Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor," IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
[CrossRef]

Ye, C. C.

Zhang, A. P.

Z. G. Guan, A. P. Zhang, M. Jiang, and S. He, "Low-coherence interrogation scheme for the multiplexed sensors based on long-period grating Mach-Zehnder interferometers, IEEE Photon. Technol. Lett. 18, 832-834 (2006).
[CrossRef]

A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
[CrossRef]

A. P. Zhang, L. Y. Shao, J. F. Ding, and S. He, "Sandwiched long-period gratings for simultaneous measurement of refractive index and temperature," IEEE Photon. Technol. Lett. 17, 2397-2399 (2005).
[CrossRef]

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, "Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor," IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (5)

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, "Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor," IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

A. P. Zhang, X. W. Chen, J. H. Yan, Z. G. Guan, S. He, and H. Y. Tam, "Optimization and fabrication of stitched long-period gratings for gain-flattening of ultrawide-band EDFAs," IEEE Photon. Technol. Lett. 17, 2559-2561 (2005).
[CrossRef]

S. D. Dyer and K. B. Rochford, "Low-coherence interferometric measurements of the dispersion of multiple fiber Bragg gratings," IEEE Photon. Technol. Lett. 13, 230-232 (2001).
[CrossRef]

Z. G. Guan, A. P. Zhang, M. Jiang, and S. He, "Low-coherence interrogation scheme for the multiplexed sensors based on long-period grating Mach-Zehnder interferometers, IEEE Photon. Technol. Lett. 18, 832-834 (2006).
[CrossRef]

A. P. Zhang, L. Y. Shao, J. F. Ding, and S. He, "Sandwiched long-period gratings for simultaneous measurement of refractive index and temperature," IEEE Photon. Technol. Lett. 17, 2397-2399 (2005).
[CrossRef]

J. Lightwave Technol. (2)

Meas. Sci. Technol. (2)

P. L. Swart, "Long-period grating Michelson refractometric sensor," Meas. Sci. Technol. 15, 1576-1580 (2004).
[CrossRef]

Y. G. Han, B. H. Lee, W. T. Han, U. C. Paek, and Y. Chung, "Fibre-optic sensing applications of a pair of long-period fibre gratings," Meas. Sci. Technol. 12, 778-781 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Rev. Sci. Instrum. (1)

T. Allsop, R. Reeves, D. J. Webb, I. Bennion, and R. Neal, "A high sensitivity refractometer based upon a long period grating Mach-Zehnder interferometer," Rev. Sci. Instrum. 73, 1702-1705 (2002).
[CrossRef]

Other (1)

H. J. Patrick, A. D. Kersey, F. Bucholtz, K. J. Ewing, J. B. Judkins, and A. M. Vengsarkar, "Chemical sensor based on long-period fibre grating response to index of refraction," in Conference on Lasers and Electro-Optics, Vol. 11 of OSA Technical Digests (Optical Society of America, 1997), pp. 420-421.

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Figures (12)

Fig. 1
Fig. 1

Schematic of the present low-coherent interrogation scheme for multiplexed sensors based on LPG pairs.

Fig. 2
Fig. 2

Typical reflectogram obtained from a numerical simulation. The inset is the whole reflectogram.

Fig. 3
Fig. 3

Transmission spectra of the LPGP (solid curve) and the Michelson interferometer (dashed curve) when the movable mirror is at the position of z = 45.0 μm.

Fig. 4
Fig. 4

Output spectra of the system when the movable mirror is at the positions of (a) 73.65 μm and (b) 147.75 μm. The dashed curve shows the spectrum of the LPGP for comparison.

Fig. 5
Fig. 5

Comparison of the reflectograms for two LPGPs with grating intervals of 80 mm (black curve) and 60 mm (gray curve). The inset shows their corresponding transmission spectra.

Fig. 6
Fig. 6

Comparison of the reflectograms for two LPGPs with grating lengths of (a) 10 mm and (b) 20 mm. The inset shows their corresponding transmission spectra.

Fig. 7
Fig. 7

Comparison of the reflectograms for two LPGPs with minimal grating transmissivities of (a) 0.25 and (b) 0.75. The gray curve is the reflectogram for a LPGP with a minimal grating transmissivity of 0.5. The insets show their transmission spectra.

Fig. 8
Fig. 8

Amplitude of the reflectogram as the strength of the grating decreases. The inset shows the evolution of the reflectogram.

Fig. 9
Fig. 9

Reconstructed spectrum from the additional part of the reflectogram. The spectrum of the LPG is given for comparison. The inset shows a comparison of the reconstructed spectrum of the whole reflectogram and the original spectrum entering the Michelson interferometer.

Fig. 10
Fig. 10

Comparison of the (a) calculated and (b) measured reflectograms for the LPGPs with a grating interval of 45 mm. The inset shows the calculated and measured transmission spectra of the LPGP.

Fig. 11
Fig. 11

Comparison of the (a) calculated and (b) measured reflectograms for the LPGPs with a grating interval of 90 mm. The inset shows the calculated and measured transmission spectra of the LPGP.

Fig. 12
Fig. 12

Comparison of the reflectograms calculated with light sources with bandwidths of 100 nm (gray curve) and 30 nm (black curve). The inset shows their transmission spectra.

Equations (7)

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

E 1 ( ω ) = ( T 0 + T j ) E 0 ( ω ) ,
T 0 = [ cos ( s l g ) i δ β 2 s sin ( s l g ) ] 2 exp [ i ( β 0 + β j + β g ) ×  l g i β 0 l e - s ] ,
T j = i κ κ * s 2 sin 2 ( s l g ) exp [ i ( β 0 + β j ) l g i β j l e - s ] ,
Δφ ( β 0 β j ) l c - c ,
Δ L = ( n eff co n eff cl , j ) l c - c ,
E 2 ( ω , z ) = [ 1 2 + exp ( 2 i ω c z ) / 2 ] exp ( i φ c ) E 1 ( ω ) ,
I ( z ) 1 4 π { + [ 1 + exp ( 2 i ω c z ) ] E 1 ( ω ) d ω } ,

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