Abstract

Gas sensing and fluid-guiding response properties of a suspended core fiber Raman analyzer with side-opened and strut microfluid-guiding array are explored. A Raman sensing model is introduced for effective mode area optimization and normalized intensity overlap enhancement between Raman sensing light and analyte. Calculations predict that there is a trade-off between the overlap and the effective mode area, while the optimal trade-off depends on the refractive index of the background material, core diameter, and strut’s thickness. Furthermore, the multi-opened-up structure ensures a fast gases diffusing into/out of each hole for real-time Raman sensing. Simulation results confirm a limited gas sensing response time of less than 6s could be feasible and, thus, a new approach to real-time gas sensing applications is identified.

© 2011 Optical Society of America

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References

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  1. M. Agah, G. R. Lambertus, R. Sacks, and K. Wise, “High-speed MEMS-based gas chromatography,” J. Microelectromech. Syst. 15, 1371–1378 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010 (4)

J. Mayrwöger, P. Hauer, W. Reichl, R. Schwödiauer, C. Krutzler, and B. Jakoby, “Modeling of infrared gas sensors using a ray tracing approach,” IEEE Sens. J. 10, 1691–1698(2010).
[CrossRef]

S. C. Eichmann, M. Weschta, J. Kiefer, T. Seeger, and A. Leipertz, “Characterization of a fast gas analyzer based on Raman scattering for the analysis of synthesis gas,” Rev. Sci. Instrum. 81, 125104 (2010).
[CrossRef]

Y. L. Hoo, L. Shujing, H. L. Hoi, and J. Wei, “Fast response microstructured optical fibre methane sensor with multiple side-openings,” IEEE Photon. Technol. Lett. 22, 296–298(2010).
[CrossRef]

T. M. Monro, S. C. Warren-Smith, E. P. Schartner, A. François, S. Heng, H. Ebendorff-Heidepriem, and S. V. Afshar, “Sensing with suspended-core optical fibres,” Opt. Fiber Technol. 16, 343–356 (2010).
[CrossRef]

2009 (1)

I. Dicaire, J.-C. Beugnot, and L. Thévenaz, “Suspended-core fibres as optical gas sensing cells: study and implementation,” Proc. SPIE 7357, 73570U-1 (2009).
[CrossRef]

2008 (3)

2007 (3)

F. M. Cox, R. Lwin, M. C. J. Large, and C. M. B. Cordeiro, “Opening up optical fibres,” Opt. Express 15, 11843–11848(2007).
[CrossRef] [PubMed]

S. V. Afshar, S. C. Warren-Smith, and T. M. Monro, “Enhancement of fluorescence-based sensing using microstructured optical fibres,” Opt. Express 15, 17891–17901 (2007).
[CrossRef] [PubMed]

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

2006 (1)

M. Agah, G. R. Lambertus, R. Sacks, and K. Wise, “High-speed MEMS-based gas chromatography,” J. Microelectromech. Syst. 15, 1371–1378 (2006).
[CrossRef]

2003 (2)

K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal,” IEEE Photon. Technol. Lett. 15, 1384–1386(2003).
[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 fibre gas sensor,” Appl. Opt. 42, 3509–3515 (2003).
[CrossRef] [PubMed]

1988 (1)

D. Marcuse, “Launching light into fibre cores from sources located in the cladding,” J. Lightwave Technol. 6, 1273–1279(1988).
[CrossRef]

1984 (1)

G. Becher, A. Haugen, and A. Bjørseth, “Multimethod determination of occupational exposure to polycyclic aromatic hydrocarbons in an aluminum plant,” Carcinogenesis 5, 647–651(1984).
[CrossRef] [PubMed]

Afshar, S. V.

Agah, M.

M. Agah, G. R. Lambertus, R. Sacks, and K. Wise, “High-speed MEMS-based gas chromatography,” J. Microelectromech. Syst. 15, 1371–1378 (2006).
[CrossRef]

Becher, G.

G. Becher, A. Haugen, and A. Bjørseth, “Multimethod determination of occupational exposure to polycyclic aromatic hydrocarbons in an aluminum plant,” Carcinogenesis 5, 647–651(1984).
[CrossRef] [PubMed]

Beugnot, J.-C.

I. Dicaire, J.-C. Beugnot, and L. Thévenaz, “Suspended-core fibres as optical gas sensing cells: study and implementation,” Proc. SPIE 7357, 73570U-1 (2009).
[CrossRef]

Bjørseth, A.

G. Becher, A. Haugen, and A. Bjørseth, “Multimethod determination of occupational exposure to polycyclic aromatic hydrocarbons in an aluminum plant,” Carcinogenesis 5, 647–651(1984).
[CrossRef] [PubMed]

Bozolan, A.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Briand, D.

D.Briand, O. Manzardo, N. F. de Rooij, J Hildenbrand, and J. Wollenstein, “Gas detection using a micromachined FTIR spectrometer,” in Sensors 2007 (IEEE, 2007), pp. 1364–1367.
[CrossRef]

Brito Cruz, C. H.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Buric, M. P.

Chen, K. P.

Chesini, G.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Cordeiro, C. M. B.

F. M. Cox, R. Lwin, M. C. J. Large, and C. M. B. Cordeiro, “Opening up optical fibres,” Opt. Express 15, 11843–11848(2007).
[CrossRef] [PubMed]

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Cox, F. M.

de Matos, C. J. S.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

de Rooij, N. F.

D.Briand, O. Manzardo, N. F. de Rooij, J Hildenbrand, and J. Wollenstein, “Gas detection using a micromachined FTIR spectrometer,” in Sensors 2007 (IEEE, 2007), pp. 1364–1367.
[CrossRef]

Dicaire, I.

I. Dicaire, J.-C. Beugnot, and L. Thévenaz, “Suspended-core fibres as optical gas sensing cells: study and implementation,” Proc. SPIE 7357, 73570U-1 (2009).
[CrossRef]

dos Santos, E. M.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Ebendorff-Heidepriem, H.

T. M. Monro, S. C. Warren-Smith, E. P. Schartner, A. François, S. Heng, H. Ebendorff-Heidepriem, and S. V. Afshar, “Sensing with suspended-core optical fibres,” Opt. Fiber Technol. 16, 343–356 (2010).
[CrossRef]

Eichmann, S. C.

S. C. Eichmann, M. Weschta, J. Kiefer, T. Seeger, and A. Leipertz, “Characterization of a fast gas analyzer based on Raman scattering for the analysis of synthesis gas,” Rev. Sci. Instrum. 81, 125104 (2010).
[CrossRef]

Facincani, T.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Falk, J.

François, A.

T. M. Monro, S. C. Warren-Smith, E. P. Schartner, A. François, S. Heng, H. Ebendorff-Heidepriem, and S. V. Afshar, “Sensing with suspended-core optical fibres,” Opt. Fiber Technol. 16, 343–356 (2010).
[CrossRef]

Hauer, P.

J. Mayrwöger, P. Hauer, W. Reichl, R. Schwödiauer, C. Krutzler, and B. Jakoby, “Modeling of infrared gas sensors using a ray tracing approach,” IEEE Sens. J. 10, 1691–1698(2010).
[CrossRef]

Haugen, A.

G. Becher, A. Haugen, and A. Bjørseth, “Multimethod determination of occupational exposure to polycyclic aromatic hydrocarbons in an aluminum plant,” Carcinogenesis 5, 647–651(1984).
[CrossRef] [PubMed]

Hautakorpi, M.

Heng, S.

T. M. Monro, S. C. Warren-Smith, E. P. Schartner, A. François, S. Heng, H. Ebendorff-Heidepriem, and S. V. Afshar, “Sensing with suspended-core optical fibres,” Opt. Fiber Technol. 16, 343–356 (2010).
[CrossRef]

Hildenbrand, J

D.Briand, O. Manzardo, N. F. de Rooij, J Hildenbrand, and J. Wollenstein, “Gas detection using a micromachined FTIR spectrometer,” in Sensors 2007 (IEEE, 2007), pp. 1364–1367.
[CrossRef]

Ho, H. L.

Hoi, H. L.

Y. L. Hoo, L. Shujing, H. L. Hoi, and J. Wei, “Fast response microstructured optical fibre methane sensor with multiple side-openings,” IEEE Photon. Technol. Lett. 22, 296–298(2010).
[CrossRef]

Hoo, Y. L.

Y. L. Hoo, L. Shujing, H. L. Hoi, and J. Wei, “Fast response microstructured optical fibre methane sensor with multiple side-openings,” IEEE Photon. Technol. Lett. 22, 296–298(2010).
[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 fibre gas sensor,” Appl. Opt. 42, 3509–3515 (2003).
[CrossRef] [PubMed]

Jakoby, B.

J. Mayrwöger, P. Hauer, W. Reichl, R. Schwödiauer, C. Krutzler, and B. Jakoby, “Modeling of infrared gas sensors using a ray tracing approach,” IEEE Sens. J. 10, 1691–1698(2010).
[CrossRef]

Jin, W.

Kiefer, J.

S. C. Eichmann, M. Weschta, J. Kiefer, T. Seeger, and A. Leipertz, “Characterization of a fast gas analyzer based on Raman scattering for the analysis of synthesis gas,” Rev. Sci. Instrum. 81, 125104 (2010).
[CrossRef]

Koshiba, M.

K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal,” IEEE Photon. Technol. Lett. 15, 1384–1386(2003).
[CrossRef]

Krutzler, C.

J. Mayrwöger, P. Hauer, W. Reichl, R. Schwödiauer, C. Krutzler, and B. Jakoby, “Modeling of infrared gas sensors using a ray tracing approach,” IEEE Sens. J. 10, 1691–1698(2010).
[CrossRef]

Lambertus, G. R.

M. Agah, G. R. Lambertus, R. Sacks, and K. Wise, “High-speed MEMS-based gas chromatography,” J. Microelectromech. Syst. 15, 1371–1378 (2006).
[CrossRef]

Large, M. C. J.

Leipertz, A.

S. C. Eichmann, M. Weschta, J. Kiefer, T. Seeger, and A. Leipertz, “Characterization of a fast gas analyzer based on Raman scattering for the analysis of synthesis gas,” Rev. Sci. Instrum. 81, 125104 (2010).
[CrossRef]

Long, D. A.

D. A. Long, Raman Spectroscopy (McGraw Hill, 1977).

Ludvigsen, H.

Lwin, R.

Manzardo, O.

D.Briand, O. Manzardo, N. F. de Rooij, J Hildenbrand, and J. Wollenstein, “Gas detection using a micromachined FTIR spectrometer,” in Sensors 2007 (IEEE, 2007), pp. 1364–1367.
[CrossRef]

Marcuse, D.

D. Marcuse, “Launching light into fibre cores from sources located in the cladding,” J. Lightwave Technol. 6, 1273–1279(1988).
[CrossRef]

Mattinen, M.

Mayrwöger, J.

J. Mayrwöger, P. Hauer, W. Reichl, R. Schwödiauer, C. Krutzler, and B. Jakoby, “Modeling of infrared gas sensors using a ray tracing approach,” IEEE Sens. J. 10, 1691–1698(2010).
[CrossRef]

Monro, T. M.

Ong, J. S. K.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Philibert, J.

J. Philibert, “One and a half century of diffusion: Fick, Einstein, before and beyond, diffusion fundamentals,” http://www.uni-leipzig.de/diffusion/powerpoint_presentations/pdf/philibert.pdf (18 July 2011).

Reichl, W.

J. Mayrwöger, P. Hauer, W. Reichl, R. Schwödiauer, C. Krutzler, and B. Jakoby, “Modeling of infrared gas sensors using a ray tracing approach,” IEEE Sens. J. 10, 1691–1698(2010).
[CrossRef]

Ruan, S. C.

Sacks, R.

M. Agah, G. R. Lambertus, R. Sacks, and K. Wise, “High-speed MEMS-based gas chromatography,” J. Microelectromech. Syst. 15, 1371–1378 (2006).
[CrossRef]

Saitoh, K.

K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal,” IEEE Photon. Technol. Lett. 15, 1384–1386(2003).
[CrossRef]

Schartner, E. P.

T. M. Monro, S. C. Warren-Smith, E. P. Schartner, A. François, S. Heng, H. Ebendorff-Heidepriem, and S. V. Afshar, “Sensing with suspended-core optical fibres,” Opt. Fiber Technol. 16, 343–356 (2010).
[CrossRef]

Schwödiauer, R.

J. Mayrwöger, P. Hauer, W. Reichl, R. Schwödiauer, C. Krutzler, and B. Jakoby, “Modeling of infrared gas sensors using a ray tracing approach,” IEEE Sens. J. 10, 1691–1698(2010).
[CrossRef]

Seeger, T.

S. C. Eichmann, M. Weschta, J. Kiefer, T. Seeger, and A. Leipertz, “Characterization of a fast gas analyzer based on Raman scattering for the analysis of synthesis gas,” Rev. Sci. Instrum. 81, 125104 (2010).
[CrossRef]

Shi, C.

Shujing, L.

Y. L. Hoo, L. Shujing, H. L. Hoi, and J. Wei, “Fast response microstructured optical fibre methane sensor with multiple side-openings,” IEEE Photon. Technol. Lett. 22, 296–298(2010).
[CrossRef]

Thévenaz, L.

I. Dicaire, J.-C. Beugnot, and L. Thévenaz, “Suspended-core fibres as optical gas sensing cells: study and implementation,” Proc. SPIE 7357, 73570U-1 (2009).
[CrossRef]

Vaz, A. R.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Wang, D. N.

Warren-Smith, S. C.

Weber, M. J.

M. J. Weber, Handbook of Optical Materials (CRC Press, 2003).

Wei, J.

Y. L. Hoo, L. Shujing, H. L. Hoi, and J. Wei, “Fast response microstructured optical fibre methane sensor with multiple side-openings,” IEEE Photon. Technol. Lett. 22, 296–298(2010).
[CrossRef]

Weschta, M.

S. C. Eichmann, M. Weschta, J. Kiefer, T. Seeger, and A. Leipertz, “Characterization of a fast gas analyzer based on Raman scattering for the analysis of synthesis gas,” Rev. Sci. Instrum. 81, 125104 (2010).
[CrossRef]

Wise, K.

M. Agah, G. R. Lambertus, R. Sacks, and K. Wise, “High-speed MEMS-based gas chromatography,” J. Microelectromech. Syst. 15, 1371–1378 (2006).
[CrossRef]

Wollenstein, J.

D.Briand, O. Manzardo, N. F. de Rooij, J Hildenbrand, and J. Wollenstein, “Gas detection using a micromachined FTIR spectrometer,” in Sensors 2007 (IEEE, 2007), pp. 1364–1367.
[CrossRef]

Woodruff, S. D.

Appl. Opt. (2)

Carcinogenesis (1)

G. Becher, A. Haugen, and A. Bjørseth, “Multimethod determination of occupational exposure to polycyclic aromatic hydrocarbons in an aluminum plant,” Carcinogenesis 5, 647–651(1984).
[CrossRef] [PubMed]

IEEE Photon. Technol. Lett. (2)

K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal,” IEEE Photon. Technol. Lett. 15, 1384–1386(2003).
[CrossRef]

Y. L. Hoo, L. Shujing, H. L. Hoi, and J. Wei, “Fast response microstructured optical fibre methane sensor with multiple side-openings,” IEEE Photon. Technol. Lett. 22, 296–298(2010).
[CrossRef]

IEEE Sens. J. (1)

J. Mayrwöger, P. Hauer, W. Reichl, R. Schwödiauer, C. Krutzler, and B. Jakoby, “Modeling of infrared gas sensors using a ray tracing approach,” IEEE Sens. J. 10, 1691–1698(2010).
[CrossRef]

J. Lightwave Technol. (1)

D. Marcuse, “Launching light into fibre cores from sources located in the cladding,” J. Lightwave Technol. 6, 1273–1279(1988).
[CrossRef]

J. Microelectromech. Syst. (1)

M. Agah, G. R. Lambertus, R. Sacks, and K. Wise, “High-speed MEMS-based gas chromatography,” J. Microelectromech. Syst. 15, 1371–1378 (2006).
[CrossRef]

Meas. Sci. Technol. (1)

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, and C. H. Brito Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18, 3075 (2007).
[CrossRef]

Opt. Express (4)

Opt. Fiber Technol. (1)

T. M. Monro, S. C. Warren-Smith, E. P. Schartner, A. François, S. Heng, H. Ebendorff-Heidepriem, and S. V. Afshar, “Sensing with suspended-core optical fibres,” Opt. Fiber Technol. 16, 343–356 (2010).
[CrossRef]

Proc. SPIE (1)

I. Dicaire, J.-C. Beugnot, and L. Thévenaz, “Suspended-core fibres as optical gas sensing cells: study and implementation,” Proc. SPIE 7357, 73570U-1 (2009).
[CrossRef]

Rev. Sci. Instrum. (1)

S. C. Eichmann, M. Weschta, J. Kiefer, T. Seeger, and A. Leipertz, “Characterization of a fast gas analyzer based on Raman scattering for the analysis of synthesis gas,” Rev. Sci. Instrum. 81, 125104 (2010).
[CrossRef]

Other (5)

D.Briand, O. Manzardo, N. F. de Rooij, J Hildenbrand, and J. Wollenstein, “Gas detection using a micromachined FTIR spectrometer,” in Sensors 2007 (IEEE, 2007), pp. 1364–1367.
[CrossRef]

M. J. Weber, Handbook of Optical Materials (CRC Press, 2003).

COMSOLA.B, www.comsol.com.

J. Philibert, “One and a half century of diffusion: Fick, Einstein, before and beyond, diffusion fundamentals,” http://www.uni-leipzig.de/diffusion/powerpoint_presentations/pdf/philibert.pdf (18 July 2011).

D. A. Long, Raman Spectroscopy (McGraw Hill, 1977).

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Figures (8)

Fig. 1
Fig. 1

Geometry of proposed multi-opened-up suspended core fiber for gas Raman sensing. (a) Three dimensional sketch; (b) cross section.

Fig. 2
Fig. 2

Intensity distribution of the fundamental mode for (a), (b) silica and (c), (d) bismuth-based fiber.

Fig. 3
Fig. 3

Normalized overlay intensity on SCF structure.

Fig. 4
Fig. 4

Confinement loss on SCF structure.

Fig. 5
Fig. 5

Effective mode area on SCF structure.

Fig. 6
Fig. 6

Raman signal intensity of silica-based and bismuth-based fibers.

Fig. 7
Fig. 7

Relationship of Raman signal intensity and strut thickness, core diameter in (a) silica-based fiber and (b) bismuth-based fiber.

Fig. 8
Fig. 8

Concentration distribution (a), (b) at 2 s . (c) Maximal Cconcentration difference versus time.

Equations (9)

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P D ( z ) = 1 2 n E holes C j ( λ ) k ( λ ) λ 4 ( ε 0 / μ 0 ) 1 / 2 | a E | 2 × holes δ E holes Re [ ( e E × h E * ) z ^ ] exp ( γ E z ) d s ,
δ E holes = total | e E | 2 d s / holes ( e E × h E * ) z ^ d s ,
d P R ( z ) = π exp [ γ R z ] 4 ω R μ 0 n R holes k R N R z 1 z 2 | e R | 2 P D ( z ) d z = π n E holes ( ε 0 / μ 0 ) 1 / 2 | a E | 2 δ E holes 8 ω R μ 0 n R holes k R N R c j ( λ ) k ( λ ) λ 4 × exp ( γ R z ) exp ( γ E z ) d z × holes | e R | 2 Re [ ( e E × h E * ) z ^ ] d s .
R = P R / P E ( 0 ) = A * NOI / A eff .
A = C j ( λ ) k ( λ ) 8 π λ 2 [ 1 exp ( 2 γ ) ] / 2 γ .
NOI = ( ε 0 μ 0 ) 1 / 2 total | e | 2 d s holes ( e × h * ) z ^ d s holes | e | 2 Re [ ( e × h * ) z ^ ] d s total | Re [ ( e × h * ) z ^ ] | 2 d s .
A eff = ( total Re [ ( e × h * ) z ^ d s ] ) 2 / total | Re [ ( e × h * ) z ^ ] | 2 d s .
L c = 8.686 . Im [ k 0 n eff ] = 8.686 * 2 π * Im [ n eff ] / λ .
( D c ( T ) + c ( T ) u ) = 0.

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