Abstract

We model and analyze coupled-resonator optical waveguide (CROW) based refractive index (RI) sensors using pixelized spatial detection. Our modeled cascaded Fabry-Perot (FP) CROWs reveal that the intra-band states mode-field distributions vary upon effective RI change at a single wavelength. The spatial Fourier transform of the CROW mode-field distributions, with each cavity field intensity integrated as a pixel, shows spatial frequency peak shift, which constitutes the basis of such a spatial domain sensor. The spatial domain sensing performance depends on the cavity number, the cavity length and the inter-cavity coupling. Our modeled 21-element CROW sensor attains a detection limit of 10−4 refractive index unit (RIU) with a sensing dynamic range of 10−3 RIU. Detailed analysis of the spatial frequency harmonic peak amplitude variation further suggests an improved detection limit. Finite-difference time-domain (FDTD) simulations of an 11-element microring CROW device shows sensitivity consistent with the FP modeling.

© 2011 OSA

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2011 (3)

Y. Sun and X. Fan, “Optical ring resonators for biochemical and chemical sensing,” Anal. Bioanal. Chem. 399(1), 205–211 (2011).
[CrossRef] [PubMed]

F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat Commun 2, 296 (2011).
[CrossRef] [PubMed]

A. Nitkowski, A. Baeumner, and M. Lipson, “On-chip spectrophotometry for bioanalysis using microring resonators,” Biomed. Opt. Express 2(2), 271–277 (2011).
[CrossRef] [PubMed]

2010 (3)

2009 (4)

T. Claes, J. G. Molera, K. De Vos, E. Schacht, R. Baets, and P. Bienstman, “Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator,” IEEE Photonics J. 1(3), 197–204 (2009).
[CrossRef]

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem. 81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 26(4), 858–866 (2009).
[CrossRef]

X. Luo and A. W. Poon, “Many-element coupled-resonator optical waveguides using gapless-coupled microdisk resonators,” Opt. Express 17(26), 23617–23628 (2009).
[CrossRef] [PubMed]

2008 (4)

D. X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16(19), 15137–15148 (2008).
[CrossRef] [PubMed]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[CrossRef] [PubMed]

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23(7), 1003–1009 (2008).
[CrossRef] [PubMed]

S. Mookherjea, J. S. Park, S. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[CrossRef]

2007 (3)

2006 (1)

2004 (2)

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. A 21(9), 1665–1673 (2004).
[CrossRef]

2003 (2)

C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[CrossRef]

S. McNab, N. Moll, and Y. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express 11(22), 2927–2939 (2003).
[CrossRef] [PubMed]

2002 (1)

1999 (1)

Absil, P. P.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Armani, A. M.

H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale 2(9), 1544–1559 (2010).
[CrossRef] [PubMed]

Assefa, S.

Baets, R.

T. Claes, J. G. Molera, K. De Vos, E. Schacht, R. Baets, and P. Bienstman, “Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator,” IEEE Photonics J. 1(3), 197–204 (2009).
[CrossRef]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15(12), 7610–7615 (2007).
[CrossRef] [PubMed]

Baeumner, A.

Bailey, R. C.

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem. 81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

Bandaru, P. R.

S. Mookherjea, J. S. Park, S. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[CrossRef]

Barrios, C. A.

Bartolozzi, I.

Bienstman, P.

T. Claes, J. G. Molera, K. De Vos, E. Schacht, R. Baets, and P. Bienstman, “Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator,” IEEE Photonics J. 1(3), 197–204 (2009).
[CrossRef]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15(12), 7610–7615 (2007).
[CrossRef] [PubMed]

Caldwell, C. W.

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23(7), 1003–1009 (2008).
[CrossRef] [PubMed]

Canciamilla, A.

F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat Commun 2, 296 (2011).
[CrossRef] [PubMed]

Casquel, R.

Chao, C.-Y.

C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[CrossRef]

Cheben, P.

Chu, S. T.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Claes, T.

T. Claes, J. G. Molera, K. De Vos, E. Schacht, R. Baets, and P. Bienstman, “Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator,” IEEE Photonics J. 1(3), 197–204 (2009).
[CrossRef]

Cooper, M. L.

De Vos, K.

T. Claes, J. G. Molera, K. De Vos, E. Schacht, R. Baets, and P. Bienstman, “Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator,” IEEE Photonics J. 1(3), 197–204 (2009).
[CrossRef]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15(12), 7610–7615 (2007).
[CrossRef] [PubMed]

Delâge, A.

Densmore, A.

DeRose, G. A.

Divliansky, I. B.

Fan, X.

Y. Sun and X. Fan, “Optical ring resonators for biochemical and chemical sensing,” Anal. Bioanal. Chem. 399(1), 205–211 (2011).
[CrossRef] [PubMed]

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23(7), 1003–1009 (2008).
[CrossRef] [PubMed]

Ferrari, C.

Gill, D.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Green, W. M. J.

Griol, A.

Gunn, L. C.

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem. 81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

Guo, L. J.

C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[CrossRef]

Gupta, G.

Gylfason, K. B.

Holgado, M.

Hryniewicz, J. V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Hunt, H. K.

H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale 2(9), 1544–1559 (2010).
[CrossRef] [PubMed]

Janz, S.

Johnson, F. G.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Katsidis, C. C.

King, O.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Lapointe, J.

Lee, R. K.

Lipson, M.

Little, B. E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Lopinski, G.

Luo, X.

Martinelli, M.

McKinnon, R.

McNab, S.

Melloni, A.

Mischki, T.

Molera, J. G.

T. Claes, J. G. Molera, K. De Vos, E. Schacht, R. Baets, and P. Bienstman, “Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator,” IEEE Photonics J. 1(3), 197–204 (2009).
[CrossRef]

Moll, N.

Mookherjea, S.

Morichetti, F.

Nitkowski, A.

Park, J. S.

M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
[CrossRef] [PubMed]

S. Mookherjea, J. S. Park, S. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[CrossRef]

Poon, A. W.

Poon, J. K. S.

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31(4), 456–458 (2006).
[CrossRef] [PubMed]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. A 21(9), 1665–1673 (2004).
[CrossRef]

Post, E.

Rooks, M.

Samarelli, A.

F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat Commun 2, 296 (2011).
[CrossRef] [PubMed]

Sánchez, B.

Schacht, E.

T. Claes, J. G. Molera, K. De Vos, E. Schacht, R. Baets, and P. Bienstman, “Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator,” IEEE Photonics J. 1(3), 197–204 (2009).
[CrossRef]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15(12), 7610–7615 (2007).
[CrossRef] [PubMed]

Scherer, A.

Scheuer, J.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. A 21(9), 1665–1673 (2004).
[CrossRef]

Schmid, J. H.

Schneider, M. A.

Seiferth, F.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Sekaric, L.

Shi, H.

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23(7), 1003–1009 (2008).
[CrossRef] [PubMed]

Siapkas, D. I.

Sohlström, H.

Sorel, M.

F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat Commun 2, 296 (2011).
[CrossRef] [PubMed]

Sun, Y.

Y. Sun and X. Fan, “Optical ring resonators for biochemical and chemical sensing,” Anal. Bioanal. Chem. 399(1), 205–211 (2011).
[CrossRef] [PubMed]

Suter, J. D.

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23(7), 1003–1009 (2008).
[CrossRef] [PubMed]

Trakalo, M.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

van, V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, V. van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16(10), 2263–2265 (2004).
[CrossRef]

Vlasov, Y.

Vlasov, Y. A.

Waldron, P.

Washburn, A. L.

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem. 81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

White, I. M.

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23(7), 1003–1009 (2008).
[CrossRef] [PubMed]

Xia, F.

Xu, D. X.

Xu, Y.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. A 21(9), 1665–1673 (2004).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[CrossRef] [PubMed]

Yang, S.

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J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23(7), 1003–1009 (2008).
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Figures (11)

Fig. 1
Fig. 1

Working principle of CROW sensors using spatial domain detection. (a) Structure of a microring CROW device. (b) In the spectral domain, the broadband transmission shifts upon effective RI change from n0 to n0+Δn. At a fixed probe wavelength λp the state transits from state 1 to 2. (c) In the spatial domain, pixelized mode patterns change corresponding to the state transitions. Through a spatial Fourier transform, the spatial frequency spectra show unique harmonic peaks. FFT: fast Fourier transform.

Fig. 2
Fig. 2

Schematic of the cascaded FP cavities model.

Fig. 3
Fig. 3

Modeling of the cascaded FP cavities with N = 11, L = 50 μm and κ = 0.4. (a) Periodic broadband transmission with states from 1 to 11. (b) Pixelized mode-field intensity distributions of states 1 to 11 at a fixed probe wavelength λp initially aligned to state 1 upon corresponding effective RI increases. (c) Spatial frequency spectra of the pixelized mode-field intensity patterns by spatial Fourier transform. Dashed arrows illustrate the shift of the dominant frequency component as the effective RI increases.

Fig. 4
Fig. 4

Schematic of the inter-state sensing windows. (a) In the red sensing window, spatial frequency peak harmonic rises (drops) upon the effective RI increases (decreases). (b) In the blue sensing window, spatial frequency peak harmonic drops (rises) upon the effective RI increases (decreases). (c) At the band-center state, the spatial frequency drops independent of the sign of the effective RI change.

Fig. 5
Fig. 5

(a) Modeled spatial frequency shifts upon effective RI increases. The red and blue shaded areas denote the two narrowed sensing windows with nearly linear responses. (b) Corresponding pixelized intensity distributions for 11-, 21- and 51-element cascaded FP cavities at the band-center states.

Fig. 6
Fig. 6

Modeled inter-state sensing scheme sensitivity as functions of (a) cavity number N, (b) cavity length L, (c) coupling coefficient κ and (d) loss coefficient α.

Fig. 7
Fig. 7

Fourier transform analysis of intra-state spatial mode patterns. (a) Spectra of the broadband transmission. The zoom-in picture shows the intra-state points labeled as A, B, C, D and E. (b) Pixelized intensity distributions at wavelengths A-E between states 4 and 5. (c) Spatial frequency spectra at intra-state positions A-E. The 4th harmonic component falls and the 5th harmonic component rises when the state transits from state 4 to state 5.

Fig. 8
Fig. 8

Intra-state sensing scheme sensitivity analysis. (a) Modeled normalized Fourier frequency amplitude as a function of effective RI increment. The red and blue curves illustrate the 4th and 5th harmonics of spatial Fourier amplitude normalized to the input I0. (b) Intra-state sensing scheme sensitivity of CROW devices with various cavity numbers and coupling coefficients.

Fig. 9
Fig. 9

FDTD simulations of an 11-element ring CROW device. (a) Schematic structure of a CROW with ring structure. (b) Periodic broadband transmission of the drop-port. (c) Simulated electric field patterns at the 11 intra-band resonance wavelengths. Zoom-in pictures show symmetric and anti-symmetric electric field distributions at the same coupling region of states 4 and 8.

Fig. 10
Fig. 10

(a) Simulated mode intensity patterns at the 11 intra-band states. (b) Pixelized intensity distributions. (c) Spatial Fourier amplitude distributions of the mode intensity patterns.

Fig. 11
Fig. 11

(a) Simulated spectra of the broadband transmission. The zoom-in picture shows the modeled intra-state positions. (b) Pixelized intensity distributions at wavelengths A-E between states 4 and 5. (c) Spatial frequency spectra of the pixelized intensity patterns.

Tables (1)

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Table 1 Comparison Between the Single-Element Microresonator-Based Spectral Domain Sensing and the CROW-Based Spatial Domain Sensing

Equations (18)

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[ a i+1 (x= j=1 i+1 L j L i+1 ) b i+1 (x= j=1 i+1 L j L i+1 ) ]= 1 κ ' i [ κ i κ ' i r i r ' i r ' i r i 1 ][ a i (x= j=1 i L j ) b i (x= j=1 i L j ) ]= Q i [ a i (x= j=1 i L j ) b i (x= j=1 i L j ) ]
Q i = 1 κ i [ 1 r i r i 1 ]
[ a i (x= j=1 i L j ) b i (x= j=1 i L j ) ]=[ exp(j β i L i α i 2 L i ) 0 0 exp(j β i L i + α i 2 L i ) ][ a i (x= j=1 i L j L i ) b i (x= j=1 i L j L i ) ]= P i [ a i (x= j=1 i L j L i ) b i (x= j=1 i L j L i ) ]
[ a 1 ( x=0 ) b 1 ( x=0 ) ]= 1 κ ' in [ κ in κ ' in r in r ' in r ' in r in 1 ][ a 0 b 0 ]= 1 κ in [ 1 r in r in 1 ][ a 0 b 0 ]= Q in [ a 0 b 0 ]
[ a N+1 b N+1 ]= 1 κ ' out [ κ out κ ' out r out r ' out r ' out r out 1 ][ a N (x= j=1 N L j ) b N (x= j=1 N L j ) ] = 1 κ out [ 1 r out r out 1 ][ a N (x= j=1 N L j ) b N (x= j=1 N L j ) ]= Q out [ a N (x= j=1 N L j ) b N (x= j=1 N L j ) ]
[ a N+1 b N+1 ]= Q out P N ( i=1 N1 Q i P i ) Q in [ a 0 b 0 ]=[ A B C D ][ a 0 b 0 ]
b 0 = C D a 0
a N+1 =(A BC D ) a 0
[ a i ( x i ) b i ( x i ) ]=[ exp(j β i x i α 2 x i ) 0 0 exp(j β i x i + α 2 x i ) ][ a i (x= j=1 i L j L i ) b i (x= j=1 i L j L i ) ]
I i ( x )/ I 0 = | a i (x)+ b i (x) | 2 / | a 0 | 2
S interstate ΔF Δ n eff = Δλ δλΔ n eff
D L interstate = 1 S interstate = δλΔ n eff Δλ
BW κ λ 0 2 πL n eff
δλ BW N1 = κ λ 0 2 (N1)πL n eff
S interstate = (N1)πL n eff Δλ κΔ n eff λ 0 2
S intrastate ΔI I 0 Δ n eff
S spectral = Δλ Δ n eff
D L spectral = LW S spectral = LWΔ n eff Δλ

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