Abstract

We present a process calibration method for designing silicon-on-insulator (SOI) contra-directional grating couplers (contra-DCs). Our method involves determining the coupling coefficients of fabricated contra-DCs by using their full-width-at-half-maximum (FWHM) bandwidths. As compared to the null method that uses the bandwidth measured at the first nulls, our FWHM method obtains more consistent results since the FWHM bandwidth is more easily determined. We also extract the coupling coefficients using curve-fitting which provide values that are in general agreement with the values obtained using our method. However, as compared to the curve-fitting method, our method does not require knowledge of the insertion loss and is easier to implement. Our method can be used to predict the FWHM bandwidths, the maximum power coupling factors, the minimum power transmission factors, and the through port group delays and dispersions of subsequent, fabricated devices, which is useful in designing filters.

© 2015 Optical Society of America

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2014 (4)

2013 (8)

P. Pintus, P. Contu, N. Andriolli, A. D’Errico, F. Di Pasquale, and F. Testa, “Analysis and design of microring-based switching elements in a silicon photonic integrated transponder aggregator,” J. Lightwave Technol. 31(24), 3943–3955 (2013).
[Crossref]

Y. Wang, J. Flueckiger, C. Lin, and L. Chrostowski, “Universal grating coupler design,” Proc. SPIE 8915, 89150Y (2013).
[Crossref]

R. Boeck, L. Chrostowski, and N. A. F. Jaeger, “Thermally tunable quadruple Vernier racetrack resonators,” Opt. Lett. 38(14), 2440–2442 (2013).
[Crossref] [PubMed]

W. Shi, H. Yun, C. Lin, M. Greenberg, X. Wang, Y. Wang, S. T. Fard, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Ultra-compact, flat-top demultiplexer using anti-reflection contra-directional couplers for CWDM networks on silicon,” Opt. Express 21(6), 6733–6738 (2013).
[Crossref] [PubMed]

R. Boeck, W. Shi, L. Chrostowski, and N. A. F. Jaeger, “FSR-eliminated Vernier racetrack resonators using grating-assisted couplers,” IEEE Photon. J. 5(5), 2202511 (2013).
[Crossref]

H. Qiu, G. Jiang, T. Hu, H. Shao, P. Yu, J. Yang, and X. Jiang, “FSR-free add-drop filter based on silicon grating-assisted contradirectional couplers,” Opt. Lett. 38(1), 1–3 (2013).
[Crossref] [PubMed]

W. Shi, H. Yun, C. Lin, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Coupler-apodized Bragg-grating add-drop filter,” Opt. Lett. 38(16), 3068–3070 (2013).
[Crossref] [PubMed]

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
[Crossref] [PubMed]

2012 (2)

W. Shi, X. Wang, W. Zhang, H. Yun, C. Lin, L. Chrostowski, and N. A. F. Jaeger, “Grating-coupled silicon microring resonators,” Appl. Phys. Lett. 100(12), 121118 (2012).
[Crossref]

R. Boeck, J. Flueckiger, H. Yun, L. Chrostowski, and N. A. F. Jaeger, “High performance Vernier racetrack resonators,” Opt. Lett. 37(24), 5199–5201 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (1)

2009 (1)

2008 (2)

A. Canciamilla, F. Morichetti, and A. Melloni, “Full characterization of integrated optical ring-resonators by phase-sensitive time-domain interferometry,” Proc. SPIE 7138, 71381L (2008).
[Crossref]

K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett. 92(20), 201111 (2008).
[Crossref]

2007 (1)

2006 (2)

S. Nacer, A. Aissat, K. Ferdjani, and M. Bensebti, “Influence of dispersion on spectral characteristics of GADC optical filters,” Opt. Quant. Electron. 38(8), 701–710 (2006).
[Crossref]

N. Zhang and J. T. Boyd, “Forward and backward grating-assisted directional couplers in silicon for wavelength-division multiplexing tunable add-drop applications,” Opt. Eng. 45(5), 054603 (2006).
[Crossref]

2004 (2)

A. Melloni, M. Martinelli, G. Cusmai, and R. Siano, “Experimental evaluation of ring resonator filters impact on the bit error rate in non return to zero transmission systems,” Opt. Commun. 234(1–6), 211–216 (2004).
[Crossref]

O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters - a tutorial overview,” J. Lightwave Technol. 22(5), 1380–1394 (2004).
[Crossref]

2003 (1)

1999 (1)

1998 (1)

M. R. Shenoy, K. Thyagarajan, V. Priye, and N. S. Madhavan, “Estimation of the characteristic parameters of fiber Bragg gratings from spectral measurements,” Proc. SPIE 3666, 94 (1998).
[Crossref]

1997 (1)

1993 (1)

J.-P. Weber, “Spectral characteristics of coupled-waveguide Bragg-reflection tunable optical filter,” IEE Proc. J. Optoelectron. 140(5), 275–284, (1993).
[Crossref]

1991 (1)

J. Willems, K. David, G. Morthier, and R. Baets, “Filter characteristics of DBR amplifier with index and gain coupling,” Electron. Lett. 27(10), 831–833 (1991).
[Crossref]

1987 (3)

R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,’ Opt. Quant. Electron. 19(5), 273–287 (1987).
[Crossref]

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating-assisted couplers,” J. Lightwave Technol. 5(2), 268–273 (1987).
[Crossref]

D. Marcuse, “Bandwidth of forward and backward coupling directional couplers,” J. Lightwave Technol. 5(12), 1773–1777 (1987).
[Crossref]

Aida, Y.

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vac. Sci. Technol. B 29(6), 06F309 (2011).
[Crossref]

Aissat, A.

S. Nacer, A. Aissat, K. Ferdjani, and M. Bensebti, “Influence of dispersion on spectral characteristics of GADC optical filters,” Opt. Quant. Electron. 38(8), 701–710 (2006).
[Crossref]

Andriolli, N.

Baehr-Jones, T.

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
[Crossref] [PubMed]

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vac. Sci. Technol. B 29(6), 06F309 (2011).
[Crossref]

Baets, R.

J. Willems, K. David, G. Morthier, and R. Baets, “Filter characteristics of DBR amplifier with index and gain coupling,” Electron. Lett. 27(10), 831–833 (1991).
[Crossref]

Bassi, P.

P. Orlandi, P. Velha, M. Gnan, P. Bassi, A. Samarelli, M. Sorel, M. J. Strain, and R. De La Rue, “Microring resonator with wavelength selective coupling in SOI,” in Proceedings of 8th IEEE International Conference on Group IV Photonics (IEEE, 2011), pp. 281–283.

Bensebti, M.

S. Nacer, A. Aissat, K. Ferdjani, and M. Bensebti, “Influence of dispersion on spectral characteristics of GADC optical filters,” Opt. Quant. Electron. 38(8), 701–710 (2006).
[Crossref]

Bergman, K.

K. Bergman, L. P. Carloni, A. Biberman, J. Chan, and G. Hendry, Photonic Network-on-Chip Design (Springer, 2014).
[Crossref]

Biberman, A.

K. Bergman, L. P. Carloni, A. Biberman, J. Chan, and G. Hendry, Photonic Network-on-Chip Design (Springer, 2014).
[Crossref]

Boeck, R.

Bojko, R.

Bojko, R. J.

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vac. Sci. Technol. B 29(6), 06F309 (2011).
[Crossref]

Boyd, J. T.

N. Zhang and J. T. Boyd, “Forward and backward grating-assisted directional couplers in silicon for wavelength-division multiplexing tunable add-drop applications,” Opt. Eng. 45(5), 054603 (2006).
[Crossref]

Boyd, R. W.

Canciamilla, A.

A. Canciamilla, F. Morichetti, and A. Melloni, “Full characterization of integrated optical ring-resonators by phase-sensitive time-domain interferometry,” Proc. SPIE 7138, 71381L (2008).
[Crossref]

Carloni, L. P.

K. Bergman, L. P. Carloni, A. Biberman, J. Chan, and G. Hendry, Photonic Network-on-Chip Design (Springer, 2014).
[Crossref]

Chan, J.

K. Bergman, L. P. Carloni, A. Biberman, J. Chan, and G. Hendry, Photonic Network-on-Chip Design (Springer, 2014).
[Crossref]

Chrostowski, L.

Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014).
[Crossref] [PubMed]

X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. F. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39(19), 5519–5522 (2014).
[Crossref] [PubMed]

R. Boeck, L. Chrostowski, and N. A. F. Jaeger, “Thermally tunable quadruple Vernier racetrack resonators,” Opt. Lett. 38(14), 2440–2442 (2013).
[Crossref] [PubMed]

W. Shi, H. Yun, C. Lin, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Coupler-apodized Bragg-grating add-drop filter,” Opt. Lett. 38(16), 3068–3070 (2013).
[Crossref] [PubMed]

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
[Crossref] [PubMed]

W. Shi, H. Yun, C. Lin, M. Greenberg, X. Wang, Y. Wang, S. T. Fard, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Ultra-compact, flat-top demultiplexer using anti-reflection contra-directional couplers for CWDM networks on silicon,” Opt. Express 21(6), 6733–6738 (2013).
[Crossref] [PubMed]

R. Boeck, W. Shi, L. Chrostowski, and N. A. F. Jaeger, “FSR-eliminated Vernier racetrack resonators using grating-assisted couplers,” IEEE Photon. J. 5(5), 2202511 (2013).
[Crossref]

Y. Wang, J. Flueckiger, C. Lin, and L. Chrostowski, “Universal grating coupler design,” Proc. SPIE 8915, 89150Y (2013).
[Crossref]

R. Boeck, J. Flueckiger, H. Yun, L. Chrostowski, and N. A. F. Jaeger, “High performance Vernier racetrack resonators,” Opt. Lett. 37(24), 5199–5201 (2012).
[Crossref] [PubMed]

W. Shi, X. Wang, W. Zhang, H. Yun, C. Lin, L. Chrostowski, and N. A. F. Jaeger, “Grating-coupled silicon microring resonators,” Appl. Phys. Lett. 100(12), 121118 (2012).
[Crossref]

W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. F. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett. 36(20), 3999–4001 (2011).
[Crossref] [PubMed]

R. Boeck, N. A. F. Jaeger, N. Rouger, and L. Chrostowski, “Series-coupled silicon racetrack resonators and the Vernier effect: theory and measurement,” Opt. Express 18(24), 25151–25157 (2010).
[Crossref] [PubMed]

W. Shi, X. Wang, W. Zhang, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Integrated microring add-drop filters with contradirectional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2012), paper JW4A.91.

L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).
[Crossref]

W. Shi, H. Yun, C. Lin, X. Wang, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Silicon CWDM demultiplexers using contra-directional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2013), paper CTu3F.5.

Chrowtowski, L.

W. Shi, X. Wang, H. Yun, W. Zhang, L. Chrowtowski, and N. A. F. Jaeger, “Add-drop filters in silicon grating-assisted asymmetric couplers,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OTh3D.3.

Contu, P.

Costa, R.

Cunningham, J. E.

Cusmai, G.

A. Melloni, M. Martinelli, G. Cusmai, and R. Siano, “Experimental evaluation of ring resonator filters impact on the bit error rate in non return to zero transmission systems,” Opt. Commun. 234(1–6), 211–216 (2004).
[Crossref]

D’Errico, A.

David, K.

J. Willems, K. David, G. Morthier, and R. Baets, “Filter characteristics of DBR amplifier with index and gain coupling,” Electron. Lett. 27(10), 831–833 (1991).
[Crossref]

De La Rue, R.

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Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014).
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W. Shi, H. Yun, C. Lin, X. Wang, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Silicon CWDM demultiplexers using contra-directional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2013), paper CTu3F.5.

W. Shi, X. Wang, W. Zhang, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Integrated microring add-drop filters with contradirectional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2012), paper JW4A.91.

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P. Orlandi, P. Velha, M. Gnan, P. Bassi, A. Samarelli, M. Sorel, M. J. Strain, and R. De La Rue, “Microring resonator with wavelength selective coupling in SOI,” in Proceedings of 8th IEEE International Conference on Group IV Photonics (IEEE, 2011), pp. 281–283.

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W. Shi, H. Yun, C. Lin, M. Greenberg, X. Wang, Y. Wang, S. T. Fard, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Ultra-compact, flat-top demultiplexer using anti-reflection contra-directional couplers for CWDM networks on silicon,” Opt. Express 21(6), 6733–6738 (2013).
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[Crossref] [PubMed]

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[Crossref]

R. Boeck, J. Flueckiger, H. Yun, L. Chrostowski, and N. A. F. Jaeger, “High performance Vernier racetrack resonators,” Opt. Lett. 37(24), 5199–5201 (2012).
[Crossref] [PubMed]

W. Shi, X. Wang, H. Yun, W. Zhang, L. Chrowtowski, and N. A. F. Jaeger, “Add-drop filters in silicon grating-assisted asymmetric couplers,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OTh3D.3.

W. Shi, H. Yun, C. Lin, X. Wang, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Silicon CWDM demultiplexers using contra-directional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2013), paper CTu3F.5.

W. Shi, X. Wang, W. Zhang, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Integrated microring add-drop filters with contradirectional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2012), paper JW4A.91.

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[Crossref]

Zhang, W.

W. Shi, X. Wang, W. Zhang, H. Yun, C. Lin, L. Chrostowski, and N. A. F. Jaeger, “Grating-coupled silicon microring resonators,” Appl. Phys. Lett. 100(12), 121118 (2012).
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W. Shi, X. Wang, H. Yun, W. Zhang, L. Chrowtowski, and N. A. F. Jaeger, “Add-drop filters in silicon grating-assisted asymmetric couplers,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OTh3D.3.

W. Shi, X. Wang, W. Zhang, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Integrated microring add-drop filters with contradirectional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2012), paper JW4A.91.

Zheng, X.

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R. Boeck, N. A. F. Jaeger, N. Rouger, and L. Chrostowski, “Series-coupled silicon racetrack resonators and the Vernier effect: theory and measurement,” Opt. Express 18(24), 25151–25157 (2010).
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Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014).
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W. Shi, H. Yun, C. Lin, X. Wang, J. Flueckiger, N. A. F. Jaeger, and L. Chrostowski, “Silicon CWDM demultiplexers using contra-directional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2013), paper CTu3F.5.

W. Shi, X. Wang, W. Zhang, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Integrated microring add-drop filters with contradirectional couplers,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2012), paper JW4A.91.

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Figures (10)

Fig. 1
Fig. 1

(a) Diagram of a contra-DC. (b) A close-up view of a portion of a contra-DC (figure was adapted from [9]).

Fig. 2
Fig. 2

(a) Diagram depicting some of the relevant contra-DC parameters as functions of Δβ. (b) Experimental drop port spectrum of one of our devices as a function of Δβ.

Fig. 3
Fig. 3

Experimental drop port spectra for the devices from (a) “run 1,” (c) “run 2,” and (e) “run 3” with gap distances equal to 140 nm, 220 nm, 340 nm, and 400 nm. Experimental through port spectra for the devices from (b) “run 1,” (d) “run 2,” and (f) “run 3” with gap distances equal to 140 nm, 220 nm, 340 nm, and 400 nm.

Fig. 4
Fig. 4

(a) Experimental bandwidth at FWHM versus gap distance and (b) extracted coupling coefficient versus gap distance using the FWHM method. (c) Experimental bandwidth at FWHM versus corrugation width and (d) extracted coupling coefficient versus corrugation width for devices from “run 1” with a fixed gap distance of 280 nm using the FWHM method.

Fig. 5
Fig. 5

Comparison between the FWHM method, the null method, and the curve-fit method to determine |κ| for (a) “run 1,” (b) “run 2,” and (c) “run 3.” (d) Drop port spectrum of a contra-DC with a gap distance of 300 nm from “run 2,” which is chosen to illustrate that there can be multiple possible choices for the location of the first null to the left of the main lobe (the red dots indicate possible choices for the null location).

Fig. 6
Fig. 6

Theoretical predicted minimum bandwidth at FWHM versus coupling length including experimental data points from the devices with gap distances of 400 nm from the three fabrication runs.

Fig. 7
Fig. 7

(a) Experimental and simulated (using the extracted |κ| obtained using the FWHM method) drop port and through port spectra for a contra-DC (from “run 2”) with a gap distance equal to 140 nm. (b) Comparison between the experimental spectra from “run 2” and the simulated spectra using the extracted |κ| of 18466 m−1 from “run 1” for contra-DCs with gap distances of 140 nm. Comparison between the experimental (c) drop port spectra and (d) through port spectra from “run 1,” “run 2,” and “run 3” and the simulated spectra using the average extracted |κ| of 18856 m−1 from the three runs for contra-DCs with gap distances of 140 nm.

Fig. 8
Fig. 8

Comparison between the experimental and the simulated (using extracted |κ|s determined from the FWHM method) (a) maximum power coupling factor and (b) minimum power transmission factor versus gap distance.

Fig. 9
Fig. 9

Comparison between the experimental through port (a) group delay response and (b) dispersion response that were determined using the Hilbert transform method and the simulated results that were determined using the extracted |κ| of 19882 m−1 as well as the measured results using the OVA.

Fig. 10
Fig. 10

Diagram depicting some of the relevant parameters used in our derivation.

Equations (28)

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| κ c | 2 = | A 2 ( 0 ) A 1 ( 0 ) | 2 = | κ | 2 sinh 2 ( s L ) s 2 cosh 2 ( s L ) + ( Δ β 2 ) 2 sinh 2 ( s L )
| t c | 2 = | A 1 ( L ) A 1 ( 0 ) | 2 = s 2 s 2 cosh 2 ( s L ) + ( Δ β 2 ) 2 sinh 2 ( s L )
κ = ω 4 ξ a * ( x , y ) ε m ( x , y ) ξ b ( x , y ) d x d y
| δ β H | = ( 2 π Δ f H c ) ( n g , a ( f 0 ) + n g , b ( f 0 ) ) = ( 2 π Δ λ L λ L λ 0 ) ( n g , a ( λ 0 ) + n g , b ( λ 0 ) )
| δ β L | = ( 2 π Δ f L c ) ( n g , a ( f 0 ) + n g , b ( f 0 ) ) = ( 2 π Δ λ H λ H λ 0 ) ( n g , a ( λ 0 ) + n g , b ( λ 0 ) )
δ β avg = | δ β H | + | δ β L | 2 = π Δ λ bw λ L λ H ( n g , a ( λ 0 ) + n g , b ( λ 0 ) )
| κ | 2 sinh 2 ( s L ) s 2 cosh 2 ( s L ) + ( δ β avg 2 ) 2 sinh 2 ( s L ) = 1 2 tanh 2 ( | κ | L )
| κ | = [ δ β avg 2 4 π 2 L 2 ] 1 2 ,
Δ λ bw min 2.783115 λ 0 2 π L [ n g , a ( λ 0 ) + n g , b ( λ 0 ) ]
δ β H = Δ β ( f H ) Δ β ( f 0 )
δ β H = β a ( f H ) + β b ( f H ) 2 π Λ β a ( f 0 ) β b ( f 0 ) + 2 π Λ
δ β H = β a ( f H ) + β b ( f H ) β a ( f 0 ) β b ( f 0 )
= ( 2 π c ) ( n a ( f H ) f H + n b ( f H ) f H n a ( f 0 ) f 0 n b ( f 0 ) f 0 )
= ( 2 π c ) ( n a ( f H ) f H n a ( f 0 ) f 0 + n b ( f H ) f H n b ( f 0 ) f 0 )
δ β H = ( 2 π Δ f H c ) ( n a ( f 0 ) + f H d n a d f | f 0 + n b ( f 0 ) + f H d n b d f | f 0 ) .
δ β H = ( 2 π Δ f H c ) ( n g , a ( f 0 ) + n g , b ( f 0 ) )
δ β L = ( 2 π Δ f L c ) ( n g , a ( f 0 ) + n g , b ( f 0 ) )
δ β avg = | δ β H | + | δ β L | 2
δ β avg = ( 2 π 2 c ) ( n g , a ( f 0 ) + n g , b ( f 0 ) ) ( Δ f H + Δ f L )
δ β avg = ( π c ) ( n g , a ( f 0 ) + n g , b ( f 0 ) ) ( f H + f L )
δ β avg = ( π c ) ( n g , a ( f 0 ) + n g , b ( f 0 ) ) ( c λ L c λ H )
δ β avg = ( π Δ λ bw λ L λ H ) ( n g , a ( λ 0 ) + n g , b ( λ 0 ) )
2 | κ | 2 sinh 2 ( s L ) tanh 2 ( | κ | L ) = s 2 cosh 2 ( s L ) + ( δ β avg 2 ) 2 sinh 2 ( s L ) .
lim κ 0 2 | κ | 2 sinh 2 ( s L ) tanh 2 ( | κ | L ) = lim κ 0 [ s 2 cosh 2 ( s L ) + ( δ β avg 2 ) 2 sinh 2 ( s L ) ]
cos ( δ β avg L ) 1 L 2 = δ β avg 2 4
cos ( δ β avg L ) + ( δ β avg L ) 2 4 1 = 0 .
Δ λ bw min = 2.783115 λ L λ H π L [ n g , a ( λ 0 ) + n g , b ( λ 0 ) ]
Δ λ bw min 2.783115 λ 0 2 π L [ n g , a ( λ 0 ) + n g , b ( λ 0 ) ] .

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