Abstract

We propose the use of interleaved graphene sections on top of a silicon waveguide to implement tunable Bragg gratings. The filter central wavelength and bandwidth can be controlled changing the chemical potential of the graphene sections. Apodization techniques are also presented.

© 2014 Optical Society of America

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References

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  1. A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [CrossRef] [PubMed]
  2. A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
    [CrossRef] [PubMed]
  3. B. Sensale-Rodriguez, R. Yan, L. Liu, D. Jena, H. G. Xing, “Graphene for Reconfigurable THz Optoelectronics,” Proc. IEEE 101(7), 1705–1716 (2013).
    [CrossRef]
  4. F. Bonnacorso, Z. Sun, T. Hasan, A. C. Ferrari, “Graphene Photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
    [CrossRef]
  5. Z. Zheng, C. Zhao, S. Lu, Y. Chen, Y. Li, H. Zhang, S. Wen, “Microwave and optical saturable absorption in graphene,” Opt. Express 20(21), 23201–23214 (2012).
    [CrossRef] [PubMed]
  6. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
    [CrossRef] [PubMed]
  7. M. Liu, X. Yin, X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
    [CrossRef] [PubMed]
  8. Z. Lu, L. Zhao, “Nanoscale electro-optic modulators based on graphene-slot waveguides,” J. Opt. Soc. Am. B 29(6), 1490–1496 (2012).
    [CrossRef]
  9. M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, A.-D. Capobianco, “Graphene-assisted critically-coupled optical ring modulator,” Opt. Express 20(21), 23144–23155 (2012).
    [CrossRef] [PubMed]
  10. L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
    [CrossRef]
  11. C. Xu, Y. Jin, L. Yang, J. Yang, X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012).
    [CrossRef] [PubMed]
  12. G. W. Hanson, “Dyadic Green’s function and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
    [CrossRef]
  13. A. Majundar, J. Kim, J. Vuckovick, F. Wan, “Graphene for Tunable Nanophotonic Resonators,” IEEE J. Sel. Top. Quantum Electron. 20(1), 68–71 (2014).
    [CrossRef]
  14. J. Feng, W. Li, X. Qian, J. Qi, L. Qi, J. Li, “Patterning of graphene,” Nanoscale 4(16), 4883–4899 (2012).
    [CrossRef] [PubMed]
  15. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [CrossRef]
  16. See http://camfr.sourceforge.net/ .
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    [CrossRef]
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    [CrossRef] [PubMed]
  19. M. Li, H. Li, Y. Painchaud, “Multi-channel notch filter based on a phase-shift phase-only-sampled fiber Bragg grating,” Opt. Express 16(23), 19388–19394 (2008).
    [CrossRef] [PubMed]

2014

A. Majundar, J. Kim, J. Vuckovick, F. Wan, “Graphene for Tunable Nanophotonic Resonators,” IEEE J. Sel. Top. Quantum Electron. 20(1), 68–71 (2014).
[CrossRef]

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

2013

B. Sensale-Rodriguez, R. Yan, L. Liu, D. Jena, H. G. Xing, “Graphene for Reconfigurable THz Optoelectronics,” Proc. IEEE 101(7), 1705–1716 (2013).
[CrossRef]

M. Burla, L. R. Cortés, M. Li, X. Wang, L. Chrostowski, J. Azaña, “Integrated waveguide Bragg gratings for microwave photonics signal processing,” Opt. Express 21(21), 25120–25147 (2013).
[CrossRef] [PubMed]

2012

2011

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[CrossRef] [PubMed]

2010

F. Bonnacorso, Z. Sun, T. Hasan, A. C. Ferrari, “Graphene Photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[CrossRef]

2008

G. W. Hanson, “Dyadic Green’s function and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[CrossRef]

M. Li, H. Li, Y. Painchaud, “Multi-channel notch filter based on a phase-shift phase-only-sampled fiber Bragg grating,” Opt. Express 16(23), 19388–19394 (2008).
[CrossRef] [PubMed]

2007

A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

1997

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

1994

Azaña, J.

Bonnacorso, F.

F. Bonnacorso, Z. Sun, T. Hasan, A. C. Ferrari, “Graphene Photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[CrossRef]

Boscolo, S.

Burla, M.

Capobianco, A.-D.

Chen, Y.

Chrostowski, L.

Cortés, L. R.

Dai, T.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

De Angelis, C.

de Sterke, C. M.

Engheta, N.

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[CrossRef] [PubMed]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

Feng, J.

J. Feng, W. Li, X. Qian, J. Qi, L. Qi, J. Li, “Patterning of graphene,” Nanoscale 4(16), 4883–4899 (2012).
[CrossRef] [PubMed]

Ferrari, A. C.

F. Bonnacorso, Z. Sun, T. Hasan, A. C. Ferrari, “Graphene Photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[CrossRef]

Geim, A. K.

A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

Geng, B.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Hanson, G. W.

G. W. Hanson, “Dyadic Green’s function and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[CrossRef]

Hasan, T.

F. Bonnacorso, Z. Sun, T. Hasan, A. C. Ferrari, “Graphene Photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[CrossRef]

Hu, T.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

Jena, D.

B. Sensale-Rodriguez, R. Yan, L. Liu, D. Jena, H. G. Xing, “Graphene for Reconfigurable THz Optoelectronics,” Proc. IEEE 101(7), 1705–1716 (2013).
[CrossRef]

Jiang, X.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

C. Xu, Y. Jin, L. Yang, J. Yang, X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012).
[CrossRef] [PubMed]

Jin, Y.

Ju, L.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Kim, J.

A. Majundar, J. Kim, J. Vuckovick, F. Wan, “Graphene for Tunable Nanophotonic Resonators,” IEEE J. Sel. Top. Quantum Electron. 20(1), 68–71 (2014).
[CrossRef]

Li, H.

Li, J.

J. Feng, W. Li, X. Qian, J. Qi, L. Qi, J. Li, “Patterning of graphene,” Nanoscale 4(16), 4883–4899 (2012).
[CrossRef] [PubMed]

Li, M.

Li, W.

J. Feng, W. Li, X. Qian, J. Qi, L. Qi, J. Li, “Patterning of graphene,” Nanoscale 4(16), 4883–4899 (2012).
[CrossRef] [PubMed]

Li, Y.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

Z. Zheng, C. Zhao, S. Lu, Y. Chen, Y. Li, H. Zhang, S. Wen, “Microwave and optical saturable absorption in graphene,” Opt. Express 20(21), 23201–23214 (2012).
[CrossRef] [PubMed]

Liu, L.

B. Sensale-Rodriguez, R. Yan, L. Liu, D. Jena, H. G. Xing, “Graphene for Reconfigurable THz Optoelectronics,” Proc. IEEE 101(7), 1705–1716 (2013).
[CrossRef]

Liu, M.

M. Liu, X. Yin, X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[CrossRef] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Locatelli, A.

Lu, S.

Lu, Z.

Majundar, A.

A. Majundar, J. Kim, J. Vuckovick, F. Wan, “Graphene for Tunable Nanophotonic Resonators,” IEEE J. Sel. Top. Quantum Electron. 20(1), 68–71 (2014).
[CrossRef]

Midrio, M.

Moresco, M.

Novoselov, K. S.

A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

Painchaud, Y.

Pei, C.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

Poladian, L.

Qi, J.

J. Feng, W. Li, X. Qian, J. Qi, L. Qi, J. Li, “Patterning of graphene,” Nanoscale 4(16), 4883–4899 (2012).
[CrossRef] [PubMed]

Qi, L.

J. Feng, W. Li, X. Qian, J. Qi, L. Qi, J. Li, “Patterning of graphene,” Nanoscale 4(16), 4883–4899 (2012).
[CrossRef] [PubMed]

Qian, X.

J. Feng, W. Li, X. Qian, J. Qi, L. Qi, J. Li, “Patterning of graphene,” Nanoscale 4(16), 4883–4899 (2012).
[CrossRef] [PubMed]

Romagnoli, M.

Sensale-Rodriguez, B.

B. Sensale-Rodriguez, R. Yan, L. Liu, D. Jena, H. G. Xing, “Graphene for Reconfigurable THz Optoelectronics,” Proc. IEEE 101(7), 1705–1716 (2013).
[CrossRef]

Shen, A.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

Sipe, J. E.

Sun, Z.

F. Bonnacorso, Z. Sun, T. Hasan, A. C. Ferrari, “Graphene Photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[CrossRef]

Ulin-Avila, E.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Vakil, A.

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[CrossRef] [PubMed]

Vuckovick, J.

A. Majundar, J. Kim, J. Vuckovick, F. Wan, “Graphene for Tunable Nanophotonic Resonators,” IEEE J. Sel. Top. Quantum Electron. 20(1), 68–71 (2014).
[CrossRef]

Wan, F.

A. Majundar, J. Kim, J. Vuckovick, F. Wan, “Graphene for Tunable Nanophotonic Resonators,” IEEE J. Sel. Top. Quantum Electron. 20(1), 68–71 (2014).
[CrossRef]

Wang, F.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Wang, X.

Wen, S.

Xing, H. G.

B. Sensale-Rodriguez, R. Yan, L. Liu, D. Jena, H. G. Xing, “Graphene for Reconfigurable THz Optoelectronics,” Proc. IEEE 101(7), 1705–1716 (2013).
[CrossRef]

Xu, C.

Yan, R.

B. Sensale-Rodriguez, R. Yan, L. Liu, D. Jena, H. G. Xing, “Graphene for Reconfigurable THz Optoelectronics,” Proc. IEEE 101(7), 1705–1716 (2013).
[CrossRef]

Yang, J.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

C. Xu, Y. Jin, L. Yang, J. Yang, X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012).
[CrossRef] [PubMed]

Yang, L.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

C. Xu, Y. Jin, L. Yang, J. Yang, X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012).
[CrossRef] [PubMed]

Yin, X.

M. Liu, X. Yin, X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[CrossRef] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Yu, H.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

Zentgraf, T.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Zhang, H.

Zhang, X.

M. Liu, X. Yin, X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[CrossRef] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Zhao, C.

Zhao, L.

Zheng, Z.

IEEE J. Sel. Top. Quantum Electron.

A. Majundar, J. Kim, J. Vuckovick, F. Wan, “Graphene for Tunable Nanophotonic Resonators,” IEEE J. Sel. Top. Quantum Electron. 20(1), 68–71 (2014).
[CrossRef]

IEEE Photon. Technol. Lett.

L. Yang, T. Hu, A. Shen, C. Pei, Y. Li, T. Dai, H. Yu, Y. Li, X. Jiang, J. Yang, “Proposal for a 2×2 Optical Switch Based on Graphene-Silicon-Waveguide Microring,” IEEE Photon. Technol. Lett. 26(3), 235–238 (2014).
[CrossRef]

J. Appl. Phys.

G. W. Hanson, “Dyadic Green’s function and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[CrossRef]

J. Lightwave Technol.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nano Lett.

M. Liu, X. Yin, X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[CrossRef] [PubMed]

Nanoscale

J. Feng, W. Li, X. Qian, J. Qi, L. Qi, J. Li, “Patterning of graphene,” Nanoscale 4(16), 4883–4899 (2012).
[CrossRef] [PubMed]

Nat. Mater.

A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

Nat. Photonics

F. Bonnacorso, Z. Sun, T. Hasan, A. C. Ferrari, “Graphene Photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[CrossRef]

Nature

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[CrossRef] [PubMed]

Opt. Express

Proc. IEEE

B. Sensale-Rodriguez, R. Yan, L. Liu, D. Jena, H. G. Xing, “Graphene for Reconfigurable THz Optoelectronics,” Proc. IEEE 101(7), 1705–1716 (2013).
[CrossRef]

Science

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[CrossRef] [PubMed]

Other

See http://camfr.sourceforge.net/ .

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Figures (7)

Fig. 1
Fig. 1

Deep silicon waveguide with a layer of graphene placed on top of it.

Fig. 2
Fig. 2

Effective index and losses (inset) for the TM fundamental mode of a deep GSW versus the chemical potential (T = 300°K and 1 / Γ = 5 x 10 13 sec).

Fig. 3
Fig. 3

Layout of the proposed silicon graphene Bragg grating.

Fig. 4
Fig. 4

Bidimensional normalized field amplitude (left) and normalized field amplitude for the X = 0 axis (right) in a grating section without (upper) and with (lower) graphene (μc = 0.58 eV) cover layer.

Fig. 5
Fig. 5

Transmission (Left) and Reflection (Right) intensity transfer function of a Silicon graphene Bragg grating with N = 3 Λ = 1250.4 nm, L = 1500 μm, μco = 0.52 eV, neffo = 1.8591, T = 300°K and 1 / Γ = 5 x 10 13 sec.

Fig. 6
Fig. 6

Central operating wavelength λD and the grating bandwidth evolution versus the chemical potential of a Silicon graphene Bragg grating with Λ = 1250.4 nm, L = 1500 μm, μco = 0.52 eV, neffo = 1.8591, T = 300°K and 1 / Γ = 5 x 10 13 sec.

Fig. 7
Fig. 7

Transmission (left) and Reflection (right) intensity transfer function of a Silicon graphene, Gaussian-apodized, Bragg grating with N = 3, Λ = 1250.48 nm, L = 1500 μm, μc = 0.62 eV, T = 300°K and 1 / Γ = 5 x 10 13 sec. taking the Gaussian window coefficient G as a parameter.

Equations (9)

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

σ ( ω , μ c ) = σ intra ( ω , μ c ) + σ inter ( ω , μ c )
σ intra ( ω , μ c ) = i e 2 π 2 ( ω + i 2 Γ ) [ μ c k B T + 2 ln ( e ( μ c / k B T ) + 1 ) ]
σ inter ( ω , μ c ) i e 2 4 π ln ( 2 | μ c | ( ω 2 i Γ ) 2 | μ c | + ( ω 2 i Γ ) )
Γ = e v F 2 μ μ c
ε g ( ω , μ c ) = 1 + i σ ( ω , μ c ) ω ε o Δ
| μ c ( V g ) | = v F π | η ( V g V o ) |
λ D ( μ c , N ) = 2 n ¯ e f f ( μ c ) Λ / N n ¯ e f f ( μ c ) = n e f f o + n e f f ( μ c ) 2
Δ λ ( μ c , N ) λ D ( μ c , N ) = Δ n e f f ( μ c , N ) n ¯ e f f ( μ c ) 1 + ( λ D ( μ c , N ) Δ n e f f ( μ c , N ) L ) 2
Δ n e f f ( μ c , z ) = Δ n e f f ( μ c , L / 2 ) exp [ G 2 ( z L 2 ) 2 ]

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