Abstract

Light traversing a hollow-core photonic band-gap fiber may experience multiple reflections and thereby a slow-down and enhanced optical path length. This offers a technologically interesting way of increasing the optical absorption of an otherwise weakly absorbing material which can infiltrate the fibre. However, in contrast to structures with a refractive index that varies along the propagation direction, like Bragg stacks, the translationally invariant structures studied here feature an intrinsic trade-off between light slow-down and filling fraction that limits the net absorption enhancement. We quantify the degree of absorption enhancement that can be achieved and its dependence on key material parameters. By treating the absorption and index on equal footing, we demonstrate the existence of an absorption-induced saturation of the group index that itself limits the maximum absorption enhancement that can be achieved.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Soljačić, S. G. Johnson, S. H. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
    [CrossRef]
  2. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
    [CrossRef]
  3. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-5-167.
    [CrossRef] [PubMed]
  4. D. B. Li, and C. Z. Ning, “Giant modal gain, amplified surface plasmon-polariton propagation, and slowing down of energy velocity in a metal-semiconductor-metal structure,” Phys. Rev. B 80, 153304 (2009).
    [CrossRef]
  5. A. V. Maslov, and C. Z. Ning, “Modal gain in a semiconductor nanowire laser with anisotropic band structure,” IEEE J. Quantum Electron. 40, 1389–1397 (2004).
    [CrossRef]
  6. R. W. Boyd, and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
    [CrossRef] [PubMed]
  7. J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
    [CrossRef]
  8. K. H. Jensen, M. N. Alam, B. Scherer, A. Lambrecht, and N. A. Mortensen, “Slow-light enhanced light-matter interactions with applications to gas sensing,” Opt. Commun. 281, 5335–5339 (2008).
    [CrossRef]
  9. N. A. Mortensen, and S. Xiao, “Slow-light enhancement of Beer–Lambert–Bouguer absorption,” Appl. Phys. Lett. 90, 141108 (2007).
    [CrossRef]
  10. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.
  11. J. Mørk, and T. R. Nielsen, “On the enhancement of absorption, phase sensitivity and light-speed control using photonic crystals,” unpublished.
  12. S. G. Johnson, and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-173.
    [CrossRef] [PubMed]
  13. J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
    [CrossRef] [PubMed]
  14. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [CrossRef] [PubMed]
  15. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College Press, 2005).
    [CrossRef]
  16. C. Jiang, M. Ibanescu, J. D. Joannopoulos, and M. Soljačić, “Zero-group-velocity modes in longitudinally uniform waveguides,” Appl. Phys. Lett. 93, 241111 (2008).
    [CrossRef]
  17. A. F. Oskooi, J. D. Joannopoulos, and S. G. Johnson, “Zero-group–velocity modes in chalcogenide holey photonic-crystal fibers,” Opt. Express 17, 10082–10090 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-10082.
    [CrossRef] [PubMed]
  18. J. Hald, J. C. Petersen, and J. Henningsen, “Saturated Optical Absorption by Slow Molecules in Hollow-Core Photonic Band-Gap Fibers,” Phys. Rev. Lett. 98, 213902 (2007).
    [CrossRef] [PubMed]
  19. J. Henningsen, J. Hald, and J. C. Petersen, “Saturated absorption in acetylene and hydrogen cyanide in hollow-core photonic bandgap fibers,” Opt. Express 13, 10475–10482 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-26-10475.
    [CrossRef] [PubMed]
  20. J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
    [CrossRef]
  21. J. Grgić, J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Group-index limitations in slow-light photonic crystals,” Photon. Nanostructures 8, 56–61 (2010).
    [CrossRef]
  22. J. Pedersen, S. Xiao, and N. A. Mortensen, “Slow-light enhanced absorption for bio-chemical sensing applications: potential of low-contrast lossy materials,” J. Eur. Opt. Soc. Rapid Publ. 3, 08007 (2008).
    [CrossRef]
  23. N. A. Mortensen, and M. D. Nielsen, “Modeling of realistic cladding structures for air-core photonic bandgap fibers,” Opt. Lett. 29, 349–351 (2004).
    [CrossRef] [PubMed]

2010 (1)

J. Grgić, J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Group-index limitations in slow-light photonic crystals,” Photon. Nanostructures 8, 56–61 (2010).
[CrossRef]

2009 (4)

A. F. Oskooi, J. D. Joannopoulos, and S. G. Johnson, “Zero-group–velocity modes in chalcogenide holey photonic-crystal fibers,” Opt. Express 17, 10082–10090 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-10082.
[CrossRef] [PubMed]

D. B. Li, and C. Z. Ning, “Giant modal gain, amplified surface plasmon-polariton propagation, and slowing down of energy velocity in a metal-semiconductor-metal structure,” Phys. Rev. B 80, 153304 (2009).
[CrossRef]

R. W. Boyd, and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
[CrossRef]

2008 (5)

K. H. Jensen, M. N. Alam, B. Scherer, A. Lambrecht, and N. A. Mortensen, “Slow-light enhanced light-matter interactions with applications to gas sensing,” Opt. Commun. 281, 5335–5339 (2008).
[CrossRef]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

J. Pedersen, S. Xiao, and N. A. Mortensen, “Slow-light enhanced absorption for bio-chemical sensing applications: potential of low-contrast lossy materials,” J. Eur. Opt. Soc. Rapid Publ. 3, 08007 (2008).
[CrossRef]

C. Jiang, M. Ibanescu, J. D. Joannopoulos, and M. Soljačić, “Zero-group-velocity modes in longitudinally uniform waveguides,” Appl. Phys. Lett. 93, 241111 (2008).
[CrossRef]

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

2007 (2)

J. Hald, J. C. Petersen, and J. Henningsen, “Saturated Optical Absorption by Slow Molecules in Hollow-Core Photonic Band-Gap Fibers,” Phys. Rev. Lett. 98, 213902 (2007).
[CrossRef] [PubMed]

N. A. Mortensen, and S. Xiao, “Slow-light enhancement of Beer–Lambert–Bouguer absorption,” Appl. Phys. Lett. 90, 141108 (2007).
[CrossRef]

2005 (1)

2004 (2)

N. A. Mortensen, and M. D. Nielsen, “Modeling of realistic cladding structures for air-core photonic bandgap fibers,” Opt. Lett. 29, 349–351 (2004).
[CrossRef] [PubMed]

A. V. Maslov, and C. Z. Ning, “Modal gain in a semiconductor nanowire laser with anisotropic band structure,” IEEE J. Quantum Electron. 40, 1389–1397 (2004).
[CrossRef]

2003 (2)

J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[CrossRef] [PubMed]

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

1999 (1)

Alam, M. N.

K. H. Jensen, M. N. Alam, B. Scherer, A. Lambrecht, and N. A. Mortensen, “Slow-light enhanced light-matter interactions with applications to gas sensing,” Opt. Commun. 281, 5335–5339 (2008).
[CrossRef]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

Boyd, R. W.

R. W. Boyd, and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Chen, Y.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
[CrossRef]

Fan, S. H.

Gauthier, D. J.

R. W. Boyd, and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Grgic, J.

J. Grgić, J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Group-index limitations in slow-light photonic crystals,” Photon. Nanostructures 8, 56–61 (2010).
[CrossRef]

Hald, J.

Henningsen, J.

Ibanescu, M.

C. Jiang, M. Ibanescu, J. D. Joannopoulos, and M. Soljačić, “Zero-group-velocity modes in longitudinally uniform waveguides,” Appl. Phys. Lett. 93, 241111 (2008).
[CrossRef]

M. Soljačić, S. G. Johnson, S. H. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
[CrossRef]

Ippen, E.

Jensen, K. H.

K. H. Jensen, M. N. Alam, B. Scherer, A. Lambrecht, and N. A. Mortensen, “Slow-light enhanced light-matter interactions with applications to gas sensing,” Opt. Commun. 281, 5335–5339 (2008).
[CrossRef]

Jiang, C.

C. Jiang, M. Ibanescu, J. D. Joannopoulos, and M. Soljačić, “Zero-group-velocity modes in longitudinally uniform waveguides,” Appl. Phys. Lett. 93, 241111 (2008).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Knight, J. C.

J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[CrossRef] [PubMed]

Lambrecht, A.

K. H. Jensen, M. N. Alam, B. Scherer, A. Lambrecht, and N. A. Mortensen, “Slow-light enhanced light-matter interactions with applications to gas sensing,” Opt. Commun. 281, 5335–5339 (2008).
[CrossRef]

Li, D. B.

D. B. Li, and C. Z. Ning, “Giant modal gain, amplified surface plasmon-polariton propagation, and slowing down of energy velocity in a metal-semiconductor-metal structure,” Phys. Rev. B 80, 153304 (2009).
[CrossRef]

Lunnemann, P.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
[CrossRef]

Maslov, A. V.

A. V. Maslov, and C. Z. Ning, “Modal gain in a semiconductor nanowire laser with anisotropic band structure,” IEEE J. Quantum Electron. 40, 1389–1397 (2004).
[CrossRef]

Mørk, J.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
[CrossRef]

Mortensen, N. A.

J. Grgić, J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Group-index limitations in slow-light photonic crystals,” Photon. Nanostructures 8, 56–61 (2010).
[CrossRef]

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

J. Pedersen, S. Xiao, and N. A. Mortensen, “Slow-light enhanced absorption for bio-chemical sensing applications: potential of low-contrast lossy materials,” J. Eur. Opt. Soc. Rapid Publ. 3, 08007 (2008).
[CrossRef]

K. H. Jensen, M. N. Alam, B. Scherer, A. Lambrecht, and N. A. Mortensen, “Slow-light enhanced light-matter interactions with applications to gas sensing,” Opt. Commun. 281, 5335–5339 (2008).
[CrossRef]

N. A. Mortensen, and S. Xiao, “Slow-light enhancement of Beer–Lambert–Bouguer absorption,” Appl. Phys. Lett. 90, 141108 (2007).
[CrossRef]

N. A. Mortensen, and M. D. Nielsen, “Modeling of realistic cladding structures for air-core photonic bandgap fibers,” Opt. Lett. 29, 349–351 (2004).
[CrossRef] [PubMed]

Nielsen, M. D.

Ning, C. Z.

D. B. Li, and C. Z. Ning, “Giant modal gain, amplified surface plasmon-polariton propagation, and slowing down of energy velocity in a metal-semiconductor-metal structure,” Phys. Rev. B 80, 153304 (2009).
[CrossRef]

A. V. Maslov, and C. Z. Ning, “Modal gain in a semiconductor nanowire laser with anisotropic band structure,” IEEE J. Quantum Electron. 40, 1389–1397 (2004).
[CrossRef]

Öhman, F.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
[CrossRef]

Oskooi, A. F.

Pedersen, J.

J. Pedersen, S. Xiao, and N. A. Mortensen, “Slow-light enhanced absorption for bio-chemical sensing applications: potential of low-contrast lossy materials,” J. Eur. Opt. Soc. Rapid Publ. 3, 08007 (2008).
[CrossRef]

Pedersen, J. G.

J. Grgić, J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Group-index limitations in slow-light photonic crystals,” Photon. Nanostructures 8, 56–61 (2010).
[CrossRef]

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

Petersen, J. C.

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Sakoda, K.

Scherer, B.

K. H. Jensen, M. N. Alam, B. Scherer, A. Lambrecht, and N. A. Mortensen, “Slow-light enhanced light-matter interactions with applications to gas sensing,” Opt. Commun. 281, 5335–5339 (2008).
[CrossRef]

Soljacic, M.

C. Jiang, M. Ibanescu, J. D. Joannopoulos, and M. Soljačić, “Zero-group-velocity modes in longitudinally uniform waveguides,” Appl. Phys. Lett. 93, 241111 (2008).
[CrossRef]

M. Soljačić, S. G. Johnson, S. H. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
[CrossRef]

van der Poel, M.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
[CrossRef]

Xiao, S.

J. Grgić, J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Group-index limitations in slow-light photonic crystals,” Photon. Nanostructures 8, 56–61 (2010).
[CrossRef]

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

J. Pedersen, S. Xiao, and N. A. Mortensen, “Slow-light enhanced absorption for bio-chemical sensing applications: potential of low-contrast lossy materials,” J. Eur. Opt. Soc. Rapid Publ. 3, 08007 (2008).
[CrossRef]

N. A. Mortensen, and S. Xiao, “Slow-light enhancement of Beer–Lambert–Bouguer absorption,” Appl. Phys. Lett. 90, 141108 (2007).
[CrossRef]

Yvind, K.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
[CrossRef]

Appl. Phys. Lett. (2)

N. A. Mortensen, and S. Xiao, “Slow-light enhancement of Beer–Lambert–Bouguer absorption,” Appl. Phys. Lett. 90, 141108 (2007).
[CrossRef]

C. Jiang, M. Ibanescu, J. D. Joannopoulos, and M. Soljačić, “Zero-group-velocity modes in longitudinally uniform waveguides,” Appl. Phys. Lett. 93, 241111 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. V. Maslov, and C. Z. Ning, “Modal gain in a semiconductor nanowire laser with anisotropic band structure,” IEEE J. Quantum Electron. 40, 1389–1397 (2004).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

J. Pedersen, S. Xiao, and N. A. Mortensen, “Slow-light enhanced absorption for bio-chemical sensing applications: potential of low-contrast lossy materials,” J. Eur. Opt. Soc. Rapid Publ. 3, 08007 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (1)

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009).
[CrossRef]

Nat. Photonics (1)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

Nature (1)

J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

K. H. Jensen, M. N. Alam, B. Scherer, A. Lambrecht, and N. A. Mortensen, “Slow-light enhanced light-matter interactions with applications to gas sensing,” Opt. Commun. 281, 5335–5339 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Photon. Nanostructures (1)

J. Grgić, J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Group-index limitations in slow-light photonic crystals,” Photon. Nanostructures 8, 56–61 (2010).
[CrossRef]

Phys. Rev. B (2)

J. G. Pedersen, S. Xiao, and N. A. Mortensen, “Limits of slow light in photonic crystals,” Phys. Rev. B 78, 153101 (2008).
[CrossRef]

D. B. Li, and C. Z. Ning, “Giant modal gain, amplified surface plasmon-polariton propagation, and slowing down of energy velocity in a metal-semiconductor-metal structure,” Phys. Rev. B 80, 153304 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

J. Hald, J. C. Petersen, and J. Henningsen, “Saturated Optical Absorption by Slow Molecules in Hollow-Core Photonic Band-Gap Fibers,” Phys. Rev. Lett. 98, 213902 (2007).
[CrossRef] [PubMed]

Science (2)

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

R. W. Boyd, and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Other (3)

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College Press, 2005).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.

J. Mørk, and T. R. Nielsen, “On the enhancement of absorption, phase sensitivity and light-speed control using photonic crystals,” unpublished.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Left panel shows a typical Ez field pattern for the considered mode, and a right panel shows the fiber geometry with the blue region indicating the symmetry-reduced calculational domain

Fig. 2.
Fig. 2.

Dispersion relation (solid curve) for a mode guided in a hollow-core photonic band gap fiber made from a high-index soft glass. The filled regions show the projected bands of the photonic crystal cladding, resulting in a band gap all the way to β = 0 (white region). Results are obtained for a lossless structure with the wave equation being solved with the aid of a plane-wave method [12].

Fig. 3.
Fig. 3.

Complex dispersion relation for the hollow core fiber being infiltrated by an absorbing gas with n = n′ + in″ with n′ = 1 and n″ ranging from 0 to 0.01. The left panel shows the dispersion while the right panel shows the corresponding absorption in dependence of the frequency (vertical axis).

Fig. 4.
Fig. 4.

Comparison of the absorption enhancement factor (left panel) and the group index (right panel), both derived from the results in Fig. 3. For the absorbing gas, n″ is varied in the range from 0.001 to 0.01. In the right panel, for comparison the dashed line represents the group index in the ideal structure, neglecting both leakage loss and absorption (calculated with the aid of a plane-wave method as in Ref. [17]).

Fig. 5.
Fig. 5.

The maximal absorption enhancement factor γmax versus intrinsic gas absorption n″ for the infiltrated gas.

Fig. 6.
Fig. 6.

Schematic illustration of light rays for a) index-guided and c) band-gap guided modes. b) and d) show the corresponding field confinement profiles. Very small inclination of wave vector with respect to the interface in the dielectric slab, panel a), implies a very weak field confinement, panel b). The multiple reflections in a 2D translationally invariant guide with transversal periodicity, panel c), may cause a more tight field confinement, panel d).

Fig. 7.
Fig. 7.

Schematic illustration of complex dispersion relation.The dashed lines mimic the lossless case while the solid lines are in the presence of moderate damping in the system. Left panel illustrates the real part of the dispersion relation; middle panel represents the imaginary part. The right panel shows the: shows the associated DOS illustrating, how the van Hove singularity is smeared in the presence of damping

Equations (3)

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

γ = α α 0
ω ( β ) = ω 0 + 1 2 ( 2 ω β 2 ) β = 0 β 2 + 𝒪 ( β 4 ) , ( β 0 )
[ t 2 + ε ( x , y ) ( ω c ) 2 + ( t 2 ε ( x , y ) ε ( x , y ) ) × t × ] H t ( x , y ) = β 2 H t ( x , y )

Metrics