## The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction

Optics Express, Vol. 15, Issue 15, pp. 9681-9691 (2007)

http://dx.doi.org/10.1364/OE.15.009681

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### Abstract

The propagation characteristics of a subwavelength plasmonic crystal are studied based on its complex Bloch band structure. Photonic crystal bands are generated with an alternative 2D Finite Element Method formulation in which the Bloch wave problem is reduced to a quadratic eigenvalue system for the Bloch wavevector amplitude *k*. This method constitutes an efficient and convenient alternative to nonlinear search methods normally employed in the calculation of photonic bands when dispersive materials are involved. The method yields complex wavevector Bloch modes that determine the wave-scattering characteristics of finite crystals. This is evidenced in a comparison between the band structure of the square-lattice plasmonic crystal and scattering transfer-functions from a corresponding finite crystal slab. We report on a wave interference effect that leads to transmission resonances similar to Fano resonances, as well as on the isotropy of the crystal’s negative index band. Our results indicate that effective propagation constants obtained from scattering simulations may not always be directly related to individual crystal Bloch bands.

© 2007 Optical Society of America

## 1. Introduction

*ω*for specific Bloch

*k*-vectors as eigenvalues of an electromagnetic wave equation defined in a unit-cell of the periodic medium. If the PC incorporates only dispersionless materials (i.e., the dielectric constant

*ε*(

*ω*) constant for all frequencies), the problem is reduced to a linear generalized eigenvalue system of the form

**A**=

_{k}x_{k}*ω*

^{2}

**B**[1–4

_{k}x_{k}4. B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, “Aplication of finite element methods to photonic crystal modelling,” IEE Proc - Sci. Meas. Technol. **149**, 293–296 (2002). [CrossRef]

*Plasmonic*Crystals (SPCs) [5

5. G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. **93**, 243902 (2004). [CrossRef]

7. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. **95**, 203901 (2005), doi:10.1103/PhysRevLett.95.203901. [CrossRef] [PubMed]

5. G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. **93**, 243902 (2004). [CrossRef]

**A**(

_{k}*ω*)

**x**=

_{k}*ω*

^{2}

**B**(

_{k}*ω*)

**x**, or, more generally,

_{k}**T**(

_{k}*ω*)

**x**=0. Solutions to this type of problem are generally obtained through iterative algorithms -normally extensions of algorithms for linear problems [8

_{k}8. A. Ruhe, “Algorithms for the nonlinear eigenvalue problem,” SIAM J. Numer. Anal. **10**, 674–689 (1973). [CrossRef]

9. A. Spence and C. Poulton, “Photonic band structure calculations using nonlinear eigenvalue techniques,” J. Comput. Phys. **204**, 65–81 (2005). [CrossRef]

*ω*are computed within a chosen convergence domain. Successful and efficient convergence usually requires a good initial guess; in addition matrix

**T**(

*ω*) and, depending on the method, its derivatives, must be recalculated at several frequencies within the search domain, causing the solution process to be time-consuming and computationally intensive. Hence, for this reason, band structure diagrams are not abundant in the metamaterials and plasmonic crystals literature.

*et al*. [4

4. B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, “Aplication of finite element methods to photonic crystal modelling,” IEE Proc - Sci. Meas. Technol. **149**, 293–296 (2002). [CrossRef]

*k*-vector magnitude; frequency in this case is a parameter, so that material dispersion is readily taken into account. While the quadratic eigenvalue equation is nonlinear, it is a more tractable problem than the general nonlinear case above, and can yield solutions more efficiently. In addition, this formulation inherently yields bands of purely imaginary- and complex-wavevector Bloch modes, which may be particularly hard to obtain with nonlinear search routines. Imaginary

*k*modes are generally available at the photonic bandgap frequencies and play an important role in representing the evanescent field inside a finite or semi-infinite PC slab under external excitation. This has been suggested in [10

10. E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B **71**, 195122 (2005). [CrossRef]

3. 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.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

*k*bandgap modes in PCs composed of nondispersive, purely dielectric materials. On the other hand, modes with complex, rather than purely imaginary,

*k*-vectors may also exist within, and even outside, bandgap regions that have considerable influence in the transmission and reflection transfer functions of finite plasmonic crystals.

4. B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, “Aplication of finite element methods to photonic crystal modelling,” IEE Proc - Sci. Meas. Technol. **149**, 293–296 (2002). [CrossRef]

5. G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. **93**, 243902 (2004). [CrossRef]

6. G. Shvets and Y. Urzhumov, “Electric and magnetic properties of sub-wavelength plasmonic crystals,” J. Opt. A: Pure Appl. Opt. **7**, S23–S31 (2005). [CrossRef]

*k*=0 is revealed. A comparison between scattering data from a finite SPC and its band diagram is also performed, in which the role of the multiple existing Bloch modes (including those with imaginary and complex wavevectors) is determined. In particular, we report on the existence of resonances in transmission through a finite crystal slab that stem from the coexistence of two SPC bands in the same frequency range, being similar in origin and character to Fano resonances [12

12. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124**, 1866–1878 (1961). [CrossRef]

## 2. Finite-element method

**H**=

*H*

_{z}**ẑ**, where

*H*is the magnetic field) or TM (

**E**=

*E*

_{z}**ẑ**, where

*E*is the electric field), with the respective

*z*-components obeying the wave equation

*ϕ*=

*H*,

_{z}*p*=1/

*ε*,

*q*=1 and

*ε*is the electric permittivity; for TM waves,

*ϕ*=

*H*,

_{z}*p*=1,

*q*=

*ε*. In Eq.(1),

*ω*/

*c*is the vacuum angular wavenumber.

*ε*(

**r**+

**T**)=

*ε*(

**r**), where

**T**is a lattice translation vector. From Bloch’s theorem we consider solutions

*ϕ*(

**r**)=

*u*(

**r**)exp(-

*i*

**k**·

**r**), with

*u*(

**r**+

**T**)=

*u*(

**r**) and

**k**in the first Brillouin zone. Along with Eq.(1), this leads to:

*w*(

**r**), and note that

*δ*

_{Γ}that covers one unit cell of the periodic medium (the integration domain may actually extend over an integer number of unit cells). Making use of the Divergence theorem, the following weak-form expression is obtained:

**n**̂ is the outward normal unit vector to the boundary

*δ*

_{Γ}. Since

*u*is periodic, the line integral vanishes. The resulting integrodifferential equation may be transformed into matrix format by following the usual FEM discretization procedure [4

**149**, 293–296 (2002). [CrossRef]

*u*is expanded in terms of such functions within each element;

*w*is taken to be each one of the local expansion functions inside each element; and the material parameters

*p*and

*q*are allowed to be constant inside each element. Then the following quadratic matrix eigenvalue equation in

*k*results:

## 3. Rectangular array plasmonic crystal

*t*) and reflection (

*r*) coefficients for plane-waves incident on a finite (i.e., with

*N*of unit cells) crystal slab were obtained using the same technique as described in [15

15. G. Shvets and Y. Urzhumov, “Negative index meta-materials based on two-dimensional metallic structures,” J. Opt. A: Pure Appl. Opt. **8**, S122–S130 (2006). [CrossRef]

*X*direction of the square-lattice Brillouin zone. Furthermone, an effective index

*n*

_{eff}is calculated using expressions from the scattering-parameter technique of Smith

*et al*. [15

15. G. Shvets and Y. Urzhumov, “Negative index meta-materials based on two-dimensional metallic structures,” J. Opt. A: Pure Appl. Opt. **8**, S122–S130 (2006). [CrossRef]

16. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E **71**, 036617 (2005). [CrossRef]

*k*-vectors exist simultaneously - for instance in systems displaying Fano-type resonances as shown below.

*k*modes extend over the entire crystal, with a phase evolution given by the

*k*(Ω) dispersion and with no amplitude variation apart from that of the periodic Bloch envelope. In Fig. 1, purely real eigenvalues are found in the ranges Ω≈0.2-0.36 and Ω≈0.54-0.6.

*k*modes play an important role in representing the decaying field inside a finite or semi-infinite PC upon wave incidence at bandgap frequencies [10

10. E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B **71**, 195122 (2005). [CrossRef]

*k*=0), or complex. Modes with purely imaginary

_{r}*k*(found in the intervals Ω=0.43-0.52 and Ω=0.6-0.61) have no phase variation across a unit-cell, and are non-propagating. Complex

*k*modes, on the other hand, present a phase variation concomitant with an exponential amplitude decay (or growth), which translate into interference effects in scattering from a finite crystal. In Fig. 1, complex

*k*modes with |

*k*| <

_{r}*π*/

*a*are found between Ω=0.36 and 0.43, and between Ω=0.52 and 0.54. Bands with

*k*marked with triangles in Fig. 1(c) have real parts at the Brillouin zone boundaries,

_{i}*k*=±

_{r}*π*/

*a*, such that the phase varies in full cycles across a unit cell. These solutions will henceforth be referred to as

*zone-boundary*modes.

*k*bands discussed above are negative refraction bands, with group velocity

*v*of opposite sign to the phase velocity

_{g}*ω*/

*k*; however, as shown in [5

**93**, 243902 (2004). [CrossRef]

6. G. Shvets and Y. Urzhumov, “Electric and magnetic properties of sub-wavelength plasmonic crystals,” J. Opt. A: Pure Appl. Opt. **7**, S23–S31 (2005). [CrossRef]

*Im*{

*k*

_{eff}} with concomitant step-transitions in

*Re*{

*k*

_{eff}}. At the same time, disregarding the resonances, both real and imaginary parts of the effective wavevector closely follow the corresponding photonic bands. Hence, this behavior has an origin similar to the resonances discussed above, given the existence of two evanescent Bloch bands, as seen in Figs. 1(a) and 1(b).

*k*bands for a lossless and a lossy structure with

*ω*/

_{c}*ω*=0.001 are compared. A noticeable difference between the two cases is the disappearance of the the small bandgap centered at Ω ≈ 0.36. Whereas in the lossless case the purely real

_{p}*k*-bands reach the Brillouin zone boundary at gap frequencies, in the lossy structure the corresponding bands do not reach

_{r}*k*=

_{r}*π*/

*a*, turning around at some

*k*

_{max}<

*π*/

*a*where the group velocity defined as

*v*=d

_{g}*ω*/

*dk*becomes infinite. Notice that in the lossless case the group velocity as defined above is infinite throughout the photonic bandgap, since

_{r}*k*=

_{r}*π*/

*a*everywhere in this range. It is apparent that the band turn-around observed in the lossy crystal case is caused by a less-efficient Bragg wave-interference in the presence of damping losses. This effect was also observed in [17

17. K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, “Nature of lossy bloch states in polaritonic photonic crystals,” Phys. Rev. B **69**, 195111 (2004). [CrossRef]

*v*>

_{g}*c*does not imply superluminal propagation [18

18. H. G. Winful, “The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities,” New J. Phys., Phys. **8**, 101 (2006). [CrossRef]

*k*exist. The degeneracy is evident from the opening of bifurcations in

_{r}*k*at Ω≈ 0.36 and splitting of

_{i}*k*bands in Ω=0.36-0.43 due to small damping losses. The singled-out band in Fig. 4(a) is a perturbed version of a zone-boundary band that exists at Ω⪷0.365 in the lossless case. Above Ω⪷0.365, the latter ceases to be a zone-boundary band, and becomes a degenerate complex-

_{i}*k*band. This transition is evident in the departure of the perturbed band from

*k*=

_{r}*π*/

*a*and its merging with a second complex-

*k*band around Ω=0.36.

*k*-vector space, Figs. 5 and 6 display, respectively, the low-frequency and two high-frequency purely real-

*k*bands. These diagrams were obtained simply by changing the direction of the unit

*k*-vector in Eq. (6).

**93**, 243902 (2004). [CrossRef]

## 4. Conclusions

*k*-space simply involves determination of the components of the unit

*k*-vector, which is convenient for the analysis of medium anisotropy.

10. E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B **71**, 195122 (2005). [CrossRef]

*n*

_{eff}determined from Eq.(8) via scattering from finite crystals will not completely coincide with effective indices of particular Bloch modes when multiple such modes exist that influence the scattering process at similar levels. At the same time, just the Bloch band structure by itself does not give an immediately clear picture of wave scattering and therefore should be used with some attention in predicting the scattering response of finite structures. At any rate, both methods may be used in conjunction to form a more complete understanding of the propagation characteristics of a metamaterial.

**71**, 195122 (2005). [CrossRef]

## Acknowledgements

## References and links

1. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

2. | K. Sakoda, |

3. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis,” Opt. Express |

4. | B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, “Aplication of finite element methods to photonic crystal modelling,” IEE Proc - Sci. Meas. Technol. |

5. | G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. |

6. | G. Shvets and Y. Urzhumov, “Electric and magnetic properties of sub-wavelength plasmonic crystals,” J. Opt. A: Pure Appl. Opt. |

7. | C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. |

8. | A. Ruhe, “Algorithms for the nonlinear eigenvalue problem,” SIAM J. Numer. Anal. |

9. | A. Spence and C. Poulton, “Photonic band structure calculations using nonlinear eigenvalue techniques,” J. Comput. Phys. |

10. | E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B |

11. | |

12. | U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. |

13. | J. Jin, |

14. | F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. |

15. | G. Shvets and Y. Urzhumov, “Negative index meta-materials based on two-dimensional metallic structures,” J. Opt. A: Pure Appl. Opt. |

16. | D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E |

17. | K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, “Nature of lossy bloch states in polaritonic photonic crystals,” Phys. Rev. B |

18. | H. G. Winful, “The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities,” New J. Phys., Phys. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(160.4670) Materials : Optical materials

(260.2030) Physical optics : Dispersion

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 1, 2007

Revised Manuscript: July 15, 2007

Manuscript Accepted: July 16, 2007

Published: July 19, 2007

**Citation**

Marcelo Davanco, Yaroslav Urzhumov, and Gennady Shvets, "The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction," Opt. Express **15**, 9681-9691 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9681

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princteon University Press, 1995).
- K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (New York: Springer, 2001).
- 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). [CrossRef] [PubMed]
- B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, "Aplication of finite element methods to photonic crystal modelling," IEE Proc - Sci. Meas. Technol. 149, 293-296 (2002). [CrossRef]
- G. Shvets and Y. A. Urzhumov, "Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances," Phys. Rev. Lett. 93, 243902 (2004). [CrossRef]
- G. Shvets and Y. Urzhumov, "Electric and magnetic properties of sub-wavelength plasmonic crystals," J. Opt. A: Pure Appl. Opt. 7, S23-S31 (2005). [CrossRef]
- C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005). [CrossRef] [PubMed]
- A. Ruhe, "Algorithms for the nonlinear eigenvalue problem," SIAM J. Numer. Anal. 10, 674-689 (1973). [CrossRef]
- A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005). [CrossRef]
- E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005). [CrossRef]
- [Online]. Available: http://www.comsol.com
- U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961). [CrossRef]
- J. Jin, The Finite Element Method in Electromagnetics, (2nd ed. Wiley, 2002).
- F. Tisseur and K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001). [CrossRef]
- G. Shvets and Y. Urzhumov, "Negative index meta-materials based on two-dimensional metallic structures," J. Opt. A: Pure Appl. Opt. 8, S122-S130 (2006). [CrossRef]
- D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005). [CrossRef]
- K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004). [CrossRef]
- H. G. Winful, "The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities," New J. Phys., Phys. 8, 101 (2006). [CrossRef]

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