## Calculation of effective permittivity, permeability, and surface impedance of negative-refraction photonic crystals

Optics Express, Vol. 15, Issue 13, pp. 8340-8345 (2007)

http://dx.doi.org/10.1364/OE.15.008340

Acrobat PDF (374 KB)

### Abstract

We consider the eigen-fields of a two-dimensional negative-refraction photonic crystal and obtain negative effective permittivity and negative effective permeability. Effective permittivity, permeability, and surface impedance are calculated by averaging the eigen-fields. The value of the surface impedance is shown to be location-dependent and is validated by finite-difference time-domain simulations. The unique power propagation mechanism in the photonic crystal is demonstrated through time-evolution of eigen-fields.

© 2007 Optical Society of America

## 1. Introduction

1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**, 4773(1996). [CrossRef] [PubMed]

5. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966 (2000). [CrossRef] [PubMed]

6. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. **10**, 509 (1968). [CrossRef]

7. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature **423**, 604 (2003). [CrossRef] [PubMed]

8. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. **92**, 127401 (2004). [CrossRef] [PubMed]

9. Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. **87**, 091907 (2005). [CrossRef]

10. Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. **95**, 153901 (2005). [CrossRef] [PubMed]

*et al*., proposed to calculate the impedance of a negative-refraction PhC by observing the reflection into its matched medium, and numerically demonstrated that the value of the impedance of a PhC strongly depends on its boundary — especially the way of truncation of the elementary cells located on this boundary [11

11. T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. **97**, 073905 (2006). [CrossRef] [PubMed]

_{r}(

**r**) = μ

_{r}(

**r**+

**R**

_{lmn}and ε

_{r}= ε

_{r}(

**r**+

**R**), where

**R**

_{lmn}represents a unit lattice translation vector. Moreover, in their work the determination of the effective parameters relies on numerically simulated results and each simulation provides the parameters for only one boundary. To address these issues, we present a method based on averaging eigen-fields to calculate the effective permittivity, permeability, and surface impedance on all possible boundaries of negative-refraction PhCs.

## 2. Calculation of effective permittivity and permeability

_{r}= 20. The lattice constant is a and the diameter of the air holes is 2

*r*= 0.7

*a*, as shown in Fig. 1(a). Subwavelength imaging by this PhC was reported in our recent work [9

9. Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. **87**, 091907 (2005). [CrossRef]

*z*-axis) modes, the electromagnetic fields have three independent components,

*E*,

_{z}*H*, and

_{x}*H*. The coordinate system as illustrated in Fig. 1(a) will be used throughout this paper. Using μ

_{y}_{r}(

**r**) = 1 in Maxwell equations, we obtain the governing function for

*E*

_{z}*E*

^{(k)}

_{z}(

**r**) =

*E*

^{(k)}

_{z}(

**r**+

**R**

_{lmn}) is a function with the same periodicity as the PhC. Plugging ε

_{r}(

**r**) = ε

_{r}(

**r**+

**R**

_{lmn}) into the governing function and using the plane-wave method [12

12. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-dormain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173 (2001). [CrossRef] [PubMed]

**k**

_{a}/2π, and the frequency to ω

*a*/2π

*c*(

*c*is the speed of light in vacuum). To avoid clutter, we only depict the first two bands and the light cone. As can be seen from Fig. 1(b), the second band is curved downward. As a result, the group velocity,

**v**

_{g}= ∇

_{k}ω(

**k**), is opposite to the phase velocity and negative refraction ensues. In this work, we focused on the intersection of the second band and the light cone because it was suggested that the corresponding effective index

*n*

_{eff}= -1. In order to see that region clearly, we project the second band in the vicinity of the intersection into a series of equi-frequency contours. As shown in Fig. 1(c), these contours are inward-growing and centered at the origin (

*k*= 0,

_{x}*k*= 0). In particular, we can see that the intersection occurs at ω = 0.236. Since the contour of ω = 0.236 is approximately circular, we can treat it as an isotropic homogenous medium. Using the plane wave method, we can also calculate the eigen-field,

_{y}*E*

^{(k)}

_{z}(

**r**), for a given wave vector. Note that the Bloch wave,

*E*(

_{z}**k**,

**r**,t) = exp(

*j*

**k**∙

**r**-

*j*ω

*t*)

*E*(

*k*)

_{z}, can be understood as the eigen-field

*E*(

*k*)

_{z}(

**r**) packaged in a plane wave, exp(

*j*

**k**∙

**r**-

*j*ω

*t*). The corresponding eigen-fields

*H*and

_{x}*H*can be calculated through

_{y}*x*-axis. We first tried (

*k*= 0.236,

_{x}*k*= 0) and found that the power propagates opposite to the x-axis, which is predicted by the dispersion of the PhC. So we switched the wave vector to (

_{y}*k*= -0.236,

_{x}*k*= 0) and repeated the computation. Figures 2 (a, b) show the amplitude and phase distributions of the corresponding eigen-field,

_{y}*E*(

*k*)

_{z}(

**r**), respectively: the amplitude distribution has the same periodicity as ε

_{r}(

**r**) = ε

_{r}(

**r**+

**R**

_{lmn}) and the phase of

*E*(

*k*)

_{z}(

**r**)

*increases*along the

*x*-axis. For comparison, the plane wave, exp(

*j*

**k**∙

**r**-

*j*ω

*t*), has longer period along the

*x*-axis than that of the PhC, and its phase

*decreases*along the

*x*-axis. See Fig. 2(c). In other words, the phase growth of the package and the carrier waves is along opposite directions, i.e. the carrier wave propagates backward while the package wave propagates forward. The phase distribution of the

*package wave*determines the power flow direction because

*S*(

*av*)

_{x}(

**r**) = -0.5Re{

*E*(

_{z}**k**,

**r**,t)[

*H*(

_{y}**k**,

**r**,

*t*)]

^{*}} = -0.5Re{

*E*(

^{(k)}_{z}**r**)[

*H*(

^{(k)}_{y}**r**]

^{*}}. As a result, the power flows forward. Compared to

*H*

^{(k)}

_{y}(

**r**),

*H*

^{(k)}

_{x}(

**r**) is very small and can be neglected. When they evolve with time, the Bloch wave moves forwards, while the carrier wave moves backwards. The supplemental movies illustrate Re{exp(

*j*

**k**∙

**r**-

*j*ω

*t*)

*E*

^{(k)}

_{z}(

**r**)} and Re{exp(

*j*

**k**∙

**r**-

*j*ω

*t*)}, respectively. Different phase growth directions of the carrier and package explain why the group velocity is opposite to the phase velocity.

*a*/4 <

*x*< √3

*a*/4 and -0.5

*a*<

*y*< 0.5

*a*. To avoid positive and negative values cancelling each other, like the case in sin-function, as well as to keep the “handness” of the electromagnetic wave, the points

**r**are grouped into region I with Re{

*H*

^{(k)}

_{y}(

**r**)} > 0 and region II with Re{

*H*

^{(k)}

_{y}(

**r**)} < 0, and the averaging is applied in these two regions separately. In region I, we obtained

*E*

^{(2Dav)}

_{z}= 0.0202 + 3×10

^{-5},

*j*,

*H*

^{(2Dav)}

_{y}= 0.0861 + 6×10

^{-5}; in region II, we obtained

*E*

^{(2Dav)}

_{z}= -0.0241 - 8×10

^{-5},

*H*

^{(2Dav)}

_{y}= -0.0822 - 2×10

^{-4}. In both regions, the imaginary part of average values is negligible. If the electric and magnetic fields of a plane wave are both known, we can in turn calculate the effective parameters

^{(I)}

_{eff}= - 4.270 and μ

^{(I)}

_{eff}= -0.235. In region II, we obtain ε

^{(II)}

_{eff}= -3.405 and μ

^{(II)}

_{eff}= -0.294. The final effective parameters can be estimated to be

*E*

^{(k)}

_{z }(

**r**)} > 0 and Re{

*E*

^{(k)}

_{z}(

**r**)} < 0, very close result can be obtained because the signs of electric and magnetic fields ensure

*S*

^{(av)}

_{x}(

**r**) > 0 at every point). Consequently, the PhC simultaneously exhibits negative effective permittivity and negative effective permeability. Note that

_{eff}μ

_{eff}= (

*k*/ω)

^{2}is always satisfied. In addition, these effective parameters are derived from the eigen-fields using effective medium theory [2

2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech **47**, 2075 (1999). [CrossRef]

11. T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. **97**, 073905 (2006). [CrossRef] [PubMed]

## 3. Calculation of effective surface impedance

*j*

**k**∙

**r**), will shift spatial frequency to

**k**. As a result, the averaging process results in the component with the spatial frequency matching that of the plane wave the best. In this fashion, the effective impedance can be calculated using

*x*-axis and obtained different values, which mean the impedance is location-dependent. So we refer to the effective impedance as effective

*surface*impedance. The imaginary part of surface impedance is negligible. As can be seen from Fig. 2(d), the surface has the maximum impedance when it goes through the center of holes (

*x*= 0), while the surface has the minimum impedance when it is located at the half way between two neighboring hole columns (

*x*= √3

*a*/4). The effective impedance is not constant, so

_{m}= ±μ

_{eff}and μ

_{m}= ±μ

_{eff}” are sufficient but not necessary. There are many possibilities to match the surface impedance of the photonic crystal only if

_{m}= 1. The selection of permeability is different from that presented in Ref. [12

12. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-dormain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173 (2001). [CrossRef] [PubMed]

*x*= 0), we apply the calculated effective impedance 0.283. As can be seen in Fig. 3(a), very high transmission is achieved. The back-refraction is less than 0.1%. In contrast, the back-reflection increases to 31.2% if the matched medium is switched to the air, as shown in Fig. 3(b). When the boundary goes along the middle line between two hole columns (

*x*= √3

*a*/4), we apply the calculated effective impedance 0.221. As can be seen in Fig. 3(c), very high transmission is achieved. The back-refraction is less than 0.1%. In contrast, the back-reflection increases to 40.7% if the matched medium is switched to the air, as shown in Fig. 3(d).

*x*-axis. The phase distribution of

*E*

^{(k)}

_{z}(

**r**) is also opposite to that of the carrier as shown in Fig. 4(c). The surface impedance strongly depends on incident angles. Figure 4(d) plots the effective impedance for the truncation surfaces located at

*x*= 0 and

*x*= √3

*a*/4 changing with respect to incident angle, respectively.

## 4. Summary

## References and links

1. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

2. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech |

3. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

4. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

5. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

6. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. |

7. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature |

8. | P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. |

9. | Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. |

10. | Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. |

11. | T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. |

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

**OCIS Codes**

(130.3130) Integrated optics : Integrated optics materials

(160.4760) Materials : Optical properties

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: March 2, 2007

Revised Manuscript: May 18, 2007

Manuscript Accepted: May 21, 2007

Published: June 19, 2007

**Citation**

Zhaolin Lu and Dennis W. Prather, "Calculation of effective permittivity, permeability, and surface impedance of negative-refraction photonic crystals," Opt. Express **15**, 8340-8345 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8340

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### References

- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773 (1996). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001). [CrossRef] [PubMed]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic wave: negative refraction by photonic crystals," Nature 423, 604 (2003). [CrossRef] [PubMed]
- P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004). [CrossRef] [PubMed]
- Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, "Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction," Appl. Phys. Lett. 87, 091907 (2005). [CrossRef]
- Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, "Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies," Phys. Rev. Lett. 95, 153901 (2005). [CrossRef] [PubMed]
- T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, "Photonic Crystal Lens: from Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006). [CrossRef] [PubMed]
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-dormain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173 (2001). [CrossRef] [PubMed]

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