## Optimization of optical transmittance of a layered metamaterial on active pairs of nanowires

Optics Express, Vol. 14, Issue 8, pp. 3389-3395 (2006)

http://dx.doi.org/10.1364/OE.14.003389

Acrobat PDF (190 KB)

### Abstract

Optical metamaterials with a negative value of the refractive index can be fabricated by means of patterning techniques developed for microelectronics. One of those is a layered metamaterial, where the electric and magnetic response comes from coupled parallel subwavelength size wires. We simulate propagation of EM waves through such a metamaterial. Its properties depend on the density of pairs of nanowires oriented in parallel in one layer. There is a tradeoff between high transmittance and large negative refractive index value *n*. The smaller is the density of nanowires; 1° – the narrower the range of frequencies, where *n* is negative; 2° – the less negative is *n*; 3° – the higher is the transmission.

© 2006 Optical Society of America

## 1. Introduction

**D**and magnetic

**B**induction are anti-parallel to the electric

**E**and magnetic

**H**fields [1

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter **10**, 4785–4809 (1999). [CrossRef]

4. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metameterials,” Opt. Lett. **30**, 3356–3358 (2005). [CrossRef]

5. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, and C. M. Soukoulis, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. **30**, 3198–3200 (2005). [CrossRef] [PubMed]

6. Y. Chen, J. Tao, X. Zhao, Z. Cui, A. S. Schwanecke, and N. I. Zheludev, “Nanoimprint and soft lithography for planar photonic meta-materials,” in *Metamaterials*,
T. Szoplik, E. Özbay, C. M. Soukoulis, and N. I. Zheludev; Eds., Proc. SPIE **5955**, 96–103 (2005). [CrossRef]

7. A. N. Lagarkov and A. K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys. Rev. B **53**, 6318–6336 (1996). [CrossRef]

8. A. K. Sarychev, R. C. McPhedran, and V. M. Shalaev, “Electrodynamics of metal-dielectric composites and electromagnetic crystals,” Phys. Rev. B **62**, 8531–8539 (2000). [CrossRef]

*et al*. [9

9. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes in metal nanowires and lefthanded materials,” J. Nonlinear Opt. Phys. Materials **11**, 65(2002). [CrossRef]

10. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes and negative refraction in metal nanowire composites,” Opt. Express **11**, 735–745 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-735 [CrossRef] [PubMed]

*et al*. [4

4. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metameterials,” Opt. Lett. **30**, 3356–3358 (2005). [CrossRef]

*et al*. [11

11. Y. Svirko, N. Zheludev, and M. Osipov, “Layered chiral metallic microstructures with inductive coupling,” Appl. Phys. Lett. **78**, 498–500 (2001). [CrossRef]

4. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metameterials,” Opt. Lett. **30**, 3356–3358 (2005). [CrossRef]

5. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, and C. M. Soukoulis, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. **30**, 3198–3200 (2005). [CrossRef] [PubMed]

*n*. The smaller is the density of nanowires; 1° – the narrower the range of frequencies, where

*n*is negative; 2° – the less negative is

*n*; 3° – the higher is the transmission. Both the first and the second points imply that the density should be high. The third and the most important from a practical point of view implication demands that the contradictory requirements must find a middle ground.

## 2. Simulation details

*b*= 60 nm thick and 2

*l*= 420 nm long. The separation of two coupled wires is

*a*= 60 nm. The accepted dimensions are consistent with the theoretical model [9

9. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes in metal nanowires and lefthanded materials,” J. Nonlinear Opt. Phys. Materials **11**, 65(2002). [CrossRef]

10. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes and negative refraction in metal nanowire composites,” Opt. Express **11**, 735–745 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-735 [CrossRef] [PubMed]

6. Y. Chen, J. Tao, X. Zhao, Z. Cui, A. S. Schwanecke, and N. I. Zheludev, “Nanoimprint and soft lithography for planar photonic meta-materials,” in *Metamaterials*,
T. Szoplik, E. Özbay, C. M. Soukoulis, and N. I. Zheludev; Eds., Proc. SPIE **5955**, 96–103 (2005). [CrossRef]

*n*= 1.51 (dielectric permeability

*ε*

_{d}= 2.28 + 0i). In an experiment it is advisable to differentiate the dielectric: between wires to choose one with high

*ε*

_{d}and fill the remainder of the cell with another one of

*ε*

_{f}<

*ε*

_{d}. In a single layer of the metamaterial pairs of wires are arranged in a rectangular grid of lattice constant ratio of 1:7 that repeats the aspect ratio of a single wire. The values of these lattice constants are chosen to achieve fill factors from 8% to 20%. Simulations are performed for one and three layers of the metamaterial. There is no relative shift of the second and third layers with respect to the first one. Four interlayer separations are considered from 400 nm to 550 nm every 50 nm.

*ω*as follows

16. W. M. Saj, “FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice,” Opt. Express **13**, 4818–4827 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4818 [CrossRef] [PubMed]

*ε*

_{∞}, = 3.70,

*ω*

_{p}= 13673 THz and Γ = 27.35 THz calculated by Sönnichsen [17] from experimental data on reflection and transmission of silver films obtained by Johnson and Christy [18

18. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*z*the simulation volume is 5,000 nm long with transversal dimensions, width

*x*and height

*y*, varying to achieve the desired fill factor. To model an infinite layer of nanowires Uniaxial Perfectly Matched Layers (UPML) are used as boundary conditions in the direction of propagation

*z*and periodic boundary conditions along the

*x*and

*y*axes. The space discretization step (spatial resolution) equals

*Δr*= 5 nm and the time step

*Δt*=

*Δr*/

*2c*= 8.34 × 10

^{-18}s, where

*c*is the speed of light. We simulate the propagation of a plane wave that is linearly polarized in the

*z*direction (along the wires) for 10,000 simulation steps and then record the field intensity, that is Poynting vector length and the discrete Fourier transform of the electric field.

## 3. Theoretical assessment of refractive index

7. A. N. Lagarkov and A. K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys. Rev. B **53**, 6318–6336 (1996). [CrossRef]

*l*is wire length, 2

*b*is its width,

*a*is separation of coupled wires,

*p*is fill factor of a layer,

*k*is wave vector, and function

*f*(Δ) depends on frequency and takes into account the skin effect of conducting wires

*γ*is dimensionless relaxation parameter,

*G*

^{2}=Ω

^{2}+2

*i*/

*γ*where g and

*Ω*are frequencies, the latter one dimensionless

*ε*

_{d}of the dielectric between wires; permittivity

*ε*

_{f}of the dielectric filling the rest of the cell volume;

*ε*

_{m}and

*μ*

_{m}, permittivity and permeability of the metal;

*σ*

_{m}conductivity of the metal; and

*J*

_{i}Bessel functions of first kind and zero and first order.

*ε*=

*ε*

_{r}+

*iε*

_{i}[Fig. 1(a)], permeability

*μ*=

*μ*

_{r}+

*iμ*

_{i}[Fig. 1(b)] and the effective refractive index

*n*of a single layer of the metamaterial calculated for the above geometry assuming a fill factor

*p*= 12%. When the condition

*ε*

_{r}|

*μ*|+

*μ*

_{r}|

*ε*| < 0 is satisfied [19

19. R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. **41**, 315–316 (2004). [CrossRef]

*p*= 12% this takes place for a range of wavelengths from 1.13-1.28 μm and is illustrated in Fig. 1(c). Increase of the fill factor value brings three consequences: 1° the spectral range of negative refractive indices widens, 2° the left zero-valued point of

*n*(

*ω*) plot shifts towards smaller wavelengths, and 3° the absolute value of the negative refraction index becomes larger.

## 4. Transmittance of a single layer vs. fill factor

*a*= 10ln(

*I*is the intensity of transmitted light integrated over the cross-section in xy plane and

*I*

_{0}is the intensity of the illuminating wave. It is calculated 270 nm behind the second wire. Lines correspond to layers with different fill factors from

*p*= 6% to 20%. For each curve, two attenuation minima are observed: the first at wavelengths λ = 2.1-2.3 μm and the second in the range λ = 1-1.75 μm.

*z*=

*2mL*

^{2}

*λ*

^{-1}behind the grating, where

*L*denotes the structure period along the

*y*axis and

*m*is integer. Periodic distribution of light intensity for λ = 0.85μm calculated behind the metamaterial layer with

*p*= 16%, that is a lattice period

*L*= 1.05μm, is shown in Fig. 3. This quasi-Talbot effect is observed for wide range of wavelengths

*Δλ*, where

*Δλ*/

*L*≈ 0.35.

*n*increases up to its maximum value. For the same spectral range the real part of

*n*is negative. For the case of considered fill factors we expect

*n*to be negative for the range λ= 1.9-2.2 μm. At that spectral range strong resonance interaction of radiation with coupled wires modifies electric D and magnetic B inductions what results in strong attenuation. For a fill factor

*p*= 8% we calculate that attenuation is about -10 dB (i.e. transmission of 35%). Layers with fill factors bigger than

*p*= 14% transmit less than 25% of incident light, thus stacking them leads to very high absorption.

## 5. Transmittance of three layers vs. fill factor

*ε*

_{d}medium. The space between layers of thickness 400, 450, 500, 550 nm is filled with

*ε*

_{f}dielectric, in our case

*ε*

_{f}=

*ε*

_{d}. The transmittance of three layers resembles that of a single one, however, the transmission minima are broader. For high fill factors three layers completely absorb light in the spectral range where the negative index is expected. For

*p*= 8% three layers have transmission greater than 10% at the spectral range

*λ*= 1.9-2.2 μm.

## 6. Phase shift

*n*the phase of light is delayed in comparison to that of transmitted through a similar dielectric layer with a positive index. The analysis of phase changes indicates whether within a narrow spectral range the metamaterial slab has a negative refractive index or not. This is done for the spectral region where resonant interaction is observed. The procedure is justified because on planes parallel to the direction of propagation for wavelengths smaller than 1.5 μm the dispersion of phase (normalized to 2π and expressed in %) exceeds 1%, which is the accepted cutoff value. Waves from the remaining range (1.5-3.0 μm) have smaller variations across phase planes and after interacting with the metamaterial they remain plane.

*Δϕ*=

*ϕ*

_{∥}−

*ϕ*

_{⊥}between

**E**field orientations parallel and perpendicular to wires in a single metamaterial layer. At the wavelength range where attenuation considerations predicted negative

*n*values we observe negative phase advancement for all fill factors. Although they are small (up to -20 deg), they increase with bigger fill factors due to an increasingly larger negative value of

*n*.

## 7. Conclusions

*n*. The smaller is the density of nanowires; 1° - the narrower the range of frequencies, where

*n*is negative; 2° - the less negative is

*n*; 3° - the higher is the transmission. Additional parameters of a metamaterial are the dielectric permittivities of a material between wires ed and that of a medium filling the remainder of a cell

*ε*

_{f}> 1.The relation

*ε*

_{f}<

*ε*

_{d}assures a large negative refractive index. Moreover, a proper choice of the geometry of a layer is crucial to eliminate diffraction effects for negatively refracted wavelengths.

## Acknowledgments

## References and Links

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

2. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter |

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. | V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metameterials,” Opt. Lett. |

5. | G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, and C. M. Soukoulis, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. |

6. | Y. Chen, J. Tao, X. Zhao, Z. Cui, A. S. Schwanecke, and N. I. Zheludev, “Nanoimprint and soft lithography for planar photonic meta-materials,” in |

7. | A. N. Lagarkov and A. K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys. Rev. B |

8. | A. K. Sarychev, R. C. McPhedran, and V. M. Shalaev, “Electrodynamics of metal-dielectric composites and electromagnetic crystals,” Phys. Rev. B |

9. | V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes in metal nanowires and lefthanded materials,” J. Nonlinear Opt. Phys. Materials |

10. | V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes and negative refraction in metal nanowire composites,” Opt. Express |

11. | Y. Svirko, N. Zheludev, and M. Osipov, “Layered chiral metallic microstructures with inductive coupling,” Appl. Phys. Lett. |

12. | T. J. Antosiewicz, W. M. Saj, J. Pniewski, and T. Szoplik, “Simulation of resonant behavior and negative refraction of metal nanowire composites,” in |

13. | F. Garwe, U. Huebner, T. Clausnitzer, E.-B. Kley, and U. Bauerschaefer, “Elongated gold nanostructures in silica for metamaterials: Technology and optical properties,” in |

14. | J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using short wire pairs,” Phys. Rev. B, |

15. | A. Taflove and S. C. Hagnes, |

16. | W. M. Saj, “FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice,” Opt. Express |

17. | C. SÖnnichsen, |

18. | P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

19. | R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. |

20. | H. Raether, |

**OCIS Codes**

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(120.5710) Instrumentation, measurement, and metrology : Refraction

(160.4760) Materials : Optical properties

(260.0260) Physical optics : Physical optics

(260.2030) Physical optics : Dispersion

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Metamaterials

**History**

Original Manuscript: February 17, 2006

Manuscript Accepted: April 3, 2006

Published: April 17, 2006

**Citation**

Tomasz J. Antosiewicz, W. M. Saj, Jacek Pniewski, and Tomasz Szoplik, "Optimization of optical transmittance of a layered metamaterial on active pairs of nanowires," Opt. Express **14**, 3389-3395 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3389

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### References

- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys. Condens. Matter 10, 4785-4809 (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, 4184-4187 (2000). [CrossRef] [PubMed]
- V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev and A. V. Kildishev, "Negative index of refraction in optical metameterials," Opt. Lett. 30, 3356-3358 (2005). [CrossRef]
- G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, and C. M. Soukoulis, "Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials," Opt. Lett. 30,3198-3200 (2005). [CrossRef] [PubMed]
- Y. Chen, J. Tao, X. Zhao, Z. Cui, A. S. Schwanecke, and N. I. Zheludev, "Nanoimprint and soft lithography for planar photonic meta-materials," in Metamaterials, T. Szoplik, E. Özbay, C. M. Soukoulis, N. I. Zheludev; Eds., Proc. SPIE 5955, 96-103 (2005). [CrossRef]
- A. N. Lagarkov and A. K. Sarychev, "Electromagnetic properties of composites containing elongated conducting inclusions," Phys. Rev. B 53, 6318-6336 (1996). [CrossRef]
- A. K. Sarychev, R. C. McPhedran and V. M. Shalaev, "Electrodynamics of metal-dielectric composites and electromagnetic crystals," Phys. Rev. B 62, 8531-8539 (2000). [CrossRef]
- V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes in metal nanowires and lefthanded materials," J. Nonlinear Opt. Phys. Materials 11, 65 (2002). [CrossRef]
- V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes and negative refraction in metal nanowire composites," Opt. Express 11, 735-745 (2003). [CrossRef] [PubMed]
- Y. Svirko, N. Zheludev and M. Osipov, "Layered chiral metallic microstructures with inductive coupling," Appl. Phys. Lett. 78, 498-500 (2001). [CrossRef]
- T. J. Antosiewicz, W. M. Saj, J. Pniewski, T. Szoplik, "Simulation of resonant behavior and negative refraction of metal nanowire composites," in Metamaterials, T. Szoplik, E. Özbay, C. M. Soukoulis, N. I. Zheludev; Eds., Proc. SPIE 5955, 109-115 (2005).
- F. Garwe, U. Huebner, T. Clausnitzer, E.-B. Kley, and U. Bauerschaefer, "Elongated gold nanostructures in silica for metamaterials: Technology and optical properties," in Metamaterials, T. Szoplik, E. Özbay, C. M. Soukoulis, N. I. Zheludev; Eds., Proc. SPIE 5955, 185-192 (2005).
- J. Zhou, L. Zhang, G. Tuttle, T. Koschny and C. M. Soukoulis, "Negative index materials using short wire pairs," Phys. Rev. B, 73, 041101 (2006). [CrossRef]
- A. Taflove and S. C. Hagnes, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artec House, Norwood, MA 2000).
- W. M. Saj, "FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice," Opt. Express 13, 4818-4827 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4818 [CrossRef] [PubMed]
- C. Sönnichsen, Plasmons in metal nanostructures, PhD Thesis (Ludwig-Maximilians-Universtät München, München, 2001).
- P. Johnson and R. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- R. A. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-316 (2004). [CrossRef]
- H. Raether, Surface Plasmons (Springer, Berlin 1988).

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