## Optical impedance of metallic nano-structures

Optics Express, Vol. 14, Issue 17, pp. 7709-7722 (2006)

http://dx.doi.org/10.1364/OE.14.007709

Acrobat PDF (280 KB)

### Abstract

Impedance matching refers to the suppression of reflected radiation from an interface and is a concept that applies right across the electromagnetic spectrum. In particular it has come to prominence in relation to the propagation of light in metallic structures and associated meta-materials. Whilst established for microwaves and electrical circuits, this concept has only very recently been observed in the optical domain, yet is not well defined or understood. We present a framework to elucidate the concept of optical impedance. We describe using a scattering matrix approach the characteristic, iterative, image and wave impedances of an optical system. With a numerical model, we explore each form of impedance matching in metal-dielectric structures. Thin gold layers may extend the concept of Brewster’s angle to normal incidence and s polarization. Optical impedance for recently realized metallic gold nano-pillars which has shown negative permeability is also explored and we show that current measurements are inconclusive to robustly state its characteristic impedance is matched to the vacuum.

© 2006 Optical Society of America

## 1. Introduction

1. 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]

2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667–669 (1998). [CrossRef]

3. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. **10**, 509 (1968). [CrossRef]

4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

5. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science **308**, 534–537 (2005). [CrossRef] [PubMed]

6. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature **438**, 335–338 (2005). [CrossRef] [PubMed]

9. S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. **68**, 449–521 (2005). [CrossRef]

10. R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. **84**, 1254–1256 (2004). [CrossRef]

11. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

12. X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E **70**, 016608 (2004). [CrossRef]

13. U. Leonhardt, “Optical Conformal Mapping,” Science **312**, 1777 (2006). [CrossRef] [PubMed]

14. J. B. Pendry, D. Schuring, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780 (2006). [CrossRef] [PubMed]

6. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature **438**, 335–338 (2005). [CrossRef] [PubMed]

## 2. Optical impedance definitions

_{ij}, of the optical device. A 2×2 scattering matrix,

**s**=

*s*

_{ij}, defines the optical interaction between a planar, homogenous or periodic, non-diffracting optical layer and an incident electromagnetic plane wave. The subscripts

*i,j*=1,2 in the scattering matrix denote the two sides of the structure. The scattering coefficients

*s*

_{11}and

*s*

_{22}correspond respectively to the reflection coefficient on side 1 respectively on side 2 whereas

*s*

_{12}and

*s*

_{21}to the transmission from side 1 to 2 and vice versa. Details of how to construct the scattering matrix for general non-symmetric and non-reciprocal can be found in the appendix.

### 2.1 Symmetric structures: Wave impedance

15. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. **67**, 717–754 (2004). [CrossRef]

*Z*

_{in}=

*Z*

_{out}=

*Z*

_{w}). In field reflectivity coefficients this translates as:

*q*= Sign(Re(1 +

_{w}is the inverse of the characteristic impedance defined in [12

12. X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E **70**, 016608 (2004). [CrossRef]

12. X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E **70**, 016608 (2004). [CrossRef]

### 2.2 Symmetric structures: Characteristic impedance

_{z}(E

_{z}) polarization as the mode having only the z-component of the magnetic (electric) field not equal to zero. With respect to the multilayer structure the H

_{z}(E

_{z}) polarization is equivalent to the p(s) polarization.

_{z}polarization, we deduce the properties of a homogenous layer that is optically equivalent to a symmetric scattering matrix with

*r*

_{w}and

*p*

_{w}as wave impedance parameters. This implies:

*k*

_{x0},

*k*

_{y0}) and a complex effective (

*k*

_{x;eff},

*k*

_{y;eff}) wavevector inside the structure. From Eq. (6), we can deduce the effective optical parameters. First we define the normal component of the effective wave vector

*h*is the total thickness of the structure and Log(

*p*

_{w}) is the principal branch of the natural logarithm. The scattering coefficients in the scattering matrix partially loose the optical path length information. This information is embedded in the argument or phase of the complex valued coefficients but its precise value is lost because the argument of a complex number is only defined modulo 2

*π*. Because of this, the effective wave vector

*k*

_{x;eff}in Eq. (7) is only defined modulo 2

*π*/

*h*.

*p*

_{w}, of a structure while varying the wavevector from (0,0) to its final value of (

*k*

_{x0},

*k*

_{y0}) we observe that its logarithm turns around the branch point a number of times. Each additional turn corresponds to an additional wavelength fitting inside the optical path length. The wavelength of the (0,0) wavevector is infinite and fits zero times in the optical path length of any finite structure. The (0,0) wavevector can be used as the starting point for counting the number of turns around the branch point. Each additional turn adds a 2

*π*to the argument of

*p*and 2

*π*/

*h*to

*k*

_{x1}. In mathematical terms, this translates into ensuring the continuity of

*k*

_{x;eff}, with respect to the parameter

*τ*∈ [0,1], while the incident wave vector is defined as (

*τk*

_{x0},

*τk*

_{y0}).

*ε*

_{eff}, permeability

*μ*

_{eff}, index of refraction

*n*and material impedance

_{eff}*Z*

_{c}are then defined with respect to this effective wave vector:

_{z}polarization can be treated exactly in the same manner leading to the exchange of

*ε*

_{eff}by

*μ*

_{eff}and vice versa. This method of defining the effective optical parameters generalizes the one used in [11

11. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

**70**, 016608 (2004). [CrossRef]

### 2.3 Periodic structures: Iterative impedance

*Z*

_{iter;1}and

*Z*

_{iter;2}. The fundamental property of the iterative impedance is that it iterates across a multi-layer structure when matched on the output side. As such the iterative impedance is useful to achieve single direction optimal transmission (see section 2.5).

*r*

_{iter;1}and

*r*

_{iter;2}) are solutions of:

*q*= Sign(Re(1 +

*s*

_{11}

*s*

_{22}-

*s*

_{12}

*s*

_{21})).

### 2.4 Non-symmetric structures: Image impedance

*Z*

_{image;1}that when terminating side 1 implies

*Z*

_{image;2}on side 2 and vice versa. The image impedance definition is useful when wanting to optimize the transmission through a non-symmetric structure in both propagation directions (see section 2.4).

### 2.5 Impedance matching

### 2.6 Transparency and impedance matching

6. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature **438**, 335–338 (2005). [CrossRef] [PubMed]

*s*

_{t}and

*s*

_{r}are respectively the sample field transmission and reflection coefficients while

*r*

_{0}is the substrate interface field reflectivity coefficient. Equation (19) is not enough by itself to deduce optical impedance matching to vacuum (

*s*

_{r}=0) and it implies the following relationship

## 3. Numerical applications

### 3.1 Symmetric structures: Characteristic and wave impedance matching

*ε*

_{eff}and permeability

*μ*

_{eff}given by Eq. (8). Remark that this replacement implies that even if the materials at an interface are identical one would generally calculate a non-zero reflectivity at this interface between two different structures. This is because the effective optical coefficients take the whole structure into account.

### 3.2 Periodic structures: Iterative impedance

16. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185-200 (1994). [CrossRef]

13. U. Leonhardt, “Optical Conformal Mapping,” Science **312**, 1777 (2006). [CrossRef] [PubMed]

14. J. B. Pendry, D. Schuring, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780 (2006). [CrossRef] [PubMed]

_{2}, 4nm gold, 24nm SiO

_{2}, 4nm gold and 80nm SiO

_{2}. For the numerical simulation, we use the experimentally determined refractive index for gold [17

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

18. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical-properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. **22**, 1099–1119 (1983). [CrossRef] [PubMed]

_{2}[19

19. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. **55**, 1205–1209 (1965). [CrossRef]

_{2}thicknesses are optimized to match the characteristic impedance to vacuum. Indeed, Fig. 2(a) shows that this is achieved from 600nm to 800nm. Further, we verify the reflection-less absorption properties of this periodic structure by calculating the reflectance and transmittance in normal incidence for a finite stack of 100 periods having a total width of 19.2μm [Fig. 2(b)].

*p*polarization and does not exist for

*s*polarization. The presence of the periodic thin gold layers extends the Brewster condition to normal incidence and the

*s*polarization (see Fig. 3).

### 3.3 Non-symmetric structures: Image impedance

**438**, 335–338 (2005). [CrossRef] [PubMed]

**438**, 335–338 (2005). [CrossRef] [PubMed]

_{∥}|) and orthogonal to this direction (|r

_{⊥}|), respectively in red and blue in Fig. 4(a).

## 4. Conclusion

**438**, 335–338 (2005). [CrossRef] [PubMed]

## Appendix: Impedance scattering decomposition

## A.1 Scattering matrix formalism

**s**´⊕

**s**´´=

**s**as the operator that combines two scattering matrices to give the total scattering of the joined structure

20. M. Mazilu, V. Donchev, and A. Miller, “A modular method for the calculation of transmission and reflection in multilayered structures,” Appl. Opt. **40**, 6670–6676 (2001). [CrossRef]

**I**given by

**s**we have

**s̅**⊕

**s**=

**s**⊕

**s̅**=

**I**and

**I**⊕

**s**=

**s**⊕

**I**=

**s**. In the following, we use this property distinguish between the surface impedance and bulk propagation.

**ŝ**of the reversed structure is:

## A.2 Symmetric structures: wave impedance scattering decomposition

**s**, can be decomposed into three successive scattering elements: two conjugated

*interfaces*

**b**

_{w},

**b̅**

_{w}and a

*bulk propagation*element,

**p**

_{w}. The interface elements are equivalent to the scattering from a refractive index discontinuity while the propagation elements to a reflection less propagation through a homogenous medium. Formally, the decomposition is

**p**

_{w}is determined by:

**b**

_{w}, depends on the optical impedance considered which in turn depends on the symmetry of the structure. In this section, we consider the symmetric case where the scattering matrix elements fulfill the relations in Eq. (1) and Eq. (2). Comparing the symmetric scattering matrix with the scattering matrix of a homogenous layer we can identify the interface and bulk scattering elements with

*r*

_{w}is given by Eq. (4) and the propagation coefficient

*p*

_{w}by

## A.3 Periodic structures: iterative impedance scattering decomposition

*r*

_{iter;1}and

*r*

_{iter;2}defined by Eq. (10), we can define the interface scattering as:

**p**

_{iter}is

*p*

_{iter;12}and

*p*

_{iter;21}are:

## A.4 Non-symmetric structures: image impedance scattering decomposition

## Acknowledgments

## References and links

1. | 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. |

2. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

3. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. |

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

5. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science |

6. | A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature |

7. | P. Lorrain and D. R. Corson, |

8. | R. Yorke, |

9. | S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. |

10. | R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. |

11. | D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

12. | X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E |

13. | U. Leonhardt, “Optical Conformal Mapping,” Science |

14. | J. B. Pendry, D. Schuring, and D. R. Smith, “Controlling electromagnetic fields,” Science |

15. | R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. |

16. | J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

17. | P. B. Johnson, “Optical constants of noble metals,” Phys. Rev. B |

18. | M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical-properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. |

19. | I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. |

20. | M. Mazilu, V. Donchev, and A. Miller, “A modular method for the calculation of transmission and reflection in multilayered structures,” Appl. Opt. |

21. | R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. |

**OCIS Codes**

(120.5700) Instrumentation, measurement, and metrology : Reflection

(260.3910) Physical optics : Metal optics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 12, 2006

Revised Manuscript: August 8, 2006

Manuscript Accepted: August 8, 2006

Published: August 21, 2006

**Citation**

M. Mazilu and K. Dholakia, "Optical impedance of metallic nano-structures," Opt. Express **14**, 7709-7722 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7709

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### References

- 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]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998). [CrossRef]
- V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- N. Fang, H. Lee, C. Sun and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005). [CrossRef] [PubMed]
- A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005). [CrossRef] [PubMed]
- .P. Lorrain and D. R. Corson, Electromagnetic fields and waves, (W. H. Freeman, 1970) Chap. 13.
- R. Yorke, Electric circuit theory (Pergamon Press, 1986) Chap. 8.
- S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005). [CrossRef]
- R. Biswas, Z. Y. Li and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004). [CrossRef]
- D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002). [CrossRef]
- X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004). [CrossRef]
- U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777 (2006). [CrossRef] [PubMed]
- J. B. Pendry, D. Schuring and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006). [CrossRef] [PubMed]
- R. J. Potton, "Reciprocity in optics," Rep. Prog. Phys. 67, 717-754 (2004). [CrossRef]
- J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114,185-200 (1994). [CrossRef]
- P. B. Johnson, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander and C. A. Ward, "Optical-properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared," Appl. Opt. 22, 1099-1119 (1983). [CrossRef] [PubMed]
- I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," J. Opt. Soc. Am. 55, 1205-1209 (1965). [CrossRef]
- M. Mazilu, V. Donchev and A. Miller, "A modular method for the calculation of transmission and reflection in multilayered structures," Appl. Opt. 40, 6670-6676 (2001). [CrossRef]
- R. Biswas, Z. Y. Li, and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004). [CrossRef]

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