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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 13 — Jun. 22, 2009
  • pp: 10800–10805
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Optical magnetic plasma in artificial flowers

Jingjing Li, Lars Thylen, Alexander Bratkovsky, Shih-Yuan Wang, and R. Stanley Williams  »View Author Affiliations


Optics Express, Vol. 17, Issue 13, pp. 10800-10805 (2009)
http://dx.doi.org/10.1364/OE.17.010800


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Abstract

We report the design of an artificial flower-like structure that supports a magnetic plasma in the optical domain. The structure is composed of alternating “petals” of conventional dielectrics (ε>0) and plasmonic materials (Re(ε)<0). The induced effective magnetic current on such a structure possesses a phase lag with respect to the incident TE-mode magnetic field, similar to the phase lag between the induced electric current and the incident TM-mode electric field on a metal wire. An analogy is thus drawn with an artificial electric plasma composed of metal wires driven by a radio frequency excitation. The effective medium of an array of flowers has a negative permeability within a certain wavelength range, thus behaving as a magnetic plasma.

© 2009 Optical Society of America

Fig. 1. (a) An artificial medium composed of flower structures. (b)A single flower with TE0 incident radiation. (c) Left: the cross section of a flower. Dark: plasmonic petals; light: dielectric petals. Right: the equivalent RLC circuit for the structure. Each side represents one period in the structure. The capacitors (C) model the dielectric sectors, the inductors (L) together with the resistors (R) model the lossy plasmonic sectors, and the voltage sources (V) represents the incident electric field.

The simplified picture of the equivalent circuit provides valuable insight into the system, the actual situation is much more complicated. A rigorous analysis requires numerical tools. For a TE incidence, the magnetic field can be expanded into the cylindrical harmonics as [10

10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, NY, 1983).

]

Hiz(r,ϕ)=𝓗0Σn=anJn(k0r)einϕ
(1)

and the scattered field can be similarly expanded as

Hsz(r,ϕ)=𝓗0Σn=(Σm=γmnam)Hn(1)(k0r)einϕ
(2)

An e -iωt convention is assumed through out the paper. Jn(·) is the Bessel function, and H (1) n (·) is the Hankel function, of the first kind of order n. 𝓗 0 is a normalization constant with a unit of magnetic field. γmn is the coefficient connecting the contribution to the n th order mode in the scattered field from the m th order mode in the incident field, which is dimensionless and is completely determined by the geometry and material properties of the structure. For our design, the coefficient γ 00 dominates the behavior of the structure and the other γ mn-s can be neglected. The 0th order mode with magnitude γ 00 a 0 in the scattered field can be considered to result from an induced effective magnetic current. An infinitely long magnetic current Im (with units of volts) radiates a TE0 mode magnetic field with magnitude -ωε 0 Im/4, where ε0 is the permittivity of the host medium[11

11. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, NJ, USA, 1994). [CrossRef]

]. It is easy to show that a 0=H loc/𝓗 0, where H loc is the local magnetic field exciting the structure (i.e. Hiz at the origin in Eq.(1)). Thus, we can model the structure as an induced magnetic current Im=αmH loc, with αm=-4γ 00/(ωε 0) completely determined by γ 00.

In the following, we first study the change of γ 00 as we change the value of ε r1 for a given geometry, and fix the operating wavelength to be 20 times the outer radius. This is not a frequency-dependent property study, but nevertheless will yield an improved understanding of this problem and will provide guidelines for the design of such a structure using realistic material properties. The structure studied here has 6 angular periods, i.e. β 1 +β 2=60° as we see in Fig. 1(c). Other periodicities are in general similar. The dielectric portion is defined with ε r2=2.2, which can be SiO2. We assume the plasmonic material is lossless at this stage, so that the expected resonance can be clearly seen. The material loss will be considered later by adding an imaginary term to the relative permittivity of the plasmonic material. The structure is simulated using the commercially available finite element method software COMSOL. We calculate the scattered field for a given incident field of known distribution, and γ 00 can then be extracted.

The magnitude of the scattering coefficient γ 00 is shown in Fig. 2 (bold solid line) for a design for which the angles of the two sections in one period are identical to each other, i.e. β 1=β 2=30°. A case when β 1β 2 is discussed later. We can see that γ 00 displays a resonance as ε r1 varies. Specifically, the magnitude of γ 00 goes to the maximum possible value |γ 00|=1 as ε r1 goes to -4.07. The phase of γ 00, which determines the phase difference between the induced effective magnetic current and the excitation, is also plotted in Fig. 2 (red bold dashed line). Notice that it indeed flips from approximately -π/2 (indicating an effective magnetization in phase with H loc) to approximately π/2 as |γ 00| achieves the maximum value. In other words, the effective magnetization of the structure can be either in phase or 180° out of phase with H loc. Thus, in analogy to the metal wire under TM excitation, such a structure can support an artificial magnetic plasma. Figure 3(a) displays the instantaneous scattered magnetic field (i.e. the total field minus the incident field for the region both inside and outside the particle) distribution, for the case of ε r1=-4.07. The magnitude of the scattered magnetic field is much higher than that of the incident field because of the resonance. The plot is a snapshot of the field distribution at a phase of π. At this instant, the scattered field inside the structure achieves its maximum value when the incident field is at the negative maximum. This verifies that the scattered field is anti-parallel to the incident excitation.

Fig. 2. Magnitude (solid lines) and phase (red dashed lines) of γ 00, for the lossless (bold) and the lossy (thin) case.

In reality, the plasmonic material is always lossy, which may influence the resonance feature. Quantitatively, such an influence is reflected in the magnitude of the γ00 coefficient at resonance: it is smaller than 1 when loss exists. To see this influence, we calculate γ 00 again, assuming an imaginary part of the relative permittivity with a magnitude 1% that of the real part for the plasmonic material. Such a ratio of the real to imaginary parts of the relative permittivity is common for many plasmonic materials in a wide frequency range of the infrared and/or visible spectra[12

12. P. B. Johnson and R.W. Christy, “Optical Constants of the Nobel Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

]. The magnitude of the scattering coefficient |γ 00| for different values of Re(ε r1) for the same structure is also shown in Fig. 2 (solid light line). Notice that it still exhibits a clear resonance feature despite the existence of material loss, at almost the same value of Re(ε r1) of the lossless case. This is most easily seen in the phase plot (thin dashed line in Fig. 2, red online). The maximum magnitude of |γ 00| is lower than that of the lossless case. Our studies show that geometries with different β 1 and β 2 values actually exhibit different material loss tolerance. In fact, when we use a larger plasmonic sector together with a smaller dielectric sector (β 1>β 2) but keep β 1+β 2=60°, the magnitude of γ 00 at resonance is larger, indicating a stronger resonance, although we are using similarly lossy plasmonic material (Im(ε r1)=0.01Re(ε r1)). Of course, the value of Re(ε r1) at which γ 00 is resonant is also different when β 1 and β 2 take values different than 30° as that in Fig. 2. An example is described in detail below.

With the insight gained above, we now examine designs using realistic material parameters. We use silver with ε r1 at different operating frequencies given by Ref.[12

12. P. B. Johnson and R.W. Christy, “Optical Constants of the Nobel Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

] and SiO2 with ε r2=2.2. A structure similar to Fig. 1 but with β 1=55° and β 2=5° is used, to achieve a strong resonance despite the existence of material loss in silver. For a structure with a=45.2nm, calculations show that the structure has a resonance at λ0=904nm, or 20 times the diameter of the cylindrical structure, as we see in the plot of γ 00 as a function of optical frequency in Fig. 3(b) (light solid line). To create an artificial magnetic plasma, we arrange the structures into a square lattice with a period d=135.6nm, or 0.15λ0 (refer to Fig. 1(a)). According to the duality principle, the formula for εr of the artificial electric plasma composed of metal wires[13

13. S. Tretyakov, “Analytical Modeling in Applied Electromagnetics,” pp. 164–175.(Artech House, INC, Norwood, MA, USA, 2003). In this reference the current coefficient αe of a wire of perfect electric conductor is used, written as a function of the radius of the wire. In our paper, no analytical formula for αm, thus the equation for µr is revised to have αm in it explicitly.

] can be revise to calculate µr for our medium. The result is

μr=12πc2d2ω21A+M
(3)

where M=0.5275 is a constant determined by the lattice shape[13

13. S. Tretyakov, “Analytical Modeling in Applied Electromagnetics,” pp. 164–175.(Artech House, INC, Norwood, MA, USA, 2003). In this reference the current coefficient αe of a wire of perfect electric conductor is used, written as a function of the radius of the wire. In our paper, no analytical formula for αm, thus the equation for µr is revised to have αm in it explicitly.

] and

A=2πωε0Im(αm1)+logωd4πc+C+i(2πωε0Re(αm1)π2)
(4)

where C=0.5772 is the Euler constant and αm=-4γ 00/(ωε 0) as discussed before. By using this formula, the response of the cylindrical structure is assumed to come from the induced magnetic current only, thus the relative permittivity is assumed to be 1, similar to the case of the wire medium where the relative permeability is assumed to be unity[13

13. S. Tretyakov, “Analytical Modeling in Applied Electromagnetics,” pp. 164–175.(Artech House, INC, Norwood, MA, USA, 2003). In this reference the current coefficient αe of a wire of perfect electric conductor is used, written as a function of the radius of the wire. In our paper, no analytical formula for αm, thus the equation for µr is revised to have αm in it explicitly.

]. Equation (3) should not be interpretted as a Drude (or Drude-Lorentz) dispersion because A is a function of ω. For the lossless case, we have Re(α -1 m)=1/(4ωε 0), determined by energy conservation[13

13. S. Tretyakov, “Analytical Modeling in Applied Electromagnetics,” pp. 164–175.(Artech House, INC, Norwood, MA, USA, 2003). In this reference the current coefficient αe of a wire of perfect electric conductor is used, written as a function of the radius of the wire. In our paper, no analytical formula for αm, thus the equation for µr is revised to have αm in it explicitly.

], so that Eq. (3) yields a real-valued µr. The real and imaginary parts of µr are shown in Fig. 3(b) as bold lines. As we expected, the real part of the effective permeability enters the negative region at frequencies above the resonance, when the phase of the induced effective magnetic current with respect to the incident magnetic field is shifted by π. Thus, the artificial medium behaves as a magnetic plasma in this frequency range. In contrast to the wire medium, here µr displays an obvious resonant feature. The real part of µr reverses sign at the resonance, when the imaginary part is maximized. Far below the resonance frequency, the structure shows obvious magnetism in this optical frequency range, when Re(µr) is greater than 1 with negligible Im(µr). This is the result of the strong scattering of the structure under TE incidence that is equivalent to an induced effective magnetization in phase with the excitation.

Fig. 3. (a)The distribution of the instantaneous scattered magnetic field (normalized to the the incident magnetic field at the origin) at resonance, for lossless plasmonic sectors. (b) γ 00 (thin solid) and γ 11 (thin dashed) of the cylindrical structure made of silver and SiO2; and Re(µr) (bold solid), Im(µr) (bold dashed) for the artificial material composed of an array of such cylindrical structures.

One assumption in making the analogy between the structure described here and the wire medium is that the scattering of the fundamental mode (TE0 mode) dominates the behavior of the structure, as in the case for metal wires. This is indeed the case, as revealed by the numerical calculations. To show this, we plot the magnitude of γ 11 in Fig. 3(b) as dashed thin lines. Over most of the frequency range we studied, γ 11 is much smaller than γ 00. Since γ 11 mostly contributes to the effective permittivity, even in the frequency range when |γ 11| is comparable with |γ 00|, it may not have a qualitative influence on the results. Because the cross section of the structure is very small compared to the free space wavelength, the higher order scattering coefficients (γnn for n>1) are orders of magnitude smaller than γ 11, as are the off-diagonal scattering coefficients (γmn for mn). This justifies the modeling of the structure as an induced effective magnetic current.

References and links

1.

M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microwaves Antennas Propagat. 1(1), 3–11 (2007). [CrossRef]

2.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

3.

W. Rotman, “Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media,” IRE Trans. Antennas Propagat. 10(1), 82–95 (1962).

4.

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, 2075–2084 (1999). [CrossRef]

5.

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557–1567 (2006). [CrossRef] [PubMed]

6.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz Magnetic Response from Artificial Materials,” Science 303, 1494–1496 (2004). [CrossRef] [PubMed]

7.

A. K. Sarychev, G. Shvets, and V. M. Shalaev, “Magnetic Plasmon Resonance,” Phys. Rev. E 73, 036,609 (2006). [CrossRef]

8.

N. Engheta, A. Salandrino, and A. Alù, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95, 095,504 (2005). [CrossRef]

9.

M. G. Silveirinha, A. Alù, J. Li, and N. Engheta, “Nanoinsulators and nanoconnectors for optical nanocircuits,” J. Appl. Phys. 103, 064,305 (2008). [CrossRef]

10.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, NY, 1983).

11.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, NJ, USA, 1994). [CrossRef]

12.

P. B. Johnson and R.W. Christy, “Optical Constants of the Nobel Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

13.

S. Tretyakov, “Analytical Modeling in Applied Electromagnetics,” pp. 164–175.(Artech House, INC, Norwood, MA, USA, 2003). In this reference the current coefficient αe of a wire of perfect electric conductor is used, written as a function of the radius of the wire. In our paper, no analytical formula for αm, thus the equation for µr is revised to have αm in it explicitly.

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: March 30, 2009
Revised Manuscript: June 2, 2009
Manuscript Accepted: June 3, 2009
Published: June 12, 2009

Citation
Jingjing Li, Lars Thylen, Alexander Bratkovsky, Shiy-Yuan Wang, and R. Stanley Williams, "Optical magnetic plasma in artificial flowers," Opt. Express 17, 10800-10805 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10800


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References

  1. M. Lapine and S. Tretyakov, "Contemporary notes on metamaterials," IET Microwaves Antennas Propagat. 1(1), 3-11 (2007). [CrossRef]
  2. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
  3. W. Rotman, "Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media," IRE Trans. Antennas Propagat. 10(1), 82-95 (1962).
  4. J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
  5. A. Alu, A. Salandrino, and N. Engheta, "Negative effective permeability and left-handed materials at optical frequencies," Opt. Express 14, 1557-1567 (2006). [CrossRef] [PubMed]
  6. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004). [CrossRef] [PubMed]
  7. A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic Plasmon Resonance," Phys. Rev. E 73, 036,609 (2006). [CrossRef]
  8. N. Engheta, A. Salandrino, and A. Al`u, "Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors," Phys. Rev. Lett. 95, 095,504 (2005). [CrossRef]
  9. M. G. Silveirinha, A. Alu, J. Li, and N. Engheta, "Nanoinsulators and nanoconnectors for optical nanocircuits," J. Appl. Phys. 103, 064,305 (2008). [CrossRef]
  10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, NY, 1983).
  11. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, NJ, USA, 1994). [CrossRef]
  12. P. B. Johnson and R. W. Christy, "Optical Constants of the Nobel Metals," Phys. Rev. B 6(12), 4370-4379 (1972). [CrossRef]
  13. S. Tretyakov, Analytical Modeling in Applied Electromagnetics, (Artech House, INC, Norwood, MA, USA, 2003), pp. 164-175. In this reference the current coefficient αe of a wire of perfect electric conductor is used, written as a function of the radius of the wire. In our paper, no analytical formula for αm, thus the equation for μr is revised to have αm in it explicitly.

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