Materials for Plasmonic Focusing


 

While surface plasmons have the capacity for deeply sub-wavelength optical focusing, they tend to be relatively inefficient. To make a useful plasmonic device, it must operate at the optimal point in both frequency and materials.

Motivation

Conventional Optics

Plasmons and Nanofocusing

Materials for Plasmonic Focusing

Efficient Optical Coupling to the Nanoscale: PhD Dissertation

Curriculum Vitae

UCLA Optoelectronics Group

SINAM efforts on Plasmons

Nanophotonics MURI

 

 

 

  

Effective nano-focusing is ultimately a function of efficiency. Losses are implicit in confining a macroscopic optical field to nanoscale dimensions. Increasing field confinement and wave-vector leads to slower group and phase velocities. This increases the interaction time of the field with the loss mechanisms of the media, which in turn creates a proportionally degraded throughput. In fact, the losses can be catastrophic for certain systems, negating any advantages of field enhancement. To ignore this fact would remove any of the real-world engineering aspects of the problem and relegate it to a mere intellectual curiosity. This puts a premium on analysis of loss mechanisms and material properties.

Surface plasmon modes tend to be very lossy and the majority of this loss is due to absorption in the metal. The poor transmission properties of plasmons are described by their decay length, which is defined as the length over which the intensity decays by e-1. Typical surface plasmon decay lengths are less than 10μm[i], while ‘long range’ surface plasmons can travel as far as hundreds of microns[ii],[iii]. These decay lengths are limited by dissipation in the metal. As the electrons collectively oscillate at these surface plasmon frequencies, they collide with the background lattice of positive ions, transferring energy which is dissipated as heat. This powerful loss mechanism strongly constrains the design of any plasmonic device. For surface plasmons to achieve even modest levels of efficiency careful selection of materials and optical frequencies is required.

Ideally, a comparison of the modal Quality Factor (Q) over frequencies and materials would yield the optimum operating point. This trade study, though, is clouded by the realities of the system. Modal Q, which will henceforth be referred to as Qmod, is determined by the details of the geometry of the plasmonic waveguide and can vary by orders of magnitude simply by altering the dimensions of the device. It must be stressed that the conductor contributes primarily to loss. Because of this, it is the intrinsic Q of the metal that is most important.

Material Q comes from classical electrodynamics. In dispersive media, the derivative of the dielectric constant takes on added importance.  The term d(ωε)/dω replaces the relative dielectric constant (εm) in determining the energy of the field and Q-factor for the plasmon modes. This is especially significant when εm is negative while thermodynamics requires a positive electric-field energy. From Landau and Lifshitz[iv] (although a more accessible derivation may be found in reference [v]) the energy stored in the field is given by:

                       Eq(a)

This quantity was derived for semitransparent dispersive media, and as the reader is well aware, good conductors tend to be highly reflective in the frequency range that supports surface plasmons (below the plasma frequency of the conductor). This is a far cry from transparency. The mathematical assumptions which lead to this definition of stored energy, however, do not strictly require transparency. They demand, instead, that the imaginary component of the wave-vector be small in comparison to the real component. This justifies the formalism, even below the plasma frequency. Therefore, the expression is valid in the regime of interest. The average heat evolved in the material per unit time in the lossy medium is52:

                      Eq(b)

In all of these equations, the prime (') denotes the real component and the double prime (")denotes the imaginary component. With these pieces in place, it remains to define the material Q-factor as a function of frequency, which we will define as the modal Q (Qm).

      Eq(c)

Equation (c) can be simplified by remembering that the imaginary component of the permeability (μ”) tends to zero at optical frequencies. We may then work from this to define a quantity known as the material Q (Qmat) which only takes the electrical energy into account, dropping the second term from the numerator of Equation (c). The magnetic energy can be disregarded because it approaches zero at large wave-vectors. Not only is this backed up by both analysis and simulations, but also follows from intuition as we enter the electrostatic regime.

Because the material Q is only integrated over a single material, it can then be simplified, canceling the integral over E2.

                                   Eq(d) 

                                  Eq(e) 

Cast in the simple form above, we now may evaluate the material Q factors for various high-conductivity metals. We used the experimentally determined dielectric constants of silver[vi],[vii],[viii],[ix], gold[x], aluminum[xi] and copper58. The results, plotted in Figure 21, illustrate why plasmon modes have such poor propagation characteristics. Silver has the highest Qmat factor, topping out around 30, while the other materials lie below 20. Various tricks can be played to keep the modal energy in the low loss dielectric, but at high k-vectors, a significant fraction of the energy must penetrate into the metal.

Figure 21: Material Q for various good conductors

 

The material Q not only places silver far above the other conventional conductors for supporting surface plasmons, but it limits the bandwidth of efficient operation. Clearly the efficiency is dimished when operating outside of the photon frequency range of 2eV-3eV. These intrinsic material properties create a fundamental barrier which limits broadband plasmonic applications.

While the optical properties of silver are favorable, the physical mechanisms that create them are a challenge to model. As is known in the field, the dielectric constant of silver cannot be adequately represented simply by the intraband transitions of a Drude character. This is because there is a mixture of free-electron states with a polarizable d-band[i], causing the plasma frequency to be pushed down from ~9eV, where it would sit in the absence of interband transitions. This is graphically illustrated in the works of Ehrenreich and Phillip[ii], which clearly illustrate the onset of interband transitions near 4eV. In the region of interest, this can be modeled with an additional term added to the Drude model with a value of approximately 5[i]. The surface plasmon parameters are very dependent on the material constants, so these approximations could not be made. In fact, the experimentally determined optical constants must be used in any thorough analysis. For our calculations, we used the tabulated values for evaporated silver. Over the region of interest (from 1.2 eV to 3.2eV), an analytical fit of the experimental data was used. In these emperical fits, ħω is the photon energy in electron Volts.

                       Eq(f)  

                               Eq(g)

A spline fit was used to interpolate between experimentally derived values outside of our range of interest. As will be discussed below, the real part of epsilon will determine the plasmon wavelength and the imaginary part determines the magnitude of absoption. Note that the magnitude of the real part of epsilon is always more than ten times greater than that of the imaginary part over the region of interest.  

These pages are all excerpts from the PhD Dissertation Efficient Optical Coupling to the Nanoscale available here


[i] N. Kroo, J.P. Thost, M. Volcker, W. Krieger and H. Walther, Europhysics Letters, vol. 15, p289 (1991).

 

[ii] D. Sarid, Physical Review Letters, vol. 47, p. 1927 (1981).

 

[iii] F.Y. Kou and T. Tamir, Optics Letters, vol. 12, p. 367 (1987).

 

[iv] L.D. Landau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinenann, Oxford, 2002).

 

[v]  R.E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).

 

[vi] L.G. Schulz and F. R. Tangherlini, Journal of the Opical Society of America, vol. 44, p. 362 (1954).

 

[vii] L.G. Schulz, Journal of the Opical Society of America, vol. 44, p. 357 (1954).

 

[viii] V.G. Padalka and I.N. Shklyarevskii, Opt. Spectr. U.S.S.R. vol. 11, p. 285 (1961).

 

[ix] R. Philip and J. Trompette, Compt. Rend., vol. 241, p. 627 (1955).

 

[x] P.B. Johnson and R.W. Christy, Physical Review B, vol. 6, p. 4370 (1972).

 

[xi] E. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1985).