INTRODUCTION

1 Introduction

Theory

Thomas Edison, known as "TheWizard of Menlo Park", invented the first revolutional light bulb in 1879. Ever since the invention, lighting devices plays an important role in our daily life. From bulb to solar powered vehicles, the applications of light became popular and popular. As everybody knows, light is a kind of electromagnetic wave. However, how electromagnetic waves interact with the surface of materials has been discussed for a long period of time. As a result, we use a short section explaining some electromagnetic phenomenon from scattering and interference as our opening.

Hyperbolic Metamaterials

After the introduction of permittivity and permeability, now we can talk about a popular and modern material—Metamaterials which have attracted many attentions. Currently, there is no universally accepted definition for metamaterials. The most basic definition of a metamaterial is any material that is simply anything other than an ordinary material [7]. Moreover, there are many similar definitions that describe metamaterials such as: engineered composites [8], material properties that are derived from their physical structure rather than their chemistry [9], exhibit properties not observed in nature [7], and exhibit properties not observed in their constituent materials [10].

The research about metamaterials used to focused about structures that possess both negative permittivity and negative permeability (known as double-negative media) because that kind of structures exhibit negative refractive index [11]. Nevertheless, more and more studies are interested in a class of metamaterials that which is highly anisotropic. These kinds of anisotropic metamaterials have hyperbolic dispersion, so they are also named as "Hyperbolic Metamaterials" [1]. Such metamaterials are kinds of uniaxial crystals, and one of the principal components of either their permittivity (ε) or permeability (μ) tensors is opposite in sign to the other two principal components. In this research, we focus on electric hyperbolic structures with μ_⊥=μ_∥>0 and either ε_∥<0 and ε_⊥>0 or ε_∥>0 and ε_⊥<0. Here, the subscripts ⊥ and ∥ means the parameter perpendicular or parallel to the anisotropy axis. Thus, it is apparent that their isofrequency surface of extraordinary waves, which is given by

(k_x^2+k_y^2)/ε_∥ +(k_z^2)/ε_⊥ =(ω/c)^2

look like Figure5.

The production of hyperbolic metamaterial is easier than that of double-negative media. Besides, The demonstration of first hyperbolic metamaterial dates back to 1969. These interesting materials are not only easier to produce but also have bunches of advantage including enhancement of spontaneous emission rates [12], diverging density of states and enhanced superlensing effect [13].

2 Principle Analysis and System Design

2.1 Principle Analysis

Although there are various of advantages, this research aims at the enhancement of spontaneous emission rate. E. M Purcell first proposed this concept at 1946, so Purcell Factor, a factor quantized the enhancement, is named after him. The Purcell Factor is generally defined as the ratio of the radiative rate in a particular electromagnetic structure to that in vacuum. The derivation of Purcell Effect is a little complicated, so we are just going to have a little discussion about them. To illustrate this effect, we know the Fermi golden rule for the radiative decay rate 1/τ in a homogeneous lossless medium indicates that transition rate for the atom-vacuum (or atom-cavity) system is proportional to the density of final states.

As the density of photon states with polarization σ equals to

ρ_σ (ω)=∑_k▒δ(ω_(k,σ)-ω)

We can know that the summation is by definition proportional to the area of the isofrequency surface ω_(k,σ)=ω, and hence it diverges in a hyperbolic medium. This lead to the divergence of the radiative decay rate at any frequency for which the isofrequency surface is hyperbolic. However, this is not a rigorous derivation. Actually, the singularity in decayed rate is cancelled because many reasons.[12][14][15][16][17] Thus, the radiative rate remains finite despite the density of state diverging. The calculated Purcell factors for different dipole orientations are presented in figure 6, which indicates that the Purcell fator ascends at the transition from the elliptic to hyperbolic regime, where ε_∥ or ε_⊥ goes through zero.

Most commonly HMM which enhances spontaneous emission rates is made of multilayers of metal and dielectric. For instance, a hyperbolic metamaterials made of Ag and Si multilayers [18](See figure 7). In contrary to metal-only surface, hyperbolic metamaterial has broader bandwidth of Purcell-enhanced spectrum. Furthermore, by adjusting the volumetric filling ratio, the peaks will shift apparently which is good for enhancement of the spontaneous emission at different wavelength.

The following (See figure 8) are the Purcell enhancement spectrum of the HMM using Ag and Si multilayers and that of using only Ag structure. We can find that the Purcell factor peaks at 600nm is about 60-fold on the Ag-Si layers, on the contrary, only nearly 10-fold on the Ag film. Because hyperbolic metamaterials are artificial, millions of application are being discussed.

2.2 System Design

First, we set the structure into a rectangular which size is 300nm*300nm, then set the Au nanowire array into hexagonal arrangement which length is 300nm, radius is 12.5nm, so the filling ratio is fixed at almost 30%. Next, we set the simulation region, FDTD Simulation, into 300nm*300nm and in order to meets the true situation or even more accurate we set the boundary condition into PML, and also set the accuracy at level 8(Max level). Then put a monitor to record the enhancement and even make it to a movie which is easier to observe. By the monitor we could collect the data and calculate the Purcell factor. The last part is to put a dipole source in the structure center below the surface 5nm.

The final procedure is running the simulation and waiting for the result, after collecting the data and calculating the Purcell factor, back to change the polarization or the position of the dipole source. By looping these procedures, we could find the relationship between the source and the Purcell factor.

3 Experiment Result

3.1 How the Polarization and Position of Dipole Affect the Purcell

Due to the fact that we are interested in the spontaneous emission rates of the mocules in HMM array not just some Purcell enhancement for dipoles at few positions, we are going to discuss the relation between position and dipole polarization.

To make analysis much easier to understand, we can set the coordinate for the nanorods. First, we assume that the origin of the coordinate is the middle of the array and surrounded by four adjacent nanorods. Then, suppose the direction

to the nearest adjacent nanorod is x axis, in contrast, the direction to the second nearest nanorod is y axis (See figure 15).

Resulted from the analytical methods done by other paper [31], the Purcell factor is relatively low for z-polarized dipole. As a result, we should first consider two case which dipole polarization is x or y oriented. At the beginning, the dipole is lacated at the origin of the array and on the plane where lower 10nm than the top of the gold nanorods. Then, we sweep the dipole of x-polarized and y-polarized from origin to near metallic nanorod along x and y coordinate. We are going to illustrate x-polarized first. If we move the x-oriented dipole emitter from origin to y = 18nm along y axis, we can find that the Purcell factor keep decreasing (See figure 16(a)). However, if we move the x-oriented dipole emitter from origin to

x = 6nm along x axis, the Purcell factor will keep increasing though (See figure 16(b)).

These observations are corresponded to the analytical results derived by local field approach for square gold nanorod array [31]. The analytical results demonstrate that for dipole which polarization perpendicular to the metallic surface, the Purcell factor increase as the dipole keep approaching the nanorod. It is so curious that we turned to find out what will happen to y-polarized dipole.

We then sweep the position of the y-oriented dipole from origin to near metallic surface along x and y axis. The Purcell spectrum which along y axis (See figure 17(a)) match our expectation that the measurements keep increasing. To our surprise, the Purcell factor spectrum along x axis (See figure 17(a)) keep ascending which is different from the analytical results derived from square gold nanorod array[31].

The results of our studies indicate that both x-oriented and y-oriented dipole can enhance the spontaneous emission rates when they move along the x-axis which is dissimilar to that in square gold nanorod array. Thus, we consider that this curious effect may enhance the spontaneous emission rates of dipoles greatly in whole structure. The calculation of average Purcell factor of gold nanorod array is needed in order to get a quantitative number of enhancement.

3.2 Average Purcell Factor Spectrum

Although we need to achieve the total Purcell factor of the whole structure, as the simulation is too tedious, we can simulate the Purcell factor in a unit cell (See figure 18(a)). To minimize the memory requirement of simulations, the symmetric properties [27] is introduced.

Symmetric properties indicate that the unit cell in hexagonal array is parallelogram and we can divide it into 8 triangular regions. Using the fact that the lattice has 60-degree rotational symmetry, this can be reduced to regions A and 1. We further recognize that the symmetric along 30-degree line, which divide region A and 1 into A, A’, 1, and 1’. The dopole in 1’ can be used to determie the other eleven regions, and dipole in A’ can be used to determine the other three regions ( See figure 18 (c) and (d)). Therefore, we can reduce the numbers of dipole position from 16 to 2 and it is a more efficient way to calculate the average Purcell factor.

The calculation process is that we can divide the nanorod array into many unit cells as figure 18(a). By symmetric properties, we only need to simulate the dipole in the region A’ and region 1 in figure 18(b) of the unit cell. After having the Purcell spectrum at each position of the region A’ and 1, we can derive the average Purcell spectrum of unit cell which will be similar to the Purcell factor of the whole structure.

We have chosen five positions in region A’ and four positions in region 1’. We simulated the dipole in three kinds of polarization which are in-plane perpendicular, parallel to metallic surface and vertical (See figure 19(a)). By using the symmetric properties, we can know 68 points in a unit cell with only simulating 9 points (See figure 19(b)).

In consequence, the Purcell factor spectrum of the three polarization is shown in Figure 20 and the average Purcell factor spectrum is shown in Figure 21. We can find that in Figure 20 dipole with perpendicular polarization has larger Purcell enhancement which is corresponded to the previous results. In addition, Purcell enhancement caused by dipole with parallel polarization is much smaller than that of perpendicular one, but larger than that of vertical one. The vertical one has the smallest Purcell enhancement is similar to the analytical results done by other research [31]. Most important of all, we find that there is an apparent peak at wavelength equals to about 525nm for these three spectrums. These peaks are corresponded to the ENZ point at Figure 11 which studies shown that the Purcell factor will increase at that point [1].

4 Conclusion

Simulations using Lumerical FDTD Solutions of hexagonal gold nanorod array has performed, which find that different polarizations of dipoles has obviously different Purcell response at different positions respect to the nanorods. It is interesting that hexagonal structure may make parallel-polarized dipole enhance their Purcell when moving toward the metallic surface of nanorod which is different with that in square structure. Besides, using simulations to prove that Purcell factor will saturate at finite size metamaterial. By calculating the average Purcell factor spectrum, we can find that there is a peak at the ENZ regime where the properties change from elliptical dispersion to hyperbolic dispersion.