We suggest optimally designed one-dimensional metal/photonic crystal structures for the excitation of optical Tamm plasmon polaritons, which show strongly enhanced electromagnetic field intensities compared to those due to conventional surface plasmon excitations. We assume that the photonic crystal is made of weakly nonlinear optical materials and calculate the reflectance and the electromagnetic field distribution precisely, using the invariant imbedding method generalized to nonlinear media. We find field intensity enhancement factors as large as 3,000 at the metal/photonic crystal interface. We verify that due to this strong enhancement, nonlinear optical effects such as optical bistability can be observed for very small values of the incident wave power. Our results imply that using our structure, very strong surface enhanced Raman scattering signals can be achieved and optical switching devices can be operated in much lower threshold light intensities.
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The strong local enhancement of the electromagnetic fields due to the excitation of surface plasmon polaritons (SPPs) is one of the key phenomena in plasmonics [1–3]. It plays a major role in various nonlinear optical applications of plasmonics. More recently, a different kind of surface electromagnetic waves called optical Tamm plasmon polaritons (TPPs) have been actively studied [4–13]. TPPs can be excited at the surface of a photonic crystal and are simple analogies of the Tamm states proposed as localized electronic states at the edge of a truncated periodic potential. They show characteristics similar to conventional propagating SPPs, but can be generated in all-dielectric multilayer structures. Unlike SPPs, they can be excited by incident s waves as well as p waves. Their dielectric loss is much smaller than that of conventional SPPs, leading to sharper coupling resonances, higher surface fields, and longer propagation distances than those for SPPs [5, 9]. These features are advantageous for current and future applications in plasmonics.
In this paper, we suggest optimally designed one-dimensional metal/photonic crystal structures for the generation of optical TPPs, which have strongly enhanced field intensities compared to those due to conventional propagating surface plasmon excitations. We assume that the photonic crystal is made of weakly nonlinear optical materials. Using the invariant imbedding method generalized to nonlinear media , we calculate the reflectance and the electromagnetic field distribution precisely. We find extremely large field intensity enhancement factors as large as 3,000 at the metal/photonic crystal interface. Due to this strong enhancement, various nonlinear optical effects including optical bistability can be observed for rather small values of the incident wave power. Our results imply that using our structure, very strong surface enhanced Raman scattering signals can be achieved and optical switching devices can be operated in much lower threshold light intensities.
2. Model and theoretical method
We consider a multilayered structure consisting of a one-dimensional photonic crystal coated with silver film. The photonic crystal is made of GaAs and AlAs multilayers the refractive indices of which are taken to be 3.7 and 3 respectively. Due to the possibility of a precise control of the deposition thicknesses by molecular beam epitaxy, GaAs and AlAs are commonly used to fabricate photonic crystals for the excitation of TPPs in the infrared region. In our calculations, the Drude model15].
We assume that linearly-polarized electromagnetic waves are obliquely incident on the structure from the metal side. The wave equation describing the propagation of these waves in nonlinear media with a Kerr nonlinearity can be solved in a numerically precise and efficient manner using a generalized version of the invariant imbedding method . Though it can be applied to both s and p waves, we focus here on the propagation of p waves of frequency ω incident on a multilayered structure, where the dielectric permittivity ε varies only in the z direction. We assume that the structure lies in 0 ≤ z ≤ L and the wave propagates in the xz plane. The magnetic field amplitude H = H(z) satisfies
The reflection coefficient r is defined by the wave function in the incident region
The invariant imbedding method can also be used in calculating the normalized magnetic field amplitude u(z) = H(z)/v. We consider the u field as a function of both z and l: u = u(z, l). Then we haveEq. (5), from l = z to l = L using the initial condition u(z, z) = 1 + r(z).
3. Numerical results
We consider the linear cases first. The number of photonic crystal periods, the thickness of one period and the metal layer thickness are denoted by N, Λ and dm respectively. Then the total thickness of the structure is L = NΛ + dm. We assume Λ = 187 nm and the optical thicknesses of GaAs and AlAs layers are equal to 309.8 nm. In the normal incidence case, the photonic bandgap is formed between h̄ω = 0.94 eV and 1.06 eV. We expect the resonance energy of the TPP, h̄ωT, lies inside this bandgap. In order to obtain the largest field enhancement, we tune dm by varying it from 40 to 60 nm. In Fig. 1, we show the profile of the real part of the refractive index, nR, along the z direction when N = 20.
In all calculations presented in this section, the incident wave is assumed to be p-polarized. In Fig. 2(a), we plot the reflectance as a function of the incident angle, when N = 20 and h̄ωT = 0.96, 0.97, 0.98, 0.99 eV. For each value of ωT, the metal layer thickness dm has been chosen to maximize the field enhancement. We find that when h̄ωT = 0.96 eV, the TPP is excited at θ = 8.26°. This angle shifts toward higher values as ωT increases. In Fig 2(b), we show the spatial distributions of the magnetic field intensity corresponding to the four reflectance minima shown in Fig. 2(a). We find that the field intensity near the the metal/photonic crystal interface is greatly enhanced with respect to that of the incident wave. This enhancement is much larger than that due to conventional propagating surface plasmon excitations.
In Fig. 3, we show the reflectance and the magnetic field distribution for four different values of N when h̄ωT = 0.96 eV. For N = 20, the TPP is excited at θ = 8.26°. The excitation angle decreases to smaller values, 5.68°, 4.96° and 4.63°, as N increases to 25, 30 and 40 respectively. We also find that the the value of dm chosen to maximize the field enhancement increases as N increases and converges to about 55 nm. In Fig. 3(b), we observe that an extremely large enhancement occurs at the interface between the photonic crystal and the metal layer. We note that the intensity of the incident wave is 1, therefore an enhancement factor as large as 3,000 can be achieved.
In Fig. 4, we plot the maximum value of the normalized magnetic field intensity, which occurs at the interface between the photonic crystal and the metal layer, versus N, when h̄ωT = 0.96 eV. The enhancement factor increases monotonically and converges to about 3,000 as N increases. This value is much larger than the field ebhancement factor due to the excitation of conventional SPPs [16, 17]. This implies that stronger nonlinear optical effects including more strongly enhanced Raman scattering signals can be obtained due to the excitation of optical TPPs. In the inset of Fig. 4, the metal layer thickness dm chosen to maximize the field enhancement is plotted versus N. We notice that dm converges to about 55 nm.
We now turn to the nonlinear case. In Fig. 5, we turn on the nonlinearity of GaAs and plot the reflectance versus incident angle for several values of βw, when h̄ωT = 0.96 eV, N = 40 and dm = 55 nm. β is a quantity proportional to χ(3) and w is a quantity proportional to the power of incident laser light. The parameter βw is the measure of the strength of optical nonlinearity. More specifically, it is straightforward to derive the identity17]. As shown in Fig. 5, we find that optical bistability begins to appear from βw ≈ 10−4. This value is much smaller than that for conventional SPPs given by βw ≈ 0.005 . This fact suggests that it is feasible to design sensitive optical switches operating at very small powers based on the giant field enhancement. When χ(3) ≈ 10−10 esu, the I value corresponding to βw ≈ 10−4 is approximately 13 MW/cm2.
In this paper, we have suggested optimally designed one-dimensional metal/photonic crystal structures for the generation of optical TPPs, which have strongly enhanced field intensities compared to those due to conventional surface plasmon excitations. Assuming that the photonic crystal is made of weakly nonlinear optical materials, we have calculated the reflectance and the electromagnetic field distribution precisely, using the invariant imbedding method generalized to nonlinear media. We have obtained extremely large field intensity enhancement factors, which are as large as 3,000 at the metal/photonic crystal interface. Due to this strong enhancement, we have shown that nonlinear optical effects such as optical bistability can be observed for rather small values of the incident wave power.
This work has been supported by the National Research Foundation of Korea Grant ( NRF-2012R1A1A2044201) funded by the Korean Government and by CNRS-Ewha International Research Center Program.
References and links
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