## Abstract

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.

© 2013 OSA

## 1. Introduction

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 [14], 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 model

with*h̄ω*= 9.01 eV and

_{p}*h̄γ*= 0.0048 eV is used to describe the dielectric permittivity of silver film, where

*ω*is the plasma frequency and

_{p}*γ*is the collision frequency. GaAs layers are considered to have a Kerr-type optical nonlinearity, the strength of which is given by the third-order nonlinear optical coefficient

*χ*

^{(3)}≈ 10

^{−10}esu [15].

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 [14]. 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

*k*is the

_{x}*x*component of the wave vector. The wave is assumed to be incident from

*z*>

*L*and transmitted to

*z*< 0.

*ε*is equal to

*ε*

_{1}and

*ε*

_{2}in the incident and transmissive regions respectively, whereas in 0 ≤

*z*≤

*L*, it is given by where

*ε*is the linear part of the dielectric permittivity and

_{L}*β*describes the strength of the Kerr-type nonlinearity.

**E**(

*z*) is the electric field amplitude.

The reflection coefficient *r* is defined by the wave function in the incident region

*v*|

^{2}(≡

*w*) is the electric field intensity of the incident wave and

*k*is the

_{z}*z*component of the wave vector.

*r*and

*w*are considered as functions of

*L*in our invariant imbedding method, using which we derive exact differential equations satisfied by them:

*θ*is the incident angle and

*ε̃*(

*l*) [≡

*ε*(

*l*)/

*ε*

_{1}] is obtained from

*r*and

*w*:

*w*

_{0}is chosen such that the final solution for

*w*(

*L*) is the same as the physical input intensity. The fact that in general, there are several

*w*

_{0}values corresponding to a given

*w*(

*L*) value gives a natural explanation of optical multistability. The reflectance

*R*is given by

*R*= |

*r*(

*L*)|

^{2}.

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 have

*z*(0 <

*z*<

*L*),

*u*(

*z*,

*L*) is obtained by integrating this equation, together with Eq. (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 *d _{m}* respectively. Then the total thickness of the structure is

*L*=

*N*Λ +

*d*. 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

_{m}*h̄ω*= 0.94 eV and 1.06 eV. We expect the resonance energy of the TPP,

*h̄ω*, lies inside this bandgap. In order to obtain the largest field enhancement, we tune

_{T}*d*by varying it from 40 to 60 nm. In Fig. 1, we show the profile of the real part of the refractive index,

_{m}*n*, along the

_{R}*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 *h̄ω _{T}* = 0.96, 0.97, 0.98, 0.99 eV. For each value of

*ω*, the metal layer thickness

_{T}*d*has been chosen to maximize the field enhancement. We find that when

_{m}*h̄ω*= 0.96 eV, the TPP is excited at

_{T}*θ*= 8.26°. This angle shifts toward higher values as

*ω*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.

_{T}*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 *h̄ω _{T}* = 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

*d*chosen to maximize the field enhancement increases as

_{m}*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 *h̄ω _{T}* = 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

*d*chosen to maximize the field enhancement is plotted versus

_{m}*N*. We notice that

*d*converges to about 55 nm.

_{m}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 *h̄ω _{T}* = 0.96 eV,

*N*= 40 and

*d*= 55 nm.

_{m}*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 identity

*χ*

^{(3)}is measured in esu and

*I*is the intensity of the incident wave measured in W/cm

^{2}[17]. 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 [17]. 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/cm

^{2}.

## 4. Conclusion

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.

## Acknowledgments

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.

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