## Abstract

We proposed a quantitative theory based on the surface plasmon polariton (SPP) coupled-mode model for SPP-Bragg reflectors composed of N periodic defects of any geometry and any refractive index profile. A SPP coupled-mode model and its recursive form were developed and shown to be equivalent. The SPP absorption loss, as well as high-order modes in each defect and possible radiation loss, is incorporated without effort. The simple recursive equations derived from the recursive model bridge the reflectance and the transmittance of *N* periodic defects to those of a single one, resulting in that the computational cost of the geometry optimization or the spectra calculation for *N* periodic defects is reduced into that for a single one. The model predictions show good agreement with fully vectorial computation data on the reflectance and the transmittance. From the recursive model, the generalized Bragg condition is proposed, which is verified by SPP-Bragg reflectors of various structures. The quantitative theory and the generalized Bragg condition proposed will greatly simplify the design of SPP-Bragg reflectors.

©2010 Optical Society of America

## 1. Introduction

Surface plasmon polaritons (SPPs) mirrors, and specially SPP-Bragg mirrors are important elements in plasmonics. Grating-like SPP reflectors of structures like periodic surface defects, either indentations or protuberances on the air-metal interface [1, 2, 3, 4, 5, 6, 7], corrugated metallic strips based on the insulator-metal-insulator (IMI) geometry [8, 9], and periodic defects in the metal-insulator-metal (MIM) structure [10, 11, 12, 13, 14], as partly illustrated in Fig. 1, have been theoretically or experimentally investigated. For SPP-Bragg mirrors based on surface defects, unidirectional [15, 16] or bidirectional [17] coupler and directional controller [18] of SPPs, and plasmonic micro-cavity used to enhance the extraordinary optical transmission (EOT) through subwavelength hole arrays [19,20] were demonstrated. Compared with IMI SPP-Bragg reflectors, MIM ones were believed to be more suitable for filters and micro-cavities at high integration levels because of light confinement.

To design high performance SPP-Bragg reflectors, there has been an explosion of interest and a large effort on the optimization in the last decade. To our knowledge, for *N* periodic defects, there is little work on the optimization of geometry parameters excluding the period. Conditions in different forms determining the optimal periods have been developed for various reflector structures. For surface grooves or metallic ridges, it has been shown that the reflectance presents maxima at the low-*λ* edges of the plasmonic bandgaps, i.e.,

where *k*
_{0} is the wave vector in the vacuum, the function “Re” refers to the real part, *n _{sp}* is the effective refractive index of the SPP mode at the air-metal interface,

*p*is the period of the array, and

*m*is the band index [4]. For rectangle dielectric ridges on the air-metal interface [1, 17], or rectangle defects in the MIM structure [12], it was believed that the Bragg condition in a different form, which results in prohibited propagation, should be used,

where *n _{eff}* is the effective refractive index of the mode in the defect region of width

*w*, and

*n*is that of the SPP mode in the other region. Note that, for reflectors composed of surface defects, the SPP mode is the mode at the air-metal interface; while for reflectors based on the MIM structure, it is the mode of the input/output single-mode MIM structure. However, many questions arise. For example, if there are high-order modes in the defect region, how to deal with them easily? Is there a generalized Bragg condition for SPP-Bragg reflectors of various structures, including those composed of Gaussian-shaped surface defects and S-shaped defects in the MIM structure [12]?

_{sp}On the other hand, given the number of defects and the optimal period, the optimization of the defect size for the maximum reflectance is still very complex, especially when *N* is relatively large. At present, efforts are focused on theoretical models such as the Rayleigh expansion [2], or numerical simulations such as finite difference time domain (FDTD) methods. However, as stated in [5] and [21], the Green dyadic technique and the discrete dipole approximation, which may provide virtually exact results, both suffer from a large (quite often prohibitive) numerical cost associated with the inversion of huge matrices and the calculation of cumbersome integrals; the Rayleigh approximation (which is valid for small scatterers) is not desirable either, because the calculation of the scattering coefficients within this approximation requires dealing with a difficult integral equation from which physical insight is not easily inferred. Moreover, many theoretical works have neglected the SPP absorption loss [1,2,4,5,6], indicating that the grating lengths considered should be shorter than the SPP absorption length. Additionally, it has been shown that the incorporation of high-order modes in the defect is not an easy task. For example, there will be *Nn _{m}* unknowns for

*N*grooves with

*n*being the number of considered modes inside each groove [5]. For fully vectorial computational techniques such as FDTD methods and finite element methods, the computational cost is high because of large numbers of unknowns when

_{m}*N*is relatively large.

Recently, a microscopic theory, which provides physical insights into the EOT through sub-wavelength hole arrays, has been proposed [22,23]. With this theory, some of the authors have largely resolved the debate on the EOT through a slit surrounded with periodic grooves, and found that the horizontal Fabry-Perot resonance effect of SPPs due to the reflection by surrounding grooves plays a key role [24].

In this work, we propose a quantitative theory at the microscopic level in forms of a SPP coupled-mode model and its recursive form (we call it the recursive model for convenience), to calculate the reflectance and the transmittance of SPP-Bragg reflectors composed of periodic defects of any geometry and any refractive index profile. In the models, the SPP absorption loss, high-order modes in the defect, and the radiation loss from the surface defect, are incorporated automatically. From the recursive model, a sets of simple recursive equations will be derived, which bridge the reflectance and the transmittance of *N* periodic defects to those of a single one, resulting in a great reduction of the computational cost. The equivalence of theses two models will be verified first. Then the generalized Bragg condition will be proposed from the recursive model. The geometry’s influences on the reflectance and the spectra calculation for various reflector structures, including surface defects and defects in the MIM structure, as shown in Fig. 1, are used to illustrate our discussions. Comparisons of the model predictions with fully vectorial aperiodic Fourier modal method (a-FMM) computation data allow us to quantitatively check the model accuracy.

The remainder of the paper is organized as follows. In Section 2, we describe the SPP coupled-mode model and the recursive model to calculate the reflectance and the transmittance for SPP-Bragg reflectors of various structures. Validations of the model predictions on the behavior of the reflectance as a function of the geometry, the operating wavelength, and the number of defects are presented in Section 3. In Section 4, we deduce and verify the generalized Bragg condition. Finally, the conclusions are summarized in Section 5.

## 2. Quantitative theoretical models

#### 2.1. The SPP coupled-mode model

Let us consider the SPP-Bragg reflector composed of *N* periodic defects, as shown in Fig. 1. To calculate the reflectance and the transmittance, a set of 2*N* equations are easily expressed based on the SPP coupled-mode model, as shown in Fig. 2(a),

where *u* = exp[*i*
*k*
_{0}
*n _{sp}*(

*p*-

*w*)],

*ρ*

_{1}

^{±}and

*τ*

_{1}

^{±}are the reflectance coefficient and the transmittance coefficient of the SPP modes at a single defect with the superscripts “+” and “-” being for the left- and right-incident modes, respectively, as shown in Figs. 2(c) and 2(d),

*A*

_{1}, ⋯,

*A*and

_{N}*B*

_{0},⋯,

*B*

_{N-1}are complex amplitudes of the left- and right-going SPP modes, respectively, and

*A*

_{0}is that of the normally incident SPP mode and is assumed to be known.

Equation (3) can be interpreted as follows. The first term and the second term on the right side of Eq. (3a) correspond to the respective contributions of the reflection of the incident SPP mode and the transmission of the left-going SPP mode at defect “1”. Eqs. (3b) to (3d) can be understood similarly. Eqs. (3e) and (3f) mean the reflection and the transmission of the right-going SPP mode at defect “*N*”, respectively.

In the model, the SPP modes only interact with their neighboring ones through reflection and transmission, indicating that Eq. (3) is a sparse linear system. The solving of this system is not complex for practical *N*. Once *B*
_{0} and *A _{N}* are obtained, the reflectance coefficient defined as

*ρ*

_{N}^{+}=

*B*

_{0}/

*A*

_{0}and the transmittance coefficient defined as

*τ*

_{N}^{+}=

*A*/

_{N}*A*

_{0}are readily achieved.

*ρ*and

_{N̄}*τ*are calculated similarly.

_{N̄}#### 2.2. The recursive SPP coupled-mode model

To achieve a more convenient calculation, we develop a recursive form of the SPP coupled-mode model, or the recursive model for short, as shown in Fig. 2(b). In the model, the reflectance coefficient *ρ*
_{N-1}
^{+} and the transmittance coefficient *τ*
_{N-1}
^{+} of *N* - 1 periodic defects, the reflectance coefficient *ρ*
_{1}
^{±} and the transmittance coefficient *τ*
_{1}
^{±} of a single defect, are assumed to be known. As a result, the *N* - 1 periodic defects on the right side (from defect “2” to defect “N”) in Fig. 2(a) are treated as a “black box” here, making the model quite simple and easy to understand. The coupled mode equations lead to

As *τ _{m}* =

*τ*

_{m}^{+}=

*τ*

_{m}^{-}for

*m*= 1,⋯ ,

*N*according to the principle of optical reversibility, from Eq. (4), we obtain recursive equations for

*ρ*

_{N}^{+}and

*τ*

_{N}Similarly, the recursive equation for *ρ*
_{N}
^{-} is given by

As a result, the reflectance *R*
^{±} = |*ρ*
_{N}
^{±}|^{2} and the transmittance *T* = |*τ _{N}*|

^{2}can be calculated recursively starting from

*ρ*

_{1}

^{±}and

*τ*

_{1}·

*ρ*

_{1}

^{±}and

*τ*

_{1}, determined by the defect geometry and the operating wavelength, can be calculated by various fully vectorial numerical techniques, or by the side-coupled cavity model specially for rectangle grooves [25]. In other words, Eqs. (5) to (7) bridge

*ρ*

_{N}

^{±}and

*τ*to

_{N}*ρ*

_{1}

^{±}and

*τ*

_{1}. Because the numerical cost of the recursive equations is negligible compared with that of the calculation of

*ρ*

_{1}

^{±}and

*τ*

_{1}, that of the geometry optimization or the spectra calculation for

*N*periodic defects is reduced into that for a single one.

We should emphasize that the SPP coupled-mode model and the recursive model are equivalent, as will be verified in Section 3.1. In these models, the SPP absorption loss is implicitly incorporated through *u* (*n _{sp}* is complex) and complex coefficients

*ρ*

_{1}

^{±}and

*τ*

_{1}. Similarly, the influences of all the modes in the defect, as well as the energy lost by coupling into radiating modes for the surface defect, are also incorporated through

*ρ*

_{1}

^{±}and

*τ*

_{1}. More importantly, there are no restrictions on the defect’s geometry and refractive index profile. As a result, these models can deal with SPP-Bragg reflectors composed of

*N*periodic defects of any geometry and any refractive index profile. Specially, for the mirror-symmetric shaped defect,

*ρ*

_{1}=

*ρ*

_{1}

^{+}=

*ρ*

_{1}

^{-}, thus

*R*=

*R*

^{+}=

*R*

^{-}.

## 3. Validations of the quantitative theoretical models

In this section, we will validate the two quantitative theoretical models. Some typical structures such as rectangle grooves and dielectric ridges, Gaussian-shaped metallic ridges and dielectric ones patterned on thick metal films, rectangle defects and triangle ones in the MIM structure, as shown in Figs. 1(b) to 1(d), are used to illustrate our discussions. For the Gaussian-shaped surface defect, whose surface is defined by the profile function *g*(*x*) = *h*exp(-*x*
^{2} /*A*
^{2}), we use height *h* and 1/*e* half-width *A* to depict the defect size (*w* is assumed to be fixed and large enough); whereas for the rectangle defect and the triangle one, *h* and *w* are used. We set *w* = 300 nm for the metallic Gaussian-shaped ridge and *w* = 670 nm for the dielectric one. For Gaussian-shaped metallic ridges, Ag at λ = 650 nm with *ε _{Ag}* = -17.03 + 1.15

*i*is used for comparisons with [6]. For the other structures, the metal is assumed to be gold, and except in Section 3.3, the analysis will be provided for λ = 800 nm. All the frequency-dependent relative permittivities of metals are tabulated in [26]. The refractive index of the dielectric ridge is assumed to be 1.46. The insulator in the MIM structure is assumed to be air and of width

*D*

_{0}= 100nm. All the

*ρ*

_{1}

^{±}and

*τ*

_{1}are calculated using the fully vectorial a-FMM [27]. Comparisons of the model predictions with fully vectorial a-FMM computational results on the influences of the period and the defect size, the reflectance and transmittance spectra, the behavior of the reflectance as a function of the number of defects, are provided to validate the accuracy of the models.

#### 3.1. On the period’s influences

First of all, let us consider the period, which is the most important parameter for the reflectance, and has been widely studied in previous works. To illustrate the accuracy of the model predictions, we randomly choose two sets of parameters for each structure under analysis. As shown in Fig. 3, the general trend of the reflectance as a function of the period, especially the optimized periods are well predicted by the SPP coupled-mode models. More importantly, the SPP coupled-mode model and the recursive model gives exactly the same results. This equivalence is due to the fact that we made no extra approximation in the recursive model. Hereafter, we only focus on the recursive model and refer to it as “the model” for short.

For defects in the MIM structure, both the reflectance and the transmittance (not shown) are predicted precisely. However, for surface defects, there are some deviations. A possible reason is that there exists cross conversion between SPPs and quasicylindrical waves (CWs) for surface defects [23,28], which is not taken into account in the pure SPP models.

It is evident that, for all the defects under analysis, there are two reflectance peaks when *p* varies in a period of λ_{sp}. However, according to Eq. (2), there should be only one peak for the dielectric surface defect and the rectangle defect in the MIM structure. This indicates that Eq. (2) gives only a half of the optimal periods. What is worse, the omitted optimized periods may present higher reflectance for some parameters, as shown by Fig. 3(b).

Moreover, one may notice the deviations of the optimal periods for rectangle surface grooves/ridges from multiple λ_{sp}/2 are small. The deviations, however, may be very large for the Gaussian-shaped dielectric ridges. As the effective index varies everywhere for the Gaussian-shaped dielectric ridge, Eq. (2) is not desirable to calculate the optimal period. Similarly, for the triangle defect in the MIM structure, it is not desirable either. As the models work well for the optimal periods of various reflector structures, there may be an underlying generalized equation.

In summary, the behavior of the reflectance as a function of the period is well captured, specially the optimized period are predicted accurately by the SPP coupled-mode model and the recursive model, which are equivalent on calculating the reflectance and the transmittance for SPP-Bragg reflectors of various structures.

#### 3.2. On the defect size’s influences

As mentioned above, given the optimal period, the optimization of the defect size for a relatively large number of defects suffers from high computational cost using previous theoretical methods or numerical techniques. With the recursive equations derived from the recursive model, which are of negligible numerical cost, however, the computational cost for *N* periodic defects is reduced into that for a single one.

Once the information of *ρ*
_{1}
^{±} and *τ*
_{1} as functions of both *w* and *h* (or *A* for Gaussian-shaped defects) is obtained, *ρ*
_{N}
^{±} is calculated recursively and efficiently according to Eqs. (5) and (7). As illustrated in Fig. 4, the recursive model quantitatively captures all the salient features of the reflectance for various reflector structures. Although the optimized defect sizes predicted by the model are not always identical to, or even not close to those by the a-FMM computations, their corresponding actual reflectance is very close to each other. Specially for SPP-Bragg reflectors based on the MIM structure, the model predictions present very high accuracy.

We notice that the optimized defect sizes may be different as *N* varies, as will be further illustrated in Section 3.4. As a result, if previous theoretical methods or numerical techniques are used, the optimization should be performed each time *N* varies, and the information of *ρ*
_{1}
^{±} as a function of both *w* and *h* (or *A*) is of limited usage on the optimization of *ρ*
_{N}
^{±}. Moreover, for dielectric surface ridges, either rectangle or Gaussian-shaped, we also noticed that the maximum reflectance increases with the ridge height.

It is known that the SPP mode in the Au-air-Au waveguide suffer from larger absorption loss than that along the air-Au interface at the same wavelength. However, we find that reflectance as high as 80% is easy to obtain for the MIM based reflector, but difficult for the reflector composed of surface defects, as clearly illustrated in Figs. 3 and 4. The reason lies in that there exists energy lost by coupling into radiating modes in the latter case. Fortunately, the radiation loss has been automatically incorporated in the model through *ρ*
_{1}
^{±} and *τ*
_{1}, which is verified by the good agreement between the model predictions and the a-FMM computation data.

In summary, the defect size’s influences on the reflectance are predicted by the model with acceptable accuracy and reduced computational cost for SPP-Bragg reflectors of any structures.

#### 3.3. On the spectra

Given the geometry, the reflectance and transmittance spectra of *N* periodic defects can also be obtained efficiently and accurately using the recursive equations by starting from those of a single one. As illustrated in Fig. 5, for surface defects and defects in the MIM structure, the
spectra features from the visible to telecom ranges, especially the wavelengths for peaks of the reflectance and dips of the transmittance are captured very well by the model. For surface defects, there are some deviations; while for defects in the MIM structure, the deviations are negligible.

#### 3.4. On the reflectance vs *N*

We study the reflectance as a function of *N* for three purposes: to check the effectiveness of the defect size optimization, to illustrate the variations of the optimized defect sizes for different *N*, and to examine the behavior of the reflectance when *N* is large enough.

For comparisons with [6], we use Gaussian-shaped metallic ridges and grooves to illustrate the discussions. The scanning parameter spaces for both *h* and *A* are assumed to be from 5 to 100 nm in steps of 5 nm. Starting from the information of *ρ*
_{1} and *τ*
_{1} as functions of both *h* and *A*, *ρ*
_{N} can be calculated efficiently using the recursive equations. Similar to Section 3.2, the optimized defect sizes for 20 Gaussian-shaped grooves/ridges are accurately predicted by the model, i.e., *h* = 70 nm, *A* = 5 nm for ridges, and *h* = 30 nm, *A* = 75 nm for grooves. As Fig. 6 illustrates clearly, the performances of the optimized structures predicted by the model are greatly improved compared with those of Fig. 3 in [6]. Although the reflectance predicted by the model may not always coincide with that by the a-FMM computation, the features especially the relative magnitudes of *R* for various sizes, which are consistent with [6], are well captured by the model. Moreover, among the defect sizes presented, the one for the largest *R* is different as *N* varies. Note that the magnitudes of *R* are a little smaller than those in [6], as the SPP absorption loss neglected there is taken into account here.

We notice that, when the period is chosen properly, the amplitude as well as the phase (not shown) of *ρ _{N}* becomes a constant as

*N*increases. This saturation is valid for any reflector structures, and is also consistent with previous works [3,4,5]. However, for other periods, the reflectance may be very low, and there may be oscillations as

*N*increases, as clearly illustrated in Fig. 7.

## 4. The Generalized Bragg condition

As shown in Figs. 6 and 7, when the period is properly chosen and *N* is large enough *ρ _{N}*

^{±}≈

*ρ*

_{N-1}

^{±}. We further take into account of |

*ρ*

_{N-1}

^{±}

*ρ*

_{1}

^{±u2}| ≪ 1, which is valid for most cases, then Eqs. (5) and (7) are reduced into

As Im(*n _{sp}*)(

*p*-

*w*) ≈ 0 for a practical SPP-Bragg reflector, the optimal periods for peaks of |

*ρ*

_{N}

^{±}| are determined by

where *m* is an integer. This equation is intuitively meaningful since it indicates that all of the reflected SPPs by each of the defects are added in-phase, as illustrated in Fig. 8. As there are no restrictions on the defect’s geometry and refractive index profile, we refer to Eq. (9) as the generalized Bragg condition for SPP-Bragg reflectors of any structures.

The generalized Bragg condition is validated with various reflector structures at given wavelengths, as illustrated in Table 1. As one can see, the relative errors are small than 5%, indicating that the optimal periods determined by Eq. (9) are of acceptable accuracy in practice. Eq. (9) is further validated for predicting the peak positions of the reflectance spectra of given structures. As shown clearly in Fig. 9, the wavelengths for reflectance peaks of surface defects and defects in the MIM structure coincide with those determined by Eq. (9).

For grooves and metallic ridges, either rectangle or Gaussian-shaped on the air-metal interface, it is evident that there is an additional term in Eq. (9) compared with Eq. (1), i.e., arg(*τ*
_{1}) - *k*
_{0}Re(*n _{sp}*)

*w*, which embodies the influences of the defect’s geometry and refractive index profile. We have performed exhaustive computations and shown that this additional term is very small (< 0.08

*π*when

*m*= 1). As a result, Eq. (9) can be reduced into, but is more accurate than Eq. (1) for air grooves (metallic ridges) in (on top of) a metallic surface.

For rectangle dielectric surface ridges and rectangle defects in the MIM structure, arg(*τ*
_{1}) in Eq. (9) is corresponding to *k*
_{0}Re(*n _{eff}*)

*w*in Eq. (2) for the influences of the defect. For the rectangle defect, where

*n*is uniform along the longitudinal direction of the defect, arg(

_{eff}*τ*

_{1}) is approximate to

*k*

_{0}Re(

*n*)

_{eff}*w*. However, compared with Eq. (9), half of solutions are omitted in Eq. (2), as has been pointed out in Section 3.1. The reason lies in that Eq. (9) is derived from constructive interference of reflection, while Eq. (2) from propagation prohibition. Moreover, the former is more favorable as it incorporates the possible extra phase shifts introduced by the conversion between modes in the defect region and the SPP mode in the other region. Specially, for the complex shaped defect, such as the Gaussian-shaped dielectric ridge on the air-metal interface and the S-shaped defect in the MIM structure [12], only Eq. (9) is applicable at present.

## 5. Conclusions

In conclusion, we have proposed a quantitative theory by developing a SPP coupled-mode model and a recursive model to analyze SPP-Bragg reflectors of any structures. It has been shown that two models give exactly the same results on the reflectance and the transmittance. With the recursive model, we derived a set of simple recursive equations, bridging *ρ*
_{N}
^{±} and *τ _{N}* to

*ρ*

_{1}

^{±}and

*τ*

_{1}and thus greatly reducing the computational cost.

Comparisons of the model predictions with fully vectorial a-FMM computation data quantitatively showed that the model is of acceptable accuracy on the behavior of the reflectance as a function of the geometry and the number of defects, and the reflectance and transmittance spectra. The quantitative models are very efficient to predict the optimized geometry parameters, including the period and the defect size for the reflectance maxima. The validations were performed for various structures at wavelengths from the visible to telecom ranges. For reflectors based on the MIM structure, both the reflectance and the transmittance are precisely predicted; while for those composed of surface defects, there are some deviations.

From the recursive equations derived from the recursive model, we proposed the generalized Bragg condition, and verified its effectiveness and accuracy for various reflector structures. We believe that the SPP coupled-mode model, specially the recursive model, and the generalized Bragg condition proposed are useful since they considerably ease the geometry optimization and the spectra calculation for the widely used SPP-Bragg reflector.

## Acknowledgements

The authors are grateful to the anonymous reviewers’ valuable suggestion. This work was supported by the National Natural Science Foundation of China under grant 60772002, and the Foundation of Yunnan University under grant 2009F31Q. This work was also supported by State Key Laboratory of Advanced Optical Communication Systems and Networks, China.

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