## Abstract

The feature of spatial dispersion in periodic layered metamaterials is theoretically investigated. An effective medium model is proposed to derive the nonlocal effective permittivity tensor, which exhibits drastic variations in the wave vector domain. Strong spatial dispersion is found in the frequency range where surface plasmon polaritons are excited. In particular, the nonlocal effect gives rise to additional waves that are identified as the bonding or antibonding modes with symmetric or antisymmetric surface charge alignments. Spatial dispersion is also manifest on the parabolic-like dispersion, a non-standard type of dispersion in the medium. The associated negative refraction and backward wave occur even when the effective permittivity components are all positive, which is considered a property not available in the local medium.

© 2013 OSA

## 1. Introduction

Spatial dispersion is a nonlocal effect that the electric polarization at a certain position is determined not only by the electric field at that position, but also by the fields at its neighbors [1–3]. An immediate consequence of spatial dispersion is that the permittivity of the underlying medium becomes wave vector dependent. Spatial dispersion is an important optical property of insulating crystals with excitons [1, 4–7]. In conducting materials, several features such as anomalous skin effect [8, 9], shift of absorption peaks [10, 11], virtual surface waves [12], longitudinal polarization waves [13–16], among others, are either due to or related to spatial dispersion. More recently, spatial dispersion also plays an important role in the properties of nanostructures [17–20]. Unusual resonances in nanoplasmonic structures are due to nonlocal response [21]. Spatial dispersion can even mimic chirality in periodic nanostructures [22].

Due to spatial dispersion, *additional* waves may appear in the nonlocal medium [23]. This is considered a consequence of extra degrees of freedom in the system, coming from the dependence of the permittivity on the wave vector. For a given frequency, there may exist two different wave vectors that satisfy the same dispersion relation. The phenomenon of wave splitting can therefore be observed in the medium [24]. Besides, the energy is transported in an additional mechanism other than the electromagnetic waves [1, 25]. The Poynting vector is different from the traditional form (**E** × **H**) [26], which is modified by an extra term related to the change rate of the permittivity with the wave vector [2]. The extra term plays a crucial role for negative refraction to occur in the nonlocal medium [27].

For periodic structures like crystals, spatial dispersion is important when the coupling of fields between adjacent unit cells cannot be ignored. This is usually the case when the wavelength of electromagnetic wave approaches the lattice period, although for some structures (e.g. wire mediums) strong spatial dispersion appears even in the long wavelength [28, 29]. Periodic metal-dielectric layered structures are considered a simple yet nontrivial example of the nonlocal medium. On the one hand, spatial dispersion is likely to occur due to the strong interactions of fields between adjacent layers inside the structure. In the optical regime, the plasmonic effect may serve as the source of spatial dispersion [30]. On the other hand, the dispersion relation of the underlying structure is analytical [31] and can be exploited to derive the *effective* permittivity that reveals the spatial dispersion or nonlocal effect in an explicit manner [32–34]. The layered structures also exhibit extraordinary features such as subwavelength imaging [35, 36] and negative refraction [37, 38].

In this study, the author investigates the feature of spatial dispersion in periodic layered meta-materials, with emphasis on the nonlocal effective permittivity. An effective medium model is proposed to derive the nonlocal effective permittivity tensor of the layered structure, with the wave vector dependence. The effective permittivity exhibits drastic variations in the wave vector domain, showing a typical feature of nonlocal resonance. In particular, strong spatial dispersion is found in the frequency range where surface plasmon polaritons are excited. Due to the nonlocal effect, additional waves appear in the medium and are identified as the bonding or antibonding modes with symmetric or antisymmetric surface charge alignments [39]. The additional wave can be a forward wave with negative refraction or a backward wave with ordinary refraction, but with a very different feature from that in a local anisotropic medium with opposite signs of the permittivity components [40]. Due to the nonlocal effect, negative refraction and backward wave occur even when the effective permittivity components are all positive.

From the viewpoint of the effective medium, the periodic layered metamaterials are considered a special type of medium with the elliptic-like, hyperbolic-like, parabolic-like, or mixed type of dispersion, depending on the frequency range. These characters come in part from the anisotropy and in part from the nonlocal nature of the layered structure, and are in correspondence with the variety of the effective permittivity components. Basic attributes of the effective permittivity tensor are categorized into four frequency ranges for the in-plane components and two ranges for the out-of-plane component, which are closely related to the dispersion characteristics of the layered structure for TM and TE polarizations, respectively. These attributes are analyzed in each frequency range to give a perspective on the spatial dispersion or nonlocal effect in the layered metamaterials.

This paper is organized as follows: In Sec. 2, the effective medium model for the periodic layered metamaterials is proposed, from which the nonlocal effective permittivity tensor is derived. In Secs. 3 and 4, the properties of the in-plane and out-of-plane effective permittivities, respectively, along with four illustrated examples, are discussed. In Sec. 5, spatial dispersion in the underlying layered structure is summarized. Finally, a conclusion of the present study is given in Sec. 6.

## 2. Effective medium model

Consider a periodic layered structure consisting of two alternating nonmagnetic materials stacked along the *x* direction, one with the dielectric constant *ε*_{1} and thickness *a*_{1}, and the other with *ε*_{2} and *a*_{2}, as schematically shown in Fig. 1. Assume that the electromagnetic wave is of the form *e*^{i(k·r−ωt)}. Let the wave vector lie on the *xz* plane, that is, **k** = (*k _{x}*, 0,

*k*), without loss of generality. The dispersion relation of the periodic layered structure is given by [31]

_{z}*a*=

*a*

_{1}+

*a*

_{2}, and

*k*

_{0}=

*ω/c*. Here, TM and TE refer to transverse magnetic, where

**E**= (

*E*, 0,

_{x}*E*) and

_{z}**H**= (0,

*H*, 0), and transverse electric, where

_{y}**E**= (0,

*E*, 0) and

_{y}**H**= (

*H*, 0,

_{x}*H*), respectively.

_{z}#### 2.1. Nonlocal effective permittivity tensor

The basic idea of the effective medium model is to extract the effective permittivity tensor from the analytical dispersion relations of the structure. The essential work is to expand the dispersion relations to higher (than the second) order of the wave vector components. Suppose that the lattice period is much less than the wavelength, *a* ≪ *λ* = 2*π/k*_{0}, so that the layered structure can be regarded as an *effective* medium, characterized by the effective permittivity tensor:

Expanding Eqs. (1) and (2) into power series of *k _{x}*,

*k*, and

_{z}*k*

_{0}up to fourth order, truncating the higher order terms, and rearranging the expansions in the forms of Eqs. (4) and (5), respectively, we have

*quasistatic*effective permittivities based on the Maxwell-Garnett mixing rule [41], and

*f*

_{1}=

*a*

_{1}/

*a*and

*f*

_{2}=

*a*

_{2}/

*a*, and the material parameters,

*ε*

_{1}and

*ε*

_{2}.

Note that
${\epsilon}_{z}^{\text{eff}}$ and
${\epsilon}_{y}^{\text{eff}}$ are no longer equal as in the quasistatic case, although the layered structure makes no difference between *y* and *z* directions. The medium properties, therefore, depend on the polarization. Note also that
${k}_{x}^{2n}$ and
${k}_{z}^{2n}$ (*n* = 1, 2) appear in the effective permittivities [cf. Eqs. (6) and (8)], meaning that the underlying effective medium is *spatially dispersive* or *nonlocal*. The nonlocal effective permittivities differ from the quasistatic ones by the factors depending on the structure and material parameters. In addition, there are no *k _{x}* and

*k*terms in the effective permittivities, which is a consequence of

_{z}*inversion symmetry*[42] of the underlying structure, an invariance property of a system when the coordinates are inverted.

The above procedure can be performed to higher orders of the wave vector components to deliver ever more accurate results, at the expense of resolving more involved coefficients. To the author’s experience, expanding the dispersion relations to eighth power of the wave vector components can yield sufficiently accurate effective permittivity tensor that recovers the dispersion of the original structure within a few percent of error. The nonlocal effective permittivities, no matter based on how higher orders of expansions, are characterized by the intrinsic frequencies of the constituent materials.

To extract the effective parameters from the dispersion relations is considered a somewhat intuitive approach. From the viewpoint of wave propagation, a homogeneous medium with the effective parameters is approximately equivalent to a finite structure with the same dispersion if the characteristic length of the structure (that is, the period) is much less than the wavelength: *a* ≪ *λ*. The benefit of the present approach is to express the effective permittivities in compact formulas. A more rigorous approach to the effective parameters may resort to homogenization based on the field averaging [33, 34]. In this approach, the effective permittivities are represented by analytical yet lengthy terms. The dispersion relation also appears as an important factor in the formulation.

#### 2.2. Characteristic frequencies

Let the materials 1 and 2 in the layered structure be the dielectric and metal, respectively. Using the Drude model (with the loss neglected) for the metal:
${\epsilon}_{2}=1-{\omega}_{p}^{2}/{\omega}^{2}$, where *ω _{p}* is the plasma frequency of the metal, the quasistatic effective permittivities are given as

*zero*frequency of ${\epsilon}_{z}^{0}$ (and ${\epsilon}_{y}^{0}$) and the

*pole*frequency of ${\epsilon}_{x}^{0}$, respectively. Note that

*ω*

_{0}(

*ω*

_{∞}) is reduced (raised) as the ratio

*f*

_{1}/

*f*

_{2}increases. The zero frequency of ${\epsilon}_{x}^{0}$ is

*ω*, which is the same as that of

_{p}*ε*

_{2}. In the quasistatic limit, the layered structure is regarded as a local anisotropic medium, which behaves like a plasmonic material in the parallel (to the metal-dielectric interface) direction, with a

*reduced*plasma frequency

*ω*

_{0}[cf. Eq. (14)], and like an ionic crystal in the perpendicular direction, with a transverse resonance frequency

*ω*

_{∞}and a longitudinal resonant frequency

*ω*[cf. Eq. (15)].

_{p}Note in Eq. (6) that
${\epsilon}_{z}^{\text{eff}}$ has a zero frequency close to *ω*_{0}, and
${\epsilon}_{x}^{\text{eff}}$ has two zero frequencies: one is *ω*_{0} and the other is close to *ω _{p}*. In addition, there are two pole frequency branches of
${\epsilon}_{x}^{\text{eff}}$: one is located at lower frequencies, with

*ω*

_{1}being the lowest frequency (at

*k*= 0), and the other is located at higher frequencies, with

_{z}*ω*

_{2}being the highest frequency (also at

*k*= 0). They are close to either

_{z}*ω*

_{0}or

*ω*

_{∞}, depending on the fraction of dielectric (or metal) in the unit cell. Figure 2(a) is an example of the variations of

*ω*

_{1}and

*ω*

_{2}, along with

*ω*

_{0}and

*ω*

_{∞}, with respect to

*f*

_{1}.

Concerning the electromagnetic waves in the periodic metal-dielectric layered structure, there are three length scales: the lattice period *a*, the wavelength *λ* = 2*π/k*_{0}, and the plasma wavelength *λ _{p}* = 2

*πc/ω*. In order for the effective medium to be valid,

_{p}*a/λ*= (

*a/λ*)(

_{p}*ω/ω*) should be, in principle, much smaller than unity. In practice, this condition is attained when

_{p}*a/λ*is small and

_{p}*ω*is below

*ω*. The parameters used in this article will be carefully arranged so that

_{p}*a/λ*is considered small enough, yet still feasible in fabrication with the modern nanotechnology.

## 3. In-plane effective permittivities

Figure 3 is an example of
${\epsilon}_{z}^{\text{eff}}$ and
${\epsilon}_{x}^{\text{eff}}$ as the functions of *ω* and *k _{x}* or

*k*for the periodic metal-dielectric layered structure with

_{z}*f*

_{1}>

*f*

_{2}, based on the effective medium model (cf. Sec. 2) with the expansions of the wave vector components to eighth order. Another example of ${\epsilon}_{z}^{\text{eff}}$ and ${\epsilon}_{x}^{\text{eff}}$ for

*f*

_{1}<

*f*

_{2}is shown in Fig. 4. In either case, the effective permittivity ${\epsilon}_{z}^{\text{eff}}$ [Figs. 3(a) and 4(a)] is basically characterized by the quasistatic effective permittivity ${\epsilon}_{z}^{0}$ [cf. Eq. (9)] and modified by the perpendicular (to the metal-dielectric interface) wave vector component

*k*as well as the frequency

_{x}*ω*. For periodic structures, the largest value of |

*k*| is determined by the lattice period:

_{x}*π/a*, and ${\epsilon}_{z}^{\text{eff}}$ deviates slightly from ${\epsilon}_{z}^{0}$ as |

*k*| increases. The nonlocal effect exhibited by ${\epsilon}_{z}^{\text{eff}}$ is usually weak.

_{x}The effective permittivity
${\epsilon}_{x}^{\text{eff}}$ [Figs. 3(b) and 4(b)], on the other hand, shows a strong non-local effect. Being dependent on the parallel wave vector component *k _{z}*, which is not restricted by the geometry,
${\epsilon}_{x}^{\text{eff}}$ differs very much from the quasistatic counterpart
${\epsilon}_{x}^{0}$. This feature is more evident near the poles of
${\epsilon}_{x}^{\text{eff}}$ (extending from the poles of
${\epsilon}_{x}^{0}$ at

*k*= 0). For a given frequency, ${\epsilon}_{x}^{\text{eff}}$ may experience a drastic change from positive to negative infinity, or the other way around, across a certain value of |

_{z}*k*|, which is considered a typical feature of

_{z}*nonlocal resonance*that occurs in a spatially dispersive medium. At

*k*= 0, the onset of resonance occurs at

_{z}*ω*

_{1}and

*ω*

_{2}. A similar feature can be observed in the quasistatic effective permittivity ${\epsilon}_{x}^{0}$ (with the resonance at

*ω*

_{∞}). As

*k*≠ 0, the resonance frequencies gradually move toward each other. In the metal-dielectric layered structure, such resonance, as will be shown later, is attributable to the excitation of surface plasmon polaritons.

_{z}In the present problem, spatial dispersion is coupled with frequency (temporal) dispersion. Basic properties of the in-plane effective permittivities are categorized into four frequency ranges:

#### 3.1. Weak spatial dispersion range: 0 < ω < ω_{1}

In this range,
${\epsilon}_{z}^{\text{eff}}<0$ and
${\epsilon}_{x}^{\text{eff}}>0$ [cf. Figs. 3 and 4], and the dispersion relation is hyperbolic-like. Here, *ω*_{1} is the lowest frequency of the lower pole branch of
${\epsilon}_{x}^{\text{eff}}$, which is close to *ω*_{0} for *f*_{1} > *f*_{2} and close to *ω*_{∞} for *f*_{1} < *f*_{2} [cf. Fig. 2(a)]. The dispersion relation [cf. Eq. (4)] is dominated by the
${k}_{z}^{2}$ and
${k}_{x}^{2}$ terms, with opposite signs of the attached coefficients (the one with
${k}_{x}^{2}$ being negative). Let the *xy* plane be an interface between vacuum and the effective medium. For a wave incident from vacuum, with the wave vector lying on the *xz* plane, a forward wave with negative refraction will occur in the medium. This feature is similar as in a uniaxially anisotropic medium with opposite signs of the permittivity components, the one normal to the interface (between vacuum and the medium) being negative [40].

#### 3.2. Strong spatial dispersion range: ω_{1} < ω < ω_{2}

In this range,
${\epsilon}_{z}^{\text{eff}}>0$ and
${\epsilon}_{x}^{\text{eff}}$ changes from positive to negative infinity by experiencing a pole for *f*_{1} > *f*_{2} [cf. Fig. 3]. Here, *ω*_{2} is the highest frequency of the higher pole branch of
${\epsilon}_{x}^{\text{eff}}$, which is close to *ω*_{∞} for *f*_{1} > *f*_{2} [cf. Fig. 2(a)]. At small |*k _{z}*|, both
${\epsilon}_{z}^{\text{eff}}$ and
${\epsilon}_{x}^{\text{eff}}$ are positive and the dispersion relation is elliptic-like. At large |

*k*|, ${\epsilon}_{x}^{\text{eff}}$ is dominated by the ${k}_{z}^{4}$ term and the dispersion relation becomes parabolic-like, which is a non-standard type of dispersion in the medium. This is the frequency range where spatial dispersion or nonlocal effect tends to be significant. In this situation, the two different relations coexist, leading to a mixed type of dispersion. Accordingly, there are two eigenwaves. This can be seen by using Eq. (6) in Eq. (4) to give

_{z}*k*, both of the allowed

_{x}*k*, either positive or negative, can be real. For a wave incident from vacuum onto the layered structure, the splitting of wave in the structure is expected to occur [24]. For

_{z}*f*

_{1}<

*f*

_{2}, on the other hand, ${\epsilon}_{z}^{\text{eff}}<0$ and ${\epsilon}_{x}^{\text{eff}}$ changes from negative to positive infinity by experiencing a pole [cf. Fig. 4]. In this case,

*ω*

_{2}is close to

*ω*

_{0}[cf. Fig. 2(a)]. At small |

*k*|, both ${\epsilon}_{z}^{\text{eff}}$ and ${\epsilon}_{x}^{\text{eff}}$ are negative and the elliptic-like dispersion does not exist. For a given

_{z}*k*, only one allowed

_{x}*k*, either positive or negative, is real. Therefore, there is only one eigenwave, with the parabolic-like dispersion. Regarding the wave propagation characteristics, this range is further divided into two subranges:

_{z}### 3.2.1. Below surface plasma frequency: *ω*_{1} < *ω* < *ω*_{sp}

_{sp}

In this range, the wave associated with the elliptic-like dispersion, if any, is ordinarily (positively) refracted. The wave with the parabolic-like dispersion, on the other hand, is negatively refracted. Both waves are forward waves. Negative refraction of the latter wave is attributable to spatial dispersion, which can be explained by the time-averaged Poynting vector for the nonlocal medium [27]:

*k*-

_{z}*k*plane, the slope of the contour normal is given as $-1/\frac{d{k}_{x}}{d{k}_{z}}=-\frac{d{k}_{z}}{d{k}_{x}}$. Based on the dispersion relation (4), the slope is implicitly solved to give

_{x}*i*,

*j*=

*x*and

*z*. The nonlocal effect is considered strong if FOM

*exceeds unity. If, in particular, FOM*

_{i}*> 1 and FOM*

_{x}*< 1, or vice versa, the slope $-\frac{d{k}_{z}}{d{k}_{x}}$ becomes negative even when both $\frac{{k}_{x}}{{\epsilon}_{z}^{\text{eff}}}$ and $\frac{{k}_{z}}{{\epsilon}_{x}^{\text{eff}}}$ are positive. This is a feature of negative refraction pertaining to the nonlocal medium. Two illustrative examples with different dielectric (metal) fractions in the unit cell are given below:*

_{z}**Example 1:** *f*_{1} > *f*_{2}. Figure 5(a) is an example showing the equifrequency contours, along with the wave vectors and Poynting vectors, for a larger dielectric fraction. The contour is characterized by a quartic curve in the *xy* plane:

*a*and

*b*are constants, and

*ε*is a small positive number. The parabolic-like character of the quartic curve comes from the

*x*

^{4}term at larger

*x*, which corresponds to the nonlocal effect in the present problem. The elliptic-like character, on the other hand, is dominant at smaller

*x*. Note that the contour normal (denoted by the green arrow) with the parabolic-like dispersion is oriented toward the same side of the incident wave vector (with respect to the interface normal), leading to negative refraction. The corresponding eigenmode, plotted in Fig. 6(a), shows a typical feature of

*surface plasmon polariton*, with the fields highly concentrated on the metal-dielectric interfaces. In particular, this mode has a symmetric alignment of surface charges and is identified as the

*bonding mode*. The colors of surface charges in the figure have been exaggerated to show more clearly the alignment pattern. Note also that this mode is located (marked by the red dot) near the pole of ${\epsilon}_{x}^{\text{eff}}$ (denoted by the vertical gray line) [cf. Fig. 5(a)]. The strong nonlocal effect is manifest on the Lorentzian resonance character of the effective permittivity ${\epsilon}_{x}^{\text{eff}}$, as shown in Fig. 7(a). As |

*k*| increases, there is a large discrepancy between ${\epsilon}_{x}^{\text{eff}}$ and ${\epsilon}_{x}^{0}$ (indicated by the dashed line). For the eigenmode with negative refraction, both ${\epsilon}_{z}^{\text{eff}}$ and ${\epsilon}_{x}^{\text{eff}}$ (marked by the red dot) are positive. This is considered a property not available in the local anisotropic medium [40].

_{z}**Example 2:** *f*_{1} < *f*_{2}. Figure 8(a) is an example for a smaller dielectric fraction. In this case, only one eigenwave with negative refraction exists. The character of the wave, however, is different from the counterpart for *f*_{1} > *f*_{2} [cf. Fig. 5(a)]. The equifrequency contour is characterized by a similar quartic curve in the *xy* plane as in Eq. (22), but with the minus sign on the right side:

*antibonding mode*, as shown in Fig. 9(a). The feature of negative refraction is consistent with that in a uniaxially anisotropic medium with opposite signs of the permittivity components: ${\epsilon}_{z}^{\text{eff}}<0$ and ${\epsilon}_{x}^{\text{eff}}>0$[40], as in the range where 0 <

*ω*<

*ω*

_{1}(cf. Sec. 3.1). The spatial dispersion associated with this mode, however, is still strong. The effective permittivity ${\epsilon}_{x}^{\text{eff}}$ in Fig. 10(a) also shows a Lorentzian resonance character, but with a reverse sense from the counterpart for

*f*

_{1}>

*f*

_{2}[cf. Fig. 7(a)]. The eigenmode is also located near the pole, where ${\epsilon}_{x}^{\text{eff}}$ changes rapidly.

### 3.2.2. Above surface plasma frequency: *ω*_{sp} < *ω* < *ω*_{2}

_{sp}

In this range, the wave associated with the elliptic-like dispersion, if any, is the same as in the range where *ω*_{1} < *ω* < *ω _{sp}* (cf. Sec. 3.2.1). The wave associated with the parabolic-like dispersion, however, is very much different. Two illustrative examples with different dielectric (metal) fractions in the unit cell are given below:

**Example 3:** *f*_{1} > *f*_{2}. Figure 5(b) is an example showing the equifrequency contours, along with the wave vectors and Poynting vectors, for a larger dielectric fraction. Due to the anomalous frequency dispersion, *k _{z}* for the parabolic-like dispersion is chosen to be negative, so that the energy flows away from the interface (+

*z*direction). Otherwise, the principle of causality will be violated. The wave with the parabolic-like dispersion is therefore a backward wave with ordinary refraction, rather than the forward wave with negative refraction as in the range below

*ω*[cf. Fig. 5(a)]. The eigenmode in Fig. 6(b) is shown to be an antibonding mode of surface plasmon polariton, with an antisymmetric alignment of surface charges on the metal-dielectric interfaces. Note that the antibonding mode has a higher frequency than the respective bonding mode [cf. Fig. 6(a)]. This feature is similar as in a metal film in vacuum [39]. The effective permittivity ${\epsilon}_{x}^{\text{eff}}$ in Fig. 7(b) looks alike as in Fig. 7(a) in the wave vector domain. They are, however, different in the frequency domain. By adding a small increment of frequency

_{sp}*δω*, the patterns of ${\epsilon}_{x}^{\text{eff}}$ move in opposite directions between the two cases [compare the gray lines in Figs. 7(a) and 7(b)]. The opposite trend in the frequency domain is responsible for the distinction between negative refraction (forward wave) and ordinary refraction (backward wave). For the eigenmode with backward wave, both ${\epsilon}_{z}^{\text{eff}}$ and ${\epsilon}_{x}^{\text{eff}}$ (marked by the red dot) are positive. This is also considered a property not available in the local anisotropic medium [40].

**Example 4:** *f*_{1} < *f*_{2}. Figure 8(b) is an example for a smaller dielectric fraction. In this case, there is only one eigenwave with the parabolic-like dispersion, as in the range below *ω _{sp}* [cf. Fig. 8(a)]. The eigenwave, however, is a backward wave with ordinary refraction, as in the range above

*ω*for

_{sp}*f*

_{1}>

*f*

_{2}[cf. Fig. 5(b)]. The character of the wave is different from either case. The eigenmode in Fig. 9(b) is shown to be a bonding mode of surface plasmon polariton, with a symmetric alignment of surface charges on the metal-dielectric interfaces. Note that the bonding mode has a higher frequency than the respective antibonding mode [cf. Fig. 9(a)]. This feature is similar as in an insulating film between two semi-infinite metals [39]. The feature of backward wave is also not available in a uniaxially anisotropic medium with opposite signs of the permittivity components, the one tangential to the interface (between vacuum and the medium) being negative [40]. In the present case, the tangential effective permittivity component is positive ( ${\epsilon}_{x}^{\text{eff}}>0$), while the normal component is negative ( ${\epsilon}_{z}^{\text{eff}}<0$). The spatial dispersion is strong as in the range below

*ω*. The effective permittivity ${\epsilon}_{x}^{\text{eff}}$ in Fig. 10(b) also shows the Lorentzian resonance character as in Fig. 10(a), but with an opposite trend with increasing the frequency [compare the gray lines in Figs. 10(a) and 10(b)].

_{sp}#### 3.3. Weak spatial dispersion range: ω_{2} < ω < ω_{p}

In this range,
${\epsilon}_{z}^{\text{eff}}>0$ and
${\epsilon}_{x}^{\text{eff}}<0$ [cf. Figs. 3 and 4], and the dispersion relation is hyperbolic-like. The dispersion curve is dominated by the
${k}_{z}^{2}$ and
${k}_{x}^{2}$ terms, with opposite signs of the attached coefficients (the one with
${k}_{z}^{2}$ being negative). The eigenwave is a backward wave with ordinary refraction, which is similar as in the range *ω _{sp}* <

*ω*<

*ω*

_{2}(cf. Sec. 3.2.2). The spatial dispersion, however, is weak in this range. The effective permittivities do not change much with the wave vector components.

#### 3.4. Weak spatial dispersion range: ω > ω_{p}

In this range, ${\epsilon}_{z}^{\text{eff}}>0$ and ${\epsilon}_{x}^{\text{eff}}>0$ [cf. Figs. 3 and 4], and the dispersion relation is elliptic-like. The dispersion curve is also dominated by the ${k}_{z}^{2}$ and ${k}_{x}^{2}$ terms, both with positive attached coefficients. In this range, the effective medium behaves like an ordinary anisotropic dielectric. The eigenwave is a forward wave with ordinary refraction. The spatial dispersion is also weak.

## 4. Out-of-plane effective permittivity

Unlike the in-plane effective permittivities
${\epsilon}_{z}^{\text{eff}}$ and
${\epsilon}_{x}^{\text{eff}}$, where the properties are very different between *f*_{1} > *f*_{2} and *f*_{1} < *f*_{2}, the out-of-plane effective permittivity
${\epsilon}_{y}^{\text{eff}}$ shows similar features for either case. For smaller |*k _{z}*| and |

*k*|, ${\epsilon}_{y}^{\text{eff}}$ is basically characterized by the quasistatic effective permittivity ${\epsilon}_{y}^{0}$, with a slight modification factor [cf. Eq. (8)]. In this situation, the effective medium behaves like an isotropic plasmonic metal with a reduced plasma frequency

_{x}*ω*

_{0}. This frequency is very close to the cutoff frequency

*ω*

_{3}for TE polarization, the lowest frequency for TE waves to propagate in the effective medium [cf. Fig. 2(b)]. For larger |

*k*| and |

_{z}*k*|, ${\epsilon}_{y}^{\text{eff}}$ may deviate much from ${\epsilon}_{y}^{0}$. Basic properties of the out-of-plane effective permittivity are divided into two frequency ranges:

_{x}#### 4.1. Below cutoff frequency: ω < ω_{3}

Figure 11(a) is an example of
${\epsilon}_{y}^{\text{eff}}$ as the functions of *k _{z}* and

*k*, based on the effective medium model [cf. Sec. 2] for

_{x}*ω*<

*ω*

_{3}. Note that ${\epsilon}_{y}^{\text{eff}}$ is negative at smaller |

*k*| and |

_{z}*k*|, which becomes more negative as |

_{x}*k*| increases and changes to positive as |

_{z}*k*| increases (the contour of ${\epsilon}_{y}^{\text{eff}}=0$ is plotted in red color). Since the frequency is below the cutoff frequency

_{x}*ω*

_{3}, there are no waves allowed to propagate in the effective medium. The spatial dispersion or nonlocal effect is not relevant.

#### 4.2. Above cutoff frequency: ω > ω_{3}

Figure 11(b) is an example of
${\epsilon}_{y}^{\text{eff}}$ for *ω* > *ω*_{3}. Note that
${\epsilon}_{y}^{\text{eff}}$ is positive at smaller |*k _{z}*| and |

*k*|, which may change to negative as |

_{x}*k*| increases and becomes more positive as |

_{z}*k*| increases. Unlike the in-plane effective permittivities, where the nonlocal effect comes mainly from

_{x}*k*, the nonlocal effect associated with the out-of-plane effective permittivity basically comes from the dependence on

_{z}*k*. This effect is usually weak as |

_{x}*k*| is bound by

_{x}*π/a*. The equifrequency contours can be characterized by a quartic curve in the

*xy*plane:

*r*is a constant and

*ε*is a small positive number. At smaller

*x*and

*y*, the contours are nearly circles, while at larger

*x*and

*y*, the contours are a bit of distorted along the

*y*axis. The dispersion relation is elliptic-like, with a normal frequency dispersion. The eigenwave is a forward wave with ordinary refraction. The spatial dispersion exhibited by ${\epsilon}_{y}^{\text{eff}}$ is not significant.

It is worthy of noting that although the effective permittivity
${\epsilon}_{y}^{\text{eff}}$ may change substantially with the wave vector components, the spatial dispersion or nonlocal effect is either not significant or irrelevant. This is because
${\epsilon}_{y}^{\text{eff}}$ always becomes negative at larger |*k _{z}*| and thus no waves exist there. The effect due to

*k*, however, is weak, as |

_{x}*k*| is bound by

_{x}*π/a*.

## 5. Summary on spatial dispersion

Based on the discussion of the effective permittivities [Secs. 3 and 4], we arrive at several identities of spatial dispersion in periodic layered metamaterials. First, strong spatial dispersion is found in the frequency range where surface plasmon polaritons occur. Therefore, spatial dispersion is only significant for TM polarization. Second, strong spatial dispersion may give rise to double eigenwaves. Subsequent phenomena such as wave splitting can be observed owing to this identity. Third, strong spatial dispersion may lead to negative refraction and backward wave even when the effective permittivity components are all positive. This identity is different from similar phenomena in the uniaxially anisotropic medium, where one of the permittivity component is negative. Fourth, strong spatial dispersion is related to the parabolic-like dispersion. This is very different from the standard dispersion types such as elliptic or hyperbolic.

## 6. Concluding remarks

In conclusion, the author has established a systematic study of spatial dispersion in periodic layered metamaterials, based on a nonlocal effective medium model. Basic features of spatial dispersion are analyzed with the help of effective permittivity tensor and dispersion characteristics. The nonlocal effect is manifest on the drastic variations of the perpendicular (to the metal-dielectric interface) effective permittivity component in the wave vector domain. Strong spatial dispersion is found in the frequency range covering the two branches of poles, giving rise to double eigenwaves, parabolic-like dispersion, negative refraction, and backward wave. These unusual features are closely related to the excitation of surface plasmon polaritons on the metal-dielectric interfaces. The collective oscillation of surface charges in the metal provides an additional mechanism other than the electromagnetic waves to transport energy, and is considered the origin of spatial dispersion in the present problem. As the resonance occurs, the strong coupling of fields between adjacent layers causes the polarization in one layer depends not only on the field in that layer, but also in neighboring layers. The periodic metal-dielectric layered structure is considered a strong nonlocal medium when the frequency is proper (around the surface plasma frequency) to excite resonance and the wavelength is long enough (compared to the lattice period) to treat the layered structure as a medium.

## Acknowledgments

This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 99-2221-E-002-121-MY3.

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