## Abstract

We present a new theoretical analysis for the light scattering at sub-wavelength dielectric structures. This analysis can provide new intuitive insights into the phase shift of the scattered light that cannot be obtained from the existing approaches. Unlike the traditional analytical (e.g. Mie formalism) and numerical (e.g. FDTD) approaches, which simulate light scattering by rigorously matching electromagnetic fields at boundaries, we consider sub-wavelength dielectric structures as leaky resonators and evaluate the light scattering as a coupling process between incident light and leaky modes of the structure. Our analysis indicates that the light scattering is fundamentally dictated by the eigenvalue of the leaky modes. It indicates that the upper limit for the scattering efficiency of a cylindrical cylinder in free space is *4n*, where *n* is the refractive index. It also indicates that the phase shift of the forward scattered light at dielectric structures can only cover half of the phase space [0, 2π], but backward scattering can provide a full phase coverage.

© 2013 OSA

## 1. Introduction

The resonant light scattering at subwavelength-sized objects constitutes a critical cornerstone for modern optics research [1–3]. Much of the significance of the resonant light scattering can be best manifested by the spectacular success of localized surface plasmon in metallic nanostructures. The plasmonic resonance, which results from the collective oscillation of free electrons, has enabled a plethora of exotic functionality, including extraordinary transmission [4], optical analogue of electromagnetic induced transparency [5, 6], super-resolution imaging [7–9], cloaking [10], and metamaterials [11, 12]. Significantly, subwavelength dielectric structures have been recently demonstrated able to provide similarly strong, tunable resonant light scattering [13–16]. The dielectric optical resonance is extremely intriguing because dielectric materials, for instance, silicon, are much less lossy than metals and because it offers a tantalizing prospect to monolithically integrate novel optical functionality into chip-scale electric or optoelectronic devices that overwhelmingly build on dielectric materials.

However, in contrast with the extensive volume of studies on plasmonic resonances, the development of optical functionality by capitalizing on dielectric resonances has remained relatively limited. One key issue lies in the lack of intuitive understanding of the resonant light scattering in dielectric structures. Compared with plasmonic resonances, which typically involves low modes such as dipole and quandrapole, the resonance in dielectric structures can be much more sophisticated with the involvement of high modes [13–15]. While the existing analytical (e.g. Mie theory) or numerical (e.g. finite difference time domain, FDTD) approaches are able to simulate the optical responses of dielectric structures [2, 3], the complicated resonance feature makes it very difficult to extract useful insights from resulting simulations. For instance, the existing approaches provide little intuitive understanding for the phase shift in the scattered wave. The intuitive insight is necessary for rationally manipulating the scattering phase to develop novel optical functionalities.

Here we present a new theoretical analysis for the light scattering at sub-wavelength dielectric structures. Unlike the traditional analytical or numerical models, which simulate light scattering by rigorously matching electromagnetic fields at boundaries [2, 3], we consider sub-wavelength dielectric structures as leaky resonators and evaluate the light scattering as a coupling between incident light and leaky modes of the structure. Our analysis indicates that the light scattering is dictated by the eigenvalue of the leaky modes. This correlation with leaky modes can provide new intuitive insights into the scattering and the phase shift in the scattered light, which cannot be obtained from the existing models. For instance, it indicates that the upper limit for the scattering efficiency of a cylindrical cylinder in free space is *4n*, where *n* is the refractive index. It also indicates that the phase shift in forward scattered light at dielectric structures can only cover half of the phase space [0, 2π], but backward scattering can provide a full phase coverage.

## 2. Calculations of leaky modes

We start with elaborating the calculation of leaky modes in one-dimensional (1D) and zero-dimensional (0D) dielectric structures.

#### 2.1 One-dimensional (1D) cylinder

For a circular cylinder located in a cylindrical coordinate (*r, ϕ, z*) as illustrated in Fig. 1
, the electric field in the z direction *E*_{z} can be generally written as,

*C*is a constant to be determined,

*β*is the propagation constant in the z direction,

*J*

_{m}() and

*H*

_{m}() are the

*m*th order of Bessel function of the first kind and Hankel function of the first kind, respectively,

*r*

_{0}is the radius of the cylinder,

*κ*and

*γ*are wave vectors in the transverse direction inside and outside the cylinder as

*n*is the refractive index of the materials, and

*k*is the wave vector in the environment that is assumed to be free space. Equations (1) and (2) can be re-written in terms of odd and even modes,

Odd mode:

*H*

_{z}can be written as

Odd mode:

*E*

_{ϕ}and

*H*

_{ϕ}can be derived from

*E*

_{z}and

*H*

_{z}as

*α*=

*κ*(

*γ*) for the field inside (outside) the cylinder,

*ω*is the frequency,

*μ*and

*ε*are the permeability and permittivity, respectively. By matching the boundary conditions at the cylinder/environment interface, we may have

*β*= 0 (and thus

*κ*=

*nk*,

*γ*=

*k*), i.e. no propagation along the cylinder. For transverse electric (TE) modes,

*C*= 0,

*C’*= 0

#### 2.2 Zero-dimensional (0D) sphere

We use the polarization of transverse electric (TE) as an example to illustrate the calculation of leaky modes in 0D spheres. The electric field is tangential to the surface of the sphere in spherical coordinate (*r, θ, ϕ*), (Fig. 2
) and can be written as

*D*and

*D’*are constant coefficients, and refer to

*odd*and

*even*modes, and

*j*

_{m}() and

*h*

_{m}() are the

*m*th order of spherical Bessel function of the first kind and spherical Hankel function of the first kind, respectively. ${F}_{e,olm}(\theta ,\varphi )$and ${G}_{e,olm}(\theta ,\varphi )$ are functions of

*θ*and

*ϕ*, and their expressions can be found out in typical textbooks on light scattering [2, 17]. Because both ${F}_{e,olm}(\theta ,\varphi )$and ${G}_{e,olm}(\theta ,\varphi )$ may be cancelled out during the calculation of leaky modes, we do not list the expressions here for the sake of simplicity. By matching the boundary conditions at the sphere/environment interface, we have

*Riccati-Bessel*functions ${\psi}_{m}(\rho )=\rho {j}_{m}(\rho )$ and ${\xi}_{m}(\rho )=\rho {h}_{m}(\rho )$ .

Solving Eqs. (17), (18), (27), and (28) can give complex values for a normalized parameter *nkr*_{0} (*nkr*_{0} = *N*_{real} - *N*_{imag}*i*). These complex values are eigenvalues of the leaky modes. We have previously calculated and tabulated the eigenvalue for typical leaky modes [14]. The real part of the eigenvalue *N*_{real} indicates the condition for the resonance with leaky modes, and the imaginary part *N*_{imag} refers to the radiative leakage of the electromagnetic energy stored in the leaky mode. For materials without intrinsic absorption loss, this imaginary part indicates spectral width of the leaky mode resonance.

## 3. Coupled leaky mode theory for the light scattering at 1D and 0D dielectric structures

We model the light scattering at dielectric structures as a coupling process between the leaky modes of the structure and external electromagnetic waves (Fig. 3
). Without losing generality, we focus on the light scattering at a circular cylinder in vacuum. For simplicity, we start from the coupling with one arbitrary leaky mode, for instance, one transverse magnetic (TM) mode TM_{ml}. The leaky mode in cylinders is characterized by an azimuthal mode number, *m*, and a radial order number, *l* [14]. By assuming no intrinsic absorption, we can describe the coupling process using formalism similar to conventional temporal coupled mode theory [14, 18–21],

*a*

_{ml}is the amplitude of the leaky mode with its squared magnitude representing the energy stored inside,

*ω*

_{ml}and

*γ*

_{ml}are the resonant frequency (eigenfrequency) and radiative decay rate of the leaky mode,

*κ*

_{ml}and

*C*

_{a,}_{ml}are the input and output coupling coefficients between the leaky mode and external waves, respectively, and

*C*

_{W,ml}is a background reflection coefficient. and are the amplitudes of incoming and outgoing waves as ${E}_{0}{W}_{ml}^{+}{H}_{m}^{(2)}(kr){e}^{im\varphi}$and ${E}_{0}{W}_{ml}^{-}{H}_{m}^{(1)}(kr){e}^{im\varphi}$, where ${H}_{m}^{(1)}$(${H}_{m}^{(2)}$) is the

*m*th order Hankel function of the first (second) kind that indicates outgoing (incoming) waves in cylindrical coordinates,

*E*

_{0}is a normalization constant to ensure the squared magnitudes of ${W}_{ml}^{+}$and ${W}_{ml}^{-}$equal to the power carried by corresponding external waves. As this model deals with the coupling of leaky modes in subwavelength structures, we thus term it as coupled leaky mode theory (CLMT). Intuitively, the incoming wave couples into the leaky mode, then propagates inside the cylinder, and eventually exits as the outgoing wave. Similar to coupled waveguide-cavity systems [22], we can set the output coupling coefficient

*C*

_{a}_{,ml}as ${C}_{a,ml}=A{e}^{i{\theta}_{ml}}$where

*A*is a coupling constant and

*θ*

_{ml}is a phase term that can be intuitively related with the light propagation inside the cylinder. By applying energy conversation and time-reversal symmetry to Eqs. (29) and (30), we can get

*A*= (2

*γ*

_{ml})

^{0.5}and ${C}_{W,ml}={e}^{i(2{\theta}_{ml}+\pi )}$.

We can derive the scattering coefficient that correlates the scattered wave with incident light from a reflection coefficient ${R}_{ml}={W}_{ml}^{-}/{W}_{ml}^{+}$. For an arbitrary incident frequency *ω*,

*R*

_{ml}. The incident plane wave can be expanded into a series of Bessel functions

*J*

_{m}. Because only one leaky mode is involved in the scattering, we may consider only the terms that can interact with the specific leaky mode. To conserve the energy carried by the incoming and outgoing waves in Fig. 3, we can have the incident and scattered wave as ${E}_{inc,ml}={E}_{0}{W}_{ml}^{+}[{H}_{m}^{(1)}(kr)+{H}_{m}^{(2)}(kr)]{e}^{im\varphi}=2{E}_{0}{W}_{ml}^{+}{J}_{m}(kr){e}^{im\varphi}$ and ${E}_{sca,ml}={E}_{0}({W}_{ml}^{-}-{W}_{ml}^{+}){H}_{m}^{(1)}(kr){e}^{im\varphi}$. The scattering coefficient

*b*

_{ml}can be written as

*θ*and Δ

_{ml}*. According to Eq. (31), Δ*

_{ml}*arises from the deviation of incident frequency*

_{ml}*ω*from the resonant frequency

*ω*

_{ml}, hence referred as off-resonance offset, and can be calculated from

*β*= (

*ω*

_{ml}-

*ω*)/

*γ*

_{ml}. Equation (33) can be solved out using the eigenvalue of the leaky mode,

*N*

_{real}-

*N*

_{imag}

*i*. The eigenfrequency

*ω*

_{ml}and radiative decay rate

*γ*

_{ml}are related with the real and imaginary parts of the eigenvalue in straightforward ways as

*ω*

_{ml}=

*c*.

*N*

_{real}/(

*n*.

*r*

_{0}), and

*γ*

_{ml}=

*c*.

*N*

_{imag}/(

*n*.

*r*

_{0}),

*c*is speed of light.

We find that *θ _{ml}* can be derived from the eigenvalue of the leaky mode as well. By assuming no incoming wave (${W}_{ml}^{+}$ = 0), we can derive ${e}^{i{\theta}_{ml}}={W}_{ml}^{-}/\sqrt{2{\gamma}_{ml}}{a}_{ml}$ from Eq. (30). With the absence of incoming waves, ${W}_{ml}^{-}$and

*a*

_{ml}are essentially related with the eigenfields of the leaky mode. The electric eigenfield of the leaky mode inside a cylinder can generally be written as${E}_{0}{W}_{ml}^{0}{J}_{m}(nkr){e}^{im\varphi}$. ${W}_{ml}^{0}$ is the amplitude of the eigenfield, and can be correlated to

*a*

_{ml}using a translation between “power picture” and “energy picture” as${\left|{a}_{ml}\right|}^{2}=A{\left|{W}_{ml}^{0}\right|}^{2}$, and$A={n}^{2}\left[{J}_{m}^{2}(nk{r}_{0})-{J}_{m-1}(nk{r}_{0}){J}_{m+1}(nk{r}_{0})\right]/2$. Additionally, with the absence of incoming waves, the outgoing wave ${W}_{ml}^{-}$ is solely contributed by the radiative decay of the energy stored inside the cylinder as${\left|{W}_{ml}^{-}\right|}^{2}=2{\gamma}_{ml}{\left|{a}_{ml}\right|}^{2}$. ${W}_{ml}^{0}$ and ${W}_{ml}^{-}$ can also be connected through the boundary condition at the cylinder/environment interface as ${W}_{ml}^{-}{H}_{m}^{(1)}(k{r}_{0})={W}_{ml}^{0}{J}_{m}(nk{r}_{0})$. By considering all these relations, we can have

*nkr*

_{0}. Apparently,

*θ*is solely determined by the eigenvalue of leaky modes, and thus we refer it as the intrinsic phase of leaky modes. Calculation results for the intrinsic phase of typical leaky modes in 1D cylinders and 0D spheres with a refractive index of 4 are listed in Table 1 and Table 2 , respectively. The listed number is in a unit of π.

_{ml}The coefficient of the light scattering that involves multiple leaky modes can be constructed from the single-mode coefficient *b*_{ml} in Eq. (32). The leaky modes with different mode number *m* are orthogonal to each other and thus may interact with incident light independently. Therefore, the scattering efficiency from a structure with multiple leaky modes *Q*_{sca} can be obtained by simply adding up the contribution from the leaky modes with different *m* as${Q}_{sca}=2/(k{r}_{0}){\displaystyle \sum _{m}\mathrm{Re}(-{b}_{m})}$ (note: *b*_{m} defined here has an opposite sign as the scattering coefficient defined in Mie Theory [2]), where *Re* means the real part. The scattering efficiency *b*_{m} includes contributions from a group of leaky modes with the same mode number *m* but different order number *l* (we refer them as group *m* leaky modes for the convenience of discussion) as ${b}_{m}={\displaystyle \sum _{l}{b}_{ml}}$. This equation can be significantly simplified with reasonable assumptions. It is reasonable to believe that incident light only interacts with neighboring leaky modes whose resonant frequency *ω*_{ml} is relatively close to the incident frequency *ω* (- π <(*ω*_{ml} – *ω*).*nr*_{0}/c< π), as illustrated in Fig. 4
. This is because the coupling efficiency of incident light to a leaky mode exponentially decreases with the difference between the incident frequency and the eigenfrequency (*ω*_{ml} – *ω*) [14]. Therefore, we can construct the scattering coefficient *b*_{m} of the group *m* with an easier way. For an arbitrary incident frequency *ω*, we consider *b*_{m} equal to one single-mode coefficient *b*_{ml} if the incident frequency *ω* is very close to the eigenfrequency of a leaky mode TM_{ml} (typically, - 0.4 <(*ω*_{ml} – *ω*).*nr*_{0}/c < 0.4). In this on-resonance case, we ignore the contribution from all other leaky modes in the group *m*. We consider *b*_{m} equal to (*b*_{ml} + *b*_{m,l + 1})/2 if *ω* is not very close to any eigenfrequency, where the subscripts *ml* and *m,l + 1* refer to the two leaky modes nearest to *ω*. Figure 5(a)
-5(b) indicates that the scattering coefficient *b*_{m} constructed from *b*_{ml} nicely matches the scattering coefficient calculated from Mie formalism as $\left[n{J}_{m}^{\text{'}}(nk{r}_{0}){J}_{m}(k{r}_{0})-{J}_{m}^{}(nk{r}_{0}){J}_{m}^{\text{'}}(k{r}_{0})\right]/\left[{J}_{m}^{}(nk{r}_{0}){H}_{m}^{\text{'}}(k{r}_{0})-n{J}_{m}^{\text{'}}(nk{r}_{0}){H}_{m}^{}(k{r}_{0})\right]$ [2]. We can also find the scattering efficiency *Q*_{sca} calculated using our model shows excellent consistence with the result calculated using the Mie formalism (Fig. 5(c)). Additionally, we calculated the scattering coefficient for 0D spheres, and again found it matching the results calculated using the Mie formalism (Fig. 6(a)
-6(b)).

The correlation of scattering coefficients with the eigenvalue of leaky modes can provide new intuitive insights into the light scattering and the phase shift in the scattered light that cannot be obtained from the existing analytical and numerical models. For instance, it suggests that the upper limit for the scattering efficiency of a circular cylinder in free space is *4n*. From the above analysis, we can find that the collective contribution to the scattering efficiency from all the modes with the same mode number *m* and different order number *l* is −2*Re*(*b*_{ml})/*kr*_{0} or -*Re*(*b*_{ml} + *b*_{m,l + 1})/*kr*_{0}, which is no larger than 2/*kr*_{0}. The overall scattering efficiency *Q*_{sca} is just a simple add-up of the contribution from the modes with different mode number *m*. As illustrated in Fig. 4, for an arbitrary incident frequency *ω*, the mode number *m* that can be involved in the light scattering bear a simple relationship with the normalized parameter of the scattering system, *m* ≤ *nkr*_{0}. By considering the dual degeneracy of typical leaky modes (even and odd modes are degenerate), we can find out the number of leaky modes that can be involved into light scattering is no larger than 2*nkr*_{0}. Therefore, the scattering efficiency *Q*_{sca} is no larger than *4n*.

The correlation with leaky modes also provides important insights into the phase shift of the scattered light. We use the forward and backward scattering of a plane wave normally impinged on a circular cylinder as an example to illustrate this notion. The phase shift is defined as the difference between the phase of scattered light and that of the light passing through without the dielectric structure (for forward scattering) or that of the light reflected from a perfect mirror located at the same position as the dielectric structure (for backward scattering). Our analysis indicates that the phase shift in the forward scattered light can only cover half of the phase space [0, 2π], but backward scattering can provide a full phase coverage that are necessary for the manipulation of phase. The scattered field of an arbitrary incident frequency *ω* can be written as ${E}_{sca}(\omega )={E}_{0}[{b}_{0}(\omega ){H}_{0}^{(1)}(kr)+2{\displaystyle \sum _{m}{b}_{m}(\omega ){(-i)}^{m}{H}_{m}^{(1)}(kr){e}^{im\varphi}}]$, where the prefactor 2 of the second term is due to the dual degeneracy of leaky modes with mode number *m* > 0. For the scattered light at far field (*kr* >>1), we can use the asymptotic approximation of the Hankel function (${H}_{m}^{(1)}(kr)\approx \sqrt{2/\pi kr}{e}^{i(kr-\pi /4-m\pi /2)}$). The scattered field can thus be written as${E}_{0}\sqrt{2/\pi kr}B{e}^{i(kr+{\theta}_{s}-\pi /4)}$, where *e ^{ikr}* indicates the optical path of the light passing without scattering (for forward scattering) or reflected from a perfect mirror (for backward scattering),

*B*is a real number, and

*θ*

_{S}- π/4 is the phase shift in the scattered light.

*B*and

*θ*

_{S}are determined by the sum of scattering coefficients as $B{e}^{i{\theta}_{s}}={b}_{0}(\omega )+2{\displaystyle \sum _{m}{b}_{m}(\omega )}$ and $B{e}^{i{\theta}_{s}}={b}_{0}(\omega )+2{\displaystyle \sum _{m}{b}_{m}(\omega )}{(-1)}^{m}$ for forward and backward scatterings, respectively. According to Eq. (32), the real part of

*b*

_{ml}can never be larger than 0. Therefore, the real part of the sum ${b}_{0}(\omega )+2{\displaystyle \sum _{m}{b}_{m}(\omega )}$ in the forward scattering can never be larger than 0, which limits the resulting phase

*θ*

_{S}in a range of [π/2, 3π/2]. But backward scattering can provide the phase shifts of [0, 2π] due to the involvement of the term (−1)

*. While this notion is illustrated with cylinders, the underlying rationale can generally apply to other one-dimensional (1D) structures with non-circular cross-section or zero-dimensional (0D) dielectric structures. Therefore, this conclusion on the phase coverage of backward and forward scatterings generally holds for 1D and 0D dielectric structures with arbitrary shapes.*

^{m}## 4. Conclusions

This new theoretical analysis evaluates the light scattering at dielectric structures from a perspective of mode coupling. It demonstrates a fundamental correlation of the light scattering with the eigenvalue of the leaky modes in dielectric structures. This correlation can offer new intuitive insights into the light scattering and the phase shift of the scattered light that cannot be obtained from existing analytical and numerical models. It may open up a new door for engineering the light-matter interaction at subwavelength dielectric structures for the development of novel functionality, such as scattering inversion, super-resolution imaging, optical cloaking, space surveillance, and beam steering.

## Acknowledgments

This work has been supported by start-up fund from North Carolina State University. L.C. acknowledges a Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities.

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