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
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 , optical analogue of electromagnetic induced transparency [5, 6], super-resolution imaging [7–9], cloaking , 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 Ez can be generally written as,Equations (1) and (2) can be re-written in terms of odd and even modes,
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 as2, 17]. Because both and 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 haveEqs. (23) and (24) to get the expression for TE leaky modes in spheresEqs. (25) and (26) as:
Solving Eqs. (17), (18), (27), and (28) can give complex values for a normalized parameter nkr0 (nkr0 = Nreal - Nimagi). These complex values are eigenvalues of the leaky modes. We have previously calculated and tabulated the eigenvalue for typical leaky modes . The real part of the eigenvalue Nreal indicates the condition for the resonance with leaky modes, and the imaginary part Nimag 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 TMml. The leaky mode in cylinders is characterized by an azimuthal mode number, m, and a radial order number, l . By assuming no intrinsic absorption, we can describe the coupling process using formalism similar to conventional temporal coupled mode theory [14, 18–21],Eqs. (29) and (30), we can get A = (2γml)0.5 and .
We can derive the scattering coefficient that correlates the scattered wave with incident light from a reflection coefficient . For an arbitrary incident frequency ω,Fig. 3, we can have the incident and scattered wave as and . The scattering coefficient bml can be written asEquation (32) indicates that the scattering coefficient contributed by a single leaky mode depends on two parameters, θml and Δml. According to Eq. (31), Δml arises from the deviation of incident frequency ω from the resonant frequency ωml, hence referred as off-resonance offset, and can be calculated fromEquation (33) can be solved out using the eigenvalue of the leaky mode, Nreal - Nimagi. The eigenfrequency ωml and radiative decay rate γml are related with the real and imaginary parts of the eigenvalue in straightforward ways as ωml = c. Nreal/(n.r0), and γml = c. Nimag/(n.r0), c is speed of light.
We find that θml can be derived from the eigenvalue of the leaky mode as well. By assuming no incoming wave ( = 0), we can derive from Eq. (30). With the absence of incoming waves, and aml 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. is the amplitude of the eigenfield, and can be correlated to aml using a translation between “power picture” and “energy picture” as, and. Additionally, with the absence of incoming waves, the outgoing wave is solely contributed by the radiative decay of the energy stored inside the cylinder as. and can also be connected through the boundary condition at the cylinder/environment interface as . By considering all these relations, we can haveTable 1 and Table 2 , respectively. The listed number is in a unit of π.
The coefficient of the light scattering that involves multiple leaky modes can be constructed from the single-mode coefficient bml 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 Qsca can be obtained by simply adding up the contribution from the leaky modes with different m as (note: bm defined here has an opposite sign as the scattering coefficient defined in Mie Theory ), where Re means the real part. The scattering efficiency bm 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 . 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 ωml is relatively close to the incident frequency ω (- π <(ωml – ω).nr0/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 (ωml – ω) . Therefore, we can construct the scattering coefficient bm of the group m with an easier way. For an arbitrary incident frequency ω, we consider bm equal to one single-mode coefficient bml if the incident frequency ω is very close to the eigenfrequency of a leaky mode TMml (typically, - 0.4 <(ωml – ω).nr0/c < 0.4). In this on-resonance case, we ignore the contribution from all other leaky modes in the group m. We consider bm equal to (bml + bm,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 bm constructed from bml nicely matches the scattering coefficient calculated from Mie formalism as . We can also find the scattering efficiency Qsca 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 −2Re(bml)/kr0 or -Re(bml + bm,l + 1)/kr0, which is no larger than 2/kr0. The overall scattering efficiency Qsca 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 ≤ nkr0. 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 2nkr0. Therefore, the scattering efficiency Qsca 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 , 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 (). The scattered field can thus be written as, where eikr 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 θS - π/4 is the phase shift in the scattered light. B and θS are determined by the sum of scattering coefficients as and for forward and backward scatterings, respectively. According to Eq. (32), the real part of bml can never be larger than 0. Therefore, the real part of the sum in the forward scattering can never be larger than 0, which limits the resulting phase θS 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)m. 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.
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.
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.
References and links
1. P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).
2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
3. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
4. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]
5. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]
7. S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3(7), 388–394 (2009). [CrossRef]
11. W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
13. L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009). [CrossRef] [PubMed]
16. O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009). [CrossRef] [PubMed]
17. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1998).
18. R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007). [CrossRef]
19. Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010). [CrossRef]
21. H. A. Haus, Wave and Fields in Optoelectronics (Prentice-Hall, 1984).
22. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).