## Plasmonic angular momentum on metal-dielectric nano-wedges in a sectorial indefinite metamaterial |

Optics Express, Vol. 21, Issue 23, pp. 28344-28358 (2013)

http://dx.doi.org/10.1364/OE.21.028344

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### Abstract

We present an analytical study in the structure-modulated plasmonic angular momentum, which is trapped in the core region of a sectorial indefinite metamaterial. This metamaterial consists of periodically arranged metal-dielectric nano-wedges along the azimuthal direction. Employing a transfer-matrix calculation and a conformal-mapping technique, our theory can deal with an arbitrary number of wedges with realistically rounded tips. We demonstrate that in the deep-subwavelength regime, strong electric fields that carry large azimuthal variations can exist only within ten-nanometer length scale around the structural center. They are naturally bounded by a characteristic radius on the order of a hundred nanometers from the center. These extreme confining properties suggest that the structure under investigation can be superior to the conventional metal-dielectric cavities in terms of nanoscale photonic manipulation.

© 2013 Optical Society of America

## 1. Introduction

1. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. **90**, 077405 (2003). [CrossRef] [PubMed]

4. X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics **6**, 450–454 (2012). [CrossRef]

2. I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. **105**, 067402 (2010). [CrossRef] [PubMed]

5. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science **336**, 205–209 (2012). [CrossRef] [PubMed]

6. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science **308**, 534–537 (2005). [CrossRef] [PubMed]

9. J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. **11**, 931–934 (2009). [CrossRef]

5. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science **336**, 205–209 (2012). [CrossRef] [PubMed]

*ϕ*-direction, and uniformly extended in both the radial

*r*-direction and axial

*z*-direction [9

9. J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. **11**, 931–934 (2009). [CrossRef]

10. J. Li, L. Thylen, A. Bratkovsky, S.-Y. Wang, and R. S. Williams, “Optical magnetic plasma in artificial flowers,” Opt. Express **17**, 10800–10805 (2009). [CrossRef] [PubMed]

*γ*

_{1}for medium 1 or

*γ*

_{2}for medium 2. The angular periodicity of one primitive unit (composed of an adjacent pair of medium 1 and medium 2) is

*γ*=

*γ*

_{1}+

*γ*

_{2}, and the total number of units is

*N*= 2

*π/γ*. This structure was once proposed by Jacob

*et al.*[7

7. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

*N*→ ∞,

*γ*→ 0, and letting the axial wavenumber

*k*→ 0). In our work, we focus on the intrinsic plasmonic edge modes [11

_{z}11. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B **6**, 3810–3815 (1972). [CrossRef]

14. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. **100**, 023901 (2008). [CrossRef] [PubMed]

15. A. Ferrando, “Discrete-symmetry vortices as angular Bloch modes,” Phys. Rev. E **72**, 036612 (2005). [CrossRef]

16. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

18. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. **96**, 163905 (2006). [CrossRef] [PubMed]

19. H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. **10**, 529–536 (2010). [CrossRef] [PubMed]

20. Z. Shen, Z. J. Hu, G. H. Yuan, C. J. Min, H. Fang, and X.-C. Yuan, “Visualizing orbital angular momentum of plasmonic vortices,” Opt. Lett. **37**, 4627–4629 (2012). [CrossRef] [PubMed]

*ω*and a fixed axial wavenumber

*k*, higher-angular-momentum modes tend to oscillate more drastically and distribute more widely in the radial direction from the structural center. Nevertheless, there always exists a characteristic bounding radius that naturally encapsulates all the field intensity into a region of just a hundred nanometers. In comparison, metal-dielectric circular cavities from conventional designs are incapable of confining this high photonic or plasmonic angular momentum in such small length scale, due to the geometrical and physical restrictions [21

_{z}21. C. Yeh and F. Shimabukuro, *The Essence of Dielectric Waveguides* (Springer, 2008). [CrossRef]

23. Q. Li and M. Qiu, “Plasmonic wave propagation in silver nanowires: guiding modes or not?” Opt. Express **21**, 8587–8595 (2013). [CrossRef] [PubMed]

24. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**, 053002 (2006). [CrossRef] [PubMed]

25. V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A **69**, 013812 (2004). [CrossRef]

## 2. General formalism

21. C. Yeh and F. Shimabukuro, *The Essence of Dielectric Waveguides* (Springer, 2008). [CrossRef]

*z*-axis but bounded in the transverse direction in the

*rϕ*-plane. (We do not consider problems of radially outgoing or incoming waves emitted from or scattered by this structure.) Employing a more convenient representation, we may fully describe the problem with two scalar potentials, Φ

_{E}and Φ

_{H}, instead of the more familiar

**and**

*E***fields. Φ**

*H*_{E}stands for the

*E*-waves (or transverse-magnetic waves), and Φ

_{z}_{H}stands for the

*H*-waves (or transverse-electric waves). They both satisfy the two-dimensional scalar Helmholtz equation, The general eigenmodes in sectorial structures are necessarily

_{z}*E*-

_{z}*H*-hybridized modes. So the Φ

_{z}_{E}-Φ

_{H}-combined electric and magnetic fields can be generated via which contain both the longitudinal and transverse components with respect to the unit vector

**e**

*. The boundary conditions across the wedge interfaces are the continuities of*

_{z}*E*,

_{z}*E*,

_{r}*εE*,

_{ϕ}*H*,

_{z}*H*, and

_{r}*μH*.

_{ϕ}*κ*must be real-valued (neglecting dissipation) in both media [12

_{r}12. A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B **24**, 5703–5712 (1981). [CrossRef]

26. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**, 539–554 (1969). [CrossRef]

*k*to lie outside the light cone of the dielectric according to Eq. (1). It turns out that the index

_{z}*ς*would have to be real-valued as well, to support the unique plasmonic edge modes [11

11. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B **6**, 3810–3815 (1972). [CrossRef]

12. A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B **24**, 5703–5712 (1981). [CrossRef]

*κ*,

_{r}*ς*,

*k*}, the scalar potentials Φ

_{z}_{E}and Φ

_{H}in a specific sector take the forms of in which we have omitted the time-harmonic factor e

^{−i}

*.*

^{ωt}*A*,

_{ς}*B*,

_{ς}*C*and

_{ς}*D*are all undetermined coefficients. K

_{ς}*is the*

_{ν}*ν*th-order modified Bessel function of the second kind. This type of Bessel function guarantees convergence as

*r*→ ∞ for arbitrarily complex-valued orders,

*ν*= i

*ς*, and arguments,

*κ*= −i

_{r}r*k*[28]. In Fig. 2, we plot K

_{r}r_{i}

*(*

_{ς}*κ*) for real-valued

_{r}r*ς*in both the

*κ*scale and ln(

_{r}r*κ*) scale. This function exhibits source-free indefinite oscillations at small argument,

_{r}r*κ*→ 0 or ln(

_{r}r*κ*) → −∞, but evanescent decay at large argument. This behavior is completely different from the function K

_{r}r*(*

_{ν}*κ*) for real-valued

_{r}r*ν*, in which case K

*(*

_{ν}*κ*) undergoes straight exponential decay (from potentially a line source at

_{r}r*r*= 0) [28]. If measured in terms of the coordinate

*r*, the oscillating and decaying regions of K

_{i}

*(*

_{ς}*κ*) are separated approximately by

_{r}r*b*defines a natural bounding radius. The waves are standing in the region

*r*≲

*b*and only weakly penetrating into the region

*r*≳

*b*. No matter how large

*b*is, these waves are always radially bounded (non-radiative), even though the material itself is radially unbounded. As we shall discuss in detail below (particularly in Sec. 4), this is a hallmark of the plasmonic edge modes at deep axial subwavelength on metal-dielectric wedges [11

11. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B **6**, 3810–3815 (1972). [CrossRef]

14. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. **100**, 023901 (2008). [CrossRef] [PubMed]

*k*in the systems containing sharp wedges is mathematically challenging. Rigorous derivation requires the complicated Kontorovich-Lebedev integral transform over the index

_{z}*ς*[29

29. A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. Lond. A **455**, 2655–2686 (1999). [CrossRef]

*κ*in both medium 1 and medium 2, In this scenario, the system is non-retarded in the

_{r}*rϕ*-plane and the boundary connection is greatly simplified. The terms explicitly carrying

*ω/ck*in Eqs. (3) and (4) can be dropped, leading to the decoupled electrostatic modes with vanishing magnetic fields, and magnetostatic modes with vanishing electric fields, The basic solutions of Φ

_{z}_{E}and Φ

_{H}in a specific sector are unchanged from Eqs. (5) and (6), except the asymptotic relation Eq. (8) replacing Eq. (1).

## 3. Spectral analysis

*ϕ*-direction. The transfer matrix traversing one angular unit can be derived as

15. A. Ferrando, “Discrete-symmetry vortices as angular Bloch modes,” Phys. Rev. E **72**, 036612 (2005). [CrossRef]

*γ*= 2

*π/N*, where

*γ*=

*γ*

_{1}+

*γ*

_{2}is the angular periodicity, and

*N*is the total number of units. The azimuthal wavenumber

*l*denotes the conserved structure-modulated angular momentum about the

_{z}*z*-axis. Its upper limit is at the boundary of the first angular Brillouin zone ±

*N*/2 determined by material design. In the continuous limit

*N*→ ∞,

*γ*→ 0,

*l*approaches the

_{z}*z*-component of the original angular momentum of plasmon polaritons, and can take however large values in this structure. The conserved angular momentum flows along the

*z*-axis through the system.

*π*circle here), If we perform a series expansion to

*l*,

_{z}γ*ςγ*

_{1}and

*ςγ*

_{2}in Eq. (13) under the continuous limit, we can find a quite appealing result, The effective permittivities from the effective medium theory automatically show up [2

2. I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. **105**, 067402 (2010). [CrossRef] [PubMed]

7. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

*η*

_{1}≡

*γ*

_{1}/

*γ*and

*η*

_{2}≡

*γ*

_{2}/

*γ*are the filling ratios. Clearly, if

*ε̃*and

_{r}*ε̃*are of opposite signs, then Eq. (14) exhibits the indefinite signature of this metamaterial, which possesses singular density of states on iso-frequency surfaces [2

_{ϕ}2. I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. **105**, 067402 (2010). [CrossRef] [PubMed]

5. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science **336**, 205–209 (2012). [CrossRef] [PubMed]

*ω*

^{2}/

*c*

^{2}because of the non-retarded regime (equivalently the

*c*→ ∞ limit) that we have chosen; however, the nontrivial frequency dependence is still implicitly enclosed in

*ε̃*and

_{r}*ε̃*.

_{ϕ}*ω*(

*ς*,

*l*) versus

_{z}*l*for several fixed values of

_{z}*ς*based on the actual medium theory, Eq. (13), and the effective medium theory, Eq. (14), respectively.

*ς*resembles a band index in the band theory of electrons in solids. For the actual spectrum shown in Fig. 4(a), we choose

*N*= 24,

*γ*=

*π*/12, as an example. The structure-modulated angular momentum

*l*is limited within the 1st Brillouin zone |

_{z}*l*| <

_{z}*π/γ*=

*N*/2 = 12. For the effective spectrum shown in Fig. 4(b), these restrictions are irrelevant (or strictly speaking,

*N*= ∞,

*γ*= 0); but for the sake of comparison, we only draw |

*l*| <

_{z}*π/γ*= 12 in the same angular momentum range. One should keep in mind that

*l*can only take discrete integers according to Eq. (12). Thus the continuous curves in the graphs just serve as visual guides for the grey dots. For every given

_{z}*ς*, there are always a lower-energy

*ω*(

*ς*,

*l*) curve and a higher-energy

_{z}*ω*(

*ς*,

*l*) curve in the two regions

_{z}*ω*<

*ω*

_{sp}and

*ω*

_{sp}<

*ω*<

*ω*

_{ϕo}respectively. The main qualitative difference between Figs. 4(a) and 4(b) is that (b) shows an energy gap in the frequency range

*ω*

_{ro}<

*ω*<

*ω*

_{ϕ∞}for however large

*ς*, whereas (a) permits the large-

*ς*curves to penetrate into the gap region and approach the

*ω*=

*ω*

_{sp}line from two sides. The reason for the existence of an energy gap in the effective medium theory can be intuitively grasped from Fig. 3, in which the frequency range

*ω*

_{ro}<

*ω*<

*ω*

_{ϕ∞}(and also

*ω*>

*ω*

_{ϕo}) embodies both positive

*ε̃*and positive

_{r}*ε̃*. This enforces

_{ϕ}*ς*to be imaginary-valued (refer to Eq. (14)), therefore eliminates the plasmonic edge modes in our structure. However, the actual medium theory from Eq. (13) implies that this picture is only approximately correct. If the band index

*ς*that controls the azimuthal confinement and the radial oscillation (for a given

*κ*) is extraordinarily large, the effective medium theory naturally breaks down. The surface plasmons on different wedge interfaces fully decouple from each other and converge independently towards the same extreme short-wavelength limit

_{r}*ω*=

*ω*

_{sp}. Furthermore, even for the small-

*ς*curves, Figs. 4(a) and 4(b) still show prominent frequency difference when

*l*approaches the Brillouin zone boundary. Only in the region where both

_{z}*ς*and

*l*are small, the two formalisms agree well with each other. All the curves in this region pass through either the (

_{z}*l*= 0,

_{z}*ω*=

*ω*

_{ro}) point or the (

*l*= 0,

_{z}*ω*=

*ω*

_{ϕ∞}) point, and are very insensitive to the value of

*ς*. This property can be deduced from Eq. (14) supposing either

*ε̃*≃ 0 or

_{r}*ε̃*≃ ∞, which is an index-near-zero (INZ) or an index-near-infinity (INI) behavior.

_{ϕ}*ω*(

*ς*,

*l*) is independent of the axial wavenumber

_{z}*k*, which seems unusual from the perspective of waveguide theory. This is a result of both the deep-subwavelength limit and the perfect wedge tips that we have assumed at

_{z}*r*= 0. As demonstrated below, once we introduce rounded wedges, even if we still keep the deep-subwavelength limit,

*ω*(

*ς*,

*l*) will become

_{z}*k*-dependent.

_{z}## 4. Rounded wedges

*r*= 0. This can be inferred from the asymptotic behavior of K

_{i}

*(*

_{ς}*κ*) at small argument (see Fig. 2) [28,33

_{r}r33. L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B **14**, 5523–5525 (1976). [CrossRef]

*κ*as

_{r}r*r*→ 0 (refer to Eq. (9)). These singular behaviors are due to the infinite charge accumulation at the infinitely sharp tips. While the strong field enhancement at the structural center is physical and is favorable for nanophotonics, the mathematical artifacts must be removed from the theory. Practically, the wedges are always rounded and can never seamlessly touch each other under fabrication. To make our theoretical study match better with the reality, we adopt a conformal coordinate mapping to conveniently achieve the rounded and gapped configurations. The divergence and indefinite oscillations can then be automatically removed. More subtle physics around the wedge tips can be revealed [12

12. A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B **24**, 5703–5712 (1981). [CrossRef]

33. L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B **14**, 5523–5525 (1976). [CrossRef]

*x*,

*y*)-coordinates, and again omit the phase factor e

^{ikzz}e

^{−i}

*, First, we define two sets of (dimensionless) complex coordinates*

^{ωt}*w*= (

*x*+i

*y*)/

*a*and

*s*=

*u*+i

*v*, where

*a*is for now a characteristic length parameter that cancels the dimension of

*x*and

*y*. Next, we connect the two coordinate systems by a conformal mapping

*w*= Λ(

*s*), where Λ is an analytical function. Thus the Helmholtz equation in the (

*u*,

*v*)-coordinates reads We introduce the following conformal mapping for any desired number of units

*N*= 1, 2, 3,..., The new (

*u*,

*v*)-coordinate system constitutes a generalized elliptic cylinder coordinate system, with

*N*sectors partitioned by

*N*−1 branch cuts. The index

*n*denotes which sector a (

*x*,

*y*)-point maps in. The length parameter

*a*is the semi-focal length measured in the old (

*x*,

*y*)-coordinate system. Far from the coordinate center (

*u*≫ 1), we have

*u*resembles the logarithm of the radial coordinate

*r*, while

*v*(together with

*n*) resembles the azimuthal coordinate

*ϕ*, in spite of some differences in the proportionality constants. Figure 5 shows the new (

*u*,

*v*)-coordinate lines for

*N*= 1 to 6. As can be seen, the conformal coordinate mapping automatically preserves the orthogonality. The shaded areas in Fig. 5 are to be filled with metal at the same filling ratio

*n*= 0, 1, 2,...,

*N*− 1. All the metal wedges near the center are naturally rounded following the coordinate lines of

*u*. The unshaded areas in Fig. 5 are to be filled with dielectric. These configurations nicely imitate the actual structures produced through nanofabrication, and are much more realistic than the one illustrated in Fig. 1.

*N*= 1 and

*N*= 2, the partial differential equation is separable, which has solutions in the form of Mathieu functions [34]. But for

*N*≥ 3, the equation is only approximately separable. For instance, in the region near the wedge tips

*u*≃ 0,

*v*≃ 0 [33

33. L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B **14**, 5523–5525 (1976). [CrossRef]

_{E}(

*u*,

*v*) ≡

*U*(

*u*)

*V*(

*v*) with an eigenvalue

*u*≫ 1 the index

*ς*introduced in this way is identical to the

*ς*that we used earlier.

*u*∈ [0, +∞) with the discrete eigenvalues denoted by an integer

*m*[28, 33

**14**, 5523–5525 (1976). [CrossRef]

*ℋ*is the

_{m}*m*th-order Hermite polynomial. For clarity, in Fig. 6 we plot the normalized Hermite function (taking 2

*k*= 1 in Eq. (27)) for a few integer values of

_{z}a/N*m*. Apparently, the lower the order

*m*is, the more trapped the plasmonic edge modes lie around the metal wedge tips

*u*≃ 0. Interestingly, Figs. 6 and 2(b) look quite alike each other, if we note the relation

*u*∼ ln(

*r/a*) from Eq. (21). Indeed,

*ς*(or simply

_{m}*m*) controls the (finite) number of radial oscillation and the bounding radius around the structural center, just as what

*ς*does in the far region. They asymptotically merge with each other.

**14**, 5523–5525 (1976). [CrossRef]

*v*is small. Applying the continuity conditions of

*V*(

*v*) and

*ε∂*(

_{v}V*v*) across the metal-dielectric interfaces at

*ω*(

*k*,

_{z}a*m*,

*l*) after considering the rounded wedges is discretized by

_{z}*m*and dependent on

*k*through Eq. (28).

_{z}a*l*is still the structure-modulated angular momentum.

_{z}*ω*(

*k*,

_{z}a*m*,

*l*) as a function of

_{z}*k*for several given values of

_{z}a*m*and

*l*. The material model used for

_{z}*ε*

_{1}of silver and

*ε*

_{2}of silicon dioxide is the same as before. In the non-retarded regime, the light speed

*c*does not enter our theory and so the semi-focal length

*a*becomes the only length parameter of the system. At this stage, we choose

*a*= 10 nm in accordance with the typical linewidth of today’s nano-lithography. The dispersion curves from our calculation intersect the light line, but in reality they may bend more quickly to zero before touching the light line [12

**24**, 5703–5712 (1981). [CrossRef]

26. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**, 539–554 (1969). [CrossRef]

**24**, 5703–5712 (1981). [CrossRef]

*k*and

_{z}a*l*, the larger the

_{z}*m*is, the more radial oscillations there are, and the higher the needed frequency

*ω*is. In general, larger-

*m*curves appear flatter, meaning a less sensitive

*k*-dependence. This is qualitatively consistent with the vanishing

_{z}a*k*-dependence in the

_{z}*ω*(

*ς*,

*l*) curves that we have obtained earlier without considering the tip effect. In the extreme case

_{z}*m*→ ∞ here (by analogy with the

*ς*→ ∞ case in Fig. 4), all the dispersion curves are pushed towards the line of surface plasma frequency

*ω*

_{sp}. To be cautious, we should remember that the quadratic expansion adopted in Eq. (24) is quantitatively correct only at a relatively small

*m*and in the near-tip regions

*u*≃ 0. For a large

*m*and

*u*≫ 1 far regions, the more accurate description is the continuous

*ς*description that we have elaborated in the prior section; the small rounded wedges tips should not induce a sizable impact there.

*N*, a higher angular momentum

*l*tends to drag all the curves downwards and so permits lower-frequency excitation of these modes. Likewise, we have also found that, for a fixed

_{z}*l*, a larger

_{z}*N*drags all the curves downwards and so permits lower-frequency excitation too. However, the high-

*l*and large-

_{z}*N*modes in our structure fall in the category of “dark” modes according to literature [35

35. H. Benisty, “Dark modes, slow modes, and coupling in multimode systems,” J. Opt. Soc. Am. B **26**, 718–724 (2009). [CrossRef]

36. F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. **82**, 209–275 (2010). [CrossRef]

## 5. Field profiles

*N*and angular momentum

*l*, we can obtain the coefficients

_{z}*A*and

_{ς}*B*in Eq. (5) or

_{ς}*A*and

_{m}*B*in Eq. (29) in all sectors. We are then able to plot the field profiles, say, on the

_{m}*z*= 0 plane, for any wanted eigenmodes, without resorting to the effective medium theory and numerical simulation. These approaches can be invalid or inaccurate for structures with a small number of sectors and sharp wedge tips.

*a*in our conformal mapping physically designates a finite gap size and tip radius of the rounded wedges, hence settles a cutoff length in our problem. For whatever specified

*ω*and

*k*, the field strength cannot oscillate at an arbitrarily small spacing and must remain finite everywhere. As we have proved, the profiles of potential field Φ

_{z}a_{E}can be approximated by the

*m*th order Hermite polynomials close to the wedge tips; on the other hand, they asymptotically approach the i

*ς*th imaginary-order modified Bessel functions away from the tips. Let us first look at the profiles around the tips for some low-

*m*modes. If we choose

*N*= 4,

*a*= 10 nm, and

*ω*= 3.54 × 10

^{15}s

^{−1}(532 nm free-space wavelength), then according to Fig. 7 we realize that only the

*m*= 0 curves have an intersection point with

*ω*= 3.54 × 10

^{15}s

^{−1}at deep subwavelength, where

*k*= 0.47 for

_{z}a*l*= 1, and

_{z}*k*= 0.77 for

_{z}a*l*= 2. Figure 8 displays the corresponding potential field profiles. The

_{z}*l*= 1 angular-momentum mode is of the cylindrical dipolar type with one sign change in the 2

_{z}*π*azimuthal circle, and the

*l*= 2 angular-momentum mode is of the cylindrical quadrupolar type with two sign changes. The electric field

_{z}**= −∇Φ**

*E*_{E}as the derivative field of Φ

_{E}gains an enormous strength within the nanoscale tip regions.

*r*> 5

*a*, i.e., if

*a*≃ 10 nm,

*r*≳ 50 nm). Figure 9 displays the

*N*= 6 and 24 two cases at several allowed values of

*l*. In the far-from-tip region and deep-subwavelength regime, the eigen-spectrum and field profiles no longer depend on

_{z}*k*or

_{z}*a*. Therefore, these plots are universal for any given

*k*. We know from Eq. (13) that for the same

_{z}*ω*, an increasing

*l*increases the index

_{z}*ς*. Then if

*k*is fixed as well, a larger

_{z}*ς*produces a larger bounding radius

*b*=

*ς/k*. In agreement with our earlier argument, although the fields prefer to localize near the structural center for a smaller

_{z}*ς*, they oscillate more drastically and spread more widely from the center for a larger

*ς*. But in any event, beyond the bounding radius, the fields always quickly fade away. For a realistic material design using today’s nano-lithography,

*N*perhaps cannot yet reach 24, like what we have demonstrated. If we take the sub-wavelength

*k*to be of the order of 0.1 nm

_{z}^{−1}, and insert the calculated

*ς*as labeled in Fig. 9,

*b*is at most of the order of 100 nm. This means high plasmonic angular momentum can indeed be tightly trapped in an impressively narrow region in this structure.

21. C. Yeh and F. Shimabukuro, *The Essence of Dielectric Waveguides* (Springer, 2008). [CrossRef]

23. Q. Li and M. Qiu, “Plasmonic wave propagation in silver nanowires: guiding modes or not?” Opt. Express **21**, 8587–8595 (2013). [CrossRef] [PubMed]

*ν*[28]. These functions either strongly converge to zero inside a diffraction-limited core, or have to be supported by a line source at

*r*= 0, which does not represent the intrinsic modes of the systems [7

7. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

## 6. Conclusion

**336**, 205–209 (2012). [CrossRef] [PubMed]

24. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**, 053002 (2006). [CrossRef] [PubMed]

25. V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A **69**, 013812 (2004). [CrossRef]

## Acknowledgments

## References and links

1. | D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. |

2. | I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. |

3. | J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. |

4. | X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics |

5. | H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science |

6. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science |

7. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express |

8. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

9. | J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. |

10. | J. Li, L. Thylen, A. Bratkovsky, S.-Y. Wang, and R. S. Williams, “Optical magnetic plasma in artificial flowers,” Opt. Express |

11. | L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B |

12. | A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B |

13. | R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B |

14. | E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. |

15. | A. Ferrando, “Discrete-symmetry vortices as angular Bloch modes,” Phys. Rev. E |

16. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

17. | L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science |

18. | L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. |

19. | H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. |

20. | Z. Shen, Z. J. Hu, G. H. Yuan, C. J. Min, H. Fang, and X.-C. Yuan, “Visualizing orbital angular momentum of plasmonic vortices,” Opt. Lett. |

21. | C. Yeh and F. Shimabukuro, |

22. | Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. |

23. | Q. Li and M. Qiu, “Plasmonic wave propagation in silver nanowires: guiding modes or not?” Opt. Express |

24. | D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. |

25. | V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A |

26. | E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. |

27. | J. D. Jackson, |

28. | M. Abramowitz and I. A. Stegun, |

29. | A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. Lond. A |

30. | N. W. Ashcroft and N. D. Mermin, |

31. | P. G. Kik, S. A. Maier, and H. A. Atwater, “Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources,” Phys. Rev. B |

32. | Y. Ma, X. Li, H. Yu, L. Tong, Y. Gu, and Q. Gong, “Direct measurement of propagation losses in silver nanowires,” Opt. Lett. |

33. | L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B |

34. | N. W. McLachlan, |

35. | H. Benisty, “Dark modes, slow modes, and coupling in multimode systems,” J. Opt. Soc. Am. B |

36. | F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(160.3918) Materials : Metamaterials

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: September 9, 2013

Revised Manuscript: October 28, 2013

Manuscript Accepted: October 28, 2013

Published: November 11, 2013

**Citation**

Dafei Jin and Nicholas X. Fang, "Plasmonic angular momentum on metal-dielectric nano-wedges in a sectorial indefinite metamaterial," Opt. Express **21**, 28344-28358 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28344

Sort: Year | Journal | Reset

### References

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