Light focusing on a stack of metal-insulator-metal waveguides sharp edge

doi:10.1364/OE.17.013615

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Light focusing on a stack of metal-insulator-metal waveguides sharp edge

W. M. Saj »View Author Affiliations

Optics Express, Vol. 17, Issue 16, pp. 13615-13623 (2009)
http://dx.doi.org/10.1364/OE.17.013615

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Abstract

Near field light focusing by two-dimensional isosceles triangle shaped stack of silver plasmon-polaritons waveguides is being investigated numerically with full-vectorial Finite Difference Time Domain method for H-polarized light and wavelength λ=500 nm. For wide angle of tip, results are in good agreement with theoretically predicted propagation constant of light in stack and while discrepancy becomes significant for smaller angle. Physical phenomena of refraction and interference, similar to ones in dielectric axicons lead to conversion of a Gaussian beam incident on the flat side of the stack into a narrow light jet behind the structure sharp edge. The beam is concentrated into long focal region of 0.37 λ width and enhancement of field amplitude is achieved in spite of significant absorption in the structure. The results are compared with bulk dielectric structure.

1. Introduction

Surface plasmon-polaritons, that is coupled light – electron oscillations at surfaces of noble metal structures are phenomena described by Maxwell’s and constitutive equations of XIX century origin and their theory was developing through the previous hundred years. Nevertheless, plasmon-polaritons based devices gained an increasing interest and research effort in recent years, due to both development of the technologies that made the fabrication of nanostructures possible as well as because of the increasing demand for computer chips operating at higher frequencies than the terahertz ones [1

S. A. Maier, Plasmonics. Fundamentals and Applications (Springer, Berlin 2007).

–3

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7–8), 20–27 (2006). [CrossRef]

Biological and chemical sensors are the branch of plasmonics applications that has been successfully explored [4

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008). [CrossRef] [PubMed]

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. 108(2), 494–521 (2008). [CrossRef] [PubMed]

]. Ability of plasmon devices to concentrate light within nanometer size spots with local field enhancement factors of orders much higher than 10² has brought a significant improvement to Surface Enhanced Raman Scattering (SERS) sensing sensibility [4

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008). [CrossRef] [PubMed]

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. 108(2), 494–521 (2008). [CrossRef] [PubMed]

However, there is a number of applications where strongly dependent on frequency light enhancement in extreme small volume achieved in nanoparticle devices is not optimal. Instead, a space spread light enhancement in a range of wavelengths is demanded e.g. for SERS particle flow sensing and energy harvesting in solar cells.

The other branch of plasmonics, guided plasmon wave devices, has focused attention with their ability to propagate light in channels of diameters significantly below the dielectric diffraction limit and promise to construct highly integrated optoelectronical devices of nanometer size [3

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7–8), 20–27 (2006). [CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006). [CrossRef]

]. The Metal-Insulator-Metal (MIM) waveguide (metal walls bounded dielectric layer) was the design extensively investigated both because of the structural simplicity and due to issues of modal structure, energy flow and refraction of light in single waveguides and stacks [6

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79(3), 035120 (2009). [CrossRef]

]. Also a variety of solutions employing plasmon waveguides for energy focusing has been presented in previous years [8

Z. Sun and H. K. Kim, “Refractive transmission of light and beam shaping with metallic nano-optic lenses,” Appl. Phys. Lett. 85(4), 642–644 (2004). [CrossRef]

–13

W. M. Saj, “Light focusing with tip formed array of plasmon-polariton waveguides,” Proc. SPIE 6641, 664120 (2007). [CrossRef]

Fig. 1. The general outlook of examined configuration and symbols used in this work. We consider interaction of H-polarized light with two dimensional, uniform in z direction stack with dielectric channel width d=50 nm, silver layer thickness p=50 nm and various angles α.

2. Plasmonic axicon

In this paper we investigate a two dimensional structure in form of isosceles triangle shaped MIM (silver-air-silver) waveguides stack illuminated by H-polarized light beam on its flat end to obtain a light jet beyond the sharp edge [13

W. M. Saj, “Light focusing with tip formed array of plasmon-polariton waveguides,” Proc. SPIE 6641, 664120 (2007). [CrossRef]

In comparison with structure from [9

H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express 13(18), 6815–6820 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6815. [CrossRef] [PubMed]

] which reproduces the Fresnel lens idea, the present work consists in single channel diameter waveguides. The recent proposal [10

R. Gordon, “Proposal for superfocusing at visible wavelengths using radiationless interference of a plasmonic array,” Phys. Rev. Lett. 102(20), 207402 (2009). [CrossRef] [PubMed]

], similar to the previous one, uses waveguides filled with various dielectrics to achieve strong surface wave coupling and near field focusing. The nanostructure investigated in [8

Z. Sun and H. K. Kim, “Refractive transmission of light and beam shaping with metallic nano-optic lenses,” Appl. Phys. Lett. 85(4), 642–644 (2004). [CrossRef]

] consists of a few waveguides and does not collect energy from a larger micrometer, size incident beam. Similar phenomena of light directing, as reported in this work, were found recently in simulations of triangle shaped finite dielectric photonic crystal [14

H. Kurt, “Limited-diffraction light propagation with axicon-shape photonic crystals,” J. Opt. Soc. Am. B 26(5), 981–986 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=josab-26-5-981. [CrossRef]

] and metamaterial tip [15

C. Rockstuhl, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Metamaterial nanotips,” Appl. Phys. Lett. 94(11), 113110 (2009). [CrossRef]

A form of the presented structure is similar to glass cone shaped axicons [16

J. H. McLeod, “The Axicon: A New Type of Optical Element,” J. Opt. Soc. Am. 44(8), 592 (1954), http://www.opticsinfobase.org/abstract.cfm?URI=josa-44-8-592. [CrossRef]

,17

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon - the Most Important Optical Element,” Opt. Photonics News 16, 34–39 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=OPN-16-4-34 [CrossRef]

]. Since its development in 1954 [15

C. Rockstuhl, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Metamaterial nanotips,” Appl. Phys. Lett. 94(11), 113110 (2009). [CrossRef]

] axicon lenses are broadly used in optical engineering due to their ability to concentrate light into the long line foci in the far field [17

] and to generate diffractionless Bessel-Gauss beams [18

R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-8-6-932. [CrossRef]

]. Recently microoptical axicon structures have been applied in near field sensing [19

Y.-J. Yu, H. Noh, M.-H. Hong, H.-R. Noh, Y. Arakawa, and W. Jhe, “Focusing characteristics of optical fiber axicon microlens for near-field spectroscopy: Dependence of tip apex angle,” Opt. Commun. 267(1), 264–270 (2006). [CrossRef]

,20

T. Grosjean, A. Fahys, M. Suarez, D. Charraut, R. Salut, and D. Courjon, “Annular nanoantenna on fibre micro-axicon,” J. Microsc. 229(2), 354–364 (2008). [CrossRef] [PubMed]

]. Nevertheless, most of theoretical descriptions of such structures (see e.g [21

A. E. Martirosyan, C. Altucci, C. de Lisio, A. Porzio, S. Solimeno, and V. Tosa, “Fringe pattern of the field diffracted by axicons,” J. Opt. Soc. Am. A 21(5), 770–776 (2004), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-5-770. [CrossRef]

,22

C. Zheng, Y. Zhang, and D. Zhao,” Calculation of the vectorial field distribution of an axicon illuminated by a linearly polarized Guassian beam,” Optik 117,3, 118–122 (2006). [CrossRef]

].) employ scalar or integral approximations, which take into account phase shift introduced to wavefronts by refraction, and deal with strong perturbation introduced by the axicon tip, neglecting more complicated mechanisms of light/structure interaction in the near field, e.g. excitation of surface waves.

Our result shows that while the light guidance through considered metallic structure is plasmon assisted, the output field shape is well explained through division of the input light by refraction at structure opposite sides into two counter refracted beams and their interference. This is a mechanism different from surface modes assisted directed emission from the slit (see e.g [23

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003). [CrossRef] [PubMed]

].) or nanojets formed by diffraction at vicinity of media interface [24

Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-7-1214. [CrossRef] [PubMed]

,25

A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-18-14200. [CrossRef] [PubMed]

]. For small tip angle the anisotropy of structure and surface waves excitation leads to decreasing diffraction in comparison to similar dielectric structures.

We expect that this two dimensional considerations are an insightful introduction to applying presented physical phenomena in three dimensional devices employing e.g. pyramidal shaped arrays of rectangular waveguides [26

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96(7), 073904 (2006). [CrossRef] [PubMed]

,27

T. T. Minh, K. Tanaka, and M. Tanaka, “Complex propagation constants of surface plasmon polariton rectangular waveguide by method of lines,” Opt. Express 16(13), 9378–9390 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9378. [CrossRef] [PubMed]

] and cone shaped coaxial multilayered fibers [28

M. S. Kushwaha and B. Djafari-Rouhani, “Plasma excitations in multicoaxial cables,” Phys. Rev. B 71(15), 153316 (2005). [CrossRef]

,29

F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74(20), 205419 (2006). [CrossRef]

3. Simulation details

The simulations of light interaction with structure as shown on Fig. 1 with various angles α are performed with home made FDTD [30

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, MA 2000).

] code. The computational area is 10 µm×8 µm large with space discretization step Δx=Δy=2 nm. Incident light is H-polarized Gaussian beam of 500 nm wavelength and with Full Width at Half Maximum (FWHM) of space intensity profile equals 2000 nm (4 λ). Time step of simulation is Δt=3.33×10^-18 s. The steady state electric E and magnetic H field distributions are obtained through Discrete Fourier Transform of simulated fields over the last wave range in the whole 20000 time steps of simulations. The Poynting vector S is found then from equation S=Re (E×H*).

Drude dependency (Eg.1) is employed to model silver permittivity

ε (ω) = ε_{\infty} (1 - ω_{p}^{2} ⁄ (ω^{2} + i ω Γ))

(1)

with parameters ε_inf=4.844, ω_p=6.541×10¹⁵ Hz, Γ=0.0755×10¹⁵ Hz calculated fitting the dispersion curve to data from [31

P. Johnson and R. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

] in wavelengths of 400–600 nm range.

Fig. 2. Dependence of primary mode propagation constant on the geometrical parameters of waveguides stack: d - dielectric channel width, p - metal layer thickne

4. Semi-analytical description of propagation in stack

For the purpose of brief analysis of propagation in stack, we make an assumption that H-polarized Gaussian beam normally incident on stack will couple most of its energy to a primary stack mode that could be described as coupled modes of single MIM waveguides, TM L+ according to notation in [6

] and TM₀ according to notation in [7

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79(3), 035120 (2009). [CrossRef]

The characteristic equation (Eq. (1) for kx propagation constant of primary mode of waveguide stack is found in a similar way as in [6

], matching solutions for single material layers and assuming periodicity of solution with period of stack d+p

0 = ε_{1} (ω) k_{2} \tanh (- {ik}_{2} p ⁄ 2) + ε_{2} (ω) k_{1} \tanh (- {ik}_{1} d ⁄ 2)

(2)

where p is metal layer thickness, d is air channel diameter, k₁ and k₂ are defined by Eq. (3)a and Eg.3b respectively

k_{1} = \sqrt{ε_{1} (ω) \frac{ω^{2}}{c^{2}} - k_{x}^{2}}

(3a)

k_{2} = \sqrt{ε_{2} (ω) \frac{ω^{2}}{c^{2}} - k_{x}^{2}}

(3b)

and where ε₁ and ε₂ are permittivity of dielectric host and silver respectively.

The effective modal index N_eff=k_x/k₀ values calculated for wavelength of 500 nm (k₀=0.0126 nm^-1) and varying geometry of stack are presented on Fig. 2. As one may suspect for large values of ratio p/d single waveguide modes are weakly coupled what manifests itself in the independence of propagation constant from the parameter p.

In further part of the work we investigate with FDTD interaction of Gaussian beam with a particular stack with dielectric channel width d=50 nm and metal layer thickness p=50 nm. For wavelength λ=500 nm the propagation constant of mode in such structure is equal k_x=0.01735+0.00005 nm^-1 what refers to N_eff=1.381+0.005i.

This approximation do not take into account the finite size of both stack and Gaussian beam. We conclude that Gaussian profile of excitation assures the field is not interacting significantly with stack edges but may lead to additional spread of refraction angles following the excitation of stack higher modes. The arbitral choice of structure for further investigations is supported by observation that while varying stack parameters and operating wavelength may influence the coupling efficiency of incident beam to stack, propagation constants and other measurable features, nevertheless, it does not affect the results qualitatively at wide range of parameters [13

W. M. Saj, “Light focusing with tip formed array of plasmon-polariton waveguides,” Proc. SPIE 6641, 664120 (2007). [CrossRef]

Fig. 3. Comparison of Snell law predictions and FDTD obtained direction of energy at points distant λ/2 from stack slope. Size of each dot for FDTD results is proportional to the total energy in observation point. Structure angle line separates points where energy is directed to and from the stack.

5. Refraction of light on the stack/air interface

According to Snell law [32

A. Hohenau, A. Drezet, M. Weissenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008). [CrossRef]

] the Total Internal Reflection is expected when the structure angle α is smaller than α_c≈87°. Relation between refraction concluded from Snell law and the observed in simulation direction of energy flow is presented on Fig. 3. The results are obtained from Poynting vector S as the angle arctan (S_y/S_x) calculated in few points at distance λ/2=250 nm from slope of the stack and distant to the tip from λ/2 to λ. The black dashed line divides points where energy is directed from and to the surface of the stack. Divergence from predicted by Snell law behavior is obvious and points out that strong output beam is obtained also for angles lower than critical α_c.

Phenomena that may have influenced the result are finite input beam width, anisotropy of stack and impact of tip singularity, generation of higher refraction orders due to structure periodicity [33

D. R. Smith, P. M. Rye, J. J. Mock, D. C. Vier, and A. F. Starr, “Enhanced diffraction from a grating on the surface of a negative-index metamaterial,” Phys. Rev. Lett. 93(13), 137405 (2004). [CrossRef] [PubMed]

] and existence of hybrid modes bounded to stack surface [34

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]

]. Nevertheless, the impact of this particular factors is hardly distinguishable in simulation results and their evaluation is postponed to future investigations.

Fig. 4. Magnetic field H_z for steady state solution: amplitude, phase and imaginary part. The phase and imaginary pictures reveal the refracted wave pattern close to the surface of structure slopes, the interference pattern with sharp phase shifts beyond the edge and the edge sourced pattern of cylindrical wave.

6. Phenomena observed in simulations

Intensity and the magnetic field distributions allow to investigate focusing properties and point a particular phenomena in FDTD simulation (Fig. 4 and Fig. 5 (Media 1)). The parallel phase plane fronts at the structure slopes are waves refracted by the stack/air interface. The long focal spot beyond the edge is created by interference of these two waves, as it is visible by the rapid shifts of phase along y-axis being a sign of a destructive interference pattern (In the far field valid description it relates to Bessel-Gauss beam creation by interference of two Gaussian beams [18

R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-8-6-932. [CrossRef]

]). Cylindrical wave with source on stack sharp edge may originate from radiative decay of stationary edge plasmon excitation of tip as well from the edge diffraction wave [35

A. Rubinowicz, “Thomas Young and the Theory of Diffraction,” Nature 180(4578), 160–162 (1957). [CrossRef]

Fig. 5. Steady state energy flow in x direction for various angles of stack slope α (top left) 80°, (top right) 120° and for comparison purposes (bottom left) incident Gaussian beam propagation without structure in area of simulation. White lines mark energy streamlines. On the bottom right, the snapshot of animation showing focusing with varying structure angle α (the constant mapping of intensity values to color is applied in all frames of animation) (MEDIA 1)

7. Focus features

The maximum enhancement of peak beam intensity equal 3.26 is observed in simulation for the structure with angle α_c (Fig. 6(a)). Focus width equals 0.5 λ and its length is 1.5 λ (Fig. 6(b)). As may be seen for flat case α=90° (Fig. 7) only around 60% of energy is transmitted beyond the structure due to reflections and absorption inside the stack.

The transmission reaches maximum for angles α around 50° because of concurrence between decreasing absorption and increasing reflection from the slope with decreasing angle and thickness of stack. The significant drop in total transmitted energy is observed for angles lower than αc. The difference between total energy in focal plane and in the focal area is connected with side lobs and scattering losses.

The focal spot size variance is presented on Fig. 6(b). The focus width and length (measured as focus peak FWHM in y and x directions, respectively) decrease monotonically with decreasing tip angle. Below angle αc focus achieves subwavelength width 0.37 λ and λ length with the peak amplitude enhancement decreasing to values around 2. Focus width is close to the half of beat length of two counter propagating waveguide stack modes L/2=πc/Re{k_x}=0.368 λ, what supports the conclusion that for small α focus is created by waves bounded to stack rather than by free space modes.

Fig. 6. Focus parameters as measured in FDTD a) intensity peak amplitude normalized to incident Gaussian beam amplitude. b) focus width and length (FWHM of focus in y and x direction, respectively).

Fig. 7. Focus parameters as measured in FDTD: intensity integrated over whole focal plane and over focus width (normalized to total energy of incident Gaussian beam).

8. Comparison with bulk dielectric structures

To compare our results with glass structures properties we simulate a blunt edged (flat angle top of width d=50 nm) dielectric structures made of medium with refractive index n=1.38. The results (Fig. 8) are similar to the one calculated for metallic structures, however vary in a few significant issues.

The transmission through dielectric structure is higher which increases overall amplitude of the field behind. Field distributions are comparable for both types of structures for wide angles of tip. However for angles α<100° the energy pattern for dielectric structures is significantly diffraction affected and, in addition to sharp main focus, includes many side lobes. E.g. for α=80° central part of focus (width_diel=168 nm) includes only 46% of total intensity in focal plane, while in case of silver structure (width_silver=204 nm) it is 58%. (As one may point out the standard diffraction limits for focus width do not apply for near fields close to material interfaces irrespective if medium is metal or dielectric [24

]).

The checkerboard interference patterns appear inside the dielectric structure as a result of interference between the beam propagating in x direction and its reflections from the slopes. Moreover, reflected from one slope light passing through the opposite dielectric/air interface forms transversal beam outside the axicon. This phenomena do not exist in stack structure due to anisotropic nature of propagation inside.

The conclusion is that a few of unwanted diffraction phenomena appearing in near field concentration are reduced in metal structures with small tip angle compared to dielectric bulk ones but at the cost of significant losses at transmission and broadening of focus area.

Fig. 8. Steady state energy flow in x direction for dielectric bulk structures (n=1.38) with (left) α=80° and (right) α=120°. White lines mark energy streamlines.

9. Conclusions

Plasmonic axicon in the form of the triangle shaped MIM waveguides stack is a simple design for effective light beam concentration from micro to nanoscale. Its performance is affected mostly by losses on transmission. In the presented examples the peak beam intensity enhancement up to 3.26 and light concentration to 0.37 λ wide and λ long focus was achieved. Ability of tip shaped metallic structures to concentrate energy to long subwavelength width focal region may be useful in sensing and as the source of light for nanoscale photonic devices.

Acknowledgments

This work was supported by Polish Ministry of Science and Higher Education (MNiSzW) Project N515 038 31/1295.

References and links

1.	S. A. Maier, Plasmonics. Fundamentals and Applications (Springer, Berlin 2007).
2.	W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
3.	R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7–8), 20–27 (2006). [CrossRef]
4.	J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008). [CrossRef] [PubMed]
5.	M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. 108(2), 494–521 (2008). [CrossRef] [PubMed]
6.	J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006). [CrossRef]
7.	S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79(3), 035120 (2009). [CrossRef]
8.	Z. Sun and H. K. Kim, “Refractive transmission of light and beam shaping with metallic nano-optic lenses,” Appl. Phys. Lett. 85(4), 642–644 (2004). [CrossRef]
9.	H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express 13(18), 6815–6820 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6815. [CrossRef] [PubMed]
10.	R. Gordon, “Proposal for superfocusing at visible wavelengths using radiationless interference of a plasmonic array,” Phys. Rev. Lett. 102(20), 207402 (2009). [CrossRef] [PubMed]
11.	X. Fan and G. P. Wang, “Nanoscale metal waveguide arrays as plasmon lenses,” Opt. Lett. 31(9), 1322–1324 (2006). [CrossRef] [PubMed]
12.	X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006). [CrossRef] [PubMed]
13.	W. M. Saj, “Light focusing with tip formed array of plasmon-polariton waveguides,” Proc. SPIE 6641, 664120 (2007). [CrossRef]
14.	H. Kurt, “Limited-diffraction light propagation with axicon-shape photonic crystals,” J. Opt. Soc. Am. B 26(5), 981–986 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=josab-26-5-981. [CrossRef]
15.	C. Rockstuhl, C. R. Simovski, S. A. Tretyakov, and F. Lederer, “Metamaterial nanotips,” Appl. Phys. Lett. 94(11), 113110 (2009). [CrossRef]
16.	J. H. McLeod, “The Axicon: A New Type of Optical Element,” J. Opt. Soc. Am. 44(8), 592 (1954), http://www.opticsinfobase.org/abstract.cfm?URI=josa-44-8-592. [CrossRef]
17.	Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon - the Most Important Optical Element,” Opt. Photonics News 16, 34–39 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=OPN-16-4-34 [CrossRef]
18.	R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-8-6-932. [CrossRef]
19.	Y.-J. Yu, H. Noh, M.-H. Hong, H.-R. Noh, Y. Arakawa, and W. Jhe, “Focusing characteristics of optical fiber axicon microlens for near-field spectroscopy: Dependence of tip apex angle,” Opt. Commun. 267(1), 264–270 (2006). [CrossRef]
20.	T. Grosjean, A. Fahys, M. Suarez, D. Charraut, R. Salut, and D. Courjon, “Annular nanoantenna on fibre micro-axicon,” J. Microsc. 229(2), 354–364 (2008). [CrossRef] [PubMed]
21.	A. E. Martirosyan, C. Altucci, C. de Lisio, A. Porzio, S. Solimeno, and V. Tosa, “Fringe pattern of the field diffracted by axicons,” J. Opt. Soc. Am. A 21(5), 770–776 (2004), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-5-770. [CrossRef]
22.	C. Zheng, Y. Zhang, and D. Zhao,” Calculation of the vectorial field distribution of an axicon illuminated by a linearly polarized Guassian beam,” Optik 117,3, 118–122 (2006). [CrossRef]
23.	L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003). [CrossRef] [PubMed]
24.	Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-7-1214. [CrossRef] [PubMed]
25.	A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-18-14200. [CrossRef] [PubMed]
26.	A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96(7), 073904 (2006). [CrossRef] [PubMed]
27.	T. T. Minh, K. Tanaka, and M. Tanaka, “Complex propagation constants of surface plasmon polariton rectangular waveguide by method of lines,” Opt. Express 16(13), 9378–9390 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9378. [CrossRef] [PubMed]
28.	M. S. Kushwaha and B. Djafari-Rouhani, “Plasma excitations in multicoaxial cables,” Phys. Rev. B 71(15), 153316 (2005). [CrossRef]
29.	F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74(20), 205419 (2006). [CrossRef]
30.	A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, MA 2000).
31.	P. Johnson and R. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
32.	A. Hohenau, A. Drezet, M. Weissenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008). [CrossRef]
33.	D. R. Smith, P. M. Rye, J. J. Mock, D. C. Vier, and A. F. Starr, “Enhanced diffraction from a grating on the surface of a negative-index metamaterial,” Phys. Rev. Lett. 93(13), 137405 (2004). [CrossRef] [PubMed]
34.	J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]
35.	A. Rubinowicz, “Thomas Young and the Theory of Diffraction,” Nature 180(4578), 160–162 (1957). [CrossRef]

OCIS Codes
(350.5500) Other areas of optics : Propagation
(250.5403) Optoelectronics : Plasmonics
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 22, 2009
Revised Manuscript: June 24, 2009
Manuscript Accepted: July 9, 2009
Published: August 3, 2009

Citation
W. M. Saj, "Light focusing on a stack of metal-insulator-metal waveguides sharp edge," Opt. Express 17, 13615-13623 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13615

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References

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Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon - the Most Important Optical Element,” Opt. Photonics News 16, 34–39 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=OPN-16-4-34 [CrossRef]
R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-8-6-932 . [CrossRef]
Y.-J. Yu, H. Noh, M.-H. Hong, H.-R. Noh, Y. Arakawa, and W. Jhe, “Focusing characteristics of optical fiber axicon microlens for near-field spectroscopy: Dependence of tip apex angle,” Opt. Commun. 267(1), 264–270 (2006). [CrossRef]
T. Grosjean, A. Fahys, M. Suarez, D. Charraut, R. Salut, and D. Courjon, “Annular nanoantenna on fibre micro-axicon,” J. Microsc. 229(2), 354–364 (2008). [CrossRef] [PubMed]
A. E. Martirosyan, C. Altucci, C. de Lisio, A. Porzio, S. Solimeno, and V. Tosa, “Fringe pattern of the field diffracted by axicons,” J. Opt. Soc. Am. A 21(5), 770–776 (2004), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-5-770 . [CrossRef]
C Zheng, Y Zhang, and D Zhao,” Calculation of the vectorial field distribution of an axicon illuminated by a linearly polarized Guassian beam,” Optik 117,3, 118–122 (2006). [CrossRef]
L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003). [CrossRef] [PubMed]
Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-7-1214 . [CrossRef] [PubMed]
A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-18-14200 . [CrossRef] [PubMed]
A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96(7), 073904 (2006). [CrossRef] [PubMed]
T. T. Minh, K. Tanaka, and M. Tanaka, “Complex propagation constants of surface plasmon polariton rectangular waveguide by method of lines,” Opt. Express 16(13), 9378–9390 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9378 . [CrossRef] [PubMed]
M. S. Kushwaha and B. Djafari-Rouhani, “Plasma excitations in multicoaxial cables,” Phys. Rev. B 71(15), 153316 (2005). [CrossRef]
F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74(20), 205419 (2006). [CrossRef]
A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, MA 2000).
P. Johnson and R. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
A. Hohenau, A. Drezet, M. Weissenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008). [CrossRef]
D. R. Smith, P. M. Rye, J. J. Mock, D. C. Vier, and A. F. Starr, “Enhanced diffraction from a grating on the surface of a negative-index metamaterial,” Phys. Rev. Lett. 93(13), 137405 (2004). [CrossRef] [PubMed]
J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]
A. Rubinowicz, “Thomas Young and the Theory of Diffraction,” Nature 180(4578), 160–162 (1957). [CrossRef]

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