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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 10 — May. 16, 2005
  • pp: 3653–3665
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Electromagnetic wave propagation through doubly dispersive subwavelength metamaterial hole

Ki Young Kim, Jeong-Hae Lee, Young Ki Cho, and Heung-Sik Tae  »View Author Affiliations


Optics Express, Vol. 13, Issue 10, pp. 3653-3665 (2005)
http://dx.doi.org/10.1364/OPEX.13.003653


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Abstract

The characteristics of the guided electromagnetic wave propagation through a subwavelength hole surrounded by a doubly dispersive metamaterial are investigated. Characteristic equations are derived for the surface polariton modes related to the subwavelength hole and mode classifications established. The surface polariton modes for two different hole-radii are numerically obtained and their electromagnetic dispersion curves and power flux characteristics analyzed and compared with each other. In particular, it was found that the border of the counter-propagation between the forward and backward Poynting vectors was located within the metamaterial, rather than at the interface between the metamaterial and the free space.

© 2005 Optical Society of America

1. Introduction

Extraordinary enhanced optical transmission though a single subwavelength aperture in metal structures has been receiving much attention [1

1. T. Thio, K. M. Pellerin, R. A. Linke., H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001). [CrossRef]

7

7. N. Bonod, E. Popov, and M. Nevière, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. 245, 355–361 (2005). [CrossRef]

] due to its unexpected behavior in relation to conventional theories and potential application in such areas as high density optical data storage, near field optical microscopy, subwavelength lithography, optical sensors, and optical displays. One plausible explanation for this phenomenon is the excitation of the surface polariton (SP) mode from the incident light at the entrance aperture, propagation of the SP mode through the subwavelength metal structure with a finite thickness or length, and de-excitation of the SP mode into the emitted light at the exit aperture [8

8. A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasma optics of nanostructures,” Phys. Solid State 45, 1327–1331 (2003). [CrossRef]

]. In principle, the efficiency of the throughput is dependent on the conversion efficiency between the SP mode and the incident (and emitted) light. In general, SP mode propagation is known to exist at the interface between metals and dielectrics [9

9. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, 1988).

], where the permittivity of the metal is negative at a given frequency and plays a crucial role in the SP mode propagation through the subwavelength metal structure. If we assume that the frequency dependent behavior of a metal at optical frequencies as the well known Drude model, i.e., εm 0(1-ωp2/ω 2), where ε 0 is the permittivity of the free space and ωp is the plasma frequency of the angular form, the permittivity of a metal is negative below the plasma frequency and also frequency dispersive. Klyuchnik et al. already showed the existence of SP modes along a subwavelength cylindrical metal cavity, and briefly examined the dispersion properties of the SP eigenmodes due to the frequency dispersive character of the metal [8

8. A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasma optics of nanostructures,” Phys. Solid State 45, 1327–1331 (2003). [CrossRef]

].

Recently, there has been a rapid growth of interest in metamaterials (MTMs), which are engineered artificial electromagnetic substances whose permittivity and permeability are simultaneously negative at a given desired frequency [10

10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

]. Ruppin also demonstrated the existence of an SP mode at the interface between an MTM and normal dielectric material [11

11. R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000). [CrossRef]

, 12

12. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 131811–1819 (2001). [CrossRef]

]. The SP modes on an MTM are expected to have multiple applications within a broad frequency spectrum, as an MTM can be artificially designed from microwave [10

10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

] to terahertz frequencies [13

13. H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005). [CrossRef] [PubMed]

] or above. Therefore, even though current fabrication techniques of isotropic MTMs are still in the early stage of development, a theoretical analysis of the electromagnetic wave propagation along an MTM subwavelength hole is worthwhile to provide the guided mode characteristics of other fundamental simple guiding structures with MTMs, such as slabs [14

14. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003). [CrossRef]

17

17. B. -I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386–9388 (2003). [CrossRef]

], grounded slabs [18

18. P. Baccarelli, P. Burghignoli, G. Lovat, and S. Paulotto, “Surface-wave suppression in a double-negative metamaterial grounded slab,” IEEE Ant. Wireless Prop. Lett. 2, 269–272 (2003). [CrossRef]

20

20. J. Schelleng, C. Monzon, P. F. Loschialpo, D. W. Forester, and L. N. Medgye-Mitschang, “Characteristics of waves guided by a grounded “left-handed” material slab of finite extent,” Phys Rev E 70, 066606 (2004). [CrossRef]

], channels [21

21. A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express 11, 2502–2510 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2502 [CrossRef] [PubMed]

], and cylinders [22

22. A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibres,” J. Opt. A: Pure Appl. Opt 7, S51–S56 (2005). [CrossRef]

24

24. K. Y. Kim, Guided and Leaky Modes of Circular Open Electromagnetic Waveguides: Dielectric, Plasma, and Metamaterial Columns, Ph.D. Thesis, Kyungpook National University, (2004), http://palgong.knu.ac.kr/~doors/PDFs/PhDThesis.pdf

].

Accordingly, the present study investigates the electromagnetic wave propagation along a subwavelength circular hole surrounded by a doubly dispersive metamaterials (DDMTM). The field components are defined and used to derive the characteristic equations. The mode classifications are then made and the electromagnetic dispersions and power flux characteristics of two different diametric holes surrounded by a DDMTM analyzed and compared.

Fig. 1. Schematic view of subwavelength DDMTM hole with diameter D=2a in cylindrical coordinate system. The inner and outer regions are the free space (region 1) and DDMTM (region 2), respectively.
Fig. 2. Material constants of DDMTM. The DNG and ENG regions are from 4 to 6 GHz and 6 to 10 GHz, respectively.

2. Field components, characteristic equations, and power fluxes

2.1 Subwavelength hole surrounded by doubly dispersive metamaterial

Figure 1 shows a schematic view of the subwavelength hole surrounded by a DDMTM. The inner and surrounding regions are the free space (region 1) and DDMTM (region 2), respectively. The diameter of the hole is depicted as D=2a, where a is the radius. The free space region is characterized by ε r1=µ r1=1.0 and the relative permittivity and permeability of the DDMTM can be respectively assumed as follows:

εr2(ω)=1ωp2ω2
(1a)
μr2(ω)=1Fω2ω2ω02
(1b)

where ω 0 is the resonant frequency of the angular form. The assumed MTM parameters are ωp /2π=10 GHz, ω 0/2π=4 GHz, and F=0.56, based on the experimental values of a structure composed of metallic rods and split ring resonators (SRRs) [10

10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

], plus these parameters have also been used in other studies on isotropic MTMs [11

11. R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000). [CrossRef]

, 12

12. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 131811–1819 (2001). [CrossRef]

, 14

14. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003). [CrossRef]

, 21

21. A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express 11, 2502–2510 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2502 [CrossRef] [PubMed]

, 22

22. A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibres,” J. Opt. A: Pure Appl. Opt 7, S51–S56 (2005). [CrossRef]

, 24

24. K. Y. Kim, Guided and Leaky Modes of Circular Open Electromagnetic Waveguides: Dielectric, Plasma, and Metamaterial Columns, Ph.D. Thesis, Kyungpook National University, (2004), http://palgong.knu.ac.kr/~doors/PDFs/PhDThesis.pdf

]. Figure 2 shows plots of the material expressions of Eq. (1). When using the selected parameters, the relative permittivity and permeability are both negative in the region from 4 to 6 GHz, called the double negative (DNG) region. Meanwhile, only the permittivity is negative in the region from 6 to 10 GHz, called the epsilon-negative (ENG) region. Since no SP mode solutions were found between f<4 GHz and f>10 GHz, those frequency regions are disregarded. Instead, the frequencies marked at A (4.714 GHz) and B (7.071 GHz) are the critical frequencies playing an important role in the dispersion relation, and will be discussed in section 3.

2.2 Field expressions

Since electromagnetic fields from an open guiding structure are vanish at infinity, a modified Bessel function of the second kind, i.e., 2 Km (k 2 r), is selected to express the behavior of the axial electric and magnetic fields along the transverse direction (r) in the DDMTM region, where m is the azimuthal eigenvalue and k 2 is the transverse propagation constant in the DDMTM region. If Km (k 2 r) is chosen as the surrounding field, k 2 is given as k 2=(β 2-k02 µ r2 ε r2)1/2 [25

25. A. Safaai-Jazi and G. L. Yip, “Classification of hybrid modes in cylindrical dielectric optical waveguides,” Radio Sci. 12, 603–609 (1977). [CrossRef]

], where β and k 0 are the axial propagation constant and free space wave number, respectively. For the DNG case, i.e., µ r2<0 and ε r2<0, the condition for the square root to be positive is β̄=β/k 0>(µ r2 ε r2)1/2, where β̄ is the normalized propagation constant. In contrast, the single negative (SNG) material constants themselves, i.e., µ r2>0 and ε r2<0 (ENG), or µ r2<0 and ε r2>0 (mu-negative (MNG)), are sufficient condition for the square root to be positive. Nonetheless, the condition β/k 0>(µ r1 ε r1)1/2 (=1.0) for the free space region (region 1) needs to be simultaneously satisfied to support the slow waves in this structure. Thus, the transverse propagation constants in the free space region can be expressed as k 1=(β 2-k02 µ r1 ε r1)1/2 and the proper choice of a Bessel function for the free space region needs to be a modified Bessel function of the first kind, i.e., Im (k 1 r) [25

25. A. Safaai-Jazi and G. L. Yip, “Classification of hybrid modes in cylindrical dielectric optical waveguides,” Radio Sci. 12, 603–609 (1977). [CrossRef]

]. Therefore, in the DNG region, the allowed SP mode solutions for β̄ must be within the region of β̄>(µ r2 ε r2)1/2 and β̄>(µ r1 ε r1)1/2 for (µ r2 ε r2)1/2>(µ r1 ε r1)1/2(=1.0) and (µ r1 ε r1)1/2 (=1.0)>(µ r2 ε r2)1/2, respectively. Resultant axial fields of the DDMTM hole are identical to those for a plasma column [26

26. V. L. Granatstein, S. P. Schlesinger, and A. Vigants, “The open plasmaguide in extreme of magnetic field,” IEEE Trans. Ant. Prop. 11, 489–496 (1963). [CrossRef]

] and the SP mode of an MTM column [22

22. A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibres,” J. Opt. A: Pure Appl. Opt 7, S51–S56 (2005). [CrossRef]

24

24. K. Y. Kim, Guided and Leaky Modes of Circular Open Electromagnetic Waveguides: Dielectric, Plasma, and Metamaterial Columns, Ph.D. Thesis, Kyungpook National University, (2004), http://palgong.knu.ac.kr/~doors/PDFs/PhDThesis.pdf

]. Figure 3 shows the allowed SP mode region of the DDMTM hole for the DNG and SNG cases. In the SNG region, only β̄>(µ r1 ε r1)1/2.

Fig. 3. Allowed SP mode region of DDMTM hole for DNG and SNG cases. (a) DNG with μr2εr2>μr1εr1, (b) DNG with μr1εr1>μr2εr2, and (c) SNG (ENG or MNG).

2.3 Characteristic equations

[εr1k1Im(k1a)Im(k1a)εr2k2Km(k2a)Km(k2a)][μr1k1Im(k1a)Im(k1a)μr2k2Km(k2a)Km(k2a)]=[mβk0a(1k121k22)]2.
(2)

Prime denotes the differentiation. For m=0, the characteristic equation (2) is split into two characteristic equations involving the TM0n and TE0n modes as follows, respectively:

εr1k1I1(k1a)I0(k1a)+εr2k2K1(k2a)K0(k2a)=0
(3a)
μr1k1I1(k1a)I0(k1a)+μr2k2K1(k2a)K0(k2a)=0.
(3b)

In a more general case, i.e., m≥1, the characteristic equation (2) can be written as follows using an empirical induction procedure:

(μr2μr1+εr2εr1)Q2±{(μr2μr1+εr2εr1)Q2}2+Rμr1εr1P=0
(4a)

where

P=1k1a(Im1(k1a)Im(k1a)mk1a)
(4b)
Q=1k2a(Km1(k2a)Km(k2a)+mk2a)
(4c)
R={mβk0a2(1k121k22)}2.
(4d)

Since the mode satisfying the characteristic equation (4) has both electric and magnetic fields in its axial components, the mode is called a hybrid mode. The “±” signs in Eq. (4a) correspond to the HEmn and EHmn modes, respectively. The mode whose axial electric (magnetic) field is dominant is traditionally referred to as the HEmn (EHmn) mode. Plus, since the HEmn (EHmn) mode is similar to the TM0n (TE0n) mode, the modes such as TM0n, HE1n, HE2n, … (TE0n, EH1n, EH2n, …) etc. are called the TM-like (TE-like) modes.

2.4 Power considerations

Backward waves can be supported in MTMs [27

27. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

] due to the intrinsic left-handedness of MTMs. Thus, to analyze the backward wave characteristics, the present study considered the normalized power flux η=(P 1+P 2)/(P 1|+|P 2|) [14

14. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003). [CrossRef]

, 21

21. A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express 11, 2502–2510 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2502 [CrossRef] [PubMed]

]. P 1 and P 2 are the fractional power fluxes in regions 1 and 2, respectively, and given as P 1=0a S z1 rdr and P 2=a S z2 rdr, respectively, where the axial components of the Poynting vectors in regions 1 and 2 can be given as S z1=E r1 H*θ1-E θ1 H*r1 and S z2=E r2 H*θ2-E θ2 H*r2, respectively. Eri and Eθi (Hri and Hθi ) (i=1, 2) are the radial and azimuthal electric (magnetic) field components. Asterisk denotes the complex conjugate. Since P 1 and P 2 are expected to have opposite signs, η>0 and η<0 are the conditions for the forward and backward waves, respectively, where η=0 means the cancellation of the power fluxes between the forward and backward waves. Moreover, a larger positive (negative) value for η implies a more tightly (loosely) bound mode in the free space region (r<a) of the subwavelength hole.

3. Numerical results and discussions

3.1 TE-like modes

Figure 4 shows the dispersion curves and their corresponding normalized power flux for the TE-like modes of the subwavelength hole with a 20.0 mm diameter. SP mode solutions existed above a frequency of 4.714 GHz, corresponding to point “A” in Fig. 2, where µ r2=1.0, and all the SP mode solutions existed in the DNG region (4 to 6 GHz) in Fig. 2. In the case of an MTM column with identical MTM parameters to Eq. (1), all the guided mode solutions existed below this critical frequency [24

24. K. Y. Kim, Guided and Leaky Modes of Circular Open Electromagnetic Waveguides: Dielectric, Plasma, and Metamaterial Columns, Ph.D. Thesis, Kyungpook National University, (2004), http://palgong.knu.ac.kr/~doors/PDFs/PhDThesis.pdf

]. It was also found that only a principal mode (n=1) existed for each azimuthal eigenvalue, whereas higher order modes existed in the case of an MTM column. The negative slopes of the dispersion curves indicated the backward waves [26

26. V. L. Granatstein, S. P. Schlesinger, and A. Vigants, “The open plasmaguide in extreme of magnetic field,” IEEE Trans. Ant. Prop. 11, 489–496 (1963). [CrossRef]

], which were also identified by the normalized power flux, i.e., η<0 as shown in Fig. 4(b). In the present study, all the TE-like SP mode solutions for the chosen parameters were backward wave type.

Fig. 4. TE-like modes of DDMTM hole with D=20.0 mm. (a) Dispersion curves and (b) normalized power flux. Corresponding operating wave numbers are also shown.

The cutoff for the TE01 (m=0) mode was on the β/k 0=1.0 line, while the cutoff frequency was 5.407 GHz, as in the case of Fig. 3(b), i.e., (µ r1 ε r1)1/2>(µ r2 ε r2)1/2. In contrast, the cutoff for the hybrid modes, such as the EH11 (m=1) and EH21 (m=2) modes, was on the β/k 0=(µ r2 ε r2)1/2 line, while the cutoff frequencies were 5.186 GHz and 4.925 GHz, respectively, as in the case of Fig. 3(a), i.e., (µ r2 ε r2)1/2>(µ r1 ε r1)1/2. Thus, the single TE-like SP mode propagation region was from 5.186 to 5.407 GHz, with a width of 0.221 GHz. When analyzing the TE-like mode of the MTM subwavelength hole with a 20.0 mm diameter, the wavelengths of the obtained TE-like SP modes ranged from 55.48 mm (5.407 GHz) to 63.64 mm (4.714 GHz), representing approximately three times the diameter of the subwavelength hole.

Fig. 5. TE-like modes of DDMTM hole with D=2.0 mm. (a) Dispersion curves and (b) normalized power flux. Insets are enlarged scales. Corresponding operating wave numbers are also shown.

When considering a smaller subwavelength DDMTM hole with a 2.0 mm diameter, the dispersion curves and normalized power flux for the TE-like modes are shown in Fig. 5. The cutoff frequency for the TE01 mode shifted higher to 5.967 GHz (from 5.407 GHz in D=20.0 mm case) and all the TE01 SP mode solutions were backward wave type. In contrast,

for the hybrid modes (m≥1), i.e., the EH11 and EH21 modes, forward waves were generated, as designated by dashed lines in Fig. 5. The slopes of the forward waves in the dispersion curves were positive, and η>0 for the corresponding region, as shown in Fig. 5(b). In spite of dissimilar geometrical configuration, forward and backward wave traveling properties associated with the respective positive and negative slopes in the dispersion curves of the MTM guiding structure were well described by Halterman et al [28

28. K. Halterman, J. M. Elson, and P. L. Overfelt, “Characteristics of bound modes in coupled dielectric waveguides containing negative index media,” Opt. Express 11, 521–529 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-521 [CrossRef] [PubMed]

]. Note that the forward waves of the EHm1 mode appeared with a small hole radius. However, in the case of a simple waveguide structure with frequency dispersive plasma media, such as a plasma slab [29

29. A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. 33, 231–233 (1962) [CrossRef]

], plasma column [26

26. V. L. Granatstein, S. P. Schlesinger, and A. Vigants, “The open plasmaguide in extreme of magnetic field,” IEEE Trans. Ant. Prop. 11, 489–496 (1963). [CrossRef]

], and plasma Goubau line [30

30. T. Tamir and S. Palócz, “Surface waves on plasma-clad metal rods,” IEEE Trans. Microwave Theory Tech. 12, 189–196 (1964). [CrossRef]

], small-sized waveguide cross sections produce backward waves from forward waves, which is the opposite to the present work.

Fig. 6. TM-like modes of DDMTM hole with D=20.0 mm. (a) Dispersion curves and (b) normalized power flux. Corresponding operating wave numbers are also shown.

Fig. 7. TM-like modes of DDMTM hole with D=2.0 mm. (a) Dispersion curves and (b) normalized power flux. The inset in (b) is an enlarged scale of the normalized power flux for the HE11 mode. At 7.32 GHz (point b), η=0. Points a, b, c, d, e, and f indicate the positions of the plots of the Ponyting vectors S z1 and S z2 in Fig. 8.

3.2 TM-like modes

Fig. 8. Spatial power distributions of Poynting vectors Sz1 and Sz2. Amplitude is an arbitrary unit.

4. Conclusions

This work investigated the electromagnetic dispersion and power flux characteristics of DDMTM holes with two different subwavelength diameters. TE-like and TM-like SP modes were both found to exist, and only the principal modes for each azimuthal eigenvalue had proper solutions. The obtained TE-like and TM-like SP modes were within the frequency region of the DNG and ENG, respectively. Two critical frequencies were identified for the lower limits of the backward TE-like and TM-like modes, where µ r2=-1.0 and ε r2=-1.0, respectively. Nonetheless, the TM01 and TE01 modes only supported backward waves, while the hybrid modes supported both forward and backward wave modes, thereby generating a frequency region where two or three orthogonal modes could coexist. In particular, forward waves appeared for the EHm1 mode when the radius of the hole was small. Meanwhile, the HEm1 mode with a small-sized hole exhibited a distinctive counter-propagation of the Poynting vectors in each region, which was the main reason for the η=+1 propagation.

The analysis presented here may find an application such as a prove tip of near-field microscope which can operate in two polarization modes, simultaneously or independently.

The present study may help in understanding the enhanced transmission phenomena, which can be observed in the subwavelength hole structure of finite length surrounded by the MTM.

Acknowledgments

This work was supported by grant No. R01-2004-000-10158-0 from the Basic Research Program of the Korea Science & Engineering Foundation.

References and links

1.

T. Thio, K. M. Pellerin, R. A. Linke., H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001). [CrossRef]

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6.

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7.

N. Bonod, E. Popov, and M. Nevière, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. 245, 355–361 (2005). [CrossRef]

8.

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10.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

11.

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000). [CrossRef]

12.

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 131811–1819 (2001). [CrossRef]

13.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005). [CrossRef] [PubMed]

14.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003). [CrossRef]

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17.

B. -I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386–9388 (2003). [CrossRef]

18.

P. Baccarelli, P. Burghignoli, G. Lovat, and S. Paulotto, “Surface-wave suppression in a double-negative metamaterial grounded slab,” IEEE Ant. Wireless Prop. Lett. 2, 269–272 (2003). [CrossRef]

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M. M. B. Suwailiam and Z. Chen, “Surface waves on a grounded double-negative (DNG) slab waveguide,” Microwave Opt. Tech. Lett. 44, 494–498 (2005). [CrossRef]

20.

J. Schelleng, C. Monzon, P. F. Loschialpo, D. W. Forester, and L. N. Medgye-Mitschang, “Characteristics of waves guided by a grounded “left-handed” material slab of finite extent,” Phys Rev E 70, 066606 (2004). [CrossRef]

21.

A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express 11, 2502–2510 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2502 [CrossRef] [PubMed]

22.

A. V. Novitsky and L. M. Barkovsky, “Guided modes in negative-refractive-index fibres,” J. Opt. A: Pure Appl. Opt 7, S51–S56 (2005). [CrossRef]

23.

H. Cory and T. Blum, Surface-wave propagation along a metamaterial cylindrical guide, MicrowaveOpt. Tech. Lett. 44, 31–35 (2005). [CrossRef]

24.

K. Y. Kim, Guided and Leaky Modes of Circular Open Electromagnetic Waveguides: Dielectric, Plasma, and Metamaterial Columns, Ph.D. Thesis, Kyungpook National University, (2004), http://palgong.knu.ac.kr/~doors/PDFs/PhDThesis.pdf

25.

A. Safaai-Jazi and G. L. Yip, “Classification of hybrid modes in cylindrical dielectric optical waveguides,” Radio Sci. 12, 603–609 (1977). [CrossRef]

26.

V. L. Granatstein, S. P. Schlesinger, and A. Vigants, “The open plasmaguide in extreme of magnetic field,” IEEE Trans. Ant. Prop. 11, 489–496 (1963). [CrossRef]

27.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

28.

K. Halterman, J. M. Elson, and P. L. Overfelt, “Characteristics of bound modes in coupled dielectric waveguides containing negative index media,” Opt. Express 11, 521–529 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-521 [CrossRef] [PubMed]

29.

A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. 33, 231–233 (1962) [CrossRef]

30.

T. Tamir and S. Palócz, “Surface waves on plasma-clad metal rods,” IEEE Trans. Microwave Theory Tech. 12, 189–196 (1964). [CrossRef]

31.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004) [CrossRef]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(240.5420) Optics at surfaces : Polaritons
(240.6680) Optics at surfaces : Surface plasmons
(260.2030) Physical optics : Dispersion
(350.5500) Other areas of optics : Propagation

ToC Category:
Research Papers

History
Original Manuscript: April 12, 2005
Revised Manuscript: May 1, 2005
Published: May 16, 2005

Citation
Ki Young Kim, Jeong-Hae Lee, Young Cho, and Heung-Sik Tae, "Electromagnetic wave propagation through doubly dispersive subwavelength metamaterial hole," Opt. Express 13, 3653-3665 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3653


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References

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  13. H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, �??Terahertz response of a microfabricated rodsplit- ring-resonator electromagnetic metamaterial,�?? Phys. Rev. Lett. 94, 063901 (2005). [CrossRef] [PubMed]
  14. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, �??Guided modes in negative-refractive-index waveguides,�?? Phys. Rev. E 67, 057602 (2003). [CrossRef]
  15. H. Dong and T. X. Wu, �??Analysis of discontinuities in double-negative (DNG) slab waveguides,�?? Microwave Opt. Tech. Lett. 39, 483-488 (2003). [CrossRef]
  16. H. Cory and A. Barger, �??Surface-wave propagation along a metamaterial slab,�?? Microwave Opt. Tech. Lett. 38, 392-395 (2003). [CrossRef]
  17. B. �??I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, �??Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,�?? J. Appl. Phys. 93, 9386-9388 (2003). [CrossRef]
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  19. M. M. B. Suwailiam, Z. Chen, �??Surface waves on a grounded double-negative (DNG) slab waveguide,�?? Microwave Opt. Tech. Lett. 44, 494-498 (2005). [CrossRef]
  20. J. Schelleng, C. Monzon, P. F. Loschialpo, D. W. Forester, and L. N. Medgye-Mitschang, �??Characteristics of waves guided by a grounded �??left-handed�?? material slab of finite extent,�?? Phys Rev E 70, 066606 (2004). [CrossRef]
  21. A. C. Peacock and N. G. R. Broderick, �??Guided modes in channel waveguides with a negative index of refraction,�?? Opt. Express 11 , 2502-2510 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2502">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2502</a> [CrossRef] [PubMed]
  22. A. V. Novitsky and L. M. Barkovsky, �??Guided modes in negative-refractive-index fibres,�?? J. Opt. A: Pure Appl. Opt 7, S51-S56 (2005). [CrossRef]
  23. H. Cory and T. Blum, "Surface-wave propagation along a metamaterial cylindrical guide", Microwave Opt. Tech. Lett. 44, 31-35 (2005). [CrossRef]
  24. K. Y. Kim, "Guided and Leaky Modes of Circular Open Electromagnetic Waveguides: Dielectric, Plasma, and Metamaterial Columns", Ph.D. Thesis, Kyungpook National University, (2004), <a href="http://palgong.knu.ac.kr/~doors/PDFs/PhDThesis.pdf">http://palgong.knu.ac.kr/~doors/PDFs/PhDThesis.pdf</a>
  25. A. Safaai-Jazi and G. L .Yip, �??Classification of hybrid modes in cylindrical dielectric optical waveguides,�?? Radio Sci. 12, 603-609 (1977). [CrossRef]
  26. V. L .Granatstein, S. P. Schlesinger, and A. Vigants, �??The open plasmaguide in extreme of magnetic field,�?? IEEE Trans. Ant. Prop. 11, 489-496 (1963). [CrossRef]
  27. V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of ε and µ,�?? Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
  28. K. Halterman, J. M. Elson, and P. L. Overfelt, �??Characteristics of bound modes in coupled dielectric waveguides containing negative index media,�?? Opt. Express 11, 521-529 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-521">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-521</a> [CrossRef] [PubMed]
  29. A. A. Oliner and T. Tamir, �??Backward waves on isotropic plasma slabs,�?? J. Appl. Phys. 33, 231-233 (1962) [CrossRef]
  30. T. Tamir and S. Palócz, �??Surface waves on plasma-clad metal rods,�?? IEEE Trans. Microwave Theory Tech. 12, 189-196 (1964). [CrossRef]
  31. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, �??Nonlinear surface waves in left-handed materials,�?? Phys. Rev. E 69, 016617 (2004) [CrossRef]

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