A corrugated perfect magnetic conductor surface supporting spoof surface magnon polaritons Liang-liang Liu, Zhuo Li,* Chang-qing Gu, Ping-ping Ning, Bing-zheng Xu, Zhen-yi Niu, and Yong-jiu Zhao Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing Univ. Aeronaut. Astronaut.), Ministry of Education, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China * [email protected]
Abstract: In this paper, we demonstrate that spoof surface magnon polaritons (SSMPs) can propagate along a corrugated perfect magnetic conductor (PMC) surface. From duality theorem, the existence of surface electromagnetic modes on corrugated PMC surfaces are manifest to be transverse electric (TE) mode compared with the transverse magnetic (TM) mode of spoof surface plasmon plaritons (SSPPs) excited on corrugated perfect electric conductor surfaces. Theoretical deduction through modal expansion method and simulation results clearly verify that SSMPs share the same dispersion relationship with the SSPPs. It is worth noting that this metamaterial will have more similar properties and potential applications as the SSPPs in large number of areas. © 2014 Optical Society of America OCIS codes: (050.6624) Subwavelength structures; (240.6690) Surface waves; (240.6680) Surface plasmons
References and links 1. W. L. Barnes, A. Dereux and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). 2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, New York, 2007). 3. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). 4. P. Nagpal, N. C. Lindquist, S. H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325, 594–597 (2009). 5. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010). 6. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces” Science 305, 847–848 (2004). 7. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental Verification of Designer Surface Plasmons,” Science 308, 670–672 (2005). 8. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A-Pure Appl. Opt. 7, S97–S101 (2005). 9. S. Maier, S. Andrews, L. Mart´ın-Moreno, and F. Garc´ıa-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805(2006). 10. A. I. Fern´andez-Dom´ınguezf, E. Moreno, L. Martin-Moreno and F. J. Garcia-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34, 2063–2065 (2009). 11. T. Jiang, L. Shen, X. Zhang, and L. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces” Progress in Electromagnetic Research M 8, 91–102 (2009). 12. D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and Esteban Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. express 18, 754–764 (2010).
#204908 - $15.00 USD (C) 2014 OSA
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10675
13. Y. J. Zhou, Q. Jiang and T. J. Cui, “Bidirectional bending splitter of designer surface plasmons,” Appl. Phys. Lett. 99, 111904 (2011). 14. T. Jiang, L. F. Shen, J. J. Wu, T. J. Yang, Z. C. Ruan and L. X. Ran, “Realization of tightly confined channel plasmon polaritons at low frequencies,” Appl. Phys. Lett. 99, 261103 (2011). 15. X. Gao, J. H. Shi, H. F. Ma, W. X. Jiang and T. J. Cui, “Dual-band spoof surface plasmon polaritons based on composite-periodic gratings,” J. Phys. D-Appl. Phys. 45, 505104 (2012). 16. H. F. Ma, X. P. Shen, Q. Cheng, W. X. Jiang and T. J. Cui, “Broadband and high-efficiency conversion from guided waves to spoof surface plasmon polaritons,” Laser Photon. Rev. 10, 00118 (2013). 17. X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, “Conformal surface plasmons propagating on ultrathin and flexible films” Proc. Natl. Acad. Sci. U.S.A. 110, 40–45 (2013). 18. X. Shen and T. J. Cui, “Planar plasmonic metamaterial on a thin film with nearly zero thickness,” Appl. Phys. Lett. 102, 211909 (2013). 19. X. Gao, J. H. Shi, X. P. Shen, H. F. Ma, W. X. Jiang, L. M. Li and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102, 151912 (2013). 20. M. G. Cottam and D. R. Tilley, Introduction to Surface and Superlattice Excitations (Cambridge University, 1989). 21. R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000). 22. A. Hartstein, E. Burstein, A. A. Maradudin, R. Brewer, and R. F. Wallis, “Surface polaritons on semi-infinite gyromagnetic media” J. Phys. C-Solid State Phys. 6, 1266–1276 (1973). 23. V. H. Arakelian, L. A. Bagdassarian, and S. G. Simonian, “Electrodynamics of bulk and surface normal magnonpolaritons in antiferromagnetic crystals,” J. Magn. Magn. Mater. 167, 149–160 (1997). 24. J. Matsuura, M. Fukui, and O. Tada, “ATR mode of surface magnon polaritons on YIG,” Solid State Commun. 45, 157–160 (1983). 25. M. Marchand and A. Caill, “Asymmetrical guided magnetic polaritons in a ferromagnetic slab,” Solid State Commun. 34, 827–831 (1980). 26. C. Shu and A. Caill´e, “Surface magnetic polaritons on uniaxial antiferromagnets,” Solid State Commun. 42, 233–238 (1982). 27. C. Thibaudeau and A. Caill´e, “The magnetic polaritons of a semi-infinite uniaxial antiferromagnet,” Solid State Commun. 87, 643–647 (1993). 28. J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,” Opt. Photon. News 15(10), 54–59 (2004). 29. Sergey I Bozhvolnyi, Plasmonic Nanoguides and Circuits (Pan Stanford Publishing Pte. Ltd, Singapore, 2009). 30. C. H. Raymond Ooi, K. C. Low, Ryota Higa and Tetsuo Ogawa, “Surface polaritons with arbitrary magnetic and dielectric materials: new regimes, effects of negative index, and superconductors” J. Opt. Soc. Am. B 29, 2691–2697 (2012).
1.
Introduction
Surface plasmon polaritons (SPPs), which are localized surface electromagnetic (EM) modes that propagate along the interface between a metal and a dielectric in optical frequencies and decay exponentially in the transverse direction [1,2], have attracted great attentions owing to their huge potential applications in large number of areas [3–5]. Thereafter, to produce highly confined surface EM waves at lower frequencies, a new plasmonic metamaterial called spoof surface plasmon polaritons (SSPPs) are proposed by Pendry et al. [6] in 2004 and has been verified experimentally in the microwave regimes by Hibbins et al. [7] in 2005. Owing to the advantages of this metamaterial, the cut-off frequency of the SSPPs can be adjusted at will by decorating the metal surfaces with one-dimensional (1D) arrays of subwavelength grooves [8], or 2D arrays of subwavelength holes [6–8], or 3D perfect electric conductor (PEC) wire with periodical array of radial grooves [9]. And due to their ability to confine EM waves in subwavelength scale with high intensity, both SPPs and SSPPs have found lots of applications from optical to microwave frequency bands [10–19]. However, a number of works have shown that magnetic materials can excite surface magnon polaritons (SMPs) in recent years, i.e., strong coupling of photons to quantized magnetization waves due to collective spins [20]. Ruppin showed that surface polariton can exist for both sand p-polarized waves in left-handed metamaterials [21]. Excitation of SMPs in gyromagnetic materials and anisotropic antiferromagnetic crystals have been discussed by Hartstein et al. [22] #204908 - $15.00 USD (C) 2014 OSA
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10676
and Arakelian et al. [23], respectively. The presence of magnon polaritons was demonstrated in a YIG slab using the attenuated total reflectance technique [24]. The dispersion relations for magnetic polaritons have been obtained for anisotropic systems of ordered ferromagnetic slabs [25] and uniaxial antiferromagnets [26,27]. It has been shown that s-polarized waves can propagate at the interface of media with negative refractive index and negative dielectric constant of low-dimensional nanowaveguides [28,29]. Then the underlying theoretical expressions of the tangential wave vectors and field distributions of s-and p-polarized waves at an interface of two media with arbitrary materials are proposed by Raymond Ooi et al. [30], making SMPs get more attentions nowdays. However, up to now, to our knowledge, the form and manner of existence and implementation of SMPs in the terahertz and microwave regimes are not realized. So, borrowing the idea of the existence of the SSPPs, in this paper, we propose a corrugated perfect magnetic conductor (PMC) surface to support spoof surface magnon polaritons (SSMPs). In duality, the existence of surface EM modes on corrugated PMC surfaces are obvious to be TE mode compared with the TM mode of the SSPPs excited on corrugated PEC surfaces. Theoretical and simulation results clearly verify that the SSMPs share the same dispersion relationship with the SSPPs and we also show that, as long as the size and spacing of the grooves are much smaller than the wavelength, a perforated PMC surface behaves as an effective medium. It is also worth noting that this metamaterial has more similar properties and applications as the SSPPs in large number of areas. 2.
Theoretical analysis
E
(a)
H
(b)
z
kx
x
h d
z
y
k
I
x εx =1 ε y =εz =d a
II a
μx =a d μ y =μ z =∞ h
Perfect Magnetic Conductor
Fig. 1. (a) A 1D periodical groove arrays of width a, depth h separated by a period d. We are interested in s-polarized surface modes propagating along the x direction with E lying in the x-y plane. (b) In the effective medium approximation the structure displayed in (a) behaves as a homogeneous but anisotropic layer of thickness h on the surface of a PMC.
2.1.
Modal expansion method
First of all, in Fig. 1(a), a corrugated PMC surface decorated by a periodic array of grooves with depth h, width a and period d is considered. According to the duality theorem in Maxwell equations, we can find the eigenmodes of the SSMPs supported by this surface with the modal expansion method [11]. We are interested in s-polarized surface waves propagating along the x ˆ x + zˆHz . In Region I above the surface (z > 0), direction with the form of E = yE ˆ y and H = xH the electric field component Ey , which is nonradiative and vanishes as z → ∞, can be written as (n)
(n)
EyI = ∑ An eikx x e−qz z ,
(1)
n
(n)
where An is a constant, kx = kx +2nπ /d (n = 0, 1, 2, 3 . . .), here, the diffraction effects are taken into account and the propagation constant of the surface wave kx lies in the first Brillouin zone, #204908 - $15.00 USD (C) 2014 OSA
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10677
(n) (n) i.e., |kx | ≤ π /d and qz = (kx )2 − k02 , k0 is the wave number in free space. The nonradiative property of the fields requires that kx > k0 . In Region II under the surface (z ≤ 0), the EM fields are zero everywhere except inside the grooves. Each groove may be viewed as a planar waveguide with length h and one end closed, and we only consider its fundamental mode in the field expansion. In this way, Ey in the groove can be written as EyII = B cos[k0 (z + h)],
(2)
where B is a constant. This field is actually a sum of two waves propagating along the ±z directions in the groove, and the tangential component of the H field vanishes at the bottom of the groove, for example, Hx = 0 at z = −h. The magnetic field components Hx and Hz can be directly obtained from Ey through Maxwell’s equations that
Hx =
i ∂ Ey , ω μ0 ∂ z
Hz = −
i ∂ Ey , ω μ0 ∂ x
(3)
(4)
where ω is the angular frequency, μ0 is the magnetic permeability in free space. The dispersion relation of the SMPs on the corrugated PMC surfaces can be obtained by imposing the matching conditions of the parallel components of the E and H fields at the interface between Regions I and II. At the interface z = 0, the E field component Ey must be continuous for |x| ≤ a/2 that (n)
∑ An eikx
x
= B cos(k0 h),
(5)
n
and by integrating Eq. (5) over this interval, we can obtain
∑ An Sn = B cos(k0 h),
(6)
n
where
Sn =
1 a
a/2
(n) (n) (n) a . eikx x e−qz z dx = sin c kx 2 −a/2
(7)
The magnetic field component Hx at the interface must be continuous over the whole period. Thus, for |x| ≤ a/2 we have (n)
(n) iqz ik0 ∑ ω μ0 An eikx x = ω μ0 B sin(k0 h), n
(8)
and for a/2 < |x| ≤ d/2
#204908 - $15.00 USD (C) 2014 OSA
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10678
(n)
(n) iqz ∑ ω μ0 An eikx x = 0. n
(9) (n)
By using the orthogonality properties of the functions eikx x , we can obtain the coefficient An = B
k0 a (n)
qz d
sin(k0 h)S0 .
(10)
Substituting Eq. (10) into Eq. (5), and neglecting the diffraction effects, we can obtain
kx2 − k02 k0
=
a 2 S tan(k0 h), d 0
(11)
which is obviously the same with the dispersion relation (9) in Ref. [8]. Considering that no material parameters are involved in Eq. (11) and only the boundary condition is changed from PEC to PMC, the duality theorem holds for the case from SSPPs to SSMPs. 2.2.
Effective medium approximation
Similarly, from the perspective of the effective medium [8], the same dispersion relation could be obtained if we replaced the array of grooves by a single homogeneous but anisotropic layer of thickness h on top of the PMC surface shown in Fig. 1(b). The homogeneous layer would have the following parameters:
μx = a/d, μy = μz = ∞.
(12)
As wave propagates in the grooves along the y and z directions with the velocity of light that √
μ x εy =
√
μ x εz ,
(13)
and thus we can obtain
εy = εz = 1/μx = d/a.
(14)
For the s-polarized plane wave impinging on the surface of a homogeneous layer of thickness h with μ and ε given by Eqs. (12) and (14), the specular reflection coefficient R can be written as Eq. (15) after some straightforward algebras, R=
kz μx kz μx
− k0 + k0 + μkzx ei2k0 h . + k0 − k0 − μkzx ei2k0 h
(15)
This effective medium implies a bound surface state when there is a divergence in the reflection coefficient of the surface for large values to the case kx > k0 and looking at the zeros of the denominator of R, we can obtain the dispersion relation of the surface modes
kx2 − k02 k0
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=
a tan(k0 h). d
(16)
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10679
1 0.9 0.8 0.7
ω/ ωc
0.6 0.5 0.4 0.3
Light line Analytical value Numerical value (SSMPs) Numerical value (SSPPs)
0.2 0.1 0
0
0.2
0.4
kd/π
0.6
0.8
1
Fig. 2. The normalized dispersion relations (ω (k)) of the SSMPs supported by a 1D array of grooves with geometrical parameters a/d = 0.4, h/d = 0.8. The green line is the theoretical result obtained with Eq. (16), the red line and blue line with open circles render the numerical results of the SSPPs and SSMPs respectively, and the dark line is light line. PEC Boundary
(a)
Bloch Boundary
o
Bloch Boundary
Vacuum
z x
a 2
PEC
a 2
h
Bloch Boundary
Vacuum
a 2
PMC
a 2
h
d
d (c)
PMC Boundary
(b)
(d)
z o
x
Fig. 3. Schematic diagrams of the simulation models. (a) and (b) are the geometries of the periodical groove arrays of width a = 0.4d, depth h = 0.8d separated by a period d on the surface of PEC and PMC with infinite in y direction separately. In which, the gray areas in (a) and (b) represent the vacuum, the red lines in (a) represent the PEC boundary conditions and the light blue lines in (b) are the PMC boundary conditions and the black lines in (a) and (b) along z direction are set as Bloch boundary conditions. (c) and (d) are the surface electric field vectors (the red arrow lines in (c) and in V/m) and magnetic field vectors (the light blue arrow lines in (d) and in A/m) of the unit cell on the xoz plane respectively.
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Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10680
3.
Results and discussions
Note that the Eq. (16) coincides with Eq. (11) in the limit kx a 1. In Fig. 2, we plot the analytical dispersion relation with Eq. (16) and numerical results for the particular case a/d = 0.4, h/d = 0.8. Though the cutoff frequency of the analytical value is slightly higher than the numerical ones due to the reason that the diffraction effects are neglected, it is worth commenting that the dispersion relations between the SSMPs and the SSPPs are identical, and the cutoff frequency ω also approaches π c0 /(2h) according to the dispersion relations and duality of Maxwell’s equations between the PEC and PMC cases. All numerical simulation results in this paper have been performed with the finite element method (FEM) [12]. The corresponding eigenvalue problem is posed in a single unit cell where Bloch boundary conditions are used. The open space is mimicked with PEC or PMC boundary conditions, and the Bloch boundary conditions are employed to model the periodic arrays in the simulations shown in Fig. 3(a) and 3(b). The surface electric and magnetic field vectors on the xoz plane of the PEC and PMC cases are illustrated in Fig. 3(c) and 3(d) with exactly identical field distributions, which further verify the essential characteristics between the SSPPs and SSMPs. 4.
Conclusions
In conclusion, we have shown that the existence of the SSMPs in corrugated PMC surfaces, and the dispersion relation is obtained through the duality of Maxwell’s equations combined with modal expansion method, and an effective medium analytical approach is used to depict the physical insight of the new metamaterial. Thanks to this new concept, a whole brunch of surface modes already known in optical region can be transformed to terahertz and microwave frequency bands. This kind of metamaterial may have a very significant impact on both fundamental and applied researches. Acknowledgments This work was supported by the National Natural Science Foundation of China for Young Scholars under Grant No. 61102033, the Fundamental Research Funds for the Central Universities under Grant NJ20140009, the Aeronautical Science Foundation of China under grant No. 20128052063 and the priority academic program development of Jiangsu Higher Education Institutions.
#204908 - $15.00 USD (C) 2014 OSA
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10681
Abstract: In this paper, we demonstrate that spoof surface magnon polaritons (SSMPs) can propagate along a corrugated perfect magnetic conductor (PMC) surface. From duality theorem, the existence of surface electromagnetic modes on corrugated PMC surfaces are manifest to be transverse electric (TE) mode compared with the transverse magnetic (TM) mode of spoof surface plasmon plaritons (SSPPs) excited on corrugated perfect electric conductor surfaces. Theoretical deduction through modal expansion method and simulation results clearly verify that SSMPs share the same dispersion relationship with the SSPPs. It is worth noting that this metamaterial will have more similar properties and potential applications as the SSPPs in large number of areas. © 2014 Optical Society of America OCIS codes: (050.6624) Subwavelength structures; (240.6690) Surface waves; (240.6680) Surface plasmons
References and links 1. W. L. Barnes, A. Dereux and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). 2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, New York, 2007). 3. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). 4. P. Nagpal, N. C. Lindquist, S. H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325, 594–597 (2009). 5. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010). 6. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces” Science 305, 847–848 (2004). 7. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental Verification of Designer Surface Plasmons,” Science 308, 670–672 (2005). 8. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A-Pure Appl. Opt. 7, S97–S101 (2005). 9. S. Maier, S. Andrews, L. Mart´ın-Moreno, and F. Garc´ıa-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805(2006). 10. A. I. Fern´andez-Dom´ınguezf, E. Moreno, L. Martin-Moreno and F. J. Garcia-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34, 2063–2065 (2009). 11. T. Jiang, L. Shen, X. Zhang, and L. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces” Progress in Electromagnetic Research M 8, 91–102 (2009). 12. D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and Esteban Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. express 18, 754–764 (2010).
#204908 - $15.00 USD (C) 2014 OSA
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10675
13. Y. J. Zhou, Q. Jiang and T. J. Cui, “Bidirectional bending splitter of designer surface plasmons,” Appl. Phys. Lett. 99, 111904 (2011). 14. T. Jiang, L. F. Shen, J. J. Wu, T. J. Yang, Z. C. Ruan and L. X. Ran, “Realization of tightly confined channel plasmon polaritons at low frequencies,” Appl. Phys. Lett. 99, 261103 (2011). 15. X. Gao, J. H. Shi, H. F. Ma, W. X. Jiang and T. J. Cui, “Dual-band spoof surface plasmon polaritons based on composite-periodic gratings,” J. Phys. D-Appl. Phys. 45, 505104 (2012). 16. H. F. Ma, X. P. Shen, Q. Cheng, W. X. Jiang and T. J. Cui, “Broadband and high-efficiency conversion from guided waves to spoof surface plasmon polaritons,” Laser Photon. Rev. 10, 00118 (2013). 17. X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, “Conformal surface plasmons propagating on ultrathin and flexible films” Proc. Natl. Acad. Sci. U.S.A. 110, 40–45 (2013). 18. X. Shen and T. J. Cui, “Planar plasmonic metamaterial on a thin film with nearly zero thickness,” Appl. Phys. Lett. 102, 211909 (2013). 19. X. Gao, J. H. Shi, X. P. Shen, H. F. Ma, W. X. Jiang, L. M. Li and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102, 151912 (2013). 20. M. G. Cottam and D. R. Tilley, Introduction to Surface and Superlattice Excitations (Cambridge University, 1989). 21. R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000). 22. A. Hartstein, E. Burstein, A. A. Maradudin, R. Brewer, and R. F. Wallis, “Surface polaritons on semi-infinite gyromagnetic media” J. Phys. C-Solid State Phys. 6, 1266–1276 (1973). 23. V. H. Arakelian, L. A. Bagdassarian, and S. G. Simonian, “Electrodynamics of bulk and surface normal magnonpolaritons in antiferromagnetic crystals,” J. Magn. Magn. Mater. 167, 149–160 (1997). 24. J. Matsuura, M. Fukui, and O. Tada, “ATR mode of surface magnon polaritons on YIG,” Solid State Commun. 45, 157–160 (1983). 25. M. Marchand and A. Caill, “Asymmetrical guided magnetic polaritons in a ferromagnetic slab,” Solid State Commun. 34, 827–831 (1980). 26. C. Shu and A. Caill´e, “Surface magnetic polaritons on uniaxial antiferromagnets,” Solid State Commun. 42, 233–238 (1982). 27. C. Thibaudeau and A. Caill´e, “The magnetic polaritons of a semi-infinite uniaxial antiferromagnet,” Solid State Commun. 87, 643–647 (1993). 28. J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,” Opt. Photon. News 15(10), 54–59 (2004). 29. Sergey I Bozhvolnyi, Plasmonic Nanoguides and Circuits (Pan Stanford Publishing Pte. Ltd, Singapore, 2009). 30. C. H. Raymond Ooi, K. C. Low, Ryota Higa and Tetsuo Ogawa, “Surface polaritons with arbitrary magnetic and dielectric materials: new regimes, effects of negative index, and superconductors” J. Opt. Soc. Am. B 29, 2691–2697 (2012).
1.
Introduction
Surface plasmon polaritons (SPPs), which are localized surface electromagnetic (EM) modes that propagate along the interface between a metal and a dielectric in optical frequencies and decay exponentially in the transverse direction [1,2], have attracted great attentions owing to their huge potential applications in large number of areas [3–5]. Thereafter, to produce highly confined surface EM waves at lower frequencies, a new plasmonic metamaterial called spoof surface plasmon polaritons (SSPPs) are proposed by Pendry et al. [6] in 2004 and has been verified experimentally in the microwave regimes by Hibbins et al. [7] in 2005. Owing to the advantages of this metamaterial, the cut-off frequency of the SSPPs can be adjusted at will by decorating the metal surfaces with one-dimensional (1D) arrays of subwavelength grooves [8], or 2D arrays of subwavelength holes [6–8], or 3D perfect electric conductor (PEC) wire with periodical array of radial grooves [9]. And due to their ability to confine EM waves in subwavelength scale with high intensity, both SPPs and SSPPs have found lots of applications from optical to microwave frequency bands [10–19]. However, a number of works have shown that magnetic materials can excite surface magnon polaritons (SMPs) in recent years, i.e., strong coupling of photons to quantized magnetization waves due to collective spins [20]. Ruppin showed that surface polariton can exist for both sand p-polarized waves in left-handed metamaterials [21]. Excitation of SMPs in gyromagnetic materials and anisotropic antiferromagnetic crystals have been discussed by Hartstein et al. [22] #204908 - $15.00 USD (C) 2014 OSA
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10676
and Arakelian et al. [23], respectively. The presence of magnon polaritons was demonstrated in a YIG slab using the attenuated total reflectance technique [24]. The dispersion relations for magnetic polaritons have been obtained for anisotropic systems of ordered ferromagnetic slabs [25] and uniaxial antiferromagnets [26,27]. It has been shown that s-polarized waves can propagate at the interface of media with negative refractive index and negative dielectric constant of low-dimensional nanowaveguides [28,29]. Then the underlying theoretical expressions of the tangential wave vectors and field distributions of s-and p-polarized waves at an interface of two media with arbitrary materials are proposed by Raymond Ooi et al. [30], making SMPs get more attentions nowdays. However, up to now, to our knowledge, the form and manner of existence and implementation of SMPs in the terahertz and microwave regimes are not realized. So, borrowing the idea of the existence of the SSPPs, in this paper, we propose a corrugated perfect magnetic conductor (PMC) surface to support spoof surface magnon polaritons (SSMPs). In duality, the existence of surface EM modes on corrugated PMC surfaces are obvious to be TE mode compared with the TM mode of the SSPPs excited on corrugated PEC surfaces. Theoretical and simulation results clearly verify that the SSMPs share the same dispersion relationship with the SSPPs and we also show that, as long as the size and spacing of the grooves are much smaller than the wavelength, a perforated PMC surface behaves as an effective medium. It is also worth noting that this metamaterial has more similar properties and applications as the SSPPs in large number of areas. 2.
Theoretical analysis
E
(a)
H
(b)
z
kx
x
h d
z
y
k
I
x εx =1 ε y =εz =d a
II a
μx =a d μ y =μ z =∞ h
Perfect Magnetic Conductor
Fig. 1. (a) A 1D periodical groove arrays of width a, depth h separated by a period d. We are interested in s-polarized surface modes propagating along the x direction with E lying in the x-y plane. (b) In the effective medium approximation the structure displayed in (a) behaves as a homogeneous but anisotropic layer of thickness h on the surface of a PMC.
2.1.
Modal expansion method
First of all, in Fig. 1(a), a corrugated PMC surface decorated by a periodic array of grooves with depth h, width a and period d is considered. According to the duality theorem in Maxwell equations, we can find the eigenmodes of the SSMPs supported by this surface with the modal expansion method [11]. We are interested in s-polarized surface waves propagating along the x ˆ x + zˆHz . In Region I above the surface (z > 0), direction with the form of E = yE ˆ y and H = xH the electric field component Ey , which is nonradiative and vanishes as z → ∞, can be written as (n)
(n)
EyI = ∑ An eikx x e−qz z ,
(1)
n
(n)
where An is a constant, kx = kx +2nπ /d (n = 0, 1, 2, 3 . . .), here, the diffraction effects are taken into account and the propagation constant of the surface wave kx lies in the first Brillouin zone, #204908 - $15.00 USD (C) 2014 OSA
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10677
(n) (n) i.e., |kx | ≤ π /d and qz = (kx )2 − k02 , k0 is the wave number in free space. The nonradiative property of the fields requires that kx > k0 . In Region II under the surface (z ≤ 0), the EM fields are zero everywhere except inside the grooves. Each groove may be viewed as a planar waveguide with length h and one end closed, and we only consider its fundamental mode in the field expansion. In this way, Ey in the groove can be written as EyII = B cos[k0 (z + h)],
(2)
where B is a constant. This field is actually a sum of two waves propagating along the ±z directions in the groove, and the tangential component of the H field vanishes at the bottom of the groove, for example, Hx = 0 at z = −h. The magnetic field components Hx and Hz can be directly obtained from Ey through Maxwell’s equations that
Hx =
i ∂ Ey , ω μ0 ∂ z
Hz = −
i ∂ Ey , ω μ0 ∂ x
(3)
(4)
where ω is the angular frequency, μ0 is the magnetic permeability in free space. The dispersion relation of the SMPs on the corrugated PMC surfaces can be obtained by imposing the matching conditions of the parallel components of the E and H fields at the interface between Regions I and II. At the interface z = 0, the E field component Ey must be continuous for |x| ≤ a/2 that (n)
∑ An eikx
x
= B cos(k0 h),
(5)
n
and by integrating Eq. (5) over this interval, we can obtain
∑ An Sn = B cos(k0 h),
(6)
n
where
Sn =
1 a
a/2
(n) (n) (n) a . eikx x e−qz z dx = sin c kx 2 −a/2
(7)
The magnetic field component Hx at the interface must be continuous over the whole period. Thus, for |x| ≤ a/2 we have (n)
(n) iqz ik0 ∑ ω μ0 An eikx x = ω μ0 B sin(k0 h), n
(8)
and for a/2 < |x| ≤ d/2
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Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10678
(n)
(n) iqz ∑ ω μ0 An eikx x = 0. n
(9) (n)
By using the orthogonality properties of the functions eikx x , we can obtain the coefficient An = B
k0 a (n)
qz d
sin(k0 h)S0 .
(10)
Substituting Eq. (10) into Eq. (5), and neglecting the diffraction effects, we can obtain
kx2 − k02 k0
=
a 2 S tan(k0 h), d 0
(11)
which is obviously the same with the dispersion relation (9) in Ref. [8]. Considering that no material parameters are involved in Eq. (11) and only the boundary condition is changed from PEC to PMC, the duality theorem holds for the case from SSPPs to SSMPs. 2.2.
Effective medium approximation
Similarly, from the perspective of the effective medium [8], the same dispersion relation could be obtained if we replaced the array of grooves by a single homogeneous but anisotropic layer of thickness h on top of the PMC surface shown in Fig. 1(b). The homogeneous layer would have the following parameters:
μx = a/d, μy = μz = ∞.
(12)
As wave propagates in the grooves along the y and z directions with the velocity of light that √
μ x εy =
√
μ x εz ,
(13)
and thus we can obtain
εy = εz = 1/μx = d/a.
(14)
For the s-polarized plane wave impinging on the surface of a homogeneous layer of thickness h with μ and ε given by Eqs. (12) and (14), the specular reflection coefficient R can be written as Eq. (15) after some straightforward algebras, R=
kz μx kz μx
− k0 + k0 + μkzx ei2k0 h . + k0 − k0 − μkzx ei2k0 h
(15)
This effective medium implies a bound surface state when there is a divergence in the reflection coefficient of the surface for large values to the case kx > k0 and looking at the zeros of the denominator of R, we can obtain the dispersion relation of the surface modes
kx2 − k02 k0
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=
a tan(k0 h). d
(16)
Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10679
1 0.9 0.8 0.7
ω/ ωc
0.6 0.5 0.4 0.3
Light line Analytical value Numerical value (SSMPs) Numerical value (SSPPs)
0.2 0.1 0
0
0.2
0.4
kd/π
0.6
0.8
1
Fig. 2. The normalized dispersion relations (ω (k)) of the SSMPs supported by a 1D array of grooves with geometrical parameters a/d = 0.4, h/d = 0.8. The green line is the theoretical result obtained with Eq. (16), the red line and blue line with open circles render the numerical results of the SSPPs and SSMPs respectively, and the dark line is light line. PEC Boundary
(a)
Bloch Boundary
o
Bloch Boundary
Vacuum
z x
a 2
PEC
a 2
h
Bloch Boundary
Vacuum
a 2
PMC
a 2
h
d
d (c)
PMC Boundary
(b)
(d)
z o
x
Fig. 3. Schematic diagrams of the simulation models. (a) and (b) are the geometries of the periodical groove arrays of width a = 0.4d, depth h = 0.8d separated by a period d on the surface of PEC and PMC with infinite in y direction separately. In which, the gray areas in (a) and (b) represent the vacuum, the red lines in (a) represent the PEC boundary conditions and the light blue lines in (b) are the PMC boundary conditions and the black lines in (a) and (b) along z direction are set as Bloch boundary conditions. (c) and (d) are the surface electric field vectors (the red arrow lines in (c) and in V/m) and magnetic field vectors (the light blue arrow lines in (d) and in A/m) of the unit cell on the xoz plane respectively.
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Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10680
3.
Results and discussions
Note that the Eq. (16) coincides with Eq. (11) in the limit kx a 1. In Fig. 2, we plot the analytical dispersion relation with Eq. (16) and numerical results for the particular case a/d = 0.4, h/d = 0.8. Though the cutoff frequency of the analytical value is slightly higher than the numerical ones due to the reason that the diffraction effects are neglected, it is worth commenting that the dispersion relations between the SSMPs and the SSPPs are identical, and the cutoff frequency ω also approaches π c0 /(2h) according to the dispersion relations and duality of Maxwell’s equations between the PEC and PMC cases. All numerical simulation results in this paper have been performed with the finite element method (FEM) [12]. The corresponding eigenvalue problem is posed in a single unit cell where Bloch boundary conditions are used. The open space is mimicked with PEC or PMC boundary conditions, and the Bloch boundary conditions are employed to model the periodic arrays in the simulations shown in Fig. 3(a) and 3(b). The surface electric and magnetic field vectors on the xoz plane of the PEC and PMC cases are illustrated in Fig. 3(c) and 3(d) with exactly identical field distributions, which further verify the essential characteristics between the SSPPs and SSMPs. 4.
Conclusions
In conclusion, we have shown that the existence of the SSMPs in corrugated PMC surfaces, and the dispersion relation is obtained through the duality of Maxwell’s equations combined with modal expansion method, and an effective medium analytical approach is used to depict the physical insight of the new metamaterial. Thanks to this new concept, a whole brunch of surface modes already known in optical region can be transformed to terahertz and microwave frequency bands. This kind of metamaterial may have a very significant impact on both fundamental and applied researches. Acknowledgments This work was supported by the National Natural Science Foundation of China for Young Scholars under Grant No. 61102033, the Fundamental Research Funds for the Central Universities under Grant NJ20140009, the Aeronautical Science Foundation of China under grant No. 20128052063 and the priority academic program development of Jiangsu Higher Education Institutions.
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Received 16 Jan 2014; revised 7 Apr 2014; accepted 18 Apr 2014; published 25 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010675 | OPTICS EXPRESS 10681
