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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7726–7740
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Plasmonic band structures and optical properties of subwavelength metal/dielectric/metal Bragg waveguides

Chao Li, Yun-Song Zhou, and Huai-Yu Wang  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7726-7740 (2012)
http://dx.doi.org/10.1364/OE.20.007726


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Abstract

In this paper, we applied the band structure theory to investigate the plasmonic band (PB) structures and optical properties of subwavelength metal/dielectric/metal Bragg waveguides in the near infrared range with either dielectric or geometric modulation. The Bloch wave vector, density of states, slowdown factor, propagation length and transmittance are calculated and analyzed. Both the modulations are in favor of manipulating surface-plasmon-polariton (SPP) waves. For the dielectric modulation, the PB structure is mainly formed by SPP modes and possesses a “regular pattern” in which the bands and gaps have a relatively even distribution. For the geometric modulation, due to the strong transverse scattering, the contributions of higher modes have to be considered and the gap widths have a significant increase compared to the dielectric modulation. A larger slowdown factor may emerge at the band edge; especially for the geometric modulation, the group velocity can be reduced to 1/100 of light, and negative group velocity is observed as well. While inside the bands, the slowdown factor is smaller and the bands are flat. The contribution of each eigenmode to the PB structure is analyzed.

© 2012 OSA

1. Introduction

Subwavelength metal/dielectric/metal waveguides, as a kind of plasmonic waveguide, are currently the subject of intensive and widespread study because of their remarkable advantages, such as strong field localization, structural simplicity, convenience of fabrication and integration into optical circuits [Refs. 1–3, and references therein]. When light propagates along a metal/dielectric interface, it will excite a collective oscillation of free electrons at the surface of the metal, generating a field that decays exponentially away from the interface. This mode is called surface plasmon polariton (SPP) [4

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

]. In a subwavelength metal/dielectric/metal waveguide, the case is somehow different, since the lowest mode is the combination of the SPP waves of the two dielectric/metal interfaces, called in-slit SPP, which decays exponentially in the metals and is flat in the dielectric [5

5. F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001). [CrossRef]

12

12. C. Li, Y. S. Zhou, and H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012). [CrossRef]

]. In this paper we mainly discuss the in-slit SPP and simply call it SPP. The propagation of this mode in the waveguide structure is essential in the subwavelength optics [1

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003). [CrossRef]

12

12. C. Li, Y. S. Zhou, and H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012). [CrossRef]

].

Inspired by the unprecedented power of controlling the propagation of light through the band structures in photonic crystals [13

13. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

19

19. Z. Y. Li and K. M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003). [CrossRef]

], periodic modulations have been introduced in subwavelength metal/dielectric/metal waveguides to manipulate the propagation of SPP waves [20

20. Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010). [CrossRef]

31

31. Z. W. Kang, W. H. Lin, and G. P. Wang, “Dual-channel broadband slow surface plasmon polaritons in metal gap waveguide superlattices,” J. Opt. Soc. Am. B 26, 1944–1945 (2009). [CrossRef]

]. Due to the waveguide configuration, the periodic modulations are usually one-dimensional and along the propagation direction. These structures are named subwavelength metal/dielectric/metal Bragg waveguides, and will be called Bragg waveguides (BWGs) in short below. In a BWG, the periodical structure scatters SPP waves, and consequently the plasmonic band (PB) and plasmonic band gap (PBG) may be produced. Within the gap, the propagation of SPP is forbidden [20

20. Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010). [CrossRef]

31

31. Z. W. Kang, W. H. Lin, and G. P. Wang, “Dual-channel broadband slow surface plasmon polaritons in metal gap waveguide superlattices,” J. Opt. Soc. Am. B 26, 1944–1945 (2009). [CrossRef]

]. While within the band, the wave can propagate, but its group velocity may be slowed down [29

29. L. Yang, C. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. 35, 4184–4186 (2010). [CrossRef] [PubMed]

31

31. Z. W. Kang, W. H. Lin, and G. P. Wang, “Dual-channel broadband slow surface plasmon polaritons in metal gap waveguide superlattices,” J. Opt. Soc. Am. B 26, 1944–1945 (2009). [CrossRef]

]. The advantages of the PB structure facilitate designing and fabricating a variety of functional plasmonic structures [21

21. G. Y. Li, L. Cai, F. Xiao, Y. J. Pei, and A. S. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express 18, 10487–10499 (2010). [CrossRef] [PubMed]

26

26. Y. Liu, Y. Liu, and J. Kim, “Characteristics of plasmonic Bragg reflectors with insulator width modulated in sawtooth profiles,” Opt. Express 18, 11589–11598 (2010). [CrossRef] [PubMed]

], such as reflectors, mirrors, filters, and microcavities.

2. Model and the periodic version of MEM

In this section, two kinds of BWGs and the periodic version of MEM are presented.

The BWG models studied in this paper are sketched in Fig. 1. In the x direction, the structures are confined between two perfectly conducting walls, and are symmetric with respect to the central lines. The propagation direction of the waveguide and the periodic modulated direction are the y direction. The basic unit consists of two layers and can be classified into two categories according to their composition: one is called dielectric modulation, see Fig. 1(a), where the slit widths are the same in all the layers but the filling materials are not; the other is called geometric modulation, see Fig. 1(b), where the filling materials are the same in all the layers but the slit widths are not. We only consider the waves with a TM polarization. The basic structural parameters are given in the caption of Fig. 1.

Fig. 1 Sketches of two kinds of BWGs. Both structures possess a central symmetry and are confined in the x direction by two perfectly conducting walls with a confined width L = 2μm. Gray, white and slashed areas represent silver, air, and dielectric, respectively. In white and slashed areas, the dielectric constants ε are 1.0 and 9.0, respectively. The BWGs are along the y direction. The basic unit, with a fixed length h = 1μm, is composed of two adjacent layers. (a) Dielectric modulation: slit width, w(p−1) = w(p) = 0.1μm. (b) Geometric modulation: slit width, w(p−1) = 0.1μm, w(p) = 0.6μm. The band structure calculation is carried out for infinitely long BWGs. The calculation of transmission is done for finitely long BWGs.

For the sake of convenience, the (p – 1)th and pth layers of the dielectric modulation shown in Fig. 1(a) are respectively called air and dielectric regions; while those of the geometric modulation shown in Fig. 1(b) are called narrower and wider regions. Please note that although the (p – 1)th layers in both modulations have different names, they are actually the same in our models.

The permittivity of silver as a function of the wavelength λ is evaluated as εAg = (3.57 – 54.33λ2)+i(−0.083λ +0.921λ3) [32

32. Y. S. Zhou, B. Y. Gu, S. Lan, and L. M. Zhao, “Time-domain analysis of mechanism of plasmon-assisted extraordinary optical transmission,” Phys. Rev. B 78, 081404 (2008). [CrossRef]

] by fitting the experimental data [33

33. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

], which is valid for 0.6 ≤ λ ≤ 1.6 μm.

Now we present the periodic version of MEM. The first step is to express the magnetic field in the lth layer by separating variables,
Hz(l)(x,y)=n=1φn(l)(x)[eikyn(l)(yQ(l1))un(l)+eikyn(l)(yQ(l))dn(l)],l=(p1)andp,
(1)
where φn(l)(x) is the eigenfunction in the x direction. The corresponding derivation and information about φn(l)(x) can be found in Ref. 11

11. C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Investigation of the wave behaviors inside a step-modulated subwavelength metal slit,” Opt. Express 19, 10073–10087 (2011). [CrossRef] [PubMed]

, and will not be readdressed here. Next, by applying the boundary continuum condition and Bloch theorem, we relate the magnetic and electric fields in the (p – 1)th and pth layers to build the coupled equations,
{[eikym(p1)q(p1)um(p1)+dm(p1)]=n=1Kmn(p1,p)[un(p)+eikyn(p)q(p)dn(p)]n=1Jmn(p,p1)[eikyn(p1)q(p1)un(p1)dn(p1)]=kym(p)[um(p)eikym(p)q(p)dm(p)]eikBh[um(p1)+eikym(p1)q(p1)dm(p1)]=n=1Kmn(p1,p)[eikyn(p)q(p)un(p)+dn(p)]eikBhn=1Jmn(p,p1)[un(p1)eikyn(p1)q(p1)dn(p1)]=kym(p)[eikym(p)q(p)um(p)dm(p)],
(2)
where
Kmn(p1,p)=L1ε(p1)φm(p1)+¯φn(p)dx,Jmn(p,p1)=L1ε(p1)φm(p)+¯φn(p1)dx.
(3)
In Eq. (3), φ̄ means the complex conjugate of φ, and φ+ is the adjoint of φ. The factor eikBh is introduced by the Bloch theorem. A numerical overflow may occur if eikBh is computed directly from Eq. (2). To avoid this overflow, the eigen problem is recast, by the S matrix [34

34. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003). [CrossRef]

], into the following form [18

18. Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003). [CrossRef]

,19

19. Z. Y. Li and K. M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003). [CrossRef]

]:
[S11(p)0S21(p)I](u(p1)d(p1))=eikBh[IS12(p)0S22(p)](u(p1)d(p1)),
(4)
where each of the four sub-matrices S11(p), S12(p), S21(p) and S22(p)contains N × N elements with N being the truncation number. Equation (4) is the eigenequation for obtaining Bloch wave vector kB.

The second concerns the precision of our numerical results. In this paper, N = 80 modes are employed in the calculation that is enough to ensure the convergence, since its relative error compared to N = 40 modes is less than 1%. However, the MEM results here are also affected by the two perfectly conducting walls. It is known that a wider confined region provides a more accurate result. Comparisons between the results of L = 2 and 4μm are shown in Figs. 2(a) and 4(a). In both cases the influences introduced by the two perfectly conducting walls are negligible. Thus, L = 2μm and N = 80 used in this paper are sufficient for providing trustworthy results. Although the implantation of perfectly matched layers in MEM is a better solution for this problem, it dramatically complicates the eigenmode calculation. By the way, a good agreement between the results of our calculation and FDTD can be obtain when the Yee’s cell is less than 1nm2 [11

11. C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Investigation of the wave behaviors inside a step-modulated subwavelength metal slit,” Opt. Express 19, 10073–10087 (2011). [CrossRef] [PubMed]

, 12

12. C. Li, Y. S. Zhou, and H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012). [CrossRef]

].

Fig. 2 Numerical results of the dielectric modulation with (a) f = 0.5 and (b) f = 0.1. In each figure, the 1st (top) panel is the DOS; the 2nd panel is the transmittances of BWGs containing 5, 10 and 15 periods; the 3rd panel is the real part (left, green) and imaginary part (right, yellow) of Bloch wave vector kB; and the 4th (bottom) panel is the slowdown factor c/vg (left, green) and propagation length Lp (right, yellow). In the bottom panels, the green dotted horizontal lines are c/vg = 1, and the green circles represent the divergences caused by the abrupt changes of the real parts of kB.

3. PB structures and optical properties of BWGs

In this section, the PB structures and optical properties of BWGs are discussed in the near infrared range.

For convenience of the following discussion, we define f = q(p−1)/h, where q(p−1) is shown in Fig. 1 and h = 1μm is the unit length, as the filling factor in both modulation models. The DOS is expressed as D(λ) =limλ→0N/▵λ = 1/π × |dRe(kB)/dλ|. The propagation length is defined as Lp = 1/(2Im(kB)). The group velocity is expressed as vg = dω/dRe(kB) = dω/dλ × dλ/dRe(kB) = −2πc/λ2 × dλ/dRe(kB), and correspondingly, the slowdown factor is defined as c/vg = −λ2/(2π) × dRe(kB)/, where c is the velocity of light in vacuum. The so-called slow-light phenomena usually refer to that c/vg is about 100. Moreover, to verify the results of the band structure calculation, we calculate the transmittance of a finitely long BWG, which is defined as the output energy divided by the input energy and calculated by the MEM developed in Ref. 11

11. C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Investigation of the wave behaviors inside a step-modulated subwavelength metal slit,” Opt. Express 19, 10073–10087 (2011). [CrossRef] [PubMed]

. In the transmittance calculation, a SPP source is placed in the air (narrower) region 2μm below the finite BWG, and the output energy is detected in the air (narrower) region 2μm above the finite BWG.

We begin with the discussion of the dielectric modulation. The results of two cases with f = 0.5 and 0.1 are presented and analyzed below. Their DOS, transmittances, Bloch wave vectors, slowdown factors and propagation lengths are plotted in Fig. 2.

As shown in Fig. 2, we can easily recognize the bands and gaps in the DOS figures. The magnitudes of DOS are nearly zero in the gaps; at the band edges, DOS peaks are observed. It is worth emphasizing that the DOS in gaps are not zero. For a nondissipative system, such as a one-dimensional photonic crystal, see Eq. (5) or (6) below in Sec. IV, a forbidden mode usually processes a pure imaginary kB, i.e. Re(kB) = 0. Then the gaps, if any (usually appear in the case described by Eq. (5)), should be the wavelength/frequency ranges with zero DOS since dRe(kB) = d0 = 0. While in the present dissipative case, kB is always a complex number, no matter whether the mode is forbidden or not. Within the gaps, as shown in Fig. 2, Re(kB) is nearly either ±π or 0, so that dRe(kB) ≠ 0, resulting a small magnitude of DOS (the abrupt changes of Re(kB) in the gaps are caused by the periodicity of eikBh). Moreover, this magnitude may become larger in the shorter wavelength or higher frequency range. Thus, the definition of gaps in a dissipative system is not as determinative as that in a nondissipative system.

As a test of our DOS curves, the transmission through a finitely long BWG is calculated. As shown in Fig. 2, the light is able to transmit in the bands and is not in the gaps. Our computation also reveals that a finite BWG with 5 periods is sufficiently long to exhibit clearly the photonic crystal effect and the ones with 15 periods shows the gaps accurately.

The imaginary part of kB, Im(kB), can be use to determine the PB structure as well. As shown in Fig. 2, the drastic changes of Im(kB), corresponding to the peaks in DOS, indicates the locations of band edges. Then the larger Im(kB) regions, standing for fast decaying waves, are the gaps, and the smaller Im(kB) regions, representing long propagating waves, are the bands. According to our calculation, Im(kB) is of order of magnitude of 10−1 (the magnitude of Im(kSPP) in the dielectric region) or less within the bands, and is larger with one or two orders within the gaps. Moreover, the bands in the shorter wavelength ranges always possess larger Im(kB), resulting in small transmittances. For example, in Fig. 2(b), among the three transmittance curves, only the one with 5 periods can be used to confirm the existence of bands in the wavelength range between 0.6 and 0.7μm, despite of its small value. This is another major difference from the nondissipative system—even in the bands waves may decay fast. The propagation length Lp is inverse to Im(kB), and its value reflects the attenuation of the wave. Within the gaps, Lp is very small.

Other important information provided by Re(kB) concerns the group velocity vg and slowdown factor c/vg. From the definitions of slowdown factor and DOS, it is known that the curves of these two physical quantities should have a similar shape, so that the slowdown factor also have the ability to determine the bands and gaps like the DOS. In the bands, the group velocities are slowed down with c/vg being 3 ∼ 10. Usually, c/vg and its changing rate become smaller as the wavelength moves from the band edge to band center. Thus, it is expected that the c/vg curve is comparatively smooth around the band center, where signals may travel without dispersion. For instance, the wavelength range 1.27 – 1.37μm of the c/vg curve in Fig. 2(a) can be considered as a flat band. On the other hand, c/vg < 1 is observed in the gaps, which means that the group velocity is superluminal. However, since the superluminal group velocity does not mean superluminal signal velocity [35

35. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

], this phenomenon is not a matter of our concern.

All the results in Fig. 2 are for the confined width L = 2μm except those marked by crosses “×” that are for L = 4μm. Since the crosses fit the solid lines perfectly, our results are reliable.

The variations of the corresponding curves in Fig. 2 for the two f values are similar, except that with the f value decreasing, the widths of both bands and gaps become narrower. It is seen from Fig. 2 that the PB structure of f = 0.1 consists of more and narrower bands and gaps. In order to give an intuitionistic view of how f influences the PB structures and optical properties of BWGs, we plot in Figs. 3(a) and 3(b) the variation of slowdown factor and propagation length as functions of f and λ, respectively.

Fig. 3 Variation of (a) slowdown factor c/vg and (b) propagation length Lp with f and λ in the dielectric modulation. The structural parameters were given in Fig. 1.

The black regions in Figs. 3(a) and 3(b) represent the PBGs. As can be seen, the bands and gaps have a relatively regular and even distribution as f and λ vary. This kind of pattern is a distinguishable feature of the dielectric modulation, and is regarded as “regular pattern”. In Sec. IV, we will show that this “regular pattern” is mainly formed by the SPP modes under a weak transverse scattering. When f and λ are small, see, the lower left corner of Figs. 3(a) and 3(b), both bands and gaps have comparatively smaller widths, showing a denser distribution; in the bands, the slowdown factor is larger but the propagation length is smaller. As f and λ become larger, towards the higher right corner of the figure, the widths of both bands and gaps gradually increase. As mentioned before, although the slowdown factor in the bands becomes smaller, its variation also becomes smoother so as to form a flat band. Especially in a broader band, a flat band may be observed. For example, when f = 0.79, the wavelength range 1.35 – 1.55μm is a flat band. Moreover, the propagation length increases with f significantly, which is less than 5μm for a small f but is nearly 50μm when f closes to 1. The main cause of this significant increase is that the air regions have a smaller absorption of the wave than the dielectric regions, thus the increase of their weight in the unit cells leads to a smaller absorption and a longer propagation.

Next, we discuss the numerical results of the geometric modulation. The DOS, transmittances, Bloch wave vectors, slowdown factors, and propagation lengths as functions of wavelength for f = 0.5 and 0.1 are plotted in Fig. 4.

Fig. 4 Numerical results of the geometric modulation with (a) f = 0.5 and (b) f = 0.1. The gray areas in (b) represent the wavelength ranges for “local modes”. Other captions are the same as those in Fig. 2.

The abrupt changes of kB cause severe oscillations of the DOS and slowdown factor. It is noticed that Fig. 4(b) shows the slow-light phenomena and negative group velocity. But what surprised us more is the gray areas labeled in Fig. 4(b), where both large DOS and large Im(kB) are observed. It seems that the DOS and transmittance curves show contradiction. The large DOS definitely means the modes are allowed in the system; while the large Im(kB) indicates the wave decay quickly. Let us recall the birth of photonic band gaps in photonic crystals, where the gaps are designed to efficiently forbid the atomic spontaneous emission [13

13. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

, 14

14. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

]. For the gray areas in Fig. 4(b), although the modes attenuate quickly, their atomic spontaneous emissions actually are accelerated, rather than forbidden, because of the large DOS. Thus, we take these regions as bands, and call the associated modes as “local modes”. Actually, in Fig. 2(b), we have already encountered this situation as the bands had small transmittances, but the concept of “local modes” have not been mentioned there, because we think their propagation lengths and transmittances are relatively large, compared with the ones here. To identify and clearly observe the “local modes”, a reference, with the propagation length and transmittance being much smaller than 1, should be used.

The variation of the slowdown factors in the bands is similar to that in the dielectric modulation, where larger values are observed at the band edges than in the bands, and they vary smoothly within the bands. Generally speaking, the slow-light phenomenon in a periodic structure is originated from the interference between the forward and backward propagating Bloch waves. However, for the geometric modulation, waves can also engender a transverse resonance in the x direction, raising the value of the slowdown factor. Here the width of the wider regions is 0.6μm. Increasing the width will further raise the slowdown factor. Besides, the propagation length here is also larger than that in the dielectric modulation. This is partly because the two regions are both air-filled which corresponds to small absorption, and partly because a wider air region corresponds to a smaller absorption.

In Figs. 5(a) and 5(b) we plot the slowdown factor and propagation length in the geometric modulation as functions of f and λ, respectively.

Fig. 5 Variation of (a) slowdown factor c/vg and (b) propagation length Lp with f and λ in the geometric modulation. The structural parameters were given in Fig. 1.

Compared with the “regular pattern” shown in Fig. 3 for the dielectric modulation, the PB structure in Fig. 5 presents an irregular distribution, which is mainly resulted from the strong transverse scattering of the SPP and higher modes. The formation of this PB structure will be analyzed in Sec. IV. Also because of this strong transverse scattering, the gaps become much wider compared to the dielectric modulation.

In Fig. 5(a), c/vg merely covers −10 ∼ 20, where the minus sign means negative group velocity. For c/vg out of this range, no matter positive or negative, their magnitudes have the same order as those shown in the insets of Fig. 4(b). For example, in Fig. 5(a), the magnitude of the bright narrow bands is between 40 and 150, while one of the black curves is between −100 and −30 (Please note that these black curves are bands). However, as revealed from the comparison of Figs. 5(a) and 5(b), these bright narrow bands and black curves that possess very large |c/vg| appear at the band edges or in the gaps and have small propagation lengths, usually smaller than 1μm, which means that they are the “local modes”. For the modes with larger propagation lengths, their c/vg curves have similar distributions with those in the dielectric modulation, but have an increase in magnitude. Unlike the Lp distribution of the dielectric modulation in Fig. 3(b), Fig. 5(b) shows an even distribution within the bands, because both the two air-filled regions have a small absorption of waves.

4. PBG formation: modal analysis

Fig. 6 (a) Imaginary part and (b) real part of the Bloch wave vector kB in the dielectric modulation for f = 0.1. Crosses: EIM; Circles: N = 1 MEM; Solid lines: N = 80 MEM.

Next, we turn to discuss the PBG formation in the geometric modulation. In contrast to the dielectric modulation, the EIM is inappropriate for the geometric modulation since the slit widths in the unit cell are not the same. Then taking kx = 0 to neglect the transverse scattering may result in imprecise results. Actually, for the given structural parameters in the present model, the effective refractive indexes in different layers are too close to form a PBG. On the other hand, as we have demonstrated in Ref. 11

11. C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Investigation of the wave behaviors inside a step-modulated subwavelength metal slit,” Opt. Express 19, 10073–10087 (2011). [CrossRef] [PubMed]

, in the geometric modulation the contributions from higher modes to the scattering cannot be neglected. Compared with the dielectric modulation, the transverse scattering here is stronger, which admits that one mode approximation is inapplicable. Thus in the following only the MEM is used to discuss the PBG formation in the cases of f = 0.5 and 0.1.

Fig. 7 Bloch wave vectors kB in the cases of (a) f = 0.5 and (b) f = 0.1 in the geometric modulation.

The curves of f = 0.1 shown in Fig. 7(b) are quite different from those in Fig. 7(a). The length of the narrower region is q(p−1) = 0.1×1μm= 0.1μm. Since it is too short, some higher modes may pass through this region in evanescent forms. Among all possible Bloch waves, only the one with the smallest positive Im(kB) can form the band (see the discussion about the selection of kB at the end of Sec. II). That is to say, at least one mode is able to propagate through the wider regions because the Bloch wave composed of evanescent waves in both layers cannot provide the smallest |Im(kB)|. Apparently, the SPP modes satisfy this condition since they are able to propagate through both layers. The second mode becomes propagating when the wavelength is within 0.6 ∼ 1.2μm as mentioned above. Thus when λ > 1.2μm the anti-symmetric modes have no contribution to the PB structure; this is verified by Fig. 7(b) where the curves of N = 1 and 2 are identical when λ > 1.2μm. However, within the range 0.6 < λ < 1.2μm where the second mode could participate in forming the Bloch wave, the two curves are also basically identical. This is because the Im(kB) of the Bloch wave formed by the symmetric modes is smaller in this wavelength range. But around λ = 0.77 and 1.064μm, the Im(kB) curve with N = 2 has dips while the N = 1 curve has not, so that these dips are mainly introduced by the second mode. The “correction” by the higher modes makes the N = 80 curve have abrupt changes around λ = 0.81 and 1.116μm. Moreover, the comparison between the Im(kB) curves of N = 1, 2 and 3 reveals that the dips at λ = 0.662 and 0.690μm are mainly caused by the third mode. Our numerical calculations confirm that it needs up to N = 5 modes to determine the basic PB structure, and the details have to be “corrected” by the higher modes, which needs at least N = 20 modes to get trustworthy results.

5. Conclusion

So far, a class of technologies has been demonstrated to produce slow light [36

36. T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008). [CrossRef]

], such as quantum interference effects or electromagnetically induced transparency, stimulated Brillouin scattering or stimulated Raman scattering, photonic crystal waveguides, and coupled resonance optical waveguides. However, only the subwavelength metal slit structures are able to make the devices as small as possible so as to easily integrate into the compact optical circuit. Furthermore, the introduction of periodical structures in the subwavelength slit generates the energy gaps and smooth energy bands. Compared with the conventional plasmonic waveguides that suffer from dispersion and losses [37

37. E. P. Fitrakis, T. Kamalakis, and T. Sphicopoulos, “Slow light in insulator-metal-insulator plasmonic waveguides,” J. Opt. Soc. Am. B 28, 2159–2164 (2011). [CrossRef]

39

39. B. Han and C. Jiang, “Plasmonic slow light waveguide and cavity,” Appl. Phys. B: Lasers Opt. 95, 97–103 (2009). [CrossRef]

], the BWG process PBGs and broad bands almost without dispersion, which is very important for practical utilities.

For dielectric modulations, since the transverse scattering in a slit with an unvaried width is weak, the contribution from the SPP modes is dominant, and those of the higher modes are negligible. This one-mode-formed PB structure has a “regular pattern” in which the widths of the bands and gaps become larger as the wavelength and filling factor increase. Therefore, one can design narrow band multi-channel devices with small filling factors, or achieve broad bands and gaps when the filling factor is large. For the model we studied, the slowdown factor in the band is between 3 ∼ 15. A slowdown factor larger than 10 is observed at the band edge; while near the band center, the slowdown factor and its variation become smaller so that the band is flat. In short, when the filling factor is large a broad band is comparatively flat. The propagation length here is mainly affected by the dielectric layers, since they have a larger absorption of waves. When the dielectric layer gets longer the propagation length will become shorter. For the mode with propagation length larger than 5μm, its slowdown factor is between 2 ∼ 8.

For geometric modulations, since the transverse scattering within the BGW is strong and the contributions of higher modes are not negligible, the PB structure is not of the feature of the “regular pattern” and the gap widths increase significantly, which provides us more spaces to design flat bands and defect modes for controlling the output signal. “Local modes” that have large DOS but small propagation length are observed. As for the slowdown factor and propagation length, the conclusion with respect to the dielectric modulation is still suitable here, but they are larger in magnitudes; especially at the band edges, the slowdown factor can be nearly ±100 while the corresponding propagation length is comparatively shorter. For the mode with propagation length larger than 5μm, its slowdown factor is between 3 ∼ 15.

Acknowledgments

This work is supported by the 973 Program of China (Grant No. 2011CB301801) and the National Natural Science Foundation of China (Grant No. 10874124, 11074145).

References and links

1.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003). [CrossRef]

2.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature (London) 440, 508–511 (2006). [CrossRef]

3.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef] [PubMed]

4.

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

5.

F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001). [CrossRef]

6.

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

7.

R. Gordon, “Light in a subwavelength slit in a metal: propagation and reflection,” Phys. Rev. B 73, 153405 (2006). [CrossRef]

8.

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007). [CrossRef]

9.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007). [CrossRef]

10.

C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B 27, 59–64 (2010). [CrossRef]

11.

C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Investigation of the wave behaviors inside a step-modulated subwavelength metal slit,” Opt. Express 19, 10073–10087 (2011). [CrossRef] [PubMed]

12.

C. Li, Y. S. Zhou, and H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012). [CrossRef]

13.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

14.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

15.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997). [CrossRef]

16.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press1995).

17.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

18.

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003). [CrossRef]

19.

Z. Y. Li and K. M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003). [CrossRef]

20.

Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010). [CrossRef]

21.

G. Y. Li, L. Cai, F. Xiao, Y. J. Pei, and A. S. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express 18, 10487–10499 (2010). [CrossRef] [PubMed]

22.

X. L. Zhong, Z. Y. Li, C. Wang, and Y. S. Zhou, “Analytical single-mode model for subwavelength metallic Bragg waveguides,” J. Appl. Phys. 109, 093115 (2011). [CrossRef]

23.

A. Hossieni and Y. Massoud, “A low-loss metal-insulator-metal plasmonic bragg reflector,” Opt. Express 14, 11318–11323 (2006). [CrossRef] [PubMed]

24.

Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007). [CrossRef]

25.

J. Q. Liu, L. L. Wang, M. D. He, W. Q. Huang, D. Y. Wang, B. S. Zou, and S. C. Wen, “A wide bandgap plasmonic Bragg reflector,” Opt. Express 16, 4888–4894 (2008). [CrossRef] [PubMed]

26.

Y. Liu, Y. Liu, and J. Kim, “Characteristics of plasmonic Bragg reflectors with insulator width modulated in sawtooth profiles,” Opt. Express 18, 11589–11598 (2010). [CrossRef] [PubMed]

27.

A. Hosseini, H. Nejati, and Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express 16, 1475–1480 (2008). [CrossRef] [PubMed]

28.

J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17, 20134–20139 (2009). [CrossRef] [PubMed]

29.

L. Yang, C. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. 35, 4184–4186 (2010). [CrossRef] [PubMed]

30.

Y. Xu, A. E. Miroshnichenko, S. Lan, Q. Guo, and L. J. Wu, “Impedance matching induce high transmissionand flat response band-pass plasmonic waveguides,” Plasmonics 6, 337–343 (2011). [CrossRef]

31.

Z. W. Kang, W. H. Lin, and G. P. Wang, “Dual-channel broadband slow surface plasmon polaritons in metal gap waveguide superlattices,” J. Opt. Soc. Am. B 26, 1944–1945 (2009). [CrossRef]

32.

Y. S. Zhou, B. Y. Gu, S. Lan, and L. M. Zhao, “Time-domain analysis of mechanism of plasmon-assisted extraordinary optical transmission,” Phys. Rev. B 78, 081404 (2008). [CrossRef]

33.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

34.

L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003). [CrossRef]

35.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

36.

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008). [CrossRef]

37.

E. P. Fitrakis, T. Kamalakis, and T. Sphicopoulos, “Slow light in insulator-metal-insulator plasmonic waveguides,” J. Opt. Soc. Am. B 28, 2159–2164 (2011). [CrossRef]

38.

D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12, 015002 (2010). [CrossRef]

39.

B. Han and C. Jiang, “Plasmonic slow light waveguide and cavity,” Appl. Phys. B: Lasers Opt. 95, 97–103 (2009). [CrossRef]

OCIS Codes
(230.1480) Optical devices : Bragg reflectors
(230.7380) Optical devices : Waveguides, channeled
(240.6680) Optics at surfaces : Surface plasmons
(290.5825) Scattering : Scattering theory

ToC Category:
Optics at Surfaces

History
Original Manuscript: January 17, 2012
Revised Manuscript: February 18, 2012
Manuscript Accepted: February 20, 2012
Published: March 20, 2012

Citation
Chao Li, Yun-Song Zhou, and Huai-Yu Wang, "Plasmonic band structures and optical properties of subwavelength metal/dielectric/metal Bragg waveguides," Opt. Express 20, 7726-7740 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7726


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References

  1. W. L. Barnes, A. Dereux, T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003). [CrossRef]
  2. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature (London) 440, 508–511 (2006). [CrossRef]
  3. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef] [PubMed]
  4. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  5. F. Villa, T. Lopez-Rios, L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001). [CrossRef]
  6. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]
  7. R. Gordon, “Light in a subwavelength slit in a metal: propagation and reflection,” Phys. Rev. B 73, 153405 (2006). [CrossRef]
  8. Y. Kurokawa, H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007). [CrossRef]
  9. B. Sturman, E. Podivilov, M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007). [CrossRef]
  10. C. Li, Y. S. Zhou, H. Y. Wang, F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B 27, 59–64 (2010). [CrossRef]
  11. C. Li, Y. S. Zhou, H. Y. Wang, F. H. Wang, “Investigation of the wave behaviors inside a step-modulated subwavelength metal slit,” Opt. Express 19, 10073–10087 (2011). [CrossRef] [PubMed]
  12. C. Li, Y. S. Zhou, H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012). [CrossRef]
  13. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]
  14. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]
  15. J. D. Joannopoulos, P. R. Villeneuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997). [CrossRef]
  16. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press1995).
  17. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
  18. Z. Y. Li, L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003). [CrossRef]
  19. Z. Y. Li, K. M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003). [CrossRef]
  20. Y. S. Zhou, B. Y. Gu, H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010). [CrossRef]
  21. G. Y. Li, L. Cai, F. Xiao, Y. J. Pei, A. S. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express 18, 10487–10499 (2010). [CrossRef] [PubMed]
  22. X. L. Zhong, Z. Y. Li, C. Wang, Y. S. Zhou, “Analytical single-mode model for subwavelength metallic Bragg waveguides,” J. Appl. Phys. 109, 093115 (2011). [CrossRef]
  23. A. Hossieni, Y. Massoud, “A low-loss metal-insulator-metal plasmonic bragg reflector,” Opt. Express 14, 11318–11323 (2006). [CrossRef] [PubMed]
  24. Z. Han, E. Forsberg, S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007). [CrossRef]
  25. J. Q. Liu, L. L. Wang, M. D. He, W. Q. Huang, D. Y. Wang, B. S. Zou, S. C. Wen, “A wide bandgap plasmonic Bragg reflector,” Opt. Express 16, 4888–4894 (2008). [CrossRef] [PubMed]
  26. Y. Liu, Y. Liu, J. Kim, “Characteristics of plasmonic Bragg reflectors with insulator width modulated in sawtooth profiles,” Opt. Express 18, 11589–11598 (2010). [CrossRef] [PubMed]
  27. A. Hosseini, H. Nejati, Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express 16, 1475–1480 (2008). [CrossRef] [PubMed]
  28. J. Liu, G. Fang, H. Zhao, Y. Zhang, S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17, 20134–20139 (2009). [CrossRef] [PubMed]
  29. L. Yang, C. Min, G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. 35, 4184–4186 (2010). [CrossRef] [PubMed]
  30. Y. Xu, A. E. Miroshnichenko, S. Lan, Q. Guo, L. J. Wu, “Impedance matching induce high transmissionand flat response band-pass plasmonic waveguides,” Plasmonics 6, 337–343 (2011). [CrossRef]
  31. Z. W. Kang, W. H. Lin, G. P. Wang, “Dual-channel broadband slow surface plasmon polaritons in metal gap waveguide superlattices,” J. Opt. Soc. Am. B 26, 1944–1945 (2009). [CrossRef]
  32. Y. S. Zhou, B. Y. Gu, S. Lan, L. M. Zhao, “Time-domain analysis of mechanism of plasmon-assisted extraordinary optical transmission,” Phys. Rev. B 78, 081404 (2008). [CrossRef]
  33. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
  34. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003). [CrossRef]
  35. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  36. T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008). [CrossRef]
  37. E. P. Fitrakis, T. Kamalakis, T. Sphicopoulos, “Slow light in insulator-metal-insulator plasmonic waveguides,” J. Opt. Soc. Am. B 28, 2159–2164 (2011). [CrossRef]
  38. D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12, 015002 (2010). [CrossRef]
  39. B. Han, C. Jiang, “Plasmonic slow light waveguide and cavity,” Appl. Phys. B: Lasers Opt. 95, 97–103 (2009). [CrossRef]

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