## Plasmonic band structures and optical properties of subwavelength metal/dielectric/metal Bragg waveguides |

Optics Express, Vol. 20, Issue 7, pp. 7726-7740 (2012)

http://dx.doi.org/10.1364/OE.20.007726

Acrobat PDF (2751 KB)

### 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

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. 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]

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

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]

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]

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]

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. 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]

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. 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]

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. 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]

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]

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]

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]

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]

**81**, 015801 (2010). [CrossRef]

**26**, 1944–1945 (2009). [CrossRef]

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]

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]

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]

**66**, 8 (2012). [CrossRef]

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]

## 2. Model and the periodic version of MEM

*p*– 1)th and

*p*th 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.

*λ*is evaluated as

*ε*= (3.57 – 54.33

_{Ag}*λ*

^{2})+

*i*(−0.083

*λ*+0.921

*λ*

^{3}) [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]

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

*μ*m.

*l*th layer by separating variables, where

*x*direction. The corresponding derivation and information about

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]

*p*– 1)th and

*p*th layers to build the coupled equations,

*φ*̄ means the complex conjugate of

*φ*, and

*φ*

^{+}is the adjoint of

*φ*. The factor

*e*

^{ikBh}is introduced by the Bloch theorem. A numerical overflow may occur if

*e*

^{ikBh}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]

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]

**67**, 165104 (2003). [CrossRef]

*N*×

*N*elements with

*N*being the truncation number. Equation (4) is the eigenequation for obtaining Bloch wave vector

*k*.

_{B}*k*. As can be seen from Eq. (4), for each given frequency (or wavelength) there are 2

_{B}*N*solutions of

*k*, among which

_{B}*N*solutions have a positive sign and the other

*N*have a negative sign, representing

*N*forward and

*N*backward propagating Bloch waves. Since these two sets of

*k*have the same absolute values, only the set with positive imaginary parts is considered. We arrange all the

_{B}*k*in a way of the ascending absolute values of their imaginary parts |Im(

_{B}*k*)|, and call the one with the smallest |Im(

_{B}*k*)| the first or the lowest mode, and so on. Then we find that the |Im(

_{B}*k*)| of the second and higher modes are larger than that of the lowest mode by at least one order of magnitude. This fact indicates that their associated Bloch waves, except the lowest mode, decay very fast so that are forbidden in the system, which is physically reasonable. Hereafter, a mode that cannot propagate in the BWG is named forbidden mode. Therefore, only the lowest mode, which has the smallest |Im(

_{B}*k*)|, is chose to analyze the PB structure. Furthermore, the lowest mode may be forbidden for certain frequencies (or wavelengths). As a result, the PBGs are formed. Further discussion about the determination of the bands and gaps in a PB structure will be carried out with specific numerical results in the next section.

_{B}*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 4

*μ*m 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*= 2

*μ*m 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 1nm

^{2}[11

**19**, 10073–10087 (2011). [CrossRef] [PubMed]

**66**, 8 (2012). [CrossRef]

## 3. PB structures and optical properties of BWGs

*f*=

*q*

^{(}

^{p}^{−1)}/

*h*, where

*q*

^{(}

^{p}^{−1)}is shown in Fig. 1 and

*h*= 1

*μ*m is the unit length, as the filling factor in both modulation models. The DOS is expressed as

*D*(

*λ*) =lim

_{▵}

_{λ}_{→0}▵

*N*/▵

*λ*= 1/

*π*× |

*d*Re(

*k*)/

_{B}*dλ|*. The propagation length is defined as

*L*= 1/(2Im(

_{p}*k*)). The group velocity is expressed as

_{B}*v*=

_{g}*dω/d*Re(

*k*) =

_{B}*dω/dλ*

*× dλ/d*Re(

*k*) = −2

_{B}*πc/λ*

^{2}×

*dλ/d*Re(

*k*), and correspondingly, the slowdown factor is defined as

_{B}*c/v*= −

_{g}*λ*

^{2}/(2

*π*) ×

*d*Re(

*k*)/

_{B}*dλ*, where

*c*is the velocity of light in vacuum. The so-called slow-light phenomena usually refer to that

*c/v*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

_{g}**19**, 10073–10087 (2011). [CrossRef] [PubMed]

*μ*m below the finite BWG, and the output energy is detected in the air (narrower) region 2

*μ*m above the finite BWG.

*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.

*k*, i.e. Re(

_{B}*k*) = 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

_{B}*d*Re(

*k*) =

_{B}*d*0 = 0. While in the present dissipative case,

*k*is always a complex number, no matter whether the mode is forbidden or not. Within the gaps, as shown in Fig. 2, Re(

_{B}*k*) is nearly either ±

_{B}*π*or 0, so that

*d*Re(

*k*) ≠ 0, resulting a small magnitude of DOS (the abrupt changes of Re(

_{B}*k*) in the gaps are caused by the periodicity of

_{B}*e*

^{ikBh}). 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.

*k*, Im(

_{B}*k*), can be use to determine the PB structure as well. As shown in Fig. 2, the drastic changes of Im(

_{B}*k*), corresponding to the peaks in DOS, indicates the locations of band edges. Then the larger Im(

_{B}*k*) regions, standing for fast decaying waves, are the gaps, and the smaller Im(

_{B}*k*) regions, representing long propagating waves, are the bands. According to our calculation, Im(

_{B}*k*) is of order of magnitude of 10

_{B}^{−1}(the magnitude of Im(

*k*) 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(

_{SPP}*k*), 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 0.7

_{B}*μ*m, 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

*L*is inverse to Im(

_{p}*k*), and its value reflects the attenuation of the wave. Within the gaps,

_{B}*L*is very small.

_{p}*k*) concerns the group velocity

_{B}*v*and slowdown factor

_{g}*c/v*. 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

_{g}*c/v*being 3 ∼ 10. Usually,

_{g}*c/v*and its changing rate become smaller as the wavelength moves from the band edge to band center. Thus, it is expected that the

_{g}*c/v*curve is comparatively smooth around the band center, where signals may travel without dispersion. For instance, the wavelength range 1.27 – 1.37

_{g}*μ*m of the

*c/v*curve in Fig. 2(a) can be considered as a flat band. On the other hand,

_{g}*c/v*< 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], this phenomenon is not a matter of our concern.

_{g}*L*= 2

*μ*m except those marked by crosses “×” that are for

*L*= 4

*μ*m. Since the crosses fit the solid lines perfectly, our results are reliable.

*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.

*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 – 1.55

*μ*m is a flat band. Moreover, the propagation length increases with

*f*significantly, which is less than 5

*μ*m for a small

*f*but is nearly 50

*μ*m 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.

*f*= 0.5 and 0.1 are plotted in Fig. 4.

*k*are no longer smooth when

_{B}*f*closes to 0. As shown in Fig. 4(b) for

*f*= 0.1, the Im(

*k*) curve has four dips at 0.662, 0.690, 0.812 and 1.116

_{B}*μ*m. At first glance, it seems that they are caused by numerical error. To verify the correctness of the results, we enlarge the transmittance figure at the corresponding wavelength range in the inset of Fig. 4(b). It is seen that the first two dips do correspond to two small transmission peaks. As for the latter two dips, no transmission peak is observed (not shown in the inset). This is because these modes, as will be proved in the next section, are composed of both symmetric and anti-symmetric modes, thus the SPP source used in the transmittance calculation cannot excite them. As a testing, we adopt

*N*= 120 modes in the modal expansion to check the precision of

*N*= 80 modes result, and the two calculated curves fit each other very well. Thus

*N*= 80 modes are trustworthy. Again, the results of

*L*= 4

*μ*m are plotted in the third panel of Fig. 4(a) to confirm that the influence brought by the two perfectly conducting walls is negligible.

*k*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(

_{B}*k*) 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(

_{B}*k*) 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

_{B}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]

*x*direction, raising the value of the slowdown factor. Here the width of the wider regions is 0.6

*μ*m. 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.

*f*and

*λ*, respectively.

*c/v*merely covers −10 ∼ 20, where the minus sign means negative group velocity. For

_{g}*c/v*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 |

_{g}*c/v*| appear at the band edges or in the gaps and have small propagation lengths, usually smaller than 1

_{g}*μ*m, which means that they are the “local modes”. For the modes with larger propagation lengths, their

*c/v*curves have similar distributions with those in the dielectric modulation, but have an increase in magnitude. Unlike the

_{g}*L*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.

_{p}## 4. PBG formation: modal analysis

**19**, 10073–10087 (2011). [CrossRef] [PubMed]

*x*direction into two groups: guided modes, of which the fields oscillate in the slit but decay in the metal, and radiation modes, of which the fields oscillate in both the slit and metal. The second is that if the eigenmodes are arranged according to the absolute values of the imaginary parts of their wave vectors |Im(

*k*)| (a greater |Im(

_{y}*k*)| means a faster attenuation), the symmetric and anti-symmetric modes appear alternatively. The SPP mode is always the lowest one, which is a guided and symmetric mode. In the case where the structure and light source are symmetrically arranged, the anti-symmetric modes are forbidden [11

_{y}**19**, 10073–10087 (2011). [CrossRef] [PubMed]

*y*direction. The former has very small |Im(

*k*)| so that can propagate for a long distance; while the latter has large |Im(

_{y}*k*)| and decays quickly. In the present paper, the SPP modes are always propagation ones. For the higher order modes, the statement that “the slit width should be equal to an integer multiple of half the wavelength” can be used as an approximate criterion of determining whether they can propagate or not [10

_{y}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]

*w*

^{(}

^{p}^{−1)}= 0.1

*μ*m only allows the SPP mode to propagate. While for the wider region with

*w*

^{(}

^{p}^{)}= 0.6

*μ*m, the second mode (the first anti-symmetric mode) can also propagate when the wavelength is in the range of 0.6 – 1.2

*μ*m.

*f*= 0.1 in the dielectric modulation. For comparison, the

*k*shown in Fig. 2(b), calculated by MEM with

_{B}*N*= 80, is redrawn in Fig. 6, along with the results of EIM and the single-mode MEM. As can be seen, the three curves agree with each other well, which indicates that the latter two approximation methods also give relatively accurate results. We would like to take a look at their theoretical basis. The EIM transfers a BWG into an equivalent one-dimensional photonic crystal according to the propagation properties of the SPP mode in each layer by

*k*

_{0}= 2

*π/λ*is the wave vector in vacuum. Without loss of generality, the dispersion relation of a one-dimensional photonic crystal is expressed by the well known Kronig–Penney relation:

*k*= 0, Eq. (5) is simplified into the following form that is commonly used in the EIM:

_{x}*k*= 0 means that the information of the transverse scattering is completely omitted. This negligence is reasonable in the case of dielectric modulation because the slit widths of all the layers are the same, and consequently the transverse scattering is very weak [11

_{x}**19**, 10073–10087 (2011). [CrossRef] [PubMed]

**19**, 10073–10087 (2011). [CrossRef] [PubMed]

*N*= 1 in Eq. (2), then the dispersion relation of the one mode MEM can be obtained analytically, which has the same expression as Eq. (5) but with

*ε̃*

^{(}

^{p}^{−1)}and

*ε̃*

^{(}

^{p}^{)}, compared to EIM results from Eq. (6), the

*N*= 1 MEM produces more accurate results, which also manifests that the PB structure in the dielectric modulation is mainly formed by the interference of forward and backward SPP modes. By the way, the slight deviation of EIM results from FDTD simulation has been mentioned in our previous work [20

**81**, 015801 (2010). [CrossRef]

*k*= 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

_{x}**19**, 10073–10087 (2011). [CrossRef] [PubMed]

*f*= 0.5 and 0.1.

*f*= 0.5, Fig. 7(a) shows the convergence with the increasing modes, which manifests that the higher modes have more or less contributions to the PBG formation. The lower modes lay down the basic PB structure, and the higher modes “correct” it gradually. Moreover, as can be seen from Fig. 7(a), the curves of

*N*= 1 and

*N*= 2 are almost identical, indicating that the second mode (the first anti-symmetric mode) has little contribution to the formation. This is because the necessary condition for a mode to participate in the PBG formation is to form a Bloch wave. For a narrower region with

*w*

^{(}

^{p}^{−1)}= 0.1

*μ*m, all the higher modes are evanescent except the SPP mode; and

*q*

^{(}

^{p}^{−1)}= 0.5 × 1

*μ*m= 0.5

*μ*m is a sufficiently long layer length for the higher modes to attenuate to negligible values. Therefore, all the antisymmetric modes are forbidden by the narrower slit (Please keep in mind that the symmetric and anti-symmetric modes are decoupled in our symmetric model, which means the SPP mode cannot excite the anti-symmetric modes). The higher symmetric modes, although evanescent in the narrower slit, participate the scattering and “correct” the PBG formation since the boundary conditions demand the SPP mode to excite them for fulfilling the continuum at the waveguide junctures. For example, the third mode (the second symmetric mode) is excited and cause the big difference between the results of

*N*= 2 (or

*N*= 1) and

*N*= 3, because the SPP mode only is insufficient to link smoothly the fields between the layer boundary. This point consists with the conclusion that the one mode approximation is inapplicable for the geometric modulation.

*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)}= 0.1×1

*μ*m= 0.1

*μ*m. 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(

*k*) can form the band (see the discussion about the selection of

_{B}*k*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(

_{B}*k*)|. 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 ∼ 1.2

_{B}*μ*m as mentioned above. Thus when

*λ*> 1.2

*μ*m 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*> 1.2

*μ*m. However, within the range 0.6 <

*λ*< 1.2

*μ*m where the second mode could participate in forming the Bloch wave, the two curves are also basically identical. This is because the Im(

*k*) of the Bloch wave formed by the symmetric modes is smaller in this wavelength range. But around

_{B}*λ*= 0.77 and 1.064

*μ*m, the Im(

*k*) curve with

_{B}*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 1.116

*μ*m. Moreover, the comparison between the Im(

*k*) curves of

_{B}*N*= 1, 2 and 3 reveals that the dips at

*λ*= 0.662 and 0.690

*μ*m 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

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]

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

*μ*m, its slowdown factor is between 2 ∼ 8.

*μ*m, its slowdown factor is between 3 ∼ 15.

## Acknowledgments

## References and links

1. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) |

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) |

3. | E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science |

4. | H. Raether, |

5. | F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B |

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 |

7. | R. Gordon, “Light in a subwavelength slit in a metal: propagation and reflection,” Phys. Rev. B |

8. | Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B |

9. | B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B |

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 |

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 |

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 |

13. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

14. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

15. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) |

16. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

17. | K. Sakoda, |

18. | Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E |

19. | Z. Y. Li and K. M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B |

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 |

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 |

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. |

23. | A. Hossieni and Y. Massoud, “A low-loss metal-insulator-metal plasmonic bragg reflector,” Opt. Express |

24. | Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. |

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 |

26. | Y. Liu, Y. Liu, and J. Kim, “Characteristics of plasmonic Bragg reflectors with insulator width modulated in sawtooth profiles,” Opt. Express |

27. | A. Hosseini, H. Nejati, and Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express |

28. | J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express |

29. | L. Yang, C. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. |

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 |

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 |

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 |

33. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

34. | L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A |

35. | L. Brillouin, |

36. | T. F. Krauss, “Why do we need slow light?” Nat. Photonics |

37. | E. P. Fitrakis, T. Kamalakis, and T. Sphicopoulos, “Slow light in insulator-metal-insulator plasmonic waveguides,” J. Opt. Soc. Am. B |

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. |

39. | B. Han and C. Jiang, “Plasmonic slow light waveguide and cavity,” Appl. Phys. B: Lasers Opt. |

**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

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

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