## Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating

Optics Express, Vol. 18, Issue 12, pp. 13258-13270 (2010)

http://dx.doi.org/10.1364/OE.18.013258

Acrobat PDF (1312 KB)

### Abstract

A metal/multi-insulator/metal waveguide plasmonic Bragg grating with a large dynamic range of index modulation is investigated analytically and numerically. Theoretical formalism of the dispersion relation for the present and general one-dimensional gratings is developed for TM waves in the vicinity of each stop band. Wide-band and narrow-band designs with their respective FWHM bandwidths of 173.4 nm and < 3.4 nm in the 1550 nm band using a grating length of < 16.0 *µ*m are numerically demonstrated. Time-average power vortexes near the silica-silicon interfaces are revealed in the stop band and are attributed to the contra-flow interaction and simultaneous satisfactions of the Bragg condition for the incident and backward-diffracted waves. An enhanced forward-propagating power is thus shown to occur over certain sections within one period due to the power coupling from the backward-diffracted waves.

© 2010 Optical Society of America

## 1. Introduction

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

2. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. **22**(7), 475–477 (1997). [CrossRef] [PubMed]

3. S. A. Maier, “Plasmonics: the promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. **12**, 1671–1677 (2006). [CrossRef]

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

6. S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express **13**(12), 4674–4682 (2005). [CrossRef] [PubMed]

9. A. Hosseini and Y. Massoud, “A low-loss metal-insulator-metal plasmonic Bragg reflector,” Opt. Express **14**(23), 318–323 (2006). [CrossRef]

13. Z. Fu, Q. Gan, K. Gao, Z. Pan, and F. J. Bartoli, “Numerical investigation of a bidirectional wave coupler based on plasmonic Bragg gratings in the near infrared domain,” J. Lightwave Technol. **26**(22), 3699–3703 (2008). [CrossRef]

6. S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express **13**(12), 4674–4682 (2005). [CrossRef] [PubMed]

7. A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact bragg gratings for long-range surface plasmon polaritons,” J. Lightwave Technol. **24**(2), 912–918 (2006). [CrossRef]

12. J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,” Opt. Express **16**(1), 413–425 (2008). [CrossRef] [PubMed]

*µ*m. More recently, we proposed a metal/multi-insulator/metal (MMIM) waveguide plasmonic grating where design examples with the grating lengths of ≤ 6.8

*µ*m and the FWHM bandwidths of approximately 9 nm in the 1310 nm band were numerically demonstrated [14

14. Y.-J. Chang and G.-Y. Lo, “A narrow band metal—multi-insulator—metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett. **22**, 634–636 (2010). [CrossRef]

## 2. Structure Description

*x*-dependence of the Si stripe width is described by

*A*

_{0}is the unperturbed Si stripe width, Λ denotes the grating period, and

*A*=

_{M}*hA*

_{0}with

*h*being the width modulation depth (0 <

*h*< 1). For simplicity, the silver boundaries are conformal to the Si stripe at a fixed gap distance

*w*

_{gap}. In between the Si stripe and the silver is the lower index region such as silica. The guided mode supported by the MMIM waveguide is symmetrically excited by a 450 nm-wide Si waveguide followed by a 700 nm-long linearly-tapered MMIM transition section. The gap width associated with the input/output linearly-tapered MMIM transition is fixed at 80 nm for achieving a high transmission efficiency.

## 3. Formulation of the Electromagnetic Problem

*ε*

_{eff}(or equivalently effective refractive index

*N*

_{eff}) of the fundamental mode associated with every single linear section, the grating geometry may be described completely in terms of the longitudinally-modulated medium,

*is the average value of*ε ¯

*ε*

_{eff}(

*x*),

*δ*is the effective permittivity modulation depth,

_{M}**K**= 2

*π*/Λ

**â**

*is the grating vector, and*

_{x}**x**=

*x*

**â**

*is the position vector with*

_{x}**â**

*being the unit vector in the*

_{x}*x*direction. The 2-D problem may thus be treated as a 1-D grating (i.e., ∂/∂

*= 0 and ∂/∂*

_{y}*= 0) with sinusoidally-stratified dielectric medium and can be analytically described by Hill’s differential equation for TM waves [15*

_{z}15. C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microw. Theory Tech. **13**(3), 297–302 (1965). [CrossRef]

*ξ*=

*πx*/Λ,

*f*(

*ξ*) = (1−

*δ*cos2

_{M}*ξ*)

^{−1/2}

*I*(

*x*),

*I*(

*x*) in the expression of

*f*(

*ξ*) denotes the longitudinal field variation associated with the

*z*-directed magnetic field and satisfies the following differential equation

*k*

_{0}is the free-space wave vector. Note that when the modulation is absent,

*δ*= 0 and all the coefficients

_{M}*c*vanish except

_{n}*c*

_{0}.

*π*)

*F*(

*κ*Λ/

*π*) is the Fourier transform of

*f*(

*ξ*) with

*κ*being the Bloch wave vector. An infinite set of homogeneous difference equations for any

*κ*may then be obtained using the fact that

*κ*Λ/

*π*=

*q*, where

*p*is a positive integer and

*q*an integer, may be obtained based on the two-mode approximation where a 2×2 truncation in the infinite set of difference equations generated from Eq. (7) is applied. For a non-trivial solution to exist, the determinant of the resultant 2×2 matrix must vanish, yielding

*u*= (

*p*−

*q*)/2,

*v*= −(

*p*+

*q*)/2,

*u*,

*v*∈ ℤ,

*c*is the Fourier series coefficient given in Eq. (4), and the shorthand notation

_{p}*D*,

_{m}*m*∈ ℤ is given by

*κ*. It is noted that in the derivation of Eq. (9), the propagation in the direction of periodic variation of the relative permittivity is assumed for simplicity. Using the fact that

*p*equals

*q*. They correspond to stop bands along

*m*= 0 unperturbed line in the

*κ*Λ/

*π*diagram. The dispersion relation in the vicinity of

*q*=

*p*. Physically, Eq. (13) represents the resonant coupling between the plane wave components

*m*= 0 and

*m*= −

*p*. Since the center of the stop band occurs at

*λ*may be estimated by

_{B}*λ*≈2√

_{B}*p*in the limit as

*δ*

^{2}

_{M}becomes negligibly small.

*κ*Λ/

*π*diagram. It is a strong function of the grating period Λ, average effective index

*, and index modulation depth*ε ¯

*δ*for a fixed operating wavelength. The largest bandgap occurs at

_{M}*p*≥ 2 becomes smaller with the decreasing higher-order coefficient

*c*. Therefore, and similar to that of the conventional dielectric grating, the narrow-band characteristic associated with the MMIM configuration can be achieved by increasing Λ, increasing

_{p}*, or minimizing*ε ¯

*δ*so as to weaken the grating coupling strength.

_{M}## 4. Results and Discussions

*E*(

_{x}*y*) is odd,

*H*(

_{z}*y*) and

*E*(

_{y}*y*) are even functions] was considered. Figure 2 illustrates the isometric plots of the real and imaginary parts of the

*N*

_{eff}associated with the fundamental odd vector parity mode for an unperturbed linear MMIM waveguide with varying

*w*

_{gap}and

*w*

_{Si}at a free-space wavelength of

*λ*

_{0}= 1550 nm. The refractive indices

*n*

_{Si}= 3.5 and

*n*

_{silica}= 1.46 were assumed and the dielectric function of the silver was taken as a complex fit to the empirical data reported in [16

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

*N*

_{eff}values were calculated based on the rigorous transmission-line network approach in conjunction with the transverse resonance condition [17

17. Y.-J. Chang, T. K. Gaylord, and G.-K. Chang, “Attenuation in waveguides on FR-4 boards due to periodic substrate undulations,” Appl. Opt. **46**(12), 2234–2243 (2007). [CrossRef] [PubMed]

*w*

_{gap}, Re[

*N*

_{eff}] increases and the magnitude of Im[

*N*

_{eff}] (|Im[

*N*

_{eff}]|) decreases with the increase in

*w*

_{Si}. This must be the case since the increase in

*w*

_{Si}comes with the increasing fractions of the total electromagnetic energy being confined in the Si stripe. On the other hand, for a fixed

*w*

_{Si}, the decrease in

*w*

_{gap}would lead to larger Re[

*N*

_{eff}] and |Im[

*N*

_{eff}]|. This is due to increasing fractions of the total electromagnetic energy entering into the metal upon decreasing

*w*

_{gap}, resulting in slightly larger Re[

*N*

_{eff}] and higher |Im[

*N*

_{eff}]| that accounts for the ohmic loss inside the metal. Hence for a fixed

*w*

_{Si}, MMIM configuration bears the same characteristic as that of conventional MIM plasmonic waveguide where Re[

*N*

_{eff}] and |Im[

*N*

_{eff}]| increase with the decreasing dielectric thickness.

*N*

_{eff}] difference for all fixed

*w*

_{Si}values in Fig. 2(a) is merely 0.1874 [2.6735 at (

*w*

_{gap},

*w*

_{Si}) = (160,100) to 2.8609 at (

*w*

_{gap},

*w*

_{Si}) = (50,100)], geometries with a fixed

*w*

_{gap}and varying

*w*

_{Si}may offer a larger dynamic range of index modulation of up to 0.7692 [from 2.6735 at (

*w*

_{gap},

*w*

_{Si}) = (160,100) to 3.4427 at (

*w*

_{gap},

*w*

_{Si}) = (160,400)]. The rate of change of Re[

*N*

_{eff}] increases rapidly with

*w*

_{Si}≤ 200 nm while the magnitude of Im[

*N*

_{eff}] remains < 5.042×10

^{−5}for

*w*

_{Si}≥ 200 nm.

*N*

_{eff}over the spans of

*w*

_{gap}and

*w*

_{Si}shown in Fig. 2, both wide-band and narrow-band designs operating in the 1550 nm band are numerically demonstrated. The Bragg wavelength is estimated using Eq. (14) and the

*N*

_{eff}calculations shown in Fig. 2, which will be elaborated in detail later for the narrow-band designs. Figure 3 shows the transmission spectra of an MMIM grating with Λ,

*A*

_{0},

*A*, and

_{M}*w*

_{gap}being 950, 250, 90, and 130 nm, respectively, and different numbers of periods. The transmission spectra were obtained based upon 2-D Finite Element Method simulations (Comsol Multiphysics) with the maximum mesh size of 20 nm in the entire computational domain so as to achieve the power convergence to the 4th decimal digit. The refractive indices of Si and silica were assumed constant across the spectrum of interest. For the 9-period case, the stop band is centered at

*λ*

_{0}= 1552.3 nm and the FWHM bandwidth is 173.4 nm. The pass band ripple becomes smaller as the operating wavelength deviates further away from the stop band.

*c*,

_{n}*n*= 1,2, ⋯. Accordingly, a higher Bragg order would lead to a narrower stop bandwidth at a sacrifice of a longer grating length for a fixed Bragg wavelength. Thus a trade-off between the Bragg order and the grating period always exists if a narrower bandwidth is desired and the constituent materials are unaltered. With the understanding of the narrow-band characteristics, three design examples for narrow-band optical notch filtering were numerically demonstrated. In these examples, the unperturbed Si width

*A*

_{0}and the constituent materials are identical to those used in the wide-bandwidth case presented above. All cases have 11 periods and provide a minimum extinction ratio of over 16.08 dB at the center of their respective stop bands. In particular, the minimum transmission in case 1 is smaller than −37.60 dB. Because of the sinusoidal apodization, the largest passband ripple among these cases is reduced to less than 0.1 dB over the wavelength span of 1500 nm to 1600 nm.

*≈ ∑*ε ¯

^{n}_{i=1}Re[

*N*

_{eff, i}] ·

*g*, where

_{i}*N*

_{eff, i}is the effective mode index of the fundamental mode associated with the

*i*-th unperturbed section of length

*l*and

_{i}*g*=

_{i}*l*/Λ is the corresponding weighting factor. The √

_{i}*value in case 2 is then found to be 3.3532. Accordingly, the 6th Bragg order is satisfied with the calculated Bragg wavelength*ε ¯

*λ*= 1570.41 nm, which is close to the simulated value of 1550.4 nm. Decreasing

_{B}*w*

_{gap}or increasing

*A*enhances the index modulation. When incorporated with the Λ adjustment,

_{M}*w*

_{gap}and

*A*add additional degrees of freedom for the Bragg wavelength design. This is demonstrated in cases 1 and 3 in Fig. 4. It is worth mentioning that the FWHM bandwidth could be fairly insensitive to the variation in the period for both the narrow-band and wide-band designs, although the rate of change is larger for the former case due to its narrow-band nature. Take the case 2 in Fig. 4 as an example, the FWHM bandwidth increases (decreases) approximately 0.5 nm (0.65 nm) for a decrease (an increase) of 50 nm in the period while in the wide-band design, it increases (decreases) by approximately 13 nm (4 nm) for a 50-nm decrease (increase) in the period.

_{M}*P*

_{norm}, where

*P*

_{norm}distributions inside the 6th unit cell of case 2 in Fig. 4 at

*λ*

_{0}= 1600 nm are shown in Fig. 5(a) as a representative case. Power interchange between the Si stripe and the silica gap regions, as indicated by the arrows in the figure, is observed and occurs repeatedly throughout the entire grating. A quantitative description of the power interchange may be obtained through the calculation of the

*x*-directed time-average power in the Si stripe and the silica gap regions over the span of one period. Figure 5(b) depicts the normalized

*x*-component of the time-average power

*P*(

_{x}*x*), where

*P*

_{in}is the input power, at

*λ*

_{0}= 1600 nm as a function of the propagation distance within the 6th unit cell. The co-flow coupling of

*P*between the Si stripe and the silica gaps is immediately apparent.

_{x}*P*in the silica gap regions (

_{x}*P*

_{x, gap}) reaches its maximum at the narrowest Si stripe width where

*P*in the Si stripe (

_{x}*P*

_{x, Si}) has the minimum. Note that the

*P*

_{x, gap}shown in Fig. 6 is the sum of the normalized

*x*-directed power in the upper and lower gap regions. Although not shown in this paper, the integration of the

*y*-directed time-average Poynting vector over

*y*at each

*x*position within one period vanishes because of the reflection symmetry of the structure with respect to the

*x*axis and the symmetric excitation mentioned earlier. Hence the summation of

*P*

_{x, gaps}and

*P*

_{x, Si}would effectively represent the normalized total time-average power at each

*x*position and remains above 0.976 inside the 6th cell, which confirms the low-loss nature in the pass band.

*P*

_{x, Si}and

*P*

_{x, gaps}exhibit forward- and backward-propagating characteristics within each unit cell. Fig. 6 shows the

*P*

_{norm}distributions and the normalized

*x*-directed time-average power within the 6th unit cell at the Bragg wavelength of 1550.4 nm. The 8 vortexes near the edges of the Si stripe in Fig. 6(a) correspond to the

*x*positions of the crests and valleys of

*P*

_{x, Si}and

*P*

_{x, gaps}curves in Fig. 6(b). The co-existence of the positive (forward-propagating)

*P*

_{x, Si}(

*P*

_{x, gaps}) and the negative (backward-propagating)

*P*

_{x, gaps}(

*P*

_{x, Si}) at the same x position indicates the occurrence of contra-flow coupling, as has been suggested by the hyperbolic form of the dispersion relation, Eq. (12). On the other hand, a one-to-one correspondence exists between the

*x*positions at which the

*P*

_{x, Si}(

*P*

_{x, gaps}) undergoes zero-crossing in Fig. 6(b) and the 4 power nulls along the center line of the Si stripe (at the silver-silica interfaces) in Fig. 6(a). The zero crossings shown in Fig. 6(b) are formed due to the cancellation of the forward- and backward-propagating

*P*at these

_{x}*x*positions. Likewise, power nulls found along the

*x*axis in the Si stripe and at the silver-silica interfaces are where the cancellations of

*x*- and

*y*-directed time-average power occur simultaneously.

*P*

_{x, Si}around

*x*= 0.4Λ exceeds unity in Fig. 6(b). The physical interpretation may be made in terms of the behavior of the backward-diffracted waves at the Bragg wavelength. The plane-wave spectrum calculated at the end of the 6th unit cell as a representative case is shown in Fig. 7. At this

*x*position, the plane-wave component with

*k*= 0 is dominant whereas another two components at

_{y}*k*/(

_{y}*k*

_{0}√

*) = ±0.14324 are also appreciable. The diffracted angles*ε ¯

*θ*associated with these components are 0° and ±8.24° (measured from the −

*x*axis), respectively. In other words, the diffracted wave vectors are either parallel to or nearly parallel to the −

*x*axis and are governed by the Floquet condition. It is worth mentioning that the plane-wave spectrum varies with the

*x*position. Specifically, slight changes in the diffracted angles and the normalized intensity associated with each plane-wave component were observed across one period. This is expected since the interference pattern produced by all the Floquet waves has to be consistent with the sinusoidally-modulated index variation that produces them in the first place.

*k*= 0 (or equivalently

_{y}*θ*= 0°), its wave vector is found to be

*σ*= |

**β**_{inc}−6

**K**| = 1.33969×10

^{7}(rad/m), where

**β**_{inc}is the incident wave vector, and the corresponding phase velocity refractive index is

*N*= 3.3057. Using these values and the fact that the grating slant angle

_{σ}*ϕ*= 0° (seen by the backward-diffracted wave) measured from the −

*x*axis, the backward-diffracted wave is found to satisfy the general Bragg condition

*λ*

_{0}= 1550.4 nm, Λ = 1405 nm, and

*m*= 5.9914 ≈ 6. Note that the refractive index of the backward-diffracted wave

*N*= 3.3057 is very close to the average effective refractive index of the grating, √

_{σ}*= 3.3069 (obtained at*ε ¯

*λ*

_{0}= 1550.4 nm and

*m*= 6). This also ensures the satisfaction of the 6th Bragg order. Hence the backward-diffracted wave satisfies the same Bragg condition as does the original incident wave (

**β**_{inc}), producing a forward-diffracted wave that is characterized by

*=*

**σ**′*−6*

**σ****K**. For other waves with diffracted angles θ≠0°, the general Bragg condition is nearly satisfied because of their small angles of diffraction with respect to the −

*x*axis. The backward-diffracted wave couples its power back to the forward-diffracted wave whose power would consequently add up to that of the original incident wave, thus increasing the normalized forward-propagating power to be over unity. This happens to all unit cells before the 7th. Beyond the 6th unit cell, the normalized

*P*

_{x, Si}and

*P*

_{x, silica}do not exceed unity due to decreasing forward-propagating power flow. Meanwhile, the total normalized power

*P*

_{x, total}does obey the power conservation law and decreases with the increasing propagation distance. Hence the formation of the

*P*

_{norm}vortexes seen in the figure is a direct result of simultaneous satisfactions of the same Bragg condition for the incident and backward-diffracted waves in the metal/multi-insulator/metal configuration. As the operating wavelength gradually deviates from

*λ*, the vortexes and

_{B}*P*

_{norm}nulls gradually fade away since the Bragg condition can no longer be satisfied completely, thus allowing more power transmission inside the Si stripe. The vortexes vanish completely in the pass band.

## 5. Summary

*µ*m, all using the MMIM configuration and the same material system (Si, silica, and silver).

## Acknowledgments

## References and links

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

2. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. |

3. | S. A. Maier, “Plasmonics: the promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. |

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

5. | B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. |

6. | S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express |

7. | A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact bragg gratings for long-range surface plasmon polaritons,” J. Lightwave Technol. |

8. | J.-W. Mu and W.-P. Huang, “A low-loss surface plasmonic Bragg grating,” J. Lightwave Technol. |

9. | A. Hosseini and Y. Massoud, “A low-loss metal-insulator-metal plasmonic Bragg reflector,” Opt. Express |

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

11. | J.-Q. Liu, L.-L. Wang, M.-D. He, W.-Q. Huang, D. Wang, B. S. Zou, and S. Wen, “A wide bandgap plasmonic Bragg reflector,” Opt. Express |

12. | J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,” Opt. Express |

13. | Z. Fu, Q. Gan, K. Gao, Z. Pan, and F. J. Bartoli, “Numerical investigation of a bidirectional wave coupler based on plasmonic Bragg gratings in the near infrared domain,” J. Lightwave Technol. |

14. | Y.-J. Chang and G.-Y. Lo, “A narrow band metal—multi-insulator—metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett. |

15. | C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microw. Theory Tech. |

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

17. | Y.-J. Chang, T. K. Gaylord, and G.-K. Chang, “Attenuation in waveguides on FR-4 boards due to periodic substrate undulations,” Appl. Opt. |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(130.3120) Integrated optics : Integrated optics devices

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: April 8, 2010

Revised Manuscript: May 25, 2010

Manuscript Accepted: June 1, 2010

Published: June 4, 2010

**Citation**

Yin-Jung Chang, "Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating," Opt. Express **18**, 13258-13270 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-12-13258

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]
- J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef] [PubMed]
- S. A. Maier, “Plasmonics: the promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. 12, 1671–1677 (2006). [CrossRef]
- E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef] [PubMed]
- B. Wang, and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87(013), 107 (2005).
- S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express 13(12), 4674–4682 (2005). [CrossRef] [PubMed]
- A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Lightwave Technol. 24(2), 912–918 (2006). [CrossRef]
- J.-W. Mu, and W.-P. Huang, “A low-loss surface plasmonic Bragg grating,” J. Lightwave Technol. 27(4), 436–439 (2009). [CrossRef]
- A. Hosseini, and Y. Massoud, “A low-loss metal-insulator-metal plasmonic Bragg reflector,” Opt. Express 14(23), 318–323 (2006). [CrossRef]
- 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]
- J.-Q. Liu, L.-L. Wang, M.-D. He, W.-Q. Huang, D. Wang, B. S. Zou, and S. Wen, “A wide bandgap plasmonic Bragg reflector,” Opt. Express 16(7), 4888–4894 (2008). [CrossRef] [PubMed]
- J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,” Opt. Express 16(1), 413–425 (2008). [CrossRef] [PubMed]
- Z. Fu, Q. Gan, K. Gao, Z. Pan, and F. J. Bartoli, “Numerical investigation of a bidirectional wave coupler based on plasmonic Bragg gratings in the near infrared domain,” J. Lightwave Technol. 26(22), 3699–3703 (2008). [CrossRef]
- Y.-J. Chang, and G.-Y. Lo, “A narrow band metal–multi-insulator–metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett. 22, 634–636 (2010). [CrossRef]
- C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microw. Theory Tech. 13(3), 297–302 (1965). [CrossRef]
- P. B. Johnson, and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
- Y.-J. Chang, T. K. Gaylord, and G.-K. Chang, “Attenuation in waveguides on FR-4 boards due to periodic substrate undulations,” Appl. Opt. 46(12), 2234–2243 (2007). [CrossRef] [PubMed]

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