## High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating

Optics Express, Vol. 16, Issue 1, pp. 413-425 (2008)

http://dx.doi.org/10.1364/OE.16.000413

Acrobat PDF (309 KB)

### Abstract

High order plasmonic Bragg reflection in the metal-insulator-metal (MIM) waveguide Bragg grating (WBG) and its applications are proposed and demonstrated numerically. With the effective index method and the standard transfer matrix method, we reveal that there exist high order plasmonic Bragg reflections in MIM WBG and corresponding Bragg wavelengths can be obtained. Contrary to the high order Bragg wavelengths in the case of the conventional dielectric slab waveguide, the results of the MIM WBG exhibit red shifts of tens of nanometers. We also propose a method to design a MIM WBG to have high order plasmonic Bragg reflection at a desired wavelength. The MIM WBG operating in visible spectral regime, which requires quite accurate fabrication process with grating period of 100 to 200 *nm* for the fundamental Bragg reflection, can be implemented by using the higher order plasmonic Bragg reflection with grating period of 400 to 600 *nm*. It is shown that the higher order plasmonic Bragg reflection can be employed to implement a narrow reflection bandwidth as well. We also address the dependence of the filling factor upon the bandgap and discuss the quarter-wave stack condition and the second bandgap closing.

© 2008 Optical Society of America

## 1. Introduction

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

3. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. **85**, 5833–5835 (2004). [CrossRef]

4. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. **23**, 413–422 (2005). [CrossRef]

5. S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. **95**, 046802 (2005). [CrossRef] [PubMed]

6. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B **61**, 10484–10503 (2000). [CrossRef]

9. R. Zia, M. D. Selker, P. B. Catrysse, and M. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A **21**, 2442–2446 (2004). [CrossRef]

*et al.*proposed compact and efficient Bragg gratings for long-range SPP operating around 1550

*nm*by introducing periodic thickness modulation of thin metal stripes embedded in a dielectric [13

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

14. B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. **87**, 013107 (2005). [CrossRef]

14. B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. **87**, 013107 (2005). [CrossRef]

15. A. Hosseini and Y. Massoud, “A low-loss metal-insulator-metal plasmonic Bragg reflector,” Opt. Express **14**, 11318–11323 (2006). [CrossRef]

*et al.*devised surface plasmon Bragg gratings formed by a periodic variation of the width of the insulator in MIM waveguide [16

16. 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]

*nm*.

*nm*. Furthermore, since the effective refractive index increases with decrease of the operating wavelength, it is required to fabricate a periodic structure with much shorter period, such as below 100

*nm*. The dielectric grating on the metal substrate can be patterned via the electron beam lithography method or the focused ion beam method, which bring about inevitable roundish edge effect [18

18. J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Letters **6**, 1928–1932 (2006). [CrossRef] [PubMed]

19. D. Z. Lin, C. K. Chang, Y. C. Chen, D. L. Yang, M. W. Lin, J. T. Yeh, J. M. Liu, C. H. Kuan, C. S. Yeh, and C. K. D. Lee, “Beaming light from a subwavelength metal slit surrounded by dielectric surface gratings,” Opt. Express **14**, 3503–3511 (2006). [CrossRef] [PubMed]

20. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A **71**, 811–818 (1981). [CrossRef]

## 2. Fundamental properties of the MIM waveguide

*t*surrounded by two half-infinite metal claddings. Since we are interested in behavior of the plasmonic wave, it is assumed that the electromagnetic (EM) field is of

*p*-polarization so that

*E*,

_{x}*H*and

_{y}*E*have non-zero values whereas

_{z}*H*,

_{x}*E*and

_{y}*H*are all zero.

_{z}*E*and

_{z}*H*along the interface between two different media, we obtain the dispersion relation as follows [11

_{y}11. 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]

*ε*is the dielectric constant of the dielectric core,

_{d}*ε*the dielectric function of metal cladding, and

_{m}*κ*and

_{d}*κ*are the transverse (

_{m}*x*-direction) wave number in the core and the cladding, respectively. To describe the dielectric function for silver, we employ the free electron model which can be expressed as follows:

*ε*

_{∞}represents the dielectric constant at infinite angular frequency and is chosen to have 3.70.

*ω*stands for the bulk plasma frequency with the value of 9eV.

_{p}*γ*means the oscillation damping of electrons and the value is 0.018eV. These parameters are chosen so that they describe the angular frequency dependency of silver with best fit for the free-space wavelength from 400 to 2000

*nm*. Note that we express the angular frequency in a form of photon energy by multiplying the Plank’s constant

*ħ*[14

14. B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. **87**, 013107 (2005). [CrossRef]

16. 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]

*E*). Hence both the transverse electric field (

_{z}*E*) and the transverse magnetic field (

_{x}*H*) exhibit the symmetric distribution at the anti-symmetric mode and the anti-symmetric distribution at the symmetric mode. The symmetric mode in MIM waveguide exhibits cut-off when its thickness goes below a cut-off thickness, which is typically shown to be hundreds of nanometers [11

_{y}11. 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]

*κ*,

_{d}*κ*and the longitudinal (

_{m}*z*-direction) wave number

*β*[1]:

*n*of the anti-symmetric mode in the MIM waveguide upon various geometrical parameters such as the operating wavelength, the type of the metal cladding, the thickness and the dielectric constant of the dielectric core. Figure 2(a) shows the dispersion relation of the anti-symmetric mode in the MIM waveguide. The dispersion relation curve of the anti-symmetric mode lies to right side of the dispersion curve of the single interface. It is known that as the thickness of the dielectric core changes from infinity to tens of nanometers, the dispersion moves from the single interface line to right side of it. Figures 2(b)–(d) illustrate

_{eff}*n*as a function of the geometrical parameters.

_{eff}*n*of the anti-symmetric mode for each case is always higher than that of the single interface. By alternatively stacking MIM waveguides with different geometrical parameters, we can build up the MIM WBG. Merits and demerits of each type – the dielectric core index modulation, the metal cladding modulation and the thickness of the dielectric core modulation – have been discussed widely in some papers [14–17

_{eff}**87**, 013107 (2005). [CrossRef]

## 3. High order plasmonic Bragg reflection in the MIM WBG

*d*

_{1}and

*d*

_{2}stand for the lengths of the MIM waveguides with the core indices of

*ε*

_{d1}and

*ε*

_{d2}, respectively. Λ is the period of the gratings in the waveguide.

*q*is an integer which represents the order of the Bragg reflection and

*λ*is the

_{B,q}*q*-th order Bragg wavelength at which reflectance from and transmittance through the waveguide gets almost unity and zero, respectively.

*ñ*is the averaged effective refractive index at the fundamental Bragg wavelength, which is defined as

_{eff}4. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. **23**, 413–422 (2005). [CrossRef]

4. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. **23**, 413–422 (2005). [CrossRef]

6. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B **61**, 10484–10503 (2000). [CrossRef]

*λ*. A point of intersection between the left-hand side of Eq. (6

6. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B **61**, 10484–10503 (2000). [CrossRef]

**61**, 10484–10503 (2000). [CrossRef]

*nm*and the dielectric function of the metal cladding is from Eq. (2

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

*n*at the operating wavelength of 1550

_{eff}*nm*is read as 1.378 and 1.656, respectively. The filling factor, or duty ratio is defined as

*f*=

*d*

_{1}/(

*d*

_{1}+

*d*

_{2})=

*d*

_{1}/Λ and its contribution to the behavior of the MIM WBG will be discussed later. For now we assume that the filling factor is 0.5, namely the ratio between the lengths of two MIM waveguides is 1:1. With the averaged effective refractive index of 1.517 at the operating wavelength of 1550

*nm*, the period of the gratings for the fundamental Bragg wavelength of 1550

*nm*is obtained as 510

*nm*. It is observed in a cyan-colored line in Fig. 4(a) that in the case of the conventional dielectric slab waveguide (DSW), the dispersion is negligible so that the high order Bragg wavelengths are obtained as about 775

*nm*and 517

*nm*, which are in good agreement with the result of Eq. (4

**23**, 413–422 (2005). [CrossRef]

*nm*and 548

*nm*, which are shifted about 14

*nm*and 31

*nm*from the result of non-dispersive assumption. This is due to the increase of the averaged effective refractive index with decrease of the operating wavelength. To check out the high order plasmonic Bragg reflection, we calculated the reflectance and transmittance spectra using the RCWA [20–23

20. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A **71**, 811–818 (1981). [CrossRef]

*nm*. It should be pointed that we have checked agreement between property of the eigenmode obtained from the RCWA and that from the analytical solution in Eq. (1). The number of cells is 20. The effect of the number of cells upon reflection and transmission spectra has been reported [17]. From Fig. 4(b), it is observed that the second and third order Bragg reflections take place at the wavelengths of 789

*nm*and 548

*nm*with reflectivity of 48.8% and 83.5%, respectively. And their bandwidths defined by the full width at half-maximum (FWHM) are obtained as 22

*nm*and 20

*nm*. It is also noteworthy that the bandwidth is narrow at the high order plasmonic Bragg reflection, which suggests an application for the design of narrow stop bandgap device. This will be discussed in the following section in this paper. Figures 5(a)–(c) illustrate the field distribution in the MIM WBG at the fundamental Bragg wavelength, the second order Bragg wavelength, and the third order Bragg wavelength, respectively. It becomes evident that the higher order plasmonic reflections occur at the higher order Bragg wavelengths of 789

*nm*and 548

*nm*. It is also observed that at the second order Bragg wavelength the reflection is quite weak and the transmission is not negligible. Thus the second order Bragg reflection does not seem to be appropriate for practical applications. Through this paper, therefore, applications utilizing the higher order plasmonic Bragg reflection mainly deal with the third order Bragg reflection.

*ω*is the angular frequency and

*K*is the Bloch wave number [24].

*t*(

*ω*) is the transmittance from unit cell.

*φ*and

_{1}*φ*are the phases introduced by two layers of a unit cell and defined as

_{2}*φ*=

_{i}*k*

_{0}

*n*

_{eff,i}*d*

_{i}(

*i*=1, 2). Figure 6 illustrates the Bloch wave number as a function of the angular frequency. Note that abscissa is normalized with

*π*/Λ and ordinate is scaled with the Bragg frequency

*ω*=

_{B}*πc*

_{0}/

*ñ*Λ, where

_{eff}*ñ*is the averaged effective refractive index at the wavelength of 1550

_{eff}*nm*. As predicted, the fundamental plasmonic bandgap takes place at the Bragg angular frequency. If there were no dispersion in this structure, the high order plasmonic bandgap would occur at the integer multiple of the Bragg angular frequency. As can be seen in Fig. 6, however, it is observed that the high order plasmonic bandgap happens at the smaller angular frequency than integer multiple. The result supports that there is red shift not only in finite Bragg grating but also in infinite bandgap structure.

## 4. Application of the higher order plasmonic Bragg reflection – MIM WBG at the visible spectral regime and MIM WBG with narrow reflection band

**23**, 413–422 (2005). [CrossRef]

*nm*and the reflection and the transmission spectra (solid blue line) are depicted in Figs. 7(a) and (b), respectively. Since the averaged effective refractive index at the wavelength of 532

*nm*is obtained as 1.623, the period of the grating is designed to be 492

*nm*for the third order Bragg reflection.

*nm*can be implemented by using the fundamental Bragg reflection with the grating period of 164

*nm*. The reflection and the transmission spectra (dashed red line) of the fundamental Bragg reflection are also illustrated in Figs. 7(a) and (b), respectively. It should be mentioned that the number of periodic cell (#) is chosen in such a way that two MIM WBGs have the same total grating length (9.84

*µm*). In Table 1, we summarize the characteristics of two MIM WBGs such as the center wavelength of Bragg reflection (

*λ*), the bandwidth (Δ

_{c}*λ*), the grating strength (Δ

*λ*/

*λ*), the reflection at

_{c}*λ*(

_{c}*R*), the transmission at

*λ*(

_{c}*T*), and the propagation loss at

*λ*(

_{c}*L*). It is observed that the grating strength of the third order Bragg reflection is weaker about threefold than that of the fundamental Bragg reflection. The third order Bragg reflection also exhibits less reflection and more propagation loss than those of the fundamental Bragg reflection. However we give attention to the fact that in the case of using the third order Bragg reflection, the reflection is more than 80% and the transmission exhibits less than 1%, which still suggests the adequate bandgap property. Considering the easiness of fabrication process, therefore, we can benefit from using the third order Bragg reflection to implement the MIM WBG having bandgap at visible spectral regime.

**87**, 013107 (2005). [CrossRef]

7. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**, 539–554 (1969). [CrossRef]

*nm*using different types of gratings. The dashed red lines in Fig. 8 correspond to results of the third order Bragg reflection. It is seen that unlike the fundamental Bragg reflection at 1550

*nm*shown in Fig. 4(b), the third order Bragg reflection exhibits the quite narrow bandwidth. Other properties are summarized in Table 2.

*nm*is 60, resulting in the total grating length of 30.660

*µm*. In the latter case, the grating period with 1530

*nm*(510 multiplied by 3) is stacked 20 times, giving rise to the total grating length of 30.600

*µm*. The solid blue lines in Fig. 8 correspond to the low index contrast. It is seen that the reflection bandwidths for the fundamental Bragg reflection with low index contrast and the third order Bragg reflection with high index contrast are obtained as 60nm and 61nm, respectively. And other properties such as reflection, transmission, and propagation loss are quite similar. It is noteworthy that the fundamental Bragg reflection with reduced cell number does not exhibit comparable narrow bandwidth. As evident from Fig. 8, the third order Bragg reflection with the high index contrast can be used to develop a narrow reflection band with similar performance for the fundamental Bragg reflection with the low index contrast. Although there is trade-off of easiness in fabrication for gratings with the long period and the high contrast against gratings with the short period and the low contrast, one thing to be emphasized is that the higher order plasmonic Bragg reflection gives us flexibility in design scheme.

## 5. Second bandgap closing and the filling factor

*nm*. Recall that the we have assumed

*n*

_{eff,1}is lower than

*n*

_{eff,2}. Since the period of the unit cell is fixed, the averaged effective refractive index decreases with increase of the filling factor, resulting in increase of the position of the bandgap in the frequency domain. This is shown in Fig. 9(a). It is also observed that at the filling factor around 0.55, the second order bandgap disappears. This is referred to as the second bandgap closing. To understand this phenomenon and get a physical insight, we investigate the ratio of the effective refractive index as a function of wavelength. The result is depicted in Fig. 9(b).

*nm*, we obtain the ratio of effective refractive index as 0.548, which lies inside the second bandgap closing region in Fig. 9(a). With simple calculation it can be shown that the filling factor of 0.548 means that the ratio between

*d*

_{1}and

*d*

_{2}is inversely proportional to the ratio of the effective refractive index at the wavelength of the second order Bragg reflection. This relation can be expressed as

*d*

_{1}:

*d*

_{2}=

*n*

_{eff,2}:

*n*

_{eff,1}, which can be regarded as condition for the quarter-wave stack. It is well known that even bandgap closing takes places at a conventional periodic structure of the quarter-wave stack. This is due to the fact that the wave reflected by the first half layer of a unit cell experiences the phase shift of 3

*π*, so that it interferes destructively with the wave reflected by the total unit cell and prevents the Bragg reflection. The result shown in Fig. 9(a) suggests that a plasmonic periodic structure of the quarter-wave stack also suppresses the second order Bragg reflection.

*nm*. It can be seen that the curves for Bloch wave near the second order bandgap adjoin to each other, giving rise to closing of the second bandgap. Based on the results above, we reason that when the ratio of the length for each region is inversely proportional to the effective refractive index at the specific high order Bragg wavelength, the gratings perform as a quarter-wave stack, leading to a second order bandgap closing. This property can be used to design a grating structure without the second order Bragg reflection.

## 6. Conclusion

*nm*and 31

*nm*from the result of non-dispersive assumption for the second and the third order plasmonic Bragg wavelengths, respectively. This was also verified with the reflection and the transmission spectra from the RCWA and the Bloch diagram from the Bloch wave analysis. As applications using the high order plasmonic Bragg reflection, we suggested the MIM WBG at the visible spectral regime without gratings of not too short period, which had the third order stop band at the wavelength of 532

*nm*with the grating period of 492

*nm*. We also proposed the MIM WBG with a narrow reflection band at the telecommunication wavelength (1550

*nm*) with the bandwidth of 61

*nm*, which was comparable with the MIM WBG with the low index contrast and the short period (60

*nm*). Finally the dependence of the filling factor upon the bandgap was addressed and the quarter-wave stack condition and the second bandgap closing were discussed.

## Acknowledgment

## References and links

1. | H. Rather, |

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

3. | T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. |

4. | A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. |

5. | S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. |

6. | P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B |

7. | E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. |

8. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B |

9. | R. Zia, M. D. Selker, P. B. Catrysse, and M. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A |

10. | J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: Frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B |

11. | J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B |

12. | I.-M. Lee, J. Jung, J. Park, H. Kim, and B. Lee, “Dispersion characteristics of channel plasmon polariton waveguides with step-trench-type grooves,” Opt. Express |

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

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

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

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

17. | A. Hosseini and Y. Massoud, “Subwavelength plasmonic Bragg reflector structures for on-chip optoelectronic applications,” International Symposium on Circuits and Systems, New Orleans, LA, 2283–2286 (2007). |

18. | J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Letters |

19. | D. Z. Lin, C. K. Chang, Y. C. Chen, D. L. Yang, M. W. Lin, J. T. Yeh, J. M. Liu, C. H. Kuan, C. S. Yeh, and C. K. D. Lee, “Beaming light from a subwavelength metal slit surrounded by dielectric surface gratings,” Opt. Express |

20. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A |

21. | M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

22. | P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A |

23. | H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A |

24. | B. E. A. Saleh and M. C. Teich, |

**OCIS Codes**

(230.1480) Optical devices : Bragg reflectors

(230.7390) Optical devices : Waveguides, planar

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 5, 2007

Revised Manuscript: December 28, 2007

Manuscript Accepted: December 28, 2007

Published: January 4, 2008

**Citation**

Junghyun Park, Hwi Kim, and Byoungho Lee, "High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating," Opt. Express **16**, 413-425 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-1-413

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### References

- H. Rather, Surface Plasmons (Springer-Verlag, Berlin, 1988).
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, "Surface plasmon polariton based modulators and switches operating at telecom wavelengths," Appl. Phys. Lett. 85, 5833-5835 (2004). [CrossRef]
- A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, "Integrated optical components utilizing long-range surface plasmon polaritons," J. Lightwave Technol. 23, 413-422 (2005). [CrossRef]
- S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005). [CrossRef] [PubMed]
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000). [CrossRef]
- E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969). [CrossRef]
- J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
- R. Zia, M. D. Selker, P. B. Catrysse, and M. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2004). [CrossRef]
- J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Planar metal plasmon waveguides: Frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005). [CrossRef]
- 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]
- I.-M. Lee, J. Jung, J. Park, H. Kim, and B. Lee, "Dispersion characteristics of channel plasmon polariton waveguides with step-trench-type grooves," Opt. Express 15, 16596-16603 (2007). [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, 912-918 (2006). [CrossRef]
- B. Wang and G. P. Wang, "Plasmon Bragg reflectors and nanocavities on flat metallic surfaces," Appl. Phys. Lett. 87, 013107 (2005). [CrossRef]
- A. Hosseini and Y. Massoud, "A low-loss metal-insulator-metal plasmonic Bragg reflector," Opt. Express 14, 11318-11323 (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]
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