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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 20 — Sep. 27, 2010
  • pp: 20939–20948
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Theoretical investigation of fabrication-related disorders on the properties of subwavelength metal-dielectric-metal plasmonic waveguides

Changjun Min and Georgios Veronis  »View Author Affiliations


Optics Express, Vol. 18, Issue 20, pp. 20939-20948 (2010)
http://dx.doi.org/10.1364/OE.18.020939


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Abstract

We theoretically investigate the effect of fabrication-related disorders on subwavelength metal-dielectric-metal plasmonic waveguides. We use a Monte Carlo method to calculate the roughness-induced excess attenuation coefficient with respect to a smooth waveguide. For small roughness height, the excess optical power loss due to disorder is small compared to the material loss in a smooth waveguide. However, for large roughness height, the excess attenuation increases rapidly with height and the propagation length of the optical mode is severely affected. We find that the excess attenuation is mainly due to reflection from the rough surfaces. However, for small roughness correlation lengths, enhanced absorption is the dominant loss mechanism due to disorder. We also find that the disorder attenuation due to reflection is approximately maximized when the power spectral density of the disordered surfaces at the Bragg spatial frequency is maximized. Finally, we show that increasing the modal confinement or decreasing the guide wavelength, increase the attenuation due to disorder.

© 2010 OSA

1. Introduction

The remainder of the paper is organized as follows. In Section 2, we describe the simulation method used for the random rough surface generation and the analysis of the rough MDM waveguides. The results obtained using this method for the rough MDM waveguides are presented in Section 3. Finally, our conclusions are summarized in Section 4.

2. Simulation method

In Fig. 1(a)
Fig. 1 (a) Schematic of the simulation configuration used to calculate the attenuation coefficient α r of a rough MDM waveguide. It consists of a section of the rough waveguide of length L sandwiched between two smooth MDM waveguides. (b) Theoretical normalized probability density of the roughness height at metal-dielectric interfaces. We also show the probability density calculated from the generated profile of a random interface realization. (c) Theoretical normalized autocorrelation function R(u) = <f(x)f(x + u)> of the roughness. Results are shown for Lc = 36nm. We also show the autocorrelation function calculated from the generated profile of a random interface realization. (d) Averaged attenuation coefficient α r as a function of the number N of random rough waveguide realizations used in the Monte Carlo method. Results are shown for a silver-air-silver MDM waveguide with w = 50nm, L = 2μm, Lc = 36nm, δ = 4nm, and λ = 1.55μm.
, we show a schematic of the simulation configuration that we use to calculate the attenuation coefficient α r of a rough MDM waveguide. It consists of a section of the rough waveguide of length L sandwiched between two smooth MDM waveguides. The roughness height function f(x) at each metal-dielectric interface is assumed to be a one-dimensional statistical homogeneous random process with zero mean. The nature of the roughness is described by the autocorrelation function of f(x) [13

13. R. Ding, L. Tsang, and H. Braunisch, “Wave Propagation in a Randomly Rough Parallel-Plate Waveguide,” IEEE Trans. Microw. Theory Tech. 57(5), 1216–1223 (2009). [CrossRef]

]
R(u)=<f(x)f(x+u)>,
where the brackets represent the ensemble average. We assume that both metal-dielectric interfaces are rough and mutually uncorrelated. We consider a realistic disorder model, in which the roughness height function f(x) obeys Gaussian statistics and has a Gaussian autocorrelation function
R(u)=δ2exp(u2/Lc2),
(1)
where δ is the roughness rms height, and L c is the correlation length [26

26. F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD Technique for Rough Surface Scattering,” IEEE Trans. Antenn. Propag. 43(11), 1183–1191 (1995).

]. We use a spectral method proposed by Thorsos [26

26. F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD Technique for Rough Surface Scattering,” IEEE Trans. Antenn. Propag. 43(11), 1183–1191 (1995).

,27

27. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988). [CrossRef]

] to randomly generate rough interface realizations. In this method, each rough interface consists of M discrete points spaced Δx apart over the surface length L = MΔx. Realizations are generated at xn = nΔx (n = 1,…, M) using
f(xn)=1Lj=M/2M/21F(kj)eikjxn,
(2)
where, for j ≥0,
F(kj)=2πLW(kj){[M(0,1)+iM(0,1)]/2,  for  j0,M/2M(0,1),  for  j=0,M/2,
(3)
and for j<0, F(kj) = F(k-j)*. In Eqs. (2) and (3), kj = 2πj/L, M(0,1) is a number sampled from a Gaussian distribution with zero mean and unity variance, and W(k) is the power spectral density of the rough surface. For a rough surface with Gaussian autocorrelation function (Eq. (1)), the power spectral density is [13

13. R. Ding, L. Tsang, and H. Braunisch, “Wave Propagation in a Randomly Rough Parallel-Plate Waveguide,” IEEE Trans. Microw. Theory Tech. 57(5), 1216–1223 (2009). [CrossRef]

,26

26. F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD Technique for Rough Surface Scattering,” IEEE Trans. Antenn. Propag. 43(11), 1183–1191 (1995).

,27

27. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988). [CrossRef]

]

W(k)12πR(u)eikudu=δ2Lc2πexp(k2Lc24).
(4)

In Figs. 1(b) and 1(c), we show the probability density and autocorrelation function, respectively, for such a random interface realization calculated directly from one generated profile by the spectral method. We observe that they are both in excellent agreement with their respective theoretical values, confirming the validity of the method that we use for random rough surface generation.

For each randomly-generated rough waveguide realization, we use a full-wave two-dimensional FDFD method [28

28. G. Veronis, and S. Fan, “Overview of Simulation Techniques for Plasmonic Devices,” in Surface Plasmon Nanophotonics, Mark L. Brongersma and Pieter G. Kik, ed. (Springer, 2007).

] to calculate the electromagnetic fields and the transmission coefficient T through the waveguide section of length L (Fig. 1(a)). This method allows us to directly use experimental data for the frequency-dependent dielectric constant of materials such as silver [29

29. E. D. Palik, Handbook of Optical Constants of Solids, (Academic, New York, 1985).

], including both the real and imaginary parts, with no approximation. Perfectly matched layer absorbing boundary conditions [30

30. J. Jin, The Finite Element Method in Electromagnetics, (Wiley, New York, 2002).

] are used at all boundaries of the simulation domain. We also note that, since we directly use a full-wave simulation method to calculate the fields in the disordered waveguides, our calculations take into account multiple scattering and localization effects [31

31. S. Mazoyer, J. P. Hugonin, and P. Lalanne, “Disorder-induced multiple scattering in photonic-crystal waveguides,” Phys. Rev. Lett. 103(6), 063903 (2009). [CrossRef] [PubMed]

].

We use a Monte Carlo method [26

26. F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD Technique for Rough Surface Scattering,” IEEE Trans. Antenn. Propag. 43(11), 1183–1191 (1995).

] to average over an ensemble of randomly-generated rough waveguide realizations. We found that, as expected [22

22. C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69(3), 731–808 (1997). [CrossRef]

,31

31. S. Mazoyer, J. P. Hugonin, and P. Lalanne, “Disorder-induced multiple scattering in photonic-crystal waveguides,” Phys. Rev. Lett. 103(6), 063903 (2009). [CrossRef] [PubMed]

], <ln(T)> undergoes a linear variation with L for large L. This allows us to define an attenuation coefficient α r for the rough MDM waveguide using [31

31. S. Mazoyer, J. P. Hugonin, and P. Lalanne, “Disorder-induced multiple scattering in photonic-crystal waveguides,” Phys. Rev. Lett. 103(6), 063903 (2009). [CrossRef] [PubMed]

]
<ln(T)>=αrL   .
(5)
We note that α r is referred to in the literature on disordered systems as the Lyapunov exponent and its inverse as the localization length [18

18. V. D. Freilikher, and S. A. Gredeskul, “Localization of waves in media with one-dimensional disorder,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1996), V. XXX, pp. 137–203.

,20

20. F. M. Izrailev and N. M. Makarov, “Onset of delocalization in quasi-one-dimensional waveguides with correlated surface disorder,” Phys. Rev. B 67(11), 113402 (2003). [CrossRef]

22

22. C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69(3), 731–808 (1997). [CrossRef]

]. In all cases we choose L large enough to ensure that Eq. (5) holds. In Fig. 1(d), we show a typical result for the convergence of the calculated attenuation coefficient α r of a rough MDM plasmonic waveguide, as a function of the number N of randomly-generated rough waveguide realizations used in the Monte Carlo method. We observe that the method converges after a few hundred waveguide realization calculations. In all cases we found that N = 1000 is sufficient for an accuracy of the results within 1%.

3. Results

We first consider the effect of the roughness rms height δ and correlation length L c on the attenuation coefficient. In Fig. 2
Fig. 2 Excess attenuation coefficient α r-α s of a rough MDM plasmonic waveguide and attenuation enhancement factor αrs with respect to a smooth waveguide as a function of roughness rms height δ for L c = 36nm (solid line) and L c = 500nm (dashed line). Also shown is the excess attenuation coefficient α r-α s of a rough MDM plasmonic waveguide as a function of δ for L c = 36nm (filled circles) and L c = 500nm (empty circles), if the metal in the MDM waveguide is lossless. All other parameters are as in Fig. 1(d).
, we show the excess attenuation coefficient α r-α s of a rough MDM plasmonic waveguide as a function of the roughness rms height δ for L c = 36nm. For convenience we also show the attenuation enhancement factor, defined as α r/α s. We observe that, as expected, the excess attenuation coefficient α r-α s increases with δ. We also found that for δ>4nm, the increase is almost exponential. For δ<4nm, we have α r/α s<1.2 for the attenuation enhancement factor. Thus, for small roughness height (δ<4nm), the excess optical power loss due to disorder is small compared to the material loss in a smooth waveguide. On the other hand, for large roughness height (δ>4nm), α r/α s increases rapidly with δ, and the propagation length of the optical mode is severely affected. Thus, for δ = 12nm the attenuation coefficient in the rough waveguide is enhanced by an order of magnitude compared to the smooth waveguide (α r/α s≈10).

In Fig. 2, we also show the excess attenuation coefficient α r-α s for a rough MDM waveguide in which the metal in the MDM waveguide is lossless (ε metal = ε metal,real, neglecting the imaginary part of the dielectric permittivity ε metal,imag). We observe that material losses in the metal do not significantly affect the excess attenuation coefficient α r-α s. In the lossless metal case, α s = 0 and the excess attenuation is only due to the reflection from the rough surfaces. Thus, we can conclude that the excess loss in a rough MDM plasmonic waveguide is mainly due to reflection from the rough surfaces. As mentioned in Section 1, this loss mechanism is connected to Anderson localization in one-dimensional disordered systems.

In Fig. 3(a)
Fig. 3 (a) Excess attenuation coefficient α r-α s of a rough MDM plasmonic waveguide and attenuation enhancement factor αrs as a function of correlation length Lc for δ = 4nm (black solid line) and δ = 8nm (black dashed line). Also shown is the excess attenuation coefficient α r-α s of a rough MDM plasmonic waveguide as a function of Lc for δ = 4nm (red solid line) and δ = 8nm (red dashed line), if the metal in the MDM waveguide is lossless. All other parameters are as in Fig. 1(d). (b) Normalized power spectral density of the disordered surfaces at the Bragg spatial frequency kBragg as a function of Lc. All other parameters are as in Fig. 1(d).
, we show the excess attenuation coefficient α r-α s and the attenuation enhancement factor α r/α s of a rough MDM plasmonic waveguide as a function of the correlation length L c for δ = 4nm, 8nm. We observe that in both cases α r-α s is maximized for L c = 120nm. In addition, the excess attenuation coefficient α r-α s increases as L c→0. In Fig. 3(a), we also show the excess attenuation coefficient α r-α s for a rough MDM waveguide in which the metal in the MDM waveguide is lossless. The results in the lossless metal case agree well with the lossy metal case for L c>20nm. Thus, we can conclude that for L c>20nm, the excess attenuation is due to reflection from the rough surfaces.

We found that the peak in excess attenuation coefficient α r-α s at L c = 120nm is associated with Bragg reflection from the rough surfaces. In a MDM plasmonic waveguide with a periodic perturbation with period P, there is strong reflection associated with Bragg Scattering [4

4. S. A. Maier, Plasmonics: fundamentals and applications, (Springer, New York, 2007).

] when
P=PBragg=λMDM/2,
where λMDM is the guide wavelength in the MDM plasmonic waveguide. The corresponding Bragg spatial frequency is
kBragg2πPBragg=4πλMDM.
(6)
Using Eqs. (4) and (6), we obtain the power spectral density W(kBragg) of the disordered surfaces at kBragg

W(kBragg)=δ2Lc2πexp[(2πLcλMDM)2].
(7)

In Fig. 3(b), we show W(kBragg) as a function of L c. We observe that the excess attenuation coefficient α r-α s due to reflection from disorders correlates very well with the power spectral density W(kBragg). Thus, the disorder attenuation due to reflection is approximately maximized when the power spectral density of the disordered surfaces at the Bragg spatial frequency W(kBragg) is maximized. Using Eq. (7), we find that W(kBragg) is maximized for Lc=λMDM22π .

For L c<20nm, there is a large difference between the lossy and lossless metal cases (Fig. 3(a)). In the lossless metal case, the excess attenuation coefficient α r-α s decreases as L c→0, due to the fact that limLc0W(kBragg)=0 . However, in the lossy metal case the excess attenuation coefficient α r-α s increases as L c→0. This is because when L c→0, the waveguide surfaces become extremely rough with disorder dimensions small compared to the skin depth, which allows light to directly pass through, and results in enhanced absorption in the metal. Thus, for L c<20nm, the reflection from the rough surfaces is small, and enhanced absorption in the metal is the dominant loss mechanism due to disorder. Finally, for very large correlation lengths (L c>400nm), the waveguide surfaces are almost flat, which results in a small reflection, hence the attenuation coefficient is only slightly enhanced with respect to the smooth waveguide (Figs. 2, 3(a)).

We further investigate the enhanced absorption in disordered MDM plasmonic waveguides by considering the electromagnetic field distribution in such waveguides. In Fig. 4(a)
Fig. 4 (a)-(b) Electric field intensity profiles for a random rough MDM plasmonic waveguide and a MDM waveguide with periodic perturbation. Results are shown for perturbation periodicity P = 200nm and amplitude A = 22δ in the sine periodic waveguide. All other parameters are as in Fig. 1(d). (c)-(d) Absorbed power density profiles for a random rough MDM waveguide and a MDM waveguide with periodic perturbation. (e)-(f) Local absorption coefficient for a random rough MDM plasmonic waveguide and a MDM waveguide with periodic perturbation. We also show the local absorption coefficient for a smooth MDM waveguide (black dashed line).
we show the electric field intensity profile for a random rough MDM plasmonic waveguide. Since metals satisfy the condition |ε metal|>>ε diel at near-infrared wavelengths [29

29. E. D. Palik, Handbook of Optical Constants of Solids, (Academic, New York, 1985).

], the electric field intensity in the metal is much smaller than the electric field intensity in the dielectric. In addition, we observe that the electric field intensity in the dielectric is enhanced at the peak of the metal roughness bumps. The enhancement is associated with the behavior of the electric field near sharp edges [32

32. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

], which is often referred to as the lightning rod effect [33

33. S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(1), 011101 (2005). [CrossRef]

]. The increased surface charge density at the metal bump peak leads to crowding of the electric field lines in the dielectric, and therefore to enhancement of the near field in the vicinity of the bump peak.

The absorbed power density depends, however, on the electric field intensity in the metal. In Fig. 4(c) we show the absorbed power density profile in the metal for the random rough MDM waveguide. Contrary to the electric field intensity in the dielectric, the electric field intensity in the metal, and therefore the absorbed power density, is maximum (minimum) at the bottom (peak) of the metal roughness bumps. Thus, in a rough MDM waveguide absorption is enhanced with respect to a smooth waveguide at the bottom of metal roughness bumps. In Fig. 4(e) we also show the local absorption coefficient α r(x) for the random rough MDM waveguide. We observe that in certain locations of the disordered waveguide, corresponding to peaks of metal roughness bumps, the absorption is actually suppressed with respect to a smooth MDM waveguide. Even though the absorption is locally suppressed at certain locations, the overall absorption in the rough waveguide is enhanced with respect to the smooth waveguide. As mentioned above, this absorption enhancement is the dominant disorder loss mechanism in the case of small roughness correlation length. We finally note that the absorption properties described above are quite general and are not limited to random rough waveguides. As an example, periodically textured waveguides exhibit similar characteristics (Figs. 4(b), 4(d), 4(f)).

4. Conclusions

We considered the effect of the roughness rms height δ and the correlation length L c on the excess attenuation coefficient α r-α s. For small roughness rms height, the reflection from the rough surfaces is small, hence the excess optical power loss due to disorder is small compared to the material loss in a smooth waveguide. For large roughness height, α r/α s increases rapidly with δ, and the propagation length of the optical mode is severely affected. We found that excess loss in rough MDM plasmonic waveguides is mainly due to reflection from the rough surfaces. However, for small roughness correlation lengths, enhanced absorption in the metal is the dominant loss mechanism due to disorder. We also found that the disorder attenuation due to reflection is approximately maximized when the power spectral density of the disordered surfaces at the Bragg spatial frequency is maximized.

We also considered the electromagnetic field distribution in disordered MDM plasmonic waveguides to further investigate the enhanced absorption in such waveguides. We found that the electric field intensity in the metal, as well as the absorbed power density, is maximum (minimum) at the bottom (peak) of the metal roughness bumps. The local absorption coefficient for the disordered waveguide can be enhanced or suppressed with respect to a smooth MDM waveguide at different locations. Even though the absorption is locally suppressed at certain locations, the overall absorption in the rough waveguide is enhanced with respect to the smooth waveguide.

Finally, we considered the effect of the MDM waveguide parameters, such as waveguide width w, operating wavelength λ, and dielectric constant ε r of the material in the waveguide, on the attenuation in rough MDM plasmonic waveguides. We found that variation of these waveguide parameters impacts several important factors that affect disorder losses, such as the ratio of disorder size to waveguide width, the guide wavelength, and the group velocity. Increasing the light confinement in a MDM plasmonic waveguide leads to increased ratio of disorder size to waveguide width, decreased guide wavelength, and decreased group velocity, and, therefore, to increased disorder losses.

Acknowledgments

This research was supported by the Louisiana Board of Regents (Contract No. LEQSF(2009-12)-RD-A-08).

References and links

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OCIS Codes
(130.2790) Integrated optics : Guided waves
(240.5770) Optics at surfaces : Roughness
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 30, 2010
Revised Manuscript: September 9, 2010
Manuscript Accepted: September 9, 2010
Published: September 17, 2010

Citation
Changjun Min and Georgios Veronis, "Theoretical investigation of fabrication-related disorders on the properties of subwavelength metal-dielectric-metal plasmonic waveguides," Opt. Express 18, 20939-20948 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20939


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References

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
  2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef] [PubMed]
  3. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7–8), 20–27 (2006). [CrossRef]
  4. S. A. Maier, Plasmonics: fundamentals and applications, (Springer, New York, 2007).
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