OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 21 — Oct. 17, 2005
  • pp: 8380–8389
« Show journal navigation

The effect of interfacial roughness on the normal incidence bandgap of one-dimensional photonic crystals

Karlene Rosera Maskaly, W. Craig Carter, Richard D. Averitt, and James L. Maxwell  »View Author Affiliations


Optics Express, Vol. 13, Issue 21, pp. 8380-8389 (2005)
http://dx.doi.org/10.1364/OPEX.13.008380


View Full Text Article

Acrobat PDF (379 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

As discussed previously, interfacial roughness in one-dimensional photonic crystals (1DPCs) can have a significant effect on their normal reflectivity at the quarter-wave tuned wavelength. We report additional finite-difference time-domain (FDTD) simulations that reveal the effect of interfacial roughness on the normal-incidence reflectivity at several other wavelengths within the photonic bandgaps of various 1DPC quarter-wave stacks. The results predict that both a narrowing and red-shifting of the bandgaps will occur due to the roughness features. These FDTD results are compared to results obtained when the homogenization approximation is applied to the same structures. The homogenization approximation reproduces the FDTD results, revealing that this approximation is applicable to roughened 1DPCs within the parameter range tested (rms roughnesses < 20% and rms wavelengths < 50% of the photonic crystal periodicity) across the entire normal incidence bandgap.

© 2005 Optical Society of America

1. Introduction

R[1L0L(yy0)2dx]12[1n1n(yy0)2]12,
(1)

Fig. 1. Closeup of a roughened interface illustrating the two interfacial roughness parameters, rms roughness and rms wavelength.

2. FDTD reflectivity results

Δr=1rroughrsmooth,
(2)

where r rough is the reflectivity of the roughened structure and r smooth is the reflectivity of the equivalent perfect (smooth interfaces) structure. In order to map out the reflectivity in the bandgap, several wavelengths within the bandgap of each structure’s perfect analogue were chosen. Approximately twenty distinct wavelengths spanning the bandgap were simulated at four different roughness values. On each of the Figs., these wavelengths are reported in normalized units (λ/a).

Approximately 400 total FDTD simulations were performed on several different 1DPC configurations. The specific refractive indices used in this study were chosen to roughly correspond to typical values for materials in the visible and near infrared regimes. The reflectivities from a TE-polarized normal incidence plane wave impinging on the roughened quarter-wave stacks were obtained. For comparison, these reflectivities were then plotted with the normal incidence reflectance spectrum for the corresponding perfect structure, which was computed with a one-dimensional transfer matrix calculation [15

15 . J. A. Kong , Electromagnetic Wave Theory ( EMW, Cambridge, Mass ., 2000 ).

]. The FDTD reflectance data were also fit with a polynomial expression in order to better observe any resulting trends.

The simulated reflectance spectra corresponding to several 4-bilayer systems are shown in Fig. 2. Each point corresponds to the calculated reflectivity results obtained from a single simulation. The curves are the polynomial fits to the reflectivity data described above, which visually reveal the resulting band structure at each roughness value. Consistent with the previously reported results, Fig. 2 shows that the reflectivity decreases with increasing roughness across the entire normal incidence band gap for all the modeled structures. However, the magnitude of this decrease is not the same for all wavelengths. The blue end of the band gap (smaller wavelengths) shows a much larger change in reflectivity than the red end (larger wavelengths). This results in a narrowing and red-shifting of the normal incidence band gap with increasing rms roughness. By using the term “red-shifting” here, we mean that the reflectivity becomes weighted towards the red end of the spectrum due to the unbalanced reflectivity change across the band gap. Although this effect is seen in all the structures presented in Fig. 2, the lowest index contrast structure (n1/n2=1.25, n1=2.25, n2=2.0) appears to be more sensitive to this red-shifting. This is illustrated by the fact that the red-shifting is already evident in the lowest index contrast structure at the smallest R value tested (3.71%), whereas the same roughness value in the other structures produces no such effect.

Fig. 2. The simulated normal incidence reflectance spectra corresponding to several 4-bilayer systems. The labels indicate the rms roughness as a percentage of the photonic crystal periodicity. In all systems, a narrowing and red-shifting of the normal incidence band gap is apparent.
Fig. 3. The relative change in reflectivity (Δr) across the entire normal incidence band gap for several 4-bilayer systems. The shading indicates the region where the reflectivity of the band gap is within 10% of its maximum value. Again, the red-shift is apparent in all systems.

Figure 3, which illustrates Δr across the bandgap, demonstrates this red-shift more clearly. Due to the steep slope of the reflectivity at the edges of the band gap, the Δr value in this region is more sensitive. As a guide to ranges of practical use, the region where the perfect band gap is within 10% of its maximum value (i.e. where the band gap reflectance is reasonably flat) is shaded in the plots. Again, Fig. 3 shows that all simulated systems experience a decrease in the percent change in reflectivity within this shaded region as the wavelength increases. However, the lowest index contrast structure shows evidence of this red-shifting at even the smallest R value. This same roughness produces a nearly uniform change in the reflectivity across the shaded region in the higher index contrast structures. Additionally, the magnitude of the reflectivity change near the center of the bandgap is larger in the lowest index contrast structure, and decreases as the index contrast increases. This is consistent with our previously reported results, which showed that the higher index contrast structures were more tolerant to interfacial roughness.

Fig. 4. The simulated normal incidence reflectance spectra corresponding to two bilayer systems with n1=2.25 and n2=1.5. The 4-bilayer structure is shown in Fig. 2. Again, a narrowing and red-shifting of the band gap is evident.

Fig. 5. The relative change in reflectivity (Δr) across the entire normal incidence band gap for two bilayer systems with n1=2.25 and n2=1.5. The 4-bilayer system is shown in Fig. 3.

3. Homogenization approximation reflectivity results

The red-shift in the normal incidence bandgap could be readily explained using classical scattering theory, which states that the magnitude of scattering increases as the incident wavelength decreases [16

16 . C. F. Bohren and D. R. Huffman , Absorption and Scattering of Light by Small Particles ( Wiley, New York , 1983 ).

]. Thus, if the decrease in reflectivity from these structures was largely due to diffuse scattering losses, then the blue end of the bandgap would show a larger reflectivity change. However, as previously reported, the fact that the homogenization approximation can accurately predict the FDTD reflectivity at the quarter-wave tuned wavelength implies that the amount of incoherent reflected power from diffuse scattering in these structures is extremely small [6

6 . K. R. Maskaly , W. C. Carter , R. D. Averitt , and J. L. Maxwell , “ Application of the homogenization approximation to roughened one-dimensional photonic crystals ,” Opt. Lett. (to be published). [PubMed]

]. In order to determine if this physical hypothesis remains valid at wavelengths other than the quarter-wave tuned wavelength, the homogenization approximation was applied to this study as well.

Fig. 6. The results of the homogenization approximation applied to the 4-bilayer structures presented in Fig. 2. Comparison of the two Figs. shows that the homogenization approximation is in good agreement with the FDTD results.

A comparison between these Figs. and Figs. 2 and 3 reveals that the homogenization approximation results match the FDTD calculations across the entire normal incidence band gap. Specifically, it reproduces the red-shifting of the FDTD data, and moreover, it accurately predicts the increased red-shifting sensitivity of the lowest index contrast structure. The validity of the homogenization approximation in this regime is further supported with the comparison of Figs. 8 and 9 with Figs. 4 and 5. Again, the red-shifting behavior is clearly predicted and the magnitude of the reflectivity change matches the FDTD calculations. There is some fluctuation present in the FDTD data that is not captured by the homogenization approximation. These fluctuations arise from variations in the amount of incoherent reflection from the rough 1DPC structures. The homogenization approximation does not capture these fluctuations because it can only provide information on the amount of coherent reflection from rough surfaces [6–9

6 . K. R. Maskaly , W. C. Carter , R. D. Averitt , and J. L. Maxwell , “ Application of the homogenization approximation to roughened one-dimensional photonic crystals ,” Opt. Lett. (to be published). [PubMed]

]. Nonetheless, it appears that the decrease in the coherent reflectivity, due to the roughness features collectively smearing-out the refractive index profile in the vicinity of the interfaces, vastly dominates the total amount of reflectivity loss in these structures. Thus, the homogenization approximation can be used to estimate the reflectivity of roughened structures within the tested parameter space (rms roughnesses < 20% and rms wavelengths < 50% of the 1DPC periodicity, index contrasts < 1.75) for all wavelengths spanning the normal incidence band gap. This presents a significant improvement in computing time over the FDTD simulations. For example, the time required to obtain the reflectivity spectrum of a roughened 1DPC is over three orders of magnitude longer using the FDTD method. Thus, these results provide a more efficient scheme for calculating roughness effects in 1DPCs. Furthermore, these results imply that reflectivity loss in these structures is dominated by the influence of the roughness features on the effective index profile of the structure, rather than by diffuse scattering from the roughened interfaces.

Fig. 7. The results of the homogenization approximation applied to the 4-bilayer structures presented in Fig. 3.

It is interesting that the homogenization approximation can also capture the red-shifting behavior obtained by the FDTD simulations. One explanation for this could be that the effective widths of the smoothed interfaces in the approximated structure are larger for bluer light than for redder light. Therefore, the interfaces are effectively sharper (and, thus, closer to the ideal 1DPC structure) for longer wavelengths. Thus, the red end of the band gap shows a smaller decrease in the reflectivity than the blue end, resulting in a red-shift. Furthermore, the increased sensitivity of low index contrast structures to roughness can be explained by the fact that the smearing of the refractive index profile at the interfaces causes low index contrast structures to deviate more from their perfect analogue than high index contrast structures. This can be seen by considering the slope of the index profile at the interfaces of the approximated structures. Perfect structures have infinite slopes at each interface for any index contrast. However, the index profiles of the approximated structures have finite slopes at the interfaces. For a given RMS roughness, the index profile at an interface will be graded over the same distance, independent of the index contrast of the structure. Thus, the slope of the index profile for a high index contrast structure will be larger (and therefore, closer to the perfect analogue) than that for a low index contrast structure at the same RMS roughness value.

Fig. 8. The results of the homogenization approximation applied to the bilayer systems shown in 4Fig. 4. The 4-bilayer structure is shown in Fig. 6. Again, comparison of the two Figs. shows that the homogenization approximation is in good agreement with the FDTD results.
Fig. 9. The results of the homogenization approximation applied to the bilayer structures presented in Fig. 5. The 4-bilayer system is shown in Fig. 7.

4. Conclusions

Several 2D FDTD simulations were done in order to determine the effect of interfacial roughness on the normal incidence band gap. Many 1D photonic crystal configurations were tested, with systematic variations in the index contrast and number of bilayers. In all systems tested, a narrowing and red-shifting of the normal incidence band gap was observed. Furthermore, the lowest index contrast system exhibited a higher sensitivity to the red-shifting. This suggests that lower index contrast structures are less tolerant to structural changes than higher index contrast systems.

Acknowledgments

This research was supported by the Los Alamos National Laboratory Directed Research and Development Program and the U.S. Army through the Institute for Soldier Nanotechnologies under contract DAAD-19-02-D-0002 with the U.S. Army Research Office. The content does not necessarily reflect the position of U.S. government, and no official endorsement should be inferred.

References and links

1 .

K. R. Maskaly , G. R. Maskaly , W. C. Carter , and J. L. Maxwell , “ Diminished normal reflectivity of one-dimensional photonic crystals due to dielectric interfacial roughness ,” Opt. Lett. 29 , 2791 – 2793 ( 2004 ). [CrossRef] [PubMed]

2 .

G. S. He , T.-C. Lin , V. K. S. Hsiao , A. N. Cartwright , P. N. Prasad , L. V. Natarajan , V. P. Tondiglia , R. Jakubiak , R. A. Vaia , and T. J. Bunning , “ Tunable two-photon pumped lasing using a holographic polymer-dispersed liquid-crystal crating as a distributed feedback element ,” Appl. Phys. Lett. 83 , 2733 – 2735 ( 2003 ). [CrossRef]

3 .

V. K. S. Hsiao , T.-C. Lin , G. S. He , A. N. Cartwright , P. N. Prasad , L. V. Natarajan , V. P. Tondiglia , and T. J. Bunning , “ Optical microfabrication of highly reflective volume Bragg gratings ,” Appl. Phys. Lett. 86 , 131113 ( 2005 ). [CrossRef]

4 .

V. Agarwal and J. A. del Rio , “ Tailoring the photonic bandgap of a porous silicon dielectric mirror ”, Appl. Phys. Lett. 82 , 1512 – 1514 ( 2003 ). [CrossRef]

5 .

S. M. Weiss , H. Ouyang , J. Zhang , and P. M. Fauchet , “ Electrical and thermal modulation of silicon photonic bandgap microcavities containing liquid crystals ,” Opt. Express 13 , 1090 – 1097 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-4-1090 . [CrossRef] [PubMed]

6 .

K. R. Maskaly , W. C. Carter , R. D. Averitt , and J. L. Maxwell , “ Application of the homogenization approximation to roughened one-dimensional photonic crystals ,” Opt. Lett. (to be published). [PubMed]

7 .

A. Sentenac , G. Toso , and M. Saillard , “ Study of coherent scattering from one-dimensional rough surfaces with a mean-field theory ,” J. Opt. Soc. Am. A 15 , 924 – 931 ( 1998 ). [CrossRef]

8 .

T. K. Gaylord , W. E. Baird , and M. G. Moharam , “ Zero-reflectivity high spatial frequency rectangular groove dielectric surface relief gratings ,” Appl. Opt. 25 , 4562 – 4567 ( 1986 ). [CrossRef] [PubMed]

9 .

G. Voronovich , Wave Scattering from Rough Surfaces ( Springer-Verlag, Berlin , 1994 ).

10 .

K. S. Yee , “ Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media ,” IEEE Trans. Antennas Propag. 14 , 302 – 307 ( 1966 ). [CrossRef]

11 .

D. M. Sullivan , Electromagnetic Simulation Using the FDTD Method ( Institute of Electrical and Electronics Engineers, Piscataway, N.J. , 2000 ). [CrossRef]

12 .

A. Taflove and S. C. Hagness , Computational Electrodynamics, 2nd ed. ( Artech House, Norwood, Mass. , 2000 ).

13 .

J. P. Berenger , “ A perfectly matched layer for the absorption of electromagnetic waves ,” J. Comput. Phys. 114 , 185 – 200 ( 1994 ). [CrossRef]

14 .

Previously, we have referred to the quantity reported in Eq. (2) as the “percent” change in reflectivity. Here, we have changed the wording to “relative” as it is a more accurate description of the quantity in Eq. (2). Please note this change when comparing this manuscript to our previous ones.

15 .

J. A. Kong , Electromagnetic Wave Theory ( EMW, Cambridge, Mass ., 2000 ).

16 .

C. F. Bohren and D. R. Huffman , Absorption and Scattering of Light by Small Particles ( Wiley, New York , 1983 ).

OCIS Codes
(230.1480) Optical devices : Bragg reflectors
(240.5770) Optics at surfaces : Roughness

ToC Category:
Research Papers

History
Original Manuscript: July 25, 2005
Revised Manuscript: September 26, 2005
Published: October 17, 2005

Citation
Karlene Maskaly, W. Carter, Richard Averitt, and James Maxwell, "The effect of interfacial roughness on the normal incidence bandgap of one-dimensional photonic crystals," Opt. Express 13, 8380-8389 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8380


Sort:  Journal  |  Reset  

References

  1. K. R. Maskaly, G. R. Maskaly, W. C. Carter, and J. L. Maxwell, �??Diminished normal reflectivity of one-dimensional photonic crystals due to dielectric interfacial roughness,�?? Opt. Lett. 29, 2791-2793 (2004). [CrossRef] [PubMed]
  2. G. S. He, T.-C. Lin, V. K. S. Hsiao, A. N. Cartwright, P. N. Prasad, L. V. Natarajan, V. P. Tondiglia, R. Jakubiak, R. A. Vaia, and T. J. Bunning, �??Tunable two-photon pumped lasing using a holographic polymer-dispersed liquid-crystal crating as a distributed feedback element,�?? Appl. Phys. Lett. 83, 2733-2735 (2003). [CrossRef]
  3. V. K. S. Hsiao, T.-C. Lin, G. S. He, A. N. Cartwright, P. N. Prasad, L. V. Natarajan, V. P. Tondiglia, and T. J. Bunning, �??Optical microfabrication of highly reflective volume Bragg gratings,�?? Appl. Phys. Lett. 86, 131113 (2005). [CrossRef]
  4. V. Agarwal and J. A. del Rio, �??Tailoring the photonic bandgap of a porous silicon dielectric mirror,�?? Appl. Phys. Lett. 82, 1512-1514 (2003). [CrossRef]
  5. S. M. Weiss, H. Ouyang, J. Zhang, and P. M. Fauchet, �??Electrical and thermal modulation of silicon photonic bandgap microcavities containing liquid crystals,�?? 13, 1090-1097 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-4-1090">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-4-1090</a>. [CrossRef] [PubMed]
  6. K. R. Maskaly, W. C. Carter, R. D. Averitt, and J. L. Maxwell, �??Application of the homogenization approximation to roughened one-dimensional photonic crystals,�?? Opt. Lett. (to be published). [PubMed]
  7. A. Sentenac, G. Toso, and M. Saillard, �??Study of coherent scattering from one-dimensional rough surfaces with a mean-field theory,�?? J. Opt. Soc. Am. A 15, 924-931 (1998). [CrossRef]
  8. T. K. Gaylord, W. E. Baird, and M. G. Moharam, �??Zero-reflectivity high spatial frequency rectangular groove dielectric surface relief gratings,�?? Appl. Opt. 25, 4562-4567 (1986). [CrossRef] [PubMed]
  9. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).
  10. K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propag. 14, 302-307 (1966). [CrossRef]
  11. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2000). [CrossRef]
  12. A. Taflove and S. C. Hagness, Computational Electrodynamics, 2nd ed. (Artech House, Norwood, Mass., 2000).
  13. J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
  14. Previously, we have referred to the quantity reported in Eq. (2) as the �??percent�?? change in reflectivity. Here, we have changed the wording to �??relative�?? as it is a more accurate description of the quantity in Eq. (2). Please note this change when comparing this manuscript to our previous ones.
  15. J. A. Kong, Electromagnetic Wave Theory (EMW, Cambridge, Mass., 2000).
  16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited