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

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References

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  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).
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  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).
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Appl. Opt. (1)

Appl. Phys. Lett. (3)

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]

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]

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]

IEEE Trans. Antennas Propag. (1)

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]

J. Comput. Phys. (1)

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

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (2)

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]

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]

Other (6)

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

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.

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

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

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

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

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Figures (9)

Fig. 1.
Fig. 1.

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

Fig. 2.
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.
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.

Fig. 4.
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.
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.

Fig. 6.
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.

Fig. 7.
Fig. 7.

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

Fig. 8.
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.
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.

Equations (2)

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R [ 1 L 0 L ( y y 0 ) 2 dx ] 1 2 [ 1 n 1 n ( y y 0 ) 2 ] 1 2 ,
Δ r = 1 r rough r smooth ,

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