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**|**Nanoscale Res Lett**|**v.7(1); 2012**|**PMC3549903

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Nanoscale Res Lett. 2012; 7(1): 391.

Published online 2012 July 13. doi: 10.1186/1556-276X-7-391

PMCID: PMC3549903

Augusto David Ariza-Flores: moc.liamg@looc1divad; Luis Manuel Gaggero-Sager: xm.meau@oreggagl; Vivechana Agarwal: xm.meau@lawragav

Received 2012 April 30; Accepted 2012 June 29.

Copyright ©2012 Ariza-Flores et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (
http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We report the theoretical comparison of the omnidirectional photonic bandgap (OPBG) of one-dimensional dielectric photonic structures, using three different refractive index profiles: sinusoidal, Gaussian, and Bragg. For different values of physical thickness (PT) and optical thickness (OT), the tunability of the OPBG of each profile is shown to depend on the maximum/minimum refractive indices. With an increase in the value of the maximum refractive index, the structures with the same PT showed a linear increment of the OPBG, in contrast to the structures with the same OT, showing an optimal combination of refractive indices for each structure to generate the maximum OPBG. An experimental verification was carried out with a multilayered dielectric porous silicon structure for all the three profiles.

Omnidirectional mirrors (OM) can reflect all the incident light independent of the incidence angle, within a certain wavelength range
[1-10]. Omnidirectional properties have been shown using one-dimensional photonic crystals
[1], cladded superlattice structures
[2], multilayered heterostructures
[3], ternary photonic bandgap materials
[4], etc. for different systems (for example, Na_{3}AlF_{6}/Ge, SiO_{2}, BaF_{2}/PbS, GaAs, etc.)
[3-5,7]. Due to their potential applications in optical telecommunications and light-emitting systems, OMs from SiO_{2}, polypropylene, Si, GaN, etc.
[11-13] have been reported. Several groups have fabricated OMs from porous silicon (PS) in the near-infrared range due to their advantage over metallic mirrors of being non-absorbing and non-dispersive
[14-18]. Usually, PS multilayered structures are designed by alternating low- and high-porosity layers like a Bragg mirror
[14] or a mechanically stable, gradually varying Gaussian-like periodic profile
[15,16]. However, for a required physical thickness and omnidirectional photonic bandgap (OPBG), the best choice of the refractive index profile and the combination of indices are still not known. In this work, we report a comparative study of the dependence of OPBG as a function of maximum refractive index for three different refractive index profiles: sinusoidal, Gaussian, and Bragg type. The comparison was carried out between the structures with the same optical thickness (OT) and physical thickness (PT). An experimental verification was performed with the help of PS multilayered photonic structures.

All PS multilayered structures were prepared through anodic etching of a (100)-oriented p-type crystalline Si wafer (resistivity 2 to 5 m*Ω*cm), under galvanostatic conditions
[19]. For the electrochemical anodization process at room temperature, the electrolyte mixture was 1:1 (*v*/*v*) of HF (48 wt.%)/ethanol (98 wt.%), respectively. The current density and the etching duration of each layer were controlled by a computer-interfaced electronic circuit where the current density varied from 8.8 to 327 mA/cm^{2}, corresponding to the refractive indices of 2.5 and 1.48, respectively. All the structures consisted of 40 periodic unit cells with a sinusoidal, Gaussian, or Bragg refractive index profile. The reflectivity measurements were carried out with a PerkinElmer Lambda 950 UV/VIS spectrophotometer with a variable angle accessory, Universal Reflectance Accessory (URA; Waltham, MA, USA), for 8° and 68°. The maximum and minimum values of the incidence angle were limited due to the angular range covered by URA.

The theoretical simulations of the reflectivity spectra were done using the transfer matrix method for a *p*-polarized electromagnetic wave
[20]. Briefly, we suppose that an incident *p*-polarized electromagnetic wave (*E*_{I} and *H*_{I}) passes through a thin multilayered structure. At the first interface (*I*), part of the light reflects and the rest is transmitted. We can relate these light beams using the contour conditions for an incident electromagnetic wave at the interface. The transmitted wave has a phase shift by the time it reaches the next surface (*E*_{II} and *H*_{II} ); then, in this new surface (*II*), we relate again the reflected and transmitted electromagnetic beams and connect each layer with a transfer matrix:

$$\left[\begin{array}{l}{E}_{I}\\ {H}_{I}\end{array}\right]=\left[\begin{array}{ll}cos\left({k}_{0}h\right)& isin\left({k}_{0}h\right)/{Y}_{I}\\ {Y}_{I}isin\left({k}_{0}h\right)& cos\left({k}_{0}h\right)\end{array}\right]\left[\begin{array}{l}{E}_{\mathrm{II}}\\ {H}_{\mathrm{II}}\end{array}\right],$$

(1)

where *k*_{0} is the magnitude of the wave vector, *h* is the optical path, and *Y*_{I} is a function of the refractive index (*n*_{I}) and the transmitted angle (*θ*_{I}):

$${Y}_{I}=\sqrt{\frac{{\epsilon}_{0}}{{\mu}_{0}}}{n}_{I}/cos\left({\theta}_{I}\right).$$

(2)

By making the same procedure, we can couple the electromagnetic field of each interface with the preceding one:

$$\left[\begin{array}{l}{E}_{I}\\ {H}_{I}\end{array}\right]={M}_{I}\left[\begin{array}{l}{E}_{\mathrm{II}}\\ {H}_{\mathrm{II}}\end{array}\right].$$

(3)

For the second interface, the electromagnetic field (*E*_{II}, *H*_{II}) can be related to the third interface (*E*_{III}, *H*_{III}) by

$$\left[\begin{array}{l}{E}_{\mathrm{II}}\\ {H}_{\mathrm{II}}\end{array}\right]={M}_{\mathrm{II}}\left[\begin{array}{l}{E}_{\mathrm{III}}\\ {H}_{\mathrm{III}}\end{array}\right].$$

(4)

Then, incident field (*E*_{I}, *H*_{I}) can be related to the third field (*E*_{III}, *H*_{III}) by multiplying the transfer matrices *M*_{I} and *M*_{II}, resulting in

$$\left[\begin{array}{l}{E}_{I}\\ {H}_{I}\end{array}\right]={M}_{I}{M}_{\mathrm{II}}\left[\begin{array}{l}{E}_{\mathrm{III}}\\ {H}_{\mathrm{III}}\end{array}\right].$$

(5)

In general, if *P* is the number of layers, each one with a specific value of refractive index *n* and optical path *h*, then the first and last interface fields are related by

$$\left[\begin{array}{l}{E}_{I}\\ {H}_{I}\end{array}\right]={M}_{I}{M}_{\mathrm{II}}\mathrm{..}{M}_{P}\left[\begin{array}{l}{E}_{(P+1)}\\ {H}_{(P+1)}\end{array}\right].$$

(6)

The characteristic matrix of the complete system is the result of multiplying each individual 2×2 matrix:

$$M={M}_{I}{M}_{\mathrm{II}}\mathrm{..}{M}_{P}=\left[\begin{array}{ll}{m}_{11}& {m}_{12}\\ {m}_{21}& {m}_{22}\end{array}\right].$$

(7)

Finally, the total transfer matrix can be reduced to the reflection and transmission coefficients, and the equation can be reformulated in terms of contour conditions. Hence, the reflectivity is given by

$$\phantom{\rule{1.5em}{0ex}}R={r}^{2},$$

(8)

where

$$\phantom{\rule{1.5em}{0ex}}r=\frac{{Y}_{0}{m}_{11}+{Y}_{0}{Y}_{s}{m}_{21}-{m}_{12}-{Y}_{s}{m}_{22}}{{Y}_{0}{m}_{11}+{Y}_{0}{Y}_{s}{m}_{21}+{m}_{12}+{Y}_{s}{m}_{22}}$$

(9)

and

$$\phantom{\rule{1.5em}{0ex}}{Y}_{s}=\sqrt{\frac{{\epsilon}_{0}}{{\mu}_{0}}}{n}_{s}/cos\left({\theta}_{s}\right).$$

(10)

We used Equation 8 to compute the reflectivity spectrum for a multilayered dielectric structure. The refractive index profiles were obtained from the following equations:

For sinusoidal,

$$\begin{array}{lll}{n}_{i}=& \frac{{n}_{\mathrm{max}}-{n}_{\mathrm{min}}}{2}sin\left(\frac{2\mathrm{\Pi P}}{N}i-\frac{\Pi}{2}\right)\phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ & +\frac{{n}_{\mathrm{max}}+{n}_{\mathrm{min}}}{2}\phantom{\rule{14.22636pt}{0ex}}i=\{0,\dots ,440\}.\phantom{\rule{2em}{0ex}}\end{array}$$

(11)

For Gaussian (for one period),

$$\phantom{\rule{-12.0pt}{0ex}}{n}_{i}=\left\{\begin{array}{ll}{n}_{\mathrm{min}}& \phantom{\rule{28.45274pt}{0ex}}i=0\\ \left({n}_{\mathrm{max}}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{n}_{\mathrm{min}}\right){e}^{-{d}^{2}{(i-11)}^{2}/{\sigma}^{2}}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{n}_{\mathrm{min}}& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}i\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\{1,\dots ,21\},\end{array}\right)$$

(12)

and for Bragg type,

$$\phantom{\rule{1.5em}{0ex}}{n}_{i}=\left\{\begin{array}{ll}{n}_{\mathrm{max}}& \phantom{\rule{28.45274pt}{0ex}}i=2k\\ {n}_{\mathrm{min}}& \phantom{\rule{28.45274pt}{0ex}}i=2k+1,\end{array}\right)$$

(13)

where *n*_{max} and *n*_{min} are the maximum and minimum refractive indices, respectively, *P* is the number of periods, *N* is the number of layers, *i* is the label representing an arbitrary layer within a certain interval, *d* is the width of each layer, and *σ*^{2} is the variance.

Figure
Figure11 shows the comparison of OPBG as a function of maximum refractive index (*n*_{max}), for the structures with sinusoidal, Gaussian, and Bragg refractive index profiles for different OT and PT.

The *n*_{max} was varied from 2.2 to 2.9, while the miminum refractive index (*n*_{min}) was adjusted to keep the OT constant as (a) 24, (b) 25, and (c) 26 *μ*m. The computed range of *n*_{max} was limited by the experimental capability to obtain high refractive indices (keeping PS as a possible reference material) and the adjusted values of *n*_{min} to keep the same OT of all the structures. Figure
Figure1a,b,c1a,b,c demonstrates that for each OT, one can find a particular value of *n*_{max} at which the profile corresponding to the higher value of OPBG changes. For example, in Figure
Figure1b,1b, the largest OPBG for *n*_{max} range of 2.25 to 2.45, the Bragg-type profile has to be the preferred choice. For 2.45 <*n*_{max} < 2.57, the sinusoidal profile has the largest OPBG, but the Gaussian profile prevails for *n*_{max} > 2.57. A similar behavior is observed for higher OTs (Figure
(Figure1c).1c). For the OT of 24 *μ*m, the Bragg-type profile fails to demonstrate any OPBG (Figure
(Figure1a).1a). Although the Gaussian structure shows the largest OPBG, the corresponding value of *n*_{max} is also very high.

Figure
Figure1d,e,f1d,e,f shows the comparison of the OPBG for the structures with the same PT, i.e., 7.76 *μ*m. The *n*_{max} was varied from 2.3 to 2.9, while the *n*_{min} was kept constant as (a) 1.1, (b) 1.35, and (c) 1.5. Figure
Figure1a,b,c1a,b,c demonstrates that the Gaussian refractive index profile always requires higher refractive index values to obtain the same OPBG as compared to the sinusoidal refractive index profile. Equivalently, the OPBG obtained for the sinusoidal profile is always higher as compared to that for the Gaussian profile for a given *n*_{max}. In spite of the failure of the Bragg-type profile to demonstrate any OPBG for *n*_{min} = 1.1 (see Figure
Figure1d),1d), the tunability to increase/decrease the OPBG for *n*_{min} = 1.35 as compared to the sinusoidal and Gaussian profiles is shown in Figure
Figure1e.1e. One can identify three particular intervals for the Bragg profile (2.35 <*n*_{max} < 2.51, 2.51 <*n*_{max} < 2.72, and 2.72 <*n*_{max} < 2.9) at which the OPBG is higher/lower as compared to the sinusoidal and Gaussian profiles (Figure
(Figure1e).1e). For a higher *n*_{min}, Figure
Figure1f1f shows a significant enhancement for the Bragg-type structure, revealing a larger OPBG as compared to the other profiles. Hence, one can obtain the tunability of the OPBG in a certain refractive index range, depending on the available refractive indices and the profile of the photonic structure.

The result shows that no particular profile can be designated as the best profile for the complete range of maximum refractive indices discussed in this work. Apart from that, one can obtain the tunability of the OPBG in a certain refractive index range, depending on the available refractive indices and the profile of the photonic structure. The vertical dashed line in Figure
Figure1b1b corresponds to *n*_{max} = 2.5 and the particular OT incorporated in the forthcoming experimental and simulated results.

Figure
Figure22 shows the experimental (fabricated with PS multilayers) and simulated reflectivity spectra for the three types of photonic structures at 8° and 68° of incidence angle. As mentioned earlier, the results are obtained for *n*_{max} = 2.5 and 25 *μ*m of OT (dashed vertical line in Figure
Figure1b).1b). OPBG is shown as a vertical gray band. Good agreement between the calculated (dashed line) and the experimental spectra (solid line) is observed. The experimental OPBG was taken with more than 90% of the reflectivity for each multilayered structure. The sinusoidal profile (Figure
(Figure2a,d)2a,d) shows a 95-nm photonic bandgap, while the Gaussian (Figure
(Figure2b,e)2b,e) and Bragg (Figure
(Figure2c,f)2c,f) profiles show 45 and 63 nm of OPBGs, respectively. Hence, for the given value of OT (25 *μ*m) and *n*_{max} (2.5), the sinusoidal profile was shown to have almost twice the OPBG than the other two profiles under discussion.

On the other hand, Figure Figure33 shows the experimental and theoretical results for the photonic structures with the same PT. A good agreement is observed between theoretical and experimental results. The overlapping of PBG for different angles was measured as 177 nm for the sinusoidal profile (Figure (Figure3a,d),3a,d), while the Gaussian (Figure (Figure3b,e)3b,e) and Bragg (Figure (Figure3c,f)3c,f) profiles show an OPBG of 130 and 80 nm, respectively. To verify the mechanical stability of such structures, the surface images of the PS multilayered structure corresponding to each profile are shown as insets. The surface fractures observed on the Bragg-type structure (see inset in Figure Figure3c)3c) are attributed to the high-porosity contrast between two consecutive layers [21-23]. For the sinusoidal and Gaussian refractive index profiles, the inset images (see inset in Figure Figure3a,b)3a,b) show a flat-uncracked surface due to the gradual variation of the porosity between consecutive layers, which helps in reducing the stress and enhances the mechanical stability [21]. Therefore, a significant reduction in the intensity of the reflectivity spectra observed for the Bragg-type photonic structure (Figure (Figure3c,f),3c,f), as compared to the theoretical simulations, is attributed to the cracked structure which provokes a higher dispersion of the incident light.

Figure
Figure44 shows the theoretical contour plots for the reflectivity spectra as a function of the wavelength and the incident angle for the sinusoidal (Figure
(Figure4a,d),4a,d), Gaussian (Figure
(Figure4b,e),4b,e), and Bragg (Figure
(Figure4c,f)4c,f) mirrors. Figure
Figure4a,b,c4a,b,c corresponds to the photonic structures with the same OT, while Figure
Figure4d,e,f4d,e,f corresponds to the photonic sutructures with the same PT. As the angle of incidence is increased, the PBG (red region) decreases for all the photonic structures. In spite of the largest PBG at 0° (over the other profiles) for the Bragg mirror, the ability for keeping a *semi-constant* stop band, independent of the incident angle, is better demonstrated for the sinusoidal and Gaussian structures, showing a more pronounced fall of the PBG (after 45°) for the Bragg structure, as compared to the other mirrors. Hence, depending on the application, the refractive index profile can be selected to have a larger PBG within a certain angular range (e.g., from 0° to 45°, Bragg mirrors are a better choice) or a small PBG but for any possible incidence angle.

We demonstrate that the width of the OPBG depends on the choice of the maximum, the minimum, and the difference of the refractive indices for any given profile (sinusoidal, Gaussian, or Bragg-type refractive index profiles). The structures with the same OT showed an optimal combination of refractive indices to generate the largest OPBG, as compared to the structures with the same PT which showed a linear increase in the OPBG. An experimental verification performed with the nanostructured porous silicon dielectric multilayered structures confirmed the superiority of the sinusoidal profile over the Gaussian profile to enhance the OPBG and reduce the structural stress compared to the Bragg structure. This study can be useful to design the required OPBG structures for photonic applications.

The authors declare that they have no competing interests.

ADA carried out the theoretical simulations, experimental fabrication, and measurements of the samples. LMGS participated in its coordination. VA conceived the study, worked on the manuscript with AD, and participated in its design and coordination. All authors read and approved the final manuscript.

ADA is a Ph.D. student (in Physics) registered at the Faculty of Sciences, UAEM and doing his research work at CIICAp-UAEM, Mexico. LMGS is a professor investigator at the Faculty of Sciences, UAEM and working on the electronic properties of semiconductors from a theoretical point of view. VA is working as a professor investigator at CIICAp UAEM in the field of nanostructured silicon (fabrication, characterization, and applications).

This work has been partially supported by CONACyT under scholarship no. 39986 and project no. 128953.

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