Background
Omnidirectional mirrors (OM) can reflect all the incident light independent of the incidence angle, within a certain wavelength range
1
2
3
4
5
6
7
8
9
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, Na3AlF6/Ge, SiO2, BaF2/PbS, GaAs, etc.)
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4
5
7
. Due to their potential applications in optical telecommunications and light-emitting systems, OMs from SiO2, polypropylene, Si, GaN, etc.
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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
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. 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
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. 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.
Methods
All PS multilayered structures were prepared through anodic etching of a (100)-oriented p-type crystalline Si wafer (resistivity 2 to 5 mΩcm), 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/cm2, 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.
Theoretical overview
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:
E
I
H
I
=
cos
(
k
0
h
)
i
sin
(
k
0
h
)
/
Y
I
Y
I
i
sin
(
k
0
h
)
cos
(
k
0
h
)
E
II
H
II
,
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
=
ε
0
μ
0
n
I
/
cos
(
θ
I
)
.
By making the same procedure, we can couple the electromagnetic field of each interface with the preceding one:
E
I
H
I
=
M
I
E
II
H
II
.
For the second interface, the electromagnetic field (E
II
, H
II
) can be related to the third interface (E
III
, H
III
) by
E
II
H
II
=
M
II
E
III
H
III
.
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
E
I
H
I
=
M
I
M
II
E
III
H
III
.
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
E
I
H
I
=
M
I
M
II
..
M
P
E
(
P
+
1
)
H
(
P
+
1
)
.
The characteristic matrix of the complete system is the result of multiplying each individual 2×2 matrix:
M
=
M
I
M
II
..
M
P
=
m
11
m
12
m
21
m
22
.
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
R
=
r
2
,
where
r
=
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
and
Y
s
=
ε
0
μ
0
n
s
/
cos
(
θ
s
)
.
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,
n
i
=
n
max
−
n
min
2
sin
2
ΠP
N
i
−
Π
2
+
n
max
+
n
min
2
i
=
{
0
,
…
,
440
}
.
For Gaussian (for one period),
n
i
=
n
min
i
=
0
n
max
−
n
min
e
−
d
2
(
i
−
11
)
2
/
σ
2
+
n
min
i
=
{
1
,
…
,
21
}
,
and for Bragg type,
n
i
=
n
max
i
=
2
k
n
min
i
=
2
k
+
1
,
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.
Results and discussion
Figure
1 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.
<p>Figure 1</p>OPBG as a function of nmax for the sinusoidal, Gaussian, and Bragg profiles
OPBG as a function of nmax for the sinusoidal, Gaussian, and Bragg profiles. Panels (a), (b), and (c) correspond to the OT of 24, 25, and 26 μm, respectively. Panels (d), (e), and (f) correspond to the PT = 7.76 μm for three different values of minimum refractive index. The OPBG was computed between 0° and 85° of incidence angle at 99% of the reflectivity.
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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 μm. 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
1a,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
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
1c). For the OT of 24 μm, the Bragg-type profile fails to demonstrate any OPBG (Figure
1a). Although the Gaussian structure shows the largest OPBG, the corresponding value of n
max is also very high.
Figure
1d,e,f shows the comparison of the OPBG for the structures with the same PT, i.e., 7.76 μm. 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
1a,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
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
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
1e). For a higher n
min, Figure
1f 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
1b corresponds to n
max = 2.5 and the particular OT incorporated in the forthcoming experimental and simulated results.
Figure
2 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 μm of OT (dashed vertical line in Figure
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
2a,d) shows a 95-nm photonic bandgap, while the Gaussian (Figure
2b,e) and Bragg (Figure
2c,f) profiles show 45 and 63 nm of OPBGs, respectively. Hence, for the given value of OT (25 μm) and n
max (2.5), the sinusoidal profile was shown to have almost twice the OPBG than the other two profiles under discussion.
<p>Figure 2</p>Theoretical (dashed line) and experimental (solid line) reflectivity spectra for structures with the same OT
Theoretical (dashed line) and experimental (solid line) reflectivity spectra for structures with the same OT. The PBG generated using the (a) sinusoidal, (b) Gaussian, and (c) Bragg profiles for 8°. The corresponding reflectivity spectra at 68° are shown in panels (d), (e), and (f), respectively. The overlapping of the experimental PBG between 8° and 68° is shown as a gray band (taken for the reflectance more than or equal to 90%).
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On the other hand, Figure
3 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
3a,d), while the Gaussian (Figure
3b,e) and Bragg (Figure
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
3c) are attributed to the high-porosity contrast between two consecutive layers
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22
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. For the sinusoidal and Gaussian refractive index profiles, the inset images (see inset in Figure
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
3c,f), as compared to the theoretical simulations, is attributed to the cracked structure which provokes a higher dispersion of the incident light.
<p>Figure 3</p>Theoretical (dashed line) and experimental (solid line) reflectivity spectra
Theoretical (dashed line) and experimental (solid line) reflectivity spectra. (a) Sinusoidal, (b) Gaussian, and (c) Bragg refractive index profiles at 8°. The corresponding reflectivity spectra at 68° are shown in panels (d), (e), and (f), respectively. The intersection of the PBG between 8° and 68° is shown as a gray band. The nmin and nmax were 1.2 and 2.4, respectively, and the total physical thickness was 7,760 nm for each structure. The inset image shows an optical microscopy surface zone for (a) sinusoidal (b) Gaussian and (c) Bragg profiles. The scale bar corresponds to 200 μm.
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Figure
4 shows the theoretical contour plots for the reflectivity spectra as a function of the wavelength and the incident angle for the sinusoidal (Figure
4a,d), Gaussian (Figure
4b,e), and Bragg (Figure
4c,f) mirrors. Figure
4a,b,c corresponds to the photonic structures with the same OT, while Figure
4d,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.
<p>Figure 4</p>Contour plot of the reflectivity spectra as a function of the angle and wavelength
Contour plot of the reflectivity spectra as a function of the angle and wavelength. (a, d) Sinusoidal, (b, e) Gaussian, and (c, f) Bragg refractive index profiles. The color scale indicates the reflectivity percentage from 0% (blue) to 100% (red).
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