We report the design, fabrication, and characterization of a porous silicon-based omnidirectional mirror for the near infrared range. The structure consists of 300 porous silicon chirped dielectric layers, optimized to have omnidirectional reflectivity response from 1000 to 2000 nm wavelength range. Measurements of reflectivity spectra are presented for non-polarized light at several incident angles (range –) with a reflectivity 95% covering a band-width. Transfer matrix method calculations were carried out to show the complete angular range for both TM and TE polarizations including a simple model to illustrate the interface scattering effects.
I. INTRODUCTION
Metallic mirrors are widely employed in optical instrumentation to manipulate the flow of light; however, as the demand of the improve performance of devices is growing, there is an increasing need for dielectric mirrors with a high reflectivity. Although metallic mirrors can reflect the light omnidirectionally, the percentage of reflectivity cannot be tuned and lies between 80% and 99% depending on the material, wavelength range, and the antireflection/protection thin films present on the first surface metallic mirrors. On the other hand, dielectric mirrors can be designed to reflect 99.9%,1,2 and the reflectivity band can be tuned; however, they are not omnidirectional in general. A few years ago, the angular-independent reflectivity was theoretically predicted only for systems with dielectric functions periodic along three orthogonal directions;3 more currently, the reflectivity of a three-dimensional system (consisting of a silicon inverse woodpile photonic crystal) has been observed to produce the omnidirectional response.4 Fink et al. reported the first fabrication and characterization of an all-dielectric one-dimensional (1D) omnidirectional reflector based on polystyrene and tellurium alternating layers, demonstrating that a 1D periodic multilayer structure can have omnidirectional reflectance, if the non-propagating states of light cover a continuous frequency range of both TM and TE polarizations within the light cone.5 Nevertheless, the omnidirectional photonic bandgap (OPBG) of periodic structures is limited by the refractive indices and the central wavelength of design, in contrast, non-periodic chirped structures can increase the OPBG several times,6,7 allowing for omnidirectional reflectance, e.g., for the complete visible range.8
Since the discovery of Fink et al., several research groups designed and fabricated omnidirectional mirrors due to the applications in tunable photonic structures,9 all-dielectric coated waveguides,10,11 dielectric mirror fibers,12 femtosecond pulsed lasers,13–15 terahertz communication systems,16 and planar waveguides17 or for the opposite case, anti-reflective omnidirectional structures, such as broadband light trapping.18 Moreover, in the last few years, investigators have sought to extend the width of the OPBG in different regions of the electromagnetic spectrum.1,8,19 In particular, porous silicon (PS) omnidirectional structures have been reported for the near-infrared telecommunication range.20–22 For example, Estevez et al. show an omnidirectional reflectance of 252 nm from 950 nm to 1456 nm, from an unbalanced mirror structure based on Gaussian sub-structures;7 Xifré-Perez et al. obtained an OPBG of 319 nm centered at by overlapping three unbalanced sub-structures;22 Bruyant et al. reported an OPBG of 340 nm centered at 1270 nm with a chirped structure of 70 layers,6 being to the best of our knowledge, the largest OPBG reported by porous silicon multilayers for the near infrared (NIR) range. While several theoretical works have proposed different designs of structures to enlarge the OPBG such as Gaussian and sinusoidal profiles,23,24 ternary periodic structures,25 negative index materials,26–28 random dielectric multilayers,29,30 the Bragg stack with a periodic gain–loss modulation,31 etc., the fabrication from porous silicon multilayers is still a challenge.
In this work, we optimize the chirping method to design, fabricate, and characterize a wide range omnidirectional structure by porous silicon dielectric multilayers, covering the near infrared range from 1 to . Section II provides the details about the fabrication of the omnidirectional mirror; Sec. III shows how reflectivity spectra is computed and the design of the structure; Sec. IV shows a discussion of the experimental results followed by the conclusions in Sec. V.
II. EXPERIMENTAL DETAILS
The PS sample was fabricated by the electrochemical anodization process of a (100)-oriented p-type Boron doped Si wafer, with resistivity. The silicon wafer is initially submerged in a solution of HF and ethanol in 1:10 volumetric proportion during 20 min in order to remove the silicon oxide native layer. Later, the wafer is placed in a teflon chamber with its polished surface exposed to an electrolyte mixture of HF (48% concentration) and ethanol (99.8% of purity) in the 2:1 volumetric proportion. A teflon mask delimits a circular surface area of the grown sample to be . The temperature of the reaction is fixed to by employing a PID control. Then, an electric current is supplied by a programmable Keithley 6220 current source through a platinum electrode immersed in the electrolyte to perform the etching process. The counter electrode is a copper plate located in the base of the wafer to close the circuit. Two alternating current densities of 2 and are supplied at different times in order to obtain the multilayer structure. The thicknesses of each layer are related to the time elapsed during the applied current, and they are calculated to obtain a chirped structure. When the electrochemical reaction ends, the sample is rinsed in ethanol and later in pentane to be finally dried with air. The oxidation process is carried out with air flux in a furnace at for 3 h, in order to coat the porous surface with a silicon oxide layer, passivating the surface of further oxidation and reducing the absorption of the structure. After the oxidation process, the porosity show 19% and 46% as compared to the initial values of 30% and 70%, respectively, with an oxidation proportion of in both cases according to simulations with the Looyenga effective medium theory. The reflectivity was measured with a PerkinElmer spectrometer Lambda 950 from 500 to 3300 nm at different angles of incidence from to with unpolarized light. The electronic microscopy images were taken by a UHR dual beam FEI Helios nanolab 600.
III. NUMERICAL SIMULATIONS
A. Calculation of reflectivity of a multilayer system
We employ the transfer matrix method (TMM) to compute the reflectivity of the chirped multilayer structure through
where is the reflection coefficient for the or polarization, is the angle of incidence on the surface of the structure, and is the wavelength of the incident light. For any polarization, the reflection coefficient is computed as
where are the elements of the transfer matrix, resulting from the multiplication of matrices that compose the multilayer structure. Therefore,
For or polarizations,
where ; , and are the permittivity and permeability of the vacuum, respectively; is the refractive index of the th layer; and is the incident angle of the light in the th layer. Moreover, sub-index 0 and refer to the incident and substrate media, respectively.
B. Chirped structure
The chirped mirror is designed as an alternating porosities structure of a Bragg reflector-like stack, where the central wavelength of the photonic bandgap is varied quasi-continuous, as a function of the th layer in the structure. The Bragg interference condition is thus set as
where and are the complex refractive index and physical thickness of the th porous layer, respectively, and
where , are the minimum and maximum wavelengths to set in the computation, and is an optimization parameter to design the OPBG structure. Figure 1 illustrates the variation of the central wavelength of the structure as a function of th layer for .
Power modulation of the Bragg-type central wavelength as a function of the th layer for .
Power modulation of the Bragg-type central wavelength as a function of the th layer for .
Complex refractive indices of the porous layers in the simulations were obtained by the Looyenga effective medium theory, which has been reported to adequately reproduce the PS optical parameters.32 We consider porous silicon as an effective medium resulting from the mixture of silicon, silicon oxide, and air with refractive index
where , , , and are the complex refractive indices of porous silicon, air, silicon oxide, and the silicon substrate, respectively. The fractions of air (porosity) and silicon oxide are represented by and , respectively. The alternating high and low porosity values, fixed by the corresponding current densities, were obtained in a previous calibration experiment through the fabrication and optical characterization of monolayers. This previous calibration experiment provides an initial approach to simulate, design, and fabricate the structure; however, the measured spectrum of the omnidirectional mirror is finally adjusted, obtaining slight variations with respect to proposed parameters. Therefore, with the help of Eqs. (5)–(7), we developed an algorithm to obtain the physical thicknesses for each porous layer that ensures a chirped mirror with the Bragg-type constructive interferences, which covers a frequency range wider than the unchirped Bragg mirror.
C. Interface roughness
The scattered light at the PS–Si interface strongly depends on the thickness of the PS layer and the doping level of the substrate, being more pronounced for large width structures.33 Moreover, the proposed mirror thickness of owns several PS–PS interfaces, increasing this effect. A simple model to take into account the scattering of light due to roughness is to consider that each ray of light travels a different length in the same layer due the different thicknesses produced by roughness. The optical reflectivity response should be the resultant of averaged spectra of each ray. Therefore, we consider a random variation of thicknesses due to roughness (of of the width of each layer), and we average 100 spectra. The random variation of widths reduces the amplitude of Fabry–Pérot fringes and lead to a better agreement between theoretical simulation and experimental data (see the supplementary information).
IV. RESULTS AND DISCUSSION
Figure 2 shows the cross-sectional SEM image of the fabricated sample revealing the multilayer structure of the mirror. It can be observed the quasi-continuous increment in the thickness of each layer as a function of the depth. The total thickness of the multilayer was measured to be in agreement with the simulated spectra (). Insets show a detailed close up in a region near the surface and the base of the mirror, evidencing the layered structure, the increasing thicknesses, and the slight roughness of interfaces. Figure 3 shows the measured reflectivity spectra of the mirror (continues red lines) for several incident angles from to along with the TMM adjusted line shapes employing the simple roughness model for both polarizations (continues gray lines). Gray bar illustrate the omnidirectional photonic bandgap for the measured angular range, with a reflectivity higher than 95% between 1000 and 2000 nm. The fabricated structure shows an OPBG of approximately 1000 nm centered at 1500 nm, with an enhancement of 2.9, 3.1, and 4 times with respect to the porous silicon multilayer structures reported by Bruyant et al.,6 Xifré-Pérez et al.,22 and Estevez et al.,7 respectively. In fact, the presented structure overwhelmingly exceeds the width of OPBG of the three-dimensional photonic crystal reported by Huisman et al., being 245 nm and centered at 1572 nm.4 TE and TM separated experimental reflectivities are not shown due to the lack of optical elements in the employed spectrometer (PerkinElmer lambda 950). Although independent measurements of TE and TM modes are more suitable than non-polarized light measurements, we believe that the OPBG or the decrease of the stop band caused by any of the modes can be detected with non-polarized light as a high reflectivity band or an intermediate reflectance, respectively, that is to say, any reduction of reflectance associated with any of the modes, results in a marked diminution of the bandgap, thus, making the measured omnidirectional reflectivity band reliable. On the other hand, a photonic band structure is not reported since we are analyzing a non-periodic system.
Cross-sectional FEI image of the chirped multilayer mirror. The insets show an amplification near the surface (upper image) and the base (lower image) of the structure.
Cross-sectional FEI image of the chirped multilayer mirror. The insets show an amplification near the surface (upper image) and the base (lower image) of the structure.
Reflectivity spectra of the fabricated sample (red lines) along with the simulation of TE and TM modes (gray lines) for several incident angles. Gray bar indicates the omnidirectional photonic bandgap.
Reflectivity spectra of the fabricated sample (red lines) along with the simulation of TE and TM modes (gray lines) for several incident angles. Gray bar indicates the omnidirectional photonic bandgap.
Figure 4 displays a simulated contour plot of the reflectivity as a function of the incident angle and wavelength, in order to show the complete angular range. Horizontal dashed lines illustrate the experimental wavelength range shown in Fig. 3 with a high reflectivity for both TM and TE polarization modes; however, the TM mode shows a slight decrease in reflectance for angles higher than . By employing the above described computation method, we realized that considerably larger OPBG structures for a low absorption frequency range can be obtained. However, we noted an increased interfacial roughness as the structure becomes larger, causing the diminution of reflectivity. This issue is even more pronounced as the angle of incidence of light increases and can be explained based on the report of Lérondel et al., who showed that scattering effects and roughness of a porous silicon layer increase as the thickness of the film increases.33 In the present case, the oblique incidence of light provokes an increment of the optical path as compared to normal incidence, which results in an increment of scattering. To overcome this obstacle, PS fabrication could be carried out at a slower rate; however, the anodization reaction could take days to fabricate one structure.
Contour plot of the reflectivity as a function of the incidence angle and wavelength. TM/TE polarization is plotted in the left/right hand side. Color-bar indicates the percentage of reflection from 0 (black) to 100% (yellow). Horizontal dashed lines illustrate the omnidirectional reflection range. Vertical solid line indicates the limit where both polarizations equal at normal incidence.
Contour plot of the reflectivity as a function of the incidence angle and wavelength. TM/TE polarization is plotted in the left/right hand side. Color-bar indicates the percentage of reflection from 0 (black) to 100% (yellow). Horizontal dashed lines illustrate the omnidirectional reflection range. Vertical solid line indicates the limit where both polarizations equal at normal incidence.
V. CONCLUSIONS
We were able to demonstrate the enlargement of the OPBG of a porous silicon all-dielectric multilayer mirror, which covers a wavelength interval from 1 to in the near infrared range and owns a reflectivity higher than 95%. The reflectivity measurements are showed for several incidence angles within an angular range from to for non-polarized light, revealing an enhancement of three times with respect to the largest OPBG reported in porous silicon.6 We showed that the chirping method through a power variation function of the Bragg-type interference rule allows us to design considerably larger OPBG structures, whose reflectivity can be limited by the roughness of the PS interfaces. This kind of structures can be applied to fabricate high-Q microcavities to study, for instance, light-mater interaction coupling.
SUPPLEMENTARY MATERIAL
See the supplementary material for a more complete explanation about the diminution of Fabry–Pérot oscillations in the measured spectra.
ACKNOWLEDGMENTS
We are grateful for the technical support provided by F. Ramírez-Jacobo and E. Ontiveros-Hernández during the course of this research. This work was partially supported by the Consejo Nacional de Ciencia y Tecnología through Grant No. 256243 and the Cátedras Conacyt Program No. 1577.
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.