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Tunability of the optical response of multilayered photonic structures has been compared with sequential (SQ) and superposition (SP) addition of refractive index profile functions. The optical response of the composite multilayered structure, formed after the SP addition of the two Bragg type refractive index profile functions has been studied as a function of percentage overlap and relative shift between the profiles. Apart from the substantial advantage in terms of the reduced physical thickness of the SP composite structures (over the SQ addition), at certain optimum values of relative shift, photonic structures with better quality factor resonant modes or a broader PBG could be designed. Similar analysis has been extended for rugate filters as well. The experimental verification of the optical response, was carried out through multilayered dielectric porous silicon structures fabricated by electrochemical anodization.
© 2013 Optical Society of America
1. Introduction
Different designs of one dimensional (1D) photonic structures have been investigated due to their ability to control the propagation of light [1–3]. In particular, dielectric reflectors are one of the most widely studied optical devices due to low absorption of the light as compared to the metallic mirrors [4–6]. Many different materials have been used to fabricate high quality 1D photonic structures, e.g., polystyrene-tellurium [6], SiO2/ZrO2[7], TiO2/SiO2[8], AlGaAs-GaAs [9], B4C/Si [10], porous silicon (PSi) [11] etc. In particular, a wide variety of photonic structures fabricated with PSi have been reported [12–18] due to its ease of fabrication [19] and relatively economical preparation method [20]. Broad band Bragg mirrors [11, 21], high quality microcavities [12], sharp rugate filters [13], omnidirectional mirrors [14], waveguides [15] etc., have been fabricated in the last decade for their possible applications as photonic devices, filters and sensors [22]. Moreover, in order to obtain a multiple-peak optical response from the same photonic structure, the fabrication of composite structures has been studied through sequential (SQ) [23] and superposition (SP) addition [24] of refractive index profile functions (corresponding examples are given in Fig. 1(a) and (b), respectively).
Fig. 1 Schematics of the refractive index profile of two Bragg mirrors designed for λ0 = 1.0 μm (i) (dotted line) and λ0 = 1.5 μm (ii) (dashed line) along with the resultant structure (iii) (solid line) for (a) superposition and (c) sequential addition, respectively. Figure (b) and (d) show the corresponding reflectivity spectra of the resultant addition in (a) and (c), respectively.
For example, Meade et al. [24], demonstrated the spectrally encoded structures through superposition of sinusoidal current-time waveforms. The SP addition method has been used to prepare optically encoded composite structures, with the similar reflectance peak corresponding to each independent rugate, and the possibility to construct an encoded library with PSi photonic structures has also been investigated. Some of the possible applications of such photonic structures have been oriented towards molecular sensing field [25], in which the physical thickness of the structures becomes a limiting factor due to the limited availability of reagents and the penetration inhomogeneity along the depth. Hence, a reduction in physical thickness of the filter amounts to its greater applicability in the sensing field. In 2009, Sailors research group [26] used the lithografic procedure to isolate similar spectrally encoded PSi structures in the form of microparticles for the simultaneous detection of multiple analytes. Recently, SP structures (double/triple rugate filters) have been found useful for the gas sensing applications due to faster recovery after vapor exposure as compared to SQ structures [27, 28].
In spite of the wide applications of such structures, no systematic study based on the percentage overlap of the superimposed refractive index profiles or their relative phase shifts has been reported. Apart from that, the study of superposition of refractive index profile (or current time waveform) has only been implemented to particular type (rugate) of optical filters [24]. The aim of the present work is to study the characteristics of the optical response of different multilayered photonic structures under SP addition. The effect of percentage overlapping and relative shift of the refractive index profiles has been investigated for Bragg mirrors and rugate filters. The experimental verification of the theoretical results has also been presented.
2. Theoretical overview
2.1. Reflectivity of dielectric multilayers
The reflectance spectrum of PSi based multilayers was simulated by the transfer matrix method [29]. Consider an electromagnetic (EM) polarized wave incident on a dielectric surface. The EM wave (having the electric and magnetic fields EI and HI, respectively) passes through a thin multilayered structure. At the first interface (I), part of the incident light is reflected and the remaining part is transmitted (considering no absorption) into the second interface (II). The transmitted wave has a phase shift by the time it reaches the next surface (II). Considering the boundary conditions of an EM wave at the interface and the appropriate phase shift, the EM field components in the first layer (i.e., EI and HI) can be written as,
where k0 is the magnitude of the wave vector, h is the optical path of the first layer and ΓI is a function of the refractive index (nI) given by This procedure relates the EM field of each layer with its proceeding layer as,For the second interface (II), the electromagnetic field (EII, HII) can be related to the third interface (EIII, HIII) by
Then incident field (EI, HI) can be related to the EM field at the third interface (i.e., EIII, HIII) by multiplying the transfer matrices MI and MII, resulting in
In general, if P is the number of layers, each one with a specific value of refractive index n and optical thickness h, then the first and last interface fields are related byThe characteristic matrix of the complete system is obtained by multiplying each individual 2 × 2 matrix
Finally, the total transfer matrix is used to calculate the reflection and transmission coefficients, in terms of the refractive indices and the physical thicknesses of the complete structure. Hence, the reflectivity is given by where and where s is the substrate, and j corresponds to the jth layer. The Eq. (8) is used to simulate the reflectivity spectra as a function of the wavelength, for all the 1D multilayered photonic structures in this article.2.2. Summation of refractive index profiles
In order to show the optical properties of the composite multilayers, and compare them with the sequential addition, the summation technique of the superposed structures is proposed as follows
where ni(x) is refractive index profile of the ith function, x is the optical thickness, and 〈c〉 is the average refractive index and can be obtained as with nH and nL the maximum and minimum values of the refractive index profiles, respectively, and p(ni) is the distribution function of ni. In Eq. (11), the average of the second, third, etc., added function was substracted from the summation to obtain only the fluctuations around the average. For the fabrication of the PSi multilayer samples 〈c〉 is kept identical for all refractive index profiles.3. Experimental details
Some of the simulated photonic structures were fabricated through anodic etching of a (100) oriented, p–type crystalline Si wafer (resistivity 0.002 – 0.005 Ω·cm), under galvanostatic conditions [14]. The electrochemical anodization process was performed at room temperature, with an electrolyte mixture of aqueous HF (concentration: 48 % of wt), glycerol (purity: 99.8 % of wt), and ethanol (purity: 99.9 % of wt) in 3:7:1 proportion of volume, respectively. The current density and the duration of the etching time of each layer was controlled by a computer interfaced electronic circuit, where the current density was varied from 1.1 to 72.5 mA/cm2, corresponding to the refractive indices of 2.05 and 1.09, respectively. After the anodization process, the samples are rinsed with ethanol (purity: 99.9 % of wt) and dried with n-pentane (purity: 99.6 % of wt). The reflectivity measurements were carried out with a Perkin Elmer Lambda 950 UV-vis-NIR spectrophotometer. The refractive indices of the samples used in this work were calculated by interferometric method at 1500 nm.
4. Results and discussion
Figure 1 shows the theoretical comparison of the refractive index profiles (n) plotted against optical thickness (x) and the reflectivity spectra (R), of the composite functions obtained by SP and SQ addition of two Bragg mirrors. Figure 1(a) and 1(c) show the schematics of the refractive index profile of each Bragg mirror (i + ii) (in dotted and dashed lines) as well as the resulting composite function (iii) (in solid line) obtained by SP and SQ addition, respectively. The refractive index profiles of the Bragg mirrors, were constituted of 10 period step function with maximum and minimum refractive indices being 2.0 and 1.47, respectively.
Figures 1(b) and 1(d) show the reflectivity spectra of the composite structures by SP and SQ addition, respectively (in solid line), along with the reflectivity response of each Bragg mirror (dotted and dashed lines). Although there exist a slight red shift and qualitatively more flat photonic bandgap (PBG) (Fig. 1(b)) in SP addition as compared to the PBGs of the SQ mirror (Fig. 1(d)), the overall reflectivity of the composite structure shows the presence of the similar PBGs as the individual structures. The total optical thickness of the composite SP structure and the SQ are 4488 nm and 7351 nm, respectively.
Figure 2 is an illustration of the SP addition for other two commonly used optical filters: rugate and microcavity. Figures 2(a) and 2(c) show the individual and composite refractive index profiles for the rugate and microcavity structures, respectively. An analogous behavior is observed for the reflectance/absorption modes in the composite structures (rugate/microcavity) as compared to the individual profiles. For example, the reflectivity peaks of each rugate filter in Fig. 2(b) (dotted and dashed lines), match with the composite spectrum (solid line). However, the two microcavities of the composite structure plotted in Fig. 2(d), show a small blue and red shift of the modes (of 5 nm and 18 nm; although not visible due to the size of the figure) as compared to the reference structures.
Fig. 2 Schematics of the refractive index (dotted/dashed line) and the resultant SP addition profile (solid line) of (a) rugate filters and (c) microcavity structures. The corresponding reflectivity spectra are shown in panels (b) and (d).
To investigate the other possible details of the SP and SQ addition, Fig. 3 shows the reflectivity spectra of the composite structures designed with two Bragg mirrors with closely located PBGs. For a particular case, the mirrors centered at λ1 = 1.5 μm and λ2 = 1.362 μm (dotted and dashed lines in Figs. 3(a) and 3(b)), with the maximum and minimum refractive indices of 1.8 and 1.45 for both mirrors, are taken. The reflectivity spectrum of the composite structure obtained by SP (solid line in Fig. 3(a)), and SQ addition (solid line in Fig. 3(b)), demonstrate a clear difference through the appearance and absence of the resonant mode in SP and SQ addition, respectively. In contrast, in Fig. 1, the distinction between SP and SQ addition is slightly noticeable due to the large separation between both Bragg mirrors obtained in SQ addition. In the case of SP addition, the appearance of a microcavity can be attributed to the close overlapping Bragg mirrors, as an intermediate state before the complete PBGs superposition of each structure. In order to demonstrate the above mentioned effect for a wider range of closed PBGs overlapping Bragg mirrors, the phenomena is showed as a contour plot (Fig. 3(c)), where the composite reflectivity response (constructed by two SP Bragg mirrors) is plotted as a function of the relative overlapping (Δλ) and wavelength (λ). The first mirror is fixed at λ1 = 1.5 μm and the operating wavelength (λ2) of the second mirror is varied, keeping λ1 as a center wavelength. The relative overlapping Δλ is calculated as Δλ = 100% × (λ2 − λ1)/λ1 for the values of Δλ within ± 30 %. Figure 3(c) demonstrates the presence of an enhanced Bragg mirror, as a result of the SP structures for the range of Δλ = (−7, 9). For the range Δλ = (−15, −7) ∪ (9, 17), a well defined resonant mode appears. Outside of these ranges (Δλ = (−30, −15) ∪ (17, 35)) two distinct Bragg mirrors are obtained. As an example of all the above mentioned regions of Δλ, some particular cases are plotted in Fig. 3(d). The reflectivity spectra of the four SP structures with λ1 = 1.36 μm combined with different values of λ2, show the tunability of the SP addition to obtain an enhanced PBG Bragg mirror (λ2 = 1.4 μm), a high quality factor microcavity (λ2 = 1.5 μm), and separated PBGs mirrors (λ2 = 1.7 μm). This behavior has been reported by Agarwal et al. [11] for the SQ added substructures, and has been used to fabricate wide PBG mirrors using the addition of several substructures [30]. Figure 3(e) shows the comparison of two resonant microcavity modes obtained by a typical half wave microcavity (dotted-dashed line) and the SP addition of Bragg mirrors (solid line), simulated with a localized mode at 1.45 μm and 10 periods. Relatively more narrow full width at half maximum (FWHM) is revealed for the structure corresponding to the SP addition, as compared to the typical half wave micro-cavity. The quality factor (Q = λ0/ΔλFWHM) obtained by the SP structure and the half wave microcavity photonic structures was computed as 227 and 97 respectively, i.e., an enhancement by a factor of 2.3 was obtained for the SP added structure as compared to the half wave microcavity.
Fig. 3 Reflectivity spectrum of a composite structure obtained by (a) SP and (b) SQ addition of two Bragg mirrors. The dotted/dashed line correspond to typical Bragg mirrors designed for λ0 = 1.362/1.5 μm. Figure (c) shows a contour plot for the reflectivity response as a function of the relative SP (Δλ/λ1) and the wavelength (λ). The first mirror is fixed at (λ2). The color scale indicates the reflectance from 0 (black) to 1 (yellow). Figure (d) shows the reflectance spectra of two SP mirrors, one is localized at 1.362 μm and the other is centered at 1.4 μm (—), 1.5 μm (- - -), 1.6 μm (···), and 1.7 μm (· – ·). Figure (e) shows the comparison of two resonant microcavity modes obtained by superposition (—) and typical half wave microcavity (· – ·) structure designed for λ0 = 1.45 μm.
In order to further explore the scope of controlling the resonant microcavity mode, the reflectivity response (R) is calculated as a function of the relative initial position (i.e., shift) between the refractive index profile functions (Fig. 4). For binary addition of refractive index profile functions of PBGs designed for wavelength at λ1 and λ2, shift (p) is defined as follows: p = (δ/lλ1) × 100, where δ is the separation of the initial points of the refractive index profile functions, lλ1 is the optical thickness of the filter designed at wavelength λ1 (with λ1 > λ2). The composite SP mirrors have been obtained by overlapping refractive index functions starting from the same position, i. e., p = 0 (0 % shift), to p = 100 shift (equivalent to the SQ addition).
Fig. 4 Schematics of the refractive index profile of two Bragg mirrors (i+ii) with maximum/minimum refractive index of 1.81/1.33, and the resultant composite structure (iii) with maximum/minimum refractive index of 1.09/2.05 for (a) 0 %, (c) 0.5 %, (e) 3.3 %, and (g) 9.5 % of shift. Figure (b), (d), (f) and (h) show the reflectivity spectra of the structures with the composite refractive index profile shown in (a), (c), (e) and (g), respectively.
In other words SQ turns out to be a special case of SP addition. The mirror corresponding to the higher wavelength is kept fixed with respect to the other mirror with the lower wavelength. Figure 4, shows the refractive index profiles of individual Bragg mirrors and composite structure with SP addition, along with the corresponding reflectivity spectra for some values between 0 and 9.5 % shift. When the structures are superposed with distinct initial point, significant change in the reflectance properties (R) can be observed. For example, Fig. 4(a) shows the functions added considering 0 % shift, and the corresponding reflectivity spectrum (Fig. 4(b)) results in a localized microcavity-like mode (with Q = 227). However, with an increase in shift to 0.5 % (Fig. 4(c)), the depth of the resonant mode decreases (Fig. 4(d)), resulting in a lower quality factor (Q = 107). A further increment to 3.3 % results in the complete elimination of the resonant mode, i.e an enhanced Bragg mirror-like reflectance (Fig. 4(f)). Finally, when the shift is increased to 9.5 %, the microcavity mode reappears. The demonstration of the fact that a small change in the relative shift between the refractive index profiles of the two Bragg mirrors, can result in the presence or the complete elimination of the resonant mode confirms “the relative shift” as a controlling parameter.
To evaluate the complete range of tunability, the reflectivity response (R) is shown as a contour plot with the percentage shift and wavelength (λ) in Fig. 5(a). With an increase in the relative shift from 0 to 100 %, a periodic appearance of resonant modes is revealed. At a higher value of % shift (more than 45 %) of the refractive index profiles, the position and the depth of the resonant mode is found to change significantly. Between the two resonant modes, an enhanced PBG with variable width can be observed. Moreover, the width of the PBG obtained with 100 % shifting (i.e., sequential addition) is found to be less than the PBG obtained at many different values of % shift (demonstrated as Fig. 5(b)). PBG as a function of percentage shift is found to reveal the existence of an optimum value to obtain the maximum.
Fig. 5 (a) Contour plot of the reflectivity spectrum, as a function of wavelength and the percentage of relative shift for two Bragg mirrors under SP addition. The color scale indicates the reflectance from 0 (black) to 1 (yellow). (b) Comparison of the PBG obtained by SP addition for three different values of shift: 4.6 % or 344 nm (· – ·), 33 % or 2474 nm (—) and 100 % shift or SQ addition (- - -), revealing an increase of PBG for SP added structures. Inset shows the PBG as a function of % shift revealing an optimum value of shift for obtaining a maximum PBG
The formation of the enhanced PBG or resonant modes can be attributed to the constructive/destructive superposition of wave-like refractive index functions, which generates an increase/decrease in the amplitude of the composite refractive index profile. Periodicity of the superposed mirrors in phase/dephase results in the enhancement/cancellation of the amplitude, generating a Bragg/microcavity-like photonic structure. The microcavity modes shown in the contour plot are equidistant, reflecting the same periodicity as the unshifted function and displays a similar reflectance in each period of the shifted functions.
In Fig. 6, the effect of “overlapping” and “shifting” has been explored for two rugate filters. The composite reflective response (R) is plotted as a function of percentage overlapping (Δλ) and wavelength (λ). The superposition of rugates is constructed by fixing the first mirror at λ1 = 1.0 μm and varying the operating wavelength (λ2) of the second rugate within ±20%. The contour plot (Fig. 6(a)), indicates the presence of two distinct rugates, except at Δλ ∼ 0, at which they coalesce to form a single rugate. In Fig. 6(b) the effect of shifting is observed between two close rugates constructed at λ1 = 1000 nm and λ2 = 946 nm. Due to the increase in the periodicity in the rugate filter (20 periods), with respect to the Bragg mirrors analyzed in the Fig. 5, with an increase in the relative shift, more number of interference modes are observed. Moreover, the inclination of the bands appearing due to the reflectance peaks confirms a periodic redshift of the reflectance spectra along the horizontal axis.
Fig. 6 (a) Contour plot of the reflectivity spectrum (R), as a function of the wavelength (λ) and the percentage of relative overlapping (Δλ) for two rugate mirrors under superposition addition. The color scale indicates the reflectance from 0 (black) to 0.9 (yellow). (b) Contour plot of the reflectivity spectrum (R), as a function of the wavelength (λ) and the percentage of relative shift for two rugate mirrors under superposition addition. The color scale indicates the reflectance from 0 (black) to 0.7 (yellow).
5. Experimental results
In this section, the major theoretical results have been verified experimentally through the porous silicon multilayered photonic structures. Figures 7(a)–7(d) show the measured reflectivity spectra of the four photonic structures, fabricated to verify the effect of shift among the two Bragg mirrors with the closely located PBGs. The mirrors were designed at λ1 = 1.5 μm and λ2 = 1.36 μm with maximum/minimum refractive index of 1.81/1.33 for each Bragg mirror, and physical thicknesses of 207/282 nm (for λ1) and 188/256 nm (for λ2), resulting in a maximum/minimum refractive index of 2.05/1.09 for the composite structure. The corresponding composite refractive index profiles are shown as Figs. 4(a), 4(c), 4(e) and 4(g), respectively. Although the experimental data is in close agreement with the corresponding theoretical data, there is a significant difference in the Q-factor/depth of the resonant mode of some of the microcavities. In Fig. 7(a), the experimental microcavity mode with the total reflectance of 40% is measured, where as the simulated data shows the complete confinement. The measured Q-factor of the experimentally obtained microcavity is 147 ± 18, as compared to the theoretical Q-value of 386. In Figs. 7(b), the microcavity mode is designed to have 40% reflectance by changing the relative shift to 0.5% (i.e., 37 nm of optical thickness), among the overlapping Bragg mirrors. The experimental microcavity mode is found to confine upto 50% of the total signal with the corresponding Q-value of 127 ± 13 as compared to the theoretical value of 206. On further increasing the relative shift to 3.3% (i.e 247 nm of optical thickness) among the refractive index profiles/waveforms of the two overlapping Bragg mirrors (Fig. 7(c)), the formation of an enhanced PBG is demonstrated. In spite of the good agreement, the width of the experimental PBG is found to be more narrow than the theoretically simulated data. Finally, in Fig. 7(d), a further change in the relative shift (9.5% or 712 nm of optical thickness) causes the reappearance of the microcavity resonant mode, with the Q-values for experimental and theoretical resonant modes as 149 ± 17 and 374, respectively. The reduction of the quality factor (and the FWHM of the mirror) in the experimental data can be attributed to the dispersion at the interfaces, experimental errors in the measured refractive indices, physical thicknesses and the resolution of the equipment.
Fig. 7 The experimental verification of “shifting” of two overlapping Bragg mirrors (λ1 = 1.5 μm and λ2 = 1.36 μm) under SP addition. The solid line and the dotted line in the reflectivity spectra plot correspond to the experimentally measured and theoretically simulated data lines respectively. The composite structure obtained with (a) 0%, (b) 0.5% (37 nm of optical thickness), (c) 3.3% (247 nm of optical thickness) and (d) 9.5% (712 nm of optical thickness), of shift and maximum/minimum refractive index of 1.09/2.05. The refractive index profiles corresponding to the panels (a), (b), (c), (d) are shown in Fig. 4 (a), (c), (e), (g)
Figure 8 compares the measured reflectivity spectra of resonant modes obtained by SP addition (solid line) and typical half-wave microcavity structures (dashed/dashed-dotted lines). Composite structure was obtained through the overlapping of two Bragg mirrors with 9.5% shift (see Fig. 4(g)), while the half-wave microcavities were designed for λ0 = 1.47 μm. The difference in the quality factor in the structures can be attributed to the resulting refractive index contrast as well as the percentage shift, i.e. with a decrease in the relative overlapping region, the refractive index intersection decreases, reducing the width of the high index contrast zone. Hence, the extent of the overlapping region of the two refractive index profiles is the factor responsible for an increase the reflectivity (see Fig. 5(b)) and contributes to the width of the PBG as well as the Q-factor of the resonant mode.
Fig. 8 Comparison of resonant microcavity (MC) modes obtained by superposition (—), and half-wave microcavity structures with Δn = 0.45 (- - -) and 0.96 (· – ·). The quality factor of SP-microcavity is 149±17 while the quality factor of half-wave microcavities are 86±18 (for Δn = 0.45) and 67±10 (for Δn = 0.96). All the microcavities are designed for λ0 = 1.47 μm.
In particular, Fig. 8 shows the resonant mode with Δn = (nmax − nmin) = 0.96 for SP structure, and Δn = 0.96 and 0.45 for half-wave microcavity structures. Although a relatively deep half-wave microcavity mode is obtained (for Δn = 0.45), the FWHM is also increased, resulting in a lower Q-factor value of 86 ± 18 as compared to 149 ± 17 for the SP structure. In addition, the Δn = 0.96 microcavity structure was observed to decrease the Q-value to 67±10 (as compared to the microcavity with Δn = 0.45 and SP structures). The decrement in the quality factor is attributed to the cracks in the PS sample generating a wider microcavity mode, leading to a reduced Q-factor value of the half-wave structure as compared to the composite filter, with the same refractive index contrast. Hence, SP addition shows a new method to increase the confinement of a resonant mode as compared to the typical microcavity structure, with the advantage of a reduced physical thickness (4.99/5.13 μm).
Figure 9 shows a cross section of four samples corresponding to the typical half wave microcavity and three other composite structures (corresponding reflectance spectra are shown as Fig. 7). In Fig. 9(a), the periodic structure and the defect layer in the middle can be observed. Figure 9(b), (c), and (d) show the composite structures with relative shift of 0%, 3.3%, and 9.5%, respectively. Presence of thin layers can be observed in the composite structures and are illustrated on the left hand side of the image as schematic refractive index profile along the depth.
Fig. 9 HRSEM cross section of (a) Halfwave microcavity (physical thickness of 5.2 μm ± 0.1 %). SP addition of two Bragg mirrors designed at λ1 = 1500 nm and λ2 = 1362 nm with (b) 0 % shift (physical thickness of 5.4 μm ± 5 %) (c) 3.3% shift (physical thickness of 5.6 μm ± 5 %) (d) 9.5% shift (physical thickness of 5.0 μm ± 5 %). The dark and clear zones correspond to the high (low) and low (high) porosity (refractive index) layers, respectively. Left hand side of the each image shows the corresponding schematic of the refractive index profile along the depth.
6. Conclusions
To obtain the required optical features with reduced physical thickness, superposition (SP) addition of refractive index profiles of 1D photonic structures has been studied for Bragg mirrors and rugate filters. SP addition of Bragg mirrors with closely located PBGs, as a function of overlapping and percentage shift, is found to generate a resonant mode or an enhanced PBG. A theoretical comparison of the resonant mode appearing in the SP structure with the typical half-wave microcavity mode shows an enhancement of quality factor by more than 2 times in the NIR range. PBG as function of percentage shift is found to reveal an optimum value of shift to obtain a maximum PBG. Experimental verification of some of the simulated results has been demonstrated through PSi multilayers. Such composite structures with high quality factor resonant modes can have possible application in enhancing the sensitivity of porous silicon based filters. Although the analysis was extended to rugate filters, the controlling parameters for the SP addition of Bragg mirrors i.e. overlapping or relative shifting of the refractive index profiles was found to show similar optical response with respect to the sequentially added rugate filters.
Acknowledgment
Anupam Mukherjee and A. David Ariza-Flores contributed equally to this work. This work has been supported by CONACyT under project 128953 and PROMEP SA/DSA/257/13. The authors acknowledge M. José Campos for SEM imaging.
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