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Abstract
This paper explores the wavelength-dependent reflectivity of alternating high and low refractive index multilayers with a thickness profile defined by a pseudo-random, maximum length sequence (MLS). An MLS contains all possible combinations of a binary sequence save one; thus, a multilayer with an MLS profile contains a superposition of a broad range of periods. The range of periodicities in an MLS multilayer should make these systems more effective broad wavelength reflectors as compared to purely periodic counterparts. We compute the reflection characteristics of MLS and periodic dielectric sequences at visible wavelengths over a range of incident angles using the transfer matrix method (TMM), a recursive multilayer calculation method. The materials SiO2 and TiO2 are chosen as the low and high refractive index materials, respectively, because these materials are commonly used in optical multilayers and because their wavelength-dependent refractive index is well known. Our results show that it is possible to create an MLS structure with high average reflectivity across the entire visible spectrum (400 nm – 700 nm) at all incident angles and polarizations. Finally, we compare the reflection characteristics of dielectric multilayers with metallic reflectors whose refractive index is based on a Brendel-Bormann (BB) model. The comparison shows that a seventh order MLS aperiodic multilayer exhibits slightly higher average reflectivity over the visible spectum than silver or aluminum metallic reflectors.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
This computational investigation examines the response of optical multilayers consisting of alternating high and low refractive index materials with a thickness profile determined by a pseudo-random maximum length sequence (MLS). The aim is to demonstrate that such multilayers have higher reflectivity over a broader bandwidth than periodic multilayers with the same index contrast and number of layers. For periodic multilayers, the forbidden transmission bandwidth, the photonic band gap of the one dimensional photonic crystal, is set by the index contrast between the two materials. Higher index contrast leads to band gaps with broader frequency coverage [1, 2]. A maximum length sequence (MLS) is a two-level pseudo-random binary sequence which has a uniformly flat frequency response except for a DC offset [3]. The flat frequency response indicates that these sequences contain equal amplitudes of all possible periodicities [4–6]. We posit that a multilayer using a thickness profile determined by an MLS will have a broader range of periods and, hence, will exhibit a wider bandgap than a periodic system of the same materials and number of layers.
The aim of this study is twofold. First, we explore the feasibility of creating a functional omnidirectional reflector across all wavelengths in the visible without the necessity of materials with extremely high index contrast. Periodic dielectric multilayers act as near perfect reflectors over a limited wavelength and angular range and provide excellent reflectivity, low absorption loss, and a high degree of robustness compared to metallic reflectors [7]. However, periodic dielectric multilayer reflectors are not highly reflective of light coming from any direction and polarization. In general, the range of reflected wavelengths shifts to the blue at angles away from normal incidence. An ideal omnidirectional dielectric multilayer reflector would have near perfect reflection for all wavelengths in a given range and at all incident angles and any polarization. In the visible range, metallic mirrors from materials such as silver and aluminum have such properties [8, 9]. The goal here is to make a metallic-like reflector from a multilayer structure using the deterministically pseudo-random maximum length sequence thickness profile. The second longer-term objective is to begin the exploration of photonic band gaps in aperiodic structures in two and three dimensions. In this work, the wavelength and angular reflection from pseudo-random MLS multilayers are compared with that for metallic mirrors and for purely periodic systems based on quarter-wavelength multilayers. Future work will explore the prospect of two- and three-dimensional structures with MLS index profiles.
Aperiodic multilayer structures with thickness profiles defined by Thue-Morse, Fibonacci, and Rudin-Shapiro algorithms have been studied before, of particular interest in these quasi-periodic systems is the phenomenon of localization [10–23].
However, to our knowledge the maximum length sequence profile multilayer has only been studied for its electromagnetic wave properties in its ability to support Bloch surface waves [10]. In acoustics there is a long history of using the perfectly random profile of an MLS defined grating to create diffusers that diffract sound equally in all directions [24].
Other methods to enhance the reflectivity of multilayers on one dimensional reflectors using random disorder and inhomogeneity have been explored [25–27]. These designs use alternate high and low refractive index layer structures with random variations of layer thicknesses. The results are thus dependent on the choice of randomization algorithm or on some optimization routine. In contrast, the MLS offers a deterministic layer thickness profile with flat Fourier response; the MLS sequence is often described as being ideally random.
2. Maximum length sequence
Maximum length sequence properties have been applied in a variety of settings from signal processing, architectural acoustics, and cellular communications [28–31]. In acoustics, the MLS is used to measure impulse responses and reverberation-decay inside a room or a theatre [28]. Higher signal−to−noise ratio (SNR) is possible in a noisy measurement system, as the MLS frequencies are distributed randomly over the entire time sequence. Low noise impulse response at the output is calculated using a cross correlation procedure [30]. Similarly, in wireless communication, using code division multiple access (CDMA), user data is spread independently with MLS (commonly called spread spectrum) over the entire bandwidth at the transmitter side. On the receiver side, the signal is despread using a synchronization replica of the MLS sequence [31].
Maximum length sequences are binary sequences also known as pseudo-random binary sequences (PRBS) that are commonly generated by a cyclic shift of m-sized linear feedback shift registers (LFSR) along with a primitive polynomial. A primitive polynomial is an irreducible polynomial that can produce all the sequences of an extension field from a base field. Figure 1 shows the shift register arrangement for the generation of a 63-element (N = 26 − 1) long sequence (which, as we show below, corresponds to 32 distinct layers) using 6 LFSR and an XOR gate. Computationally, these sequences can be generated using the recursive formulation Sk+3 = Sk+1 ⊕ Sk. Here, the ⊕ symbol represents XOR (modulo-2 sum) operation [3–5].
Fig. 1 (a) The MLS generation using shift register and XOR gate. (b) Flat Fourier spectra for 6th order MLS multilayer structure.
All MLS sequences produce a flat frequency response. The frequency response for N=6 is illustrated in Fig. 1(b). The MLS exhibits very low values of correlation except at zero offset, a feature that is used in the acoustic impulse and spread spectrum communication applications. Figure 1 shows the flat Fourier spectrum, which is the relevant feature in our applications. This feature shows that all periodicities are equally contained in the sequence.
To make the generation process clear we present a simple example for the N=3 MLS sequence in Fig. 2. The process begins with the seed for N=3 which consists of three elements equal to one. The next element in the sequence is generated by taking the XOR of the first two elements; the taps in this example are at positions 1 and 2. The result 1 ⊕ 1 = 0 is highlighted in blue. The sequence shifts to the left and the fifth element of the sequence is the ⊕ of elements 2 and 3. The process continues until the sequence replicates the seed at which point the sequence will repeat. The existence of all periods but one in the generated sequence can be shown by going through the sequence grouping the elements in sets of three. This process shows that every possible permutation of three elements occurs once in the sequence except for [0 0 0]. Each 1 and 0 in the generated sequence represents high and low refractive index materials respectively. Grouping the like elements leads to the values [3 2 1 1] for N=3. To convert this set of values into a multilayer involves selecting a minimum thickness for the high and low index layers dH and dL. Typically for the visible reflector we chose dH = λ/4nH and dL = λ/4nL where nH and nL are the refractive index values of the two materials respectively. The multilayer thicknesses would then be given by [3dH 2dL 1dH 1dL]. For these simulations we chose λ to be 550 nm resulting in high reflectivity across the visible. Table 1 defines the MLS sequence in terms of order, tap, and number of distinct layers. The 6th order MLS multilayer structure along with taps at 1 and 2 has 32 distinct alternate high and low refractive index materials derived from the 63 sequence elements.
Table 1. Orders of MLS sequence Generation
As for the simple N=3 case illustrated in Fig. 2, to convert the MLS sequence to multilayer thicknesses the 1 and 0 are taken to correspond to the two different refractive index materials. Thus, the final 7 element sequence for N=3 is expressed as [3 2 1 1] with 4 distinct layers. The 6th order MLS multilayer structure thickness is determined by following alternate high and low refractive index sequence: [6, 5, 1, 4, 2, 3, 1, 1, 1, 2, 4, 1, 1, 3, 3, 2, 1, 2, 1, 1, 2, 1, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1. This sequence length is 63 (26 -1) which converts to 32 distinct high and low index layers. ]
For the calculations in this paper we chose SiO2 and TiO2 as the low and high refractive index materials respectively because these materials are compatible and commonly used to create multilayer structures. The wavelength dependent refractive index of SiO2 and TiO2 are given below. The expression for the refractive index for SiO2 and TiO2 with λ ranges from 0.2 µm to 4.0 µm is expressed as [32, 33]
and respectively. In these relations the wavelength is expressed in micrometers.The loss factor is very low and constant in both SiO2 and TiO2 over the entire visible frequency range from 400 nm to 700 nm. In TiO2 the loss increases rapidly for wavelengths shorter than 350 nm [34]. We accounted for the loss in SiO2 and TiO2 by adding an imaginary term to the dielectric constant of 0.0007i for TiO2 and 0.0001i for SiO2. These values have been shown to give good agreement between simulation and experimental studies of Bloch surface wave generation in SiO2 and TiO2 multilayers [2].
3. MLS multilayer structures
The dielectric multilayer reflector is a structure built of alternating layers of high and low refractive index materials with a periodicity assumed here to be along the z-direction. Figure 3(a) shows the general layout of a one-dimensional photonic crystal or Bragg stack, which consists of a dielectric multilayer having high refractive index ηH with length LH, low refractive index ηL with length LL, left medium ηa and right medium ηb. The substrate is assumed to be semi-infinite. The optical thickness for the Bragg stack is taken as a quarter wavelength that satisfies where λ is the wavelength in vacuum. The incident wave Ei+, and reflected wave Ei− observed at the left end while transmitted wave Ei+ at right boundary. The amplitude reflection can be calculated using recursive propagation of impedances or reflection responses at interfaces.
Here, the reflection coefficient is calculated using following parameters: N = 15, layers (2N + 1 = 31), ηH(TiO2) = 2.6479, ηL(SiO2) = 1.4599, ηa = 1.0, ηb = 1.52 and λ = 550nm. Therefore, the generalized impedance is expressed asFig. 3 (a) The dielectric alternate high and low refractive index multilayer structure. (b) The 6th order MLS multilayer structure model.
The most common multilayer in optical reflection applications is the periodic Bragg reflector built with repeated identical quarter-wave bilayers of low and high refractive index on a substrate (ηa) [35, 36]. Now using , the generalized reflection coefficient can be expressed as
We obtain amplitude reflectivity Γ1 = −0.9999 and as the intensity reflection at the target wavelength. The simplified expression for Δλ is expressed as:
With this expression we find ρ = 0.2894 and Δλ=213.11 nm for given ηH and ηL. Therefore, this dielectric acts as broadband reflector and is, in principle, capable of reflecting wavelengths ranging from 443.5 nm to 656.6 nm (nearly equal to visible frequency range). The shortcoming of this analytical approach is that the wavelength dependence of the refractive indices of the constituent materials is not taken into account and the calculation is valid only for normal incidence. The calculations in the next section show the reflection of the Bragg stack for wavelength-dependent indices, s- and p-polarization, and for angles away from normal incidence.
In contrast to the quarter-wavelength structure, the MLS multilayer is designed using high and low refractive index materials with thicknesses set by an MLS sequence. There is no easy analytical method to derive the reflection as in the case of the periodic quarter-wave stack and thus, we used a computational approach as described in the following section. Figure 3(b) illustrates the 6th order dielectric MLS multilayer structure designed using the recursive formulation Sk+3 = Sk+1 ⊕ Sk.
4. Simulation results and discussions
We conducted our numerical simulations both using the transfer matrix method (TMM) and verified the results independently using a recursive calculational model, both commonly used techniques to analyze the electromagnetic propagation through multilayer films. TMM is based on the boundary condition of Maxwell’s equations where the electric field at the end of the layer can be derived using matrix operations for a given field at the beginning of a layer. The reflection from a multilayer structure is the transformation of the overall system matrix, which is the product of each layered matrix. Here, we implemented the simulations using Matlab R 2017 and Python 3.6.
Simulations were performed for both N=6 and N=7 MLS multilayer structures. For N=6 the number of layers is 34 with a total multilayer thickness of 5.06 µm. For N=7 the number of layers is 66 with a total multilayer thickness of 9.63 µm. Current fabrication methods allow multilayers of up to 600 layers and a total thickness of 23 µm [37]. Figure 4 illustrates the reflection characteristics for the 6th order MLS multilayer for both s- and p-polarized light incident with 0°, 30°, and 60° respectively. As indicated from Table 1, this multilayer contains 32 layers consisting of alternating SiO2 and TiO2 layers with thicknesses scaled by the MLS sequence. In this case, the smallest layer thickness is 519 nm for TiO2 and 941 nm for SiO2. Clearly, the reflection curves contain significant structure with many reflectivity dips as a function of wavelength. However, exclusive of the narrow dips, there is high reflectivity at all incident angles across all the wavelengths in the visible (400 nm to 700 nm).
Fig. 4 Reflection characteristics for 6th order MLS multilayer structure for both s- and p-polarized light incident with 0°, 30°, and 60° respectively.
Many of the dips in reflectivity are very narrow, on the order of 2 nm or less, and thus have little effect on the overall reflectivity across all visible wavelengths. To assess the effectiveness of the multilayer as an omni-directional reflector, the metric we chose to compare the reflectivity of the MLS systems with periodic multilayers and with metallic reflectors is the average reflectivity taken over the wavelength range from 400 nm to 700 nm. For s-polarized light, the average reflection in visible frequency range of 400 nm to 700 nm is more than 95 percent at all incident angles. For p-polarized light, the average reflectivity is lower, particularly in the vicinity of Brewster’s angle, as expected, at around 60°. The specific average reflection values are summarized in Table 2. The first point of comparison for the MLS reflectivity for N=6 is to contrast the results with the reflectivity of a periodic multilayer with the same number of layers and index transitions. Figure 5 illustrates the characteristics for 32 layer alternate high and low index multilayer structure for both s- and p-polarized light incident at 0°, 30°, and 60° respectively.
Table 2. Average Reflection Coefficient for Different 6th Order MLS for both s- and p-Polarized Light
Fig. 5 Reflection characteristics for 32 layer alternate high and low index multilayer structure for both s- and p-polarized light incident with 0°, 30°, and 60° respectively.
These graphs indicate that the periodic multilayers are essentially perfect reflectors for a narrow range centered about the design wavelength of 550 nm. However, the high reflectivity does not encompass the entire visible range and falls off in such a way that the average reflectivity across the entire visible range is notably lower compared to 6th order MLS multilayer structure. Table 3 summarizes the average reflection for the periodic multilayer structure. As the table indicates, the average reflectivity for the MLS multilayer is between 10% and 15% higher than the periodic multilayer for s-polarized light and between 15% and 30% higher than the periodic multilayer for p-polarized light. Figure 6 plots the average reflection in visible frequency range as a function of incident angle for the 6th order (N = 6) MLS and the 32 layer alternate high and low index multilayer structure for both s- and p-polarized light. This plot verifies the omni-directional reflection characteristic of the MLS multilayer. For s-polarized light in the visible range, the reflection at all angles is greater than 95%. The p-polarized light has lower reflectivity at larger angles compared to s-polarized light for both MLS and alternate high and low index multilayer, but it is still above 80%. To answer the question of whether the MLS system should be improved, we next calculated the MLS reflectivity for the N=7 MLS sequence.
Table 3. Average Reflection Coefficient for 32 Layer Alternate High and Low Index Multilayer for both s- and p-Polarized Light
Fig. 6 Average reflection as function of incident angle in the visible frequency range for 6th order (N = 6) MLS and 32 layer alternate high and low index multilayer structure for both sand p-polarized light.
Figure 7 shows the reflection characteristics for 7th order MLS multilayer structure for both sand p-polarized light incident with 0°, 30°, and 60° respectively. The 7th order MLS multilayer structure thickness is determined by following an alternate high and low refractive index sequence: [7, 4, 3, 1, 4, 2, 1, 1, 2, 2, 1, 2, 1, 6, 1, 3, 1, 2, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 2, 2, 1, 1, 1, 1, 2, 3, 2, 4, 1, 2, 1, 1, 4, 6, 5, 1, 1, 2, 1, 1, 1, 3, 2, 1, 3, 3]. This sequence length is 127 (=27 -1) which converts to 64 distinct high and low index layers. Simulations for the 7th order MLS again show significant structure with many very narrow dips in the reflection curve. However, the average reflection in visible frequency range of 400 nm to 700 nm is more than 96% for s-polarized light at all angles. The reflectivity at 60° and 30° respectively is 98.39% and 96.29% for s-polarized light and 91.83% and 95.58% for p-polarized light. Table 4 summarizes the average reflection for 7th order MLS multilayer structure for both s- and p-polarized light. The N=6 and N=7 MLS multilayers would function better in practice than the plots imply if the incident light being reflected had even a small angular spread in which case the effect of the narrow dips at specific angles would be blurred because of the angular shift of the narrow minima.
Fig. 7 Reflection characteristics for 7th order MLS multilayer structure for both s- and p-polarized light incident with 0°, 30°, and 60° respectively.
Table 4. Average Reflection Coefficient for Different 7th Order MLS for both s- and p-Polarized Light
For a better point of reference of the MLS system as an omni-directional reflector, we compare the average reflection results with those for two metals, silver and aluminum, with high reflectivity throughout the visible. The wavelength-dependent refractive index of the two metals in the visible frequency range is determined using a Brendel-Bormann (BB) model. The BB model provides the most accurate real and imaginary permittivity values for the materials by using a Gaussian complex error function [9]. For the aluminum metallic reflector, we calculated s-polarized reflectivities of 95.76% and 92.73% and p-polarized reflectivities of 84.66% and 96.74% for light incident at 60° and 30° respectively. Similarly, for silver, we found 98.12% and 95.64% reflectivity for s-polarized light and 93.22% and 95.64% for p-polarized light incident at 60° and 30° respectively. Further, 91.63% and 96.22% reflection is achieved for light at normal incidence respectively for aluminum and silver metallic reflectors. Table 5 summarizes the average reflection for metallic reflectors for both s- and p-polarized light. Figure 8 plots the reflection characteristics as a function of wavelength for metallic reflectors in the visible frequency range for incident angles of 0°, 30°, and 60°.
Table 5. Average Reflection Coefficient for Metallic Reflectors for both s- and p-Polarized Light
Fig. 8 Reflection characteristics for metallic reflectors in visible frequency range for both sand p-polarized light incident at an angle of 0°, 30°, and 60°.
Figure 9 plots the average reflectivity for the N=7 MLS multilayer for the two metals for both sand p-polarized waves. The plots demonstrate that the average reflection for the 7th order MLS is higher than that of both metallic reflectors. Clearly, the MLS has slightly higher reflectivity than the silver and much better reflectivity than aluminum for s-polarized wave. The graphical illustration in Fig. 9 indicates that the MLS reflector is effectively achieving omni-directional reflection for all range of angles in the visible frequency range of 400 to 700 nm.
Fig. 9 Average reflection as function of incident angle in visible frequency region for 7th order (N = 7) MLS and metallic reflectors for both s- and p-polarized light.
Moreover, we studied the the reflection characteristics for different random layer arrangements by keeping the total number of layers constant at 64 in all simulations. We use the randomness formulation of and [25], where r is the inhomogeneous parameter, the term 1 + r1 (i) is a random factor added to high refractive index material thickness, H(i) and L(i) represent the thickness of high and low refractive index in the ith layers respectively. Similarly, and , where λ0 is the center wavelength of 550 nm; ηH and ηL represent the index of the high and low refractive index materials respectively. The first randomness is generated by using r = 0 and r1 = 0. This choice is equivalent to a 64 layer quarter wave alternate high and low index structure. Similarly, a second randomness experiment is simulated by r = 0.1 and r1 = 0, we found slightly better reflection performance.
Finally, extensive randomness is generated by varying the r1i as: 0.005,0.01, 0.015, 0.020,0.025, 0.030, 0.033, 0.036, 0.039, 0.042, 0.045, 0.048, −0.048, −0.045, −0.039, 0.036,−0.033, −0.030, 0.025,−0.020, −0.017, −0.014, −0.011, −0.008, −0.005,−0.002, 0.048, 0.045, 0.042, 0.039, 0.036, 0.033. Table 6 summarizes the average s-polarized reflectivity for these three different random order inhomogeneous multilayer structures. The average reflectivity for 7th order MLS illustrated in the first column of Table 4, has higher average reflection than those of these different random layer multilayer structures.
Table 6. Average Reflection Coefficient for Different Random Order Multilayer Structures
Clearly, the parameter space of random multilayers is very large and it is not easy to exhaustively search this space. However, to make a broadband reflector that works at all angles, the key factor is the existence of a wide range of periodic high-low index transitions of uniform amplitude. The Fourier spectrum of the transitions of such a system would be broad and flat. The MLS sequence is deterministic and possesses these characteristics in a mathematically perfect way. It was this characteristic of MLS sequences that led to the design and demonstration of acoustic grating diffusers [24], one of the earliest applications of the MLS.
Finally, the reflection characteristics of a multilayer depends on the refractive index contrast ratio of the high index material to the low index material. Higher contrast ratio leads to better reflector performance. We analyze the reflection characteristics for commonly used refractive index materials such as ZnS(2.3862) -M gF2(1.384) and ZnS(2.3862)-Cryolite(1.33). Table 7 illustrates the average reflection coefficient over a broad range of incident angles in the visible frequency regime for 6th order (N = 6) MLS multilayer structure design using these different dielectric materials. The MLS multilayer design with GaP(3.31) -MgF2(1.38), having refractive index contrast ratio of 2.493, has higher reflection (more than 99%) than any other material for all ranges of incident angles. For the wave incident at 0°, the GaP -MgF2, ZnS-Cryolite, and ZnS -MgF2 with refractive index ratio 2.493, 1.76, and 1.724 has reflection of 99.83%, 96.19%, and 95.22% respectively.
Table 7. Average Reflection Coefficient in Visible Frequency Range for 6th Order MLS Designed Using Different Materials
5. Conclusion
In conclusion, this work has explored the reflection performance of MLS multilayer structures at visible wavelengths. The MLS system is an aperiodic system that has been little studied for applications in optics. It is a system that is ripe for investigation as a medium to support optical localization and surface waves, and as an ultra-wide photonic band gap material in two and three dimensions. The wide range of periodicities contained in an MLS structure leads to broadband reflection characteristics for all incident angles. The analysis here shows that multilayers with an MLS profile can function as effective omni-directional reflectors. The narrow reflectivity dips mean that such reflectors would not be good for monochromatic laser reflection but rather for incoherent broad bandwidth light. The comparative analysis presented was made both to periodic multilayers and to the metallic reflectors silver and aluminum, whose refractive index was modeled based on the Brendel-Bormann (BB) model in the visible frequency region. We demonstrated broadband reflection for the MLS multilayer design by varying the number of layers or the refractive index of the constituent layers. The qualitative design and analysis of such quasi-periodic and aperiodic structures shows significant promise for future applications in optics.
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