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In this paper, we propose a narrowband DWDM filter structure, whose reflection band characteristics, meets the ITU-T standard. The proposed filter structure is based on Fibonacci quasi-periodic structures composed of multilayers with large index differences. Studying the effects of the optical and geometrical parameters of Fibonacci quasi-periodic structures on its filtering properties, we have realized that to achieve the ITU-T standard, we need to cascade two successive structures both with the same generation numbers j=4 and orders n=25 and apodized refractive indices. The apodization process helps to minimize the stop band sidelobes. We have also demonstrated that beside Fibonacci’s order, n, the layers dimensions, and their refractive index ratios are the main design parameters.
©2007 Optical Society of America
1. Introduction
Current advancements in dense wavelength division multiplexing (DWDM) systems make narrow and multiband optical filters attractive to optical device engineers and designers. DWDM filters have tight and narrow stopbands, near-square amplitude responses and extremely low group delay (GD) variations in the passband. Available technologies are multilayer dielectric thin film filters (TFFs) [1], arrayed waveguide gratings (AWGs) and fiber Bragg gratings (FBGs) [2].
DWDM filters are multichannel filters with several stop bands with adjacent channels covering large range of wavelengths [1]. Interleaver/deinterleaver is an example of DWDM filter separating odd and even channels [3–5].
Multilayer structures are one of the most suitable alternatives for DWDM filters. Fiber gratings (FG’s) are being widely used as wavelength selective reflective filters. High selectivity, low insertion loss, no polarization sensitivity, sharp, and well-defined filter amplitude responses (ideal rectangular filters) are critical characteristics for FG’s in WDM communication systems [6].
Multiplexing/demultiplexing a large number of channels, with a single wavelength multilayer filter, in a DWDM system, would require a large number of filters in a series. Reducing the number of filters in the cascade is of prime concern. Number of filters might be reduced if one replaces the cascaded single band filters with a multiband filter.
Large multiband periodic filters based on superimposed chirped FBGs have already been designed for DWDM to have flat top responses [7].
Gratings with periodic refractive index having single stopband behave as a filter. Designing a multi narrowband filter is possible by cascading a given number of periodic structures with different periods. Here, we introduce quasi-periodic gratings based on Fibonacci sequences as multiband filters, where a single structure has several stopbands. Fibonacci optical multilayer structures were first introduced by Kohmoto et. al., in a optical system capable of localizing photons [8]. Then, Sibilia et. al has demonstrated that the transmission spectrum, produced by such structures, are dense in wavelengths, displaying a self-similar pattern [9]. Next, Gellermann et al. has shown the existence of bandgaps in the spectrum of these structures, experimentally [10]. Omnidirectional bandgaps, using Fibonacci quasi-periodic structures, were also reported by Lusk et.al., [11]. Peng et. al., have reported resonant transmission of light in a symmetric Fibonacci multilayers, characterized by many perfect transmission peaks, useful for narrow-band multiwavelength optical filtering applications [12].
Fibonacci sequences have been used either as reflectors or as transmitters in designing microcavity structures [13–15]. Useful mathematical expressions concerning such structures have already been derived and reported [16–17].
In this paper, we show the possibility of designing DWDM filters with two Fibonacci multilayer structures. The DWDM filter is designed according to the ITU grid, where the neighboring channels are separated by 0.8 nm. Our aim in this paper is to replace the cascaded single band filters by a multiband Fibonacci class FC 4(n) filter.
Our numerical evaluation of reflectance is done by Transfer Matrix Method (TMM). The numerical results show that reflection from FC 4(n) structure includes a large series of narrowband reflections surrounded by two large omnidirectional reflections. We demonstrate that the narrowband reflections from these structures can be used as DWDM filters. For such application, the number of layers in the structure should be large.
We also demonstrate various effects of the refractive index difference, the number of layers and their thicknesses on the FC 4(n) filter properties. To eliminate the sidelobes of each band, we have used refractive index apodization.
Organization of the paper is as follows. In Sec. 2 we present the theoretical background for the Fibonacci quasi-periodic structure, where we briefly describe the general theoretical model based on TMM used for the calculation of the structure’s response. In Section 3, we show the possibility of designing multiband reflective filter, using FC4(n) structure. We have also demonstrated dependence of the filter properties on its geometry and physical properties such as refractive index profile. Properties, design and simulation of DWDM filters based on these structures are presented in Sec. 4. Section 5 presents an example for the proposed device structure. Finally, we close our discussions by the conclusions presented in Sec. 6.
2. Fibonacci quasi-periodic structure
In this section the structure of the DWDM filter is introduced. After realization of the structure, we will declare relations which are used in calculation of the reflection and transmission. According to mathematical recursive relations the Fibonacci-class quasiperiodic structures can be generated using the following two main substitutions done for two basic elements A and B [16].
and
where, n is a positive integer. Based on the proposed substitutions, the following relations show the recursive relations for a general case.
For the purpose of illustration, an explicit form of the proposed general relation for the Fibonacci class FC 4(n) (for j=4) is presented, net:
Now, for realization of the Fibonacci class in optical domain, we assume that A and B are optical dielectric layers with constant index of refractions na and nb and thicknesses da and db, respectively. The order of layers is such that the structure is neither periodic nor disordered, but there is a recursive relation between the layers following Fibonacci numbers/law, so, these structures are called quasi-periodic structures. The general case of an arbitrary number of dielectric layers of arbitrary thicknesses is shown in Fig. 1. There are M layers, M+1 interface and M+2 dielectric media, including the left and right semi-infinite media. In this figure, the parameters li, ki, and ρi are the i th layer thickness, the light wavenumber and the Fresnel coefficient in that layer, respectively. E i+ and E i- are the incident and reflected electric fields at the i th interface, while, i=1, 2, 3,…, M.
We calculate light transmission and reflection through these structures using the transfer matrix method (TMM). In order to illustrate the ability of the TMM in such a calculation we assume a general structure with arbitrary order of layers. The key point behind the applicability of the TMM to calculate the reflectivity of such a multilayer structure is the assumption that the final layer is semi-infinite, in extent. With this assumption, the reflectivity at the last interface can be determined. Knowing this value, using the matrix relations across all previous interfaces and the corresponding material thicknesses, one can calculate the total reflectivity at the incident interface, Γ1 [18].
The incident and reflected fields are considered at the left of each interface. The overall reflection response, Γ1=E 1-/E 1+, can be obtained recursively by the propagation of the reflection responses.
The reflection coefficient ρi at the left hand side of the i th interface is defined in terms of the refractive indices, ni and n i-1, as follows:
The forward and backward fields at the left of interface i are related to those at the left of interface i+1 by:
where Ti=1+ ρi, and kili is the phase thickness of the ith layer, which can be expressed in terms of its optical thickness nili and the operating free-space wavelength by kili=2π(nili)/λ. Reflection responses Γi=Ei -/Ei + will satisfy the recursions:
Equation (6) is the building block of the TMM. It enables the calculation of the reflectivity of any arbitrary multilayer structure regardless of whether it is periodic or not.
Equation (6) shows the possibility of calculation Γi from the Γi+1 through a recursive relation. This recursions are initialized with ΓM+1=ρM+1 and ends to Γ1. Arrangement of layers is specified by Eq. (3), shows the complexity of the structure and difficulty of calculation using Eq. (6). We used Eq. (6) to calculate the reflection from each interface, separating layers A and B. In the next section we present the reflection response, R=| Γ1|2=|E 1-/E 1+|2, from a FC 4(n) multilayer structure, and investigate the filtering properties of such structure with various number of layers and various dimensions.
In our numerical calculations we assumed the material used in the structure to have insignificant absorption coefficients.
3. Reflective filters with FC4(n)
Here, we first investigate the general shape of the reflection from a FC 4(n) multilayer structure. Figure 2(a) shows the reflection spectrum for n=30, na=1.9, nb=1.6, da=1µm, and db=2µm, in a given range of wavelengths. As shown in the figure, the reflection spectrum of FC 4(30) contains two types of stop bands; a wide band called omnidirectional band, followed by 29 narrow stop bands. Such arrangements of the wide and narrow stop bands are repeated periodically with respect to λ. Since the reflection spectrum is the reciprocal lattice of the geometrical structure, it strongly depends on the arrangement of layers A and B. Examining Eq. (3) one can realize the narrowbands are the reciprocal of layer B and their numbers are related to the Fibonacci’s order by (n-1), whereas the wide bands are reciprocal of layer A. Each reflection stop band is equivalent to a photonic bandgap. Since the structure is quasiperiodic the bandgaps are repeated in a regular manner, which is neither periodic nor disordered.
Fig. 2. (a). Reflection spectrum from FC 4(30) with parameters: na=1.9, nb=1.6, da=1µm, db=2µm, L=53.2 mm, Δλ=0.13nm and FSR=1.3nm; (b). Central portion of the spectrum expanded over two stop bands.
Narrowband reflections from these structures can be used as wavelength selective filters in optical communication systems. Widths of the reflection bands which are in the middle of the reflection spectrum are narrower than those which are closer to the large bands. It is possible to choose a range of wavelengths in the spectrum in which the adjacent bands have almost equal bandwidths and hence the difference between the bandwidths of the adjacent bands can be neglected.
Some properties of the reflection filters such as bandwidths (Δλ), free spectral range (FSR), reflection amplitude (A), structure geometrical length (L), and sidelobes depend on the parameters of the FC 4(n) such as n, na, nb, da and db. In order to use these reflection multiband filters in optical telecommunication systems, the stop bands must have appropriate properties. In order to design such applicable filters, we investigate effect of the physical and geometrical parameters on properties of the multiband filter, next.
To enable the reader to see the details of the stop band and its sidelobes, we have expanded the central portion of the spectrum which is illustrated in Fig. 2(b)
3.1. Effects of the Fibonacci’s order (n)
We first investigate the effects of the structure’s order (n) on the various properties of multiband filter. From Eq. (3) we see that n determines the number of the seeds (FC 3(n)) which are repeated in the structure of the FC 4(n). The Structure’s order has a considerable influence on its properties. Table 1 shows the dependence of the various properties of the middle band on the values of n. The bandwidth of the flitter (Δλ) as well as its FSR decreases considerably as n increases. The latter property enables one to design dense multi channel filter with large n. Another property of the reflection band which depends up on n is its amplitude (A) which increases by an increase in n. This produces a flat top filter which is in favor of an optical communication system. The table also shows that as n increases the amplitude of the sidelobes of the middle band also increases. Such an effect can cause crosstalk between the adjacent channels deteriorating the optical filter and hence the system performance. As one can see from the table, the length of the filter, L, strongly increases by n. This parameter L is of prime importance in an industrial context. It corresponds to the quantity of the matter used in the fabrication of the filter and is directly linked to the duration of the fabrication process.
The data presented in Table 1 demonstrate that there should be some tradeoff between the Fibonacci filter parameters.
Table 1. Dependence of the various properties of a multiband filter FC4(n) on the values of n, for na=1.9, nb=1.6, da=0.75µm, db=2µm.
3.2. Effects of the layers thicknesses, da and db
Thicknesses of the layers A (da) and B (db) could also affect the various properties of the Fibonacci stop band filter.
We first, vary the thickness of layer A from 0.25 to 1.25µm. Table 2 shows the dependence of the various parameters of the reflection bands of a FC 4(30) on such variations. We see that the length of the structure (L), reflection bandwidth (Δλ), and FSR have little dependence on da. However, such variations displaces the mid band central wavelength, λc, considerably. This results show that λc varies linearly with da. Hence by choosing a right value for da one can tune the central wavelength of the filter.
Table 2. Dependence of the various properties of a multiband filter FC4(30) on the values of da, for na=1.9, nb=1.6, db=2µm.
Next we vary the layer B thickness, 0.5µm<db <2µm. Table 3 shows the dependence of the different parameters of the reflection bands on db. In this case, the reflection band parameters are affected strongly by db variations. The length of the structure (L), reflection bandwidths (Δλ), and FSR vary considerably with db. Although, L increases linearly with db both Δλ and FSR decrease in nonlinear fashions as db increases. This shows that using db as a design parameter is not simple as da.
Table 3. Dependence of the various properties of a multiband filter FC4(30) on the values of db, for na=1.9, nb=1.6, da=0.75µm.
3.3. Effects of the layers refractive indices, na and nb
The layers refractive indices (na and nb) also affect the reflection bands of the Fibonacci filter. Tables 4 and 5 show the effects of na and nb on the reflection band properties, respectively.
Table 4. Dependence of the various properties of a multiband filter FC4(30) on the values of na, for nb=1.6, da=0.75µm, db=2µm, λc=1567.05 nm, L=53 mm and FSR=1.4 nm.
Table 5. Dependence of the various properties of a multiband filter FC4(30) on the values of na, for na=1.9, da=0.75µm, db=2µm, λc=1567.05 nm, L=53 mm and FSR=1.4 nm
The results demonstrate that varying one of the two indices while keeping the other one constant the reflection band central wavelength (λc) and FSR do not vary while other parameters such as bandwidth (Δλ), the reflectivity amplitude (A), and the sidelobes vary in a similar manner as long as Δn=na-nb is kept constant. That is, the reflection band properties depend upon Δn rather than na and nb, individually. Note that, effects of an increase in na is same as a decrease in nb, for a fix Δn.
4. Designing a multichannel DWDM filter
DWDM filter is one of the critical components in modern ultra high speed communication systems and is also one of the most challenging components to design and fabricate. The filter stop bands must have sharp slope, no sidelobes, low insertion loss, standard stop band width (Δλ=0.1 nm at-3dB), and temperature independent central wavelengths. Such filters can be designed as add/drop multiplexers with 50 and 100GHz spacing whose central wavelengths can be selected from any standard ITU grid. In comparison with the other thin film filters, the quasi-periodic FC j(n) structures promise to deliver steep skirted profiles necessary to provide high adjacent channel isolation as the channel spacing in DWDM systems gets smaller and smaller.
The aim of this work is to design a 32-channel DWDM filter with 100GHz spacing (FSR=0.8 nm) using FC 4(n). To achieve such design, based on ITU standard, we introduce the following design rules:
1) Adjust da to obtain a desired central wavelength, λc.
2) Adjust db and n to obtain desired values for bandwidth and FSR.
3) Adjust Δn to obtain desired values sharpness and sidelobes for the multichannel filter.
As we have mentioned earlier, in Sec. 3, there must be some tradeoff between choosing design parameters such as n, da, db, and Δn, to achieve a standard DWDM filter using FC 4(n). With such tradeoff and the above design rules, we could not achieve an ITU-T standard 32-channel DWDM filter, using a single FC 4(n) quasi-periodic structure. However, cascading two successive FC 4(25) gave us enough room for maneuver with the design rules and tradeoff. One of the two FC 4(25) structures reflects the odd channels and the second one reflects the even channels superimposed on the first one. In the new cascaded filters, all parameters of the two FC 4(25) structures are the same except in their layer A thickness, da. The one reflecting the odd channels has d a1=2.4µm while the one with even channels has d a2=2.97µm. One should note that cascading more than two structures will produce rejection bands with low FSR. In this case the sidelobes of each band overlap the neighboring bands, deteriorating the filter characteristics.
In our numerical calculations of reflection spectrum for Two cascaded FC 4(25), we have simply assumed two juxtaposed structures with no coupling layer between them.
Figure 3 illustrate the reflection spectrum of the cascaded structure. In order to obtain an appropriate reflection response, we had to use a trail and error process. The reflection spectrum contains 32 narrow width reflection bands situated within the C band suitable for modern optical communication systems.
Fig. 3. (a). Reflection spectrum from two cascaded FC 4(25) as a 32 channel DWDM filter, design parameters are: d a1=2.4µm, d a2=2.97µm, db=2.4µm, na=1.75, nb=1.6, Δλ≈0.11 nm (at -3 dB); (b). Central portion of the spectrum, expanded over two stop bands.
To see the details of stop band and its sidelobe, an enlarged portion of the reflection spectrum is shown in Fig. 3(b). This figure shows the resulting sidelobes amplitudes are still too high for optical communication applications.
A critical parameter for such filters is their reflection phase shift as a function of the wavelength, λ. The group velocity dispersion affects the pulse width. To quantify this effect one should calculate the group-delay (τg) defined by
where φ is the phase of the reflection coefficient, Γ1, ω=2πc/λ is the radian frequency of the light beam and c is the speed of the light in free space. By taking the derivative of Eq. (7) with respect to the wavelength, one can obtain a relation for the group velocity dispersion:
Using a sample reflection band as shown in Fig. 4(a), we have calculated τg and D numerically. The wavelength dependence of the last two parameters are illustrated in Figs. 4(b) and 4(c). As is illustrated in Fig. 4(b), the group delay variation in the bandwidth of the filter is insignificant, whereas the variations at the band edges are large. Figure 4(c) demonstrates the nondispersive behavior of the filter in the reflection band which is consistent with the insignificant variation of the group delay in the same region. Note that there is an asymmetry in the behavior of the group delay at the band edges of the filter with respect to its center wavelength. This behavior is due to the asymmetry existing in the reflection phase which in turn is due to the quasi-periodicity of the filter layers.
The sidelobes in the reflection bands cause high crosstalk between the adjacent channels, so they should be eliminated in multichannel filters. These large sidelobes are due to the fact that a nonuniform index variation with large Δn causes resonance in the reflectivity occurring on both sides of each stop band in the spectrum. An appropriate method of sidelobe reduction in the reflection spectrum is to apodize the refractive index profile [19]. The apodization is done by multiplying the index profile by an envelope function such as Gaussian:
Fig. 4. (a). A sample reflection band of a FC 4(25) with parameters da=2.4 µm, db=2.4 µm, na=1.75, nb=1.6; (b) The group delay of the reflection band illustrated in (a); (c) The group velocity dispersion for the same sample band
where z is the coordinate along which the apodization is performed, L is the length of the structure, and h and σ are the amplitude and the standard deviation of the apodization function, respectively. We have assumed that the apodized indices na(z) and nb(z) have the general form of
in which n 0 is the reference refractive index, Δn a,b, is the apodized portion of the refractive index of layer A or B.
Figure 5 shows the result of the apodization applied to the two successive cascaded FC 4(25). Figure 4(b) illustrates an expanded portion of the reflection bands. This figure shows how the sidelobes have been minimized by the apodization process.
Fig. 5. (a). Reflection spectrum from two cascaded FC 4(25) apodized by Eq. (9), the parameters are: d a1=2.4µm, d a2=2.97µm, db=2.4µm, n 0=1.3, Δna=0.45, Δnb=0.3, σ=0.3, h=1.5, Δλ≈0.1 nm; (b). Central portion of the spectrum., expanded over two stop bands.
The illustrated results also demonstrate that the amplitude of middle bands of the apodized filter has been reduced, in comparison with that of unapodized one. Since the sidelobes have been eliminated the crosstalk between adjacent channels is not large. The apodized structure is more suitable for optical communication applications.
Apodization techniques were demonstrated to be feasible experimentally [20–21]. However, using such a process in fabricating an already complicated structure such as FC 4(25) complicates the fabrication process even further.
5. The Proposed waveguide structure
Fig. 6. (a). Cross sectional view of the proposed waveguide structure, sandwiched between two layers of SiO2; (b) A 3-D view of the proposed waveguide structure, from the top. Layer B (blue) nb=1.6, and Layer A (pink) with na=1.75.
We have already seen that for DWDM applications Δn=(na-nb) should be large. Therefore, the most suitable technique to achieve a desirable waveguide structure is to use a standard microelectronics planar technology. Figure 6(a) illustrates a cross sectional view of the
waveguide sandwiched between layers of SiO2 deposited on top of a Si substrate. Figure 6(b) shows a three dimensional view of the waveguide. As shown in the figure, layers A and B are materials with refractive indices of na=1.75 and nb=1.6, respectively. For example, refractive indices of SiON [22] and oxidized AlAs [23] are both 1.6, whereas refractive indices of specific liquid crystals [24], particular optical glasses [25] and some photonic polymers [26] are all equal to 1.75. Obviously materials used in fabrication of the proposed filter should be compatible with standard microelectronic fabrication technology. It is obvious that fabrication of such structure involves a two-level photolithographic process. The first masking step is needed to etch a large window in the SiO2 layer grown on top of the Si substrate. The z-dimension of the window equals the length of the waveguide, L. After the window is filled by the layer B material, using one of the standard deposition techniques, the second mask is used for etching the desired windows in the layer B. To say the least, the second mask dimensions along the z-direction should be designed according to the Fibonacci’s quasi-periodic dimensions, da and db. These windows should be filled by the layer A material. At last, a layer of SiO2 should be deposited on top of the waveguide structure, using a compatible deposition technique.. Notice that the refractive index for a high quality SiO2 is n SiO2~1.5< nb=1.6<na=1.75.
For a couple of reasons we have considered photolithographic processes to manufacture the proposed filter structure. First of all, the geometrical thickness of a FC 4(25) filter is about L=37mm. Standard deposition technologies like thermal evaporation, ion-assisted deposition or sputtering are not suitable for manufacturing such filters due to their low deposition rate (5–10nm/s for the better case). Moreover, the control of the thickness of the layers during deposition will be very difficult as the layers are non quarterwave layers and standard optical monitoring is only well adapted for quarterwave layers. Hence, these technologies are not adapted in the case of Fibonacci filters.
Since the fabrication tolerances could have some impacts on our theoretical analysis, we have investigated such effects on the calculated characteristics of the filter. We have realized that for a deviation of ±1nm in the size of da, the center wavelengths of a FC 4(25) filter shift about ±10pm from their original values, whereas the same deviation in the size of db displaces the stop bands about ±7.5pm towards the center of the band. In contrary to the former case, in the latter case value of the λc remains unaltered. These error analyses demonstrate that Fibonacci filters are quite sensitive to the fabrication tolerances, and hence a high precession technology is needed.
Another important tolerance which should be taken into account for experimental purposes is the variation of the refractive indices, due to the variations in the stoichiometry of the materials used in the fabrication. We have found that due to the high index ratio of layers A and B, variations on n a,b in the order of 10-3 and less have little effects on the properties of the proposed filter structure.
Meanwhile, wavelength dependence of the layers refractive indices for the proposed materials, for the C-band, is in the order of ∂n/∂λ~10-5. Therefore, effect of the variations of the refractive indices due to ∂n/∂λ on FC 4(25) characteristics is insignificant.
An important practical parameter for DWDM filter is its thermal sensitivity (∂λ/∂T). We have used Takashashi model [27], in order to calculate thermal sensitivity of FC4(25) filter. As a result we have observed two different effects on the DWDM filter characteristics. One of these two is the shift of the center wavelength of each band by ∂λ/∂T~7pm/°C. This effect is due to the variation of the thickness of the layer A with temperature (dda/dT). The second effect observed was the compression the total reflection band of the filter. In fact each stopband moved towards the center of the band by ∂λ/∂T~4.5pm/°C. This effect is due to the thermal variation of the Layer B thickness (ddb/dT). These data were obtained using the thermal expansion coefficients and the temperature coefficients for SiON and liquid crystal. This considerable thermal sensitivity makes a temperature control system necessary.
6. Conclusion
In this paper, we have proposed a scheme to design a narrowband DWDM filter structure based on Fibonacci’s quasi-periodic structures, FCj(n). We have realized that to achieve a narrowband filter suitable for DWDM optical communications according to the ITU-T standard, one needs to use a couple of apodized FC 4(25) structures cascaded in series. The cascading schemes made it possible to control and tune both the stopband 3dB bandwidth of the filter and its FSR to meet the ITU-T standard, while the apodization process helps to minimize the stop band sidelobes to desired values.
For achieving very narrow multi-band filter using quasi-periodic structures there is a tradeoff between filter length and the bandwidth. Quality of the designed filters can be improved using precise apodization techniques.
The proposed filter has good compatibility with fiber optic communication systems. It comprises passive multilayer configuration which can be adapted in fiber layout. Different properties of DWDM filter can be adjusted by one of the structures parameters such as n, da, db, and Δn, independently. Another advantage of this filter is its narrowbands of about 1nm bandwidth. Furthermore, nearly unity reflection amplitudes are achievable. Nonetheless, there are a few drawbacks for experimental applications. The main drawback of such filters are their complicated fabrication process due to the large thicknesses of the structure and the large ratio of the refractive indices of the layers A and B. In addition, thermal sensitivity of such filters as well as their sensitivity to error in fabrication are considerable.
As we have mentioned in Sec. 2, the material considered for fabricating the proposed structure are assumed to have small absorption coefficients. This is true for materials such as SiO2, SiON. However, if one uses materials with considerable absorption effect, the material loss should be taken into account in the filter design.
Acknowledgments
This work was supported in part by Iran Telecommunication Research Center (ITRC) under grant T500-3679.
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