Multi-channel Si-liquid crystal filter with fine tuning capability of individual channels for compensation of fabrication tolerances

In this study, a technique for the optimization of the optical characteristics of multi-channel filters after fabrication is proposed. The multi-channel filter under consideration is based on a Si photonic crystal (PhC), tunable liquid crystal and opto-fluidic technologies. By filling air grooves in the one-dimensional, Si-Air PhC with a nematic liquid crystal, an efficiently coupled multi-channel filter can be realised in which a wide stop band is used for channel separation over a wide frequency range. By selectively tuning the refractive index in various coupled cavities, continuous individual tuning of the central channel (or edge channels) up to 25% of the total channel spacing is demonstrated. To our knowledge, this is the first report on the electro-optical solution for the compensation of fabrication tolerances in an integrated platform.


Background
Although Si fabrication technology has significantly developed over the last 20 years, one of the main problems for optical, nano-scale periodic structures, such as Fabry-Perot interferometers and multi-channel photonic crystal (PhC) filters, remains in defining of the critical dimensions precisely in the system [1][2][3]. In general, structural deviations and non-uniformities present in the patterned features on the wafer occur for three principal reasons. First of all, there is the fundamental diffraction limit of the projection optics. Secondly, the mask pattern differs from the original design due to limitations in the mask fabrication process. Finally, random and systematic variations inevitably occur in a multitude of lithographic process parameters, such as focus and exposure [4]. Fabrication tolerances for modern e-beam lithography are usually assumed to be a minimum of about 5% to 10% of the nominal target dimensions. This crucially affects the optical characteristics of multi-channel filter devices [5][6][7]. It changes the precise wavelength position of individual channels; it increases the out-of-band reflection and causes an attenuation of the maximum intensity, resulting in a lowering of the quality factor, Q. One of the most promising solutions to these problems may be a filter device with low-power and low-loss capability to compensate for the optical filter deviations. This would allow fine tuning of individual channels in the filter system by varying the temperature or by applying an electric field. However, one of the main challenges in the realisation of tunable multi-channel devices remains the fixed channel spacing or free spectral range which cannot be easily tuned due to the strong coupling between channels. One of the more successful attempts to tackle this problem was presented in [8], where the authors demonstrated a solution for an integrated platform. This approach requires the incorporation of metallic micro-heaters into large 245 μm resonators, demanding precise temperature control of the device during operation.
To our knowledge, there have been no reports in the literature on addressing this problem using electrotuning for the individual channels in a multi-channel system that is compatible with complementary metal oxide semiconductor technology. The design of the tunable multi-channel filter proposed in this work is based on a one-dimensional (1D) Si photonic crystal (PhC) using opto-fluidics and liquid crystal (LC) technologies [2,5,[9][10][11][12][13][14]. An LC is one of the most attractive tuning material for Si-based integrated devices, enabling tuning of the resonance modes using low applied voltages (from 1.5 V) with negligible absorption during device operation. By creating optical cavities within the periodic structure by infiltration with LC, the PhC mirror can be transformed into a highly efficient coupled multi-channel filter [15]. A coupled multi-cavity PhC system has significant advantages over other types of coupled resonators in terms of device simplicity, ease of integration on a Si chip, and power consumption. We extended the continuous fine-tuning capability of LC single microcavity to a system of individually tunable coupled multi-cavities. The application of an electric field to each cavity was done individually, and its continuous selective variation across all cavities allows to overcome the problems related to the strong coupling between channels. Using an example of a coupled triple-cavity PhC filter operated using the first SBs, we have developed a simple model for easier manipulation of the LC within individual cavities, enabling the independent fine tuning of each channel in the overall system. We note that the model suggested can be extended to a higher number of coupled cavities (defects) and, therefore, to a higher number of resonances, with an improved Q value.

Methods
A free-spaced, microstructured, grooved Si structure, with quarter-wavelength Si-Air layers of optical thicknesses n Si d Si ¼ n air d air ¼ mλ m =4 (where m is the order of the stop bands, SBs), is considered here as a model for an ordinary 1D PhC mirror. The refractive index of air, n air , is 1, while the refractive index of Si is taken as n Si = 3.48. The coupled resonators-induced transparency around the wavelength λ m occurs when optical or geometrical defects are introduced into the perfect periodic structure ( Figure 1) [16,17].
The creation of a microcavity resonator requires a doubling of the optical thickness of certain grooves, , where i is the number of the groove. This can be achieved, for example, by infiltration of the groove with filler with a suitable refractive index. By varying the optical thicknesses of the defects simultaneously, the resonance (or channel) peak positions, λ c , and the Q factor of the coupled resonances obtained can be continuously tuned within the stop band (SB) [5,18,19]. For this purpose, the LC is a promising candidate material, with its capability of continuous reverse tuning of the refractive index over a wide range of values by the application of a voltage, V [20]. In this study, we use the commercially available nematic LC E7 available by Merck KGaA, Darmstadt, Germany, which demonstrates a high anisotropy of the refractive index [11][12][13][14]. For clarity, we consider an initial switching position that can achieve the same intermediate orientation of the LC molecules in all cavities. That is, it produces the average refractive index n LC ¼ 1:7472 þ 1:5217 ð Þ =2 ¼ 1:63 at a switching voltage of approximately 10 V.
All modelled results are obtained using numerical simulations based on a transfer matrix method (TMM) [21]. The optimal design parameters for the LC-coupled microcavities, d Si , d air and d def = d LC , which give the highest Q factor and the narrowest equal wavelength spacings between the channels, λ s , are determined using a graphical engineering approach, based on a Gap Map (GM) presentation [22]. The methodology of this approach is quite general and is described in detail in references [5,15,20]. In brief, optimal design parameters are determined by certain range of filling fractions,  Table 1 gives the TMM-simulated optimal design parameters, together with the optical characteristics of the sample triple-channel device [5].

Results and discussion
Optimal parameters taken from Table 1 were used for the fabrication of integrated triple-channel devices on a <100 > p-type silicon-on-insulator (SOI) wafer with a device layer thickness of 4.5 μm. The fabrication process involved used e-beam lithography to define the patterns followed by plasma etching down to the buried oxide layer. Scanning electron microscopy (SEM) investigation of the devices fabricated revealed the presence of a random deviation of the Si wall thicknesses, δd Si , of up to 10.8% of the target thickness. In Figure 2a, this deviation is demonstrated on the SEM image of the structure with a = 900 nm, which operates within the second order SB (m = 2, Table 1). This thickness deviation, δd Si , directly influences the optical thicknesses of the cavities, Þ , and also the optical thicknesses of the spacings between the cavities (or optical thickness of the air grooves), n air d air ¼ n air d air AE δd Si ð Þ. Variation of the geometrical parameters of a coupledcavity device results in random shifts, splitting and attenuation of each of the cavity resonances [14,[23][24][25].
Simulated transmission spectra for the devices originally designed and fabricated are demonstrated in Figure 2b,c, d. Although the attenuation of the maximum transmission intensities is only about 3% to 5% resulting in a negligible reduction in the parameter Q, the precise wavelength position of each channel, λ c , was affected significantly. All of the channels are unequally blue/redshifted by approximately 0.3% to 1.2% from the originally designed operational wavelengths, which is more than the target channel bandwidth (Figure 2b,c,d, red lines). A summary of the TMM analysis for the three fabricated structures, operating within the first three SBs along with data on the SEM registered thickness deviations, ±δd Si , and the corresponding random shifts of the resonant peak wavelengths, ±Δλ c1,2,3 , are reported in Table 2.
The simplest way to compensate for these errors, postfabrication (d air ± δd Si ), is to adjust the refractive indices of the coupled cavities individually to their optimised values, n Þ def d air ¼ λ m =2 decreases which leads to a blue shift of its resonance peaks [15]. Let us monitor the positions of the triple resonances during these refractive index variations. We use normalised wavelength units, NW = λ/a, and therefore, the results obtained can be applied to a variety of nano-and micro-structures, operating over a wide wavelength range. In Figure 3b, we demonstrate transmission spectra for Δn = 0.013 (thin blue line) and Δn = 0.026 (thick blue line). The central channel is blue-shifted, while the edge resonances have retained their positions. For the example considered, the maximum refractive index difference, 2Δn = 0.052 (which is only 24% of the birefringence value for the LC E7, Δn E7 = 0.2255), allows us to achieve a maximum relative independent shift of the central channel, Δλ s , of up to 25% of the channel spacing, λ s . However, further  increases in the overall difference, 2Δn, will break the symmetry of the system and initiate a coupling effect between all of the cavities, resulting in a shift of all the resonances simultaneously. The opposite manipulation of the refractive indices, i.e. reducing n     Figure 4b for Δn = 0, 0.02, 0.052 and 0.11 for the structure under consideration. The increase of the channel spacing, λ s + Δλ s , results in a decrease of the channel spacing between the central channel and the right edge channel, λ s2,3 = λ s − Δλ s , lowering the out-of-band reflection between them.
As in the previous model, the overall increase in the refractive index, Δn up to 0.052, for the example considered, leads to a linear change in the channel spacings of up to 25% or λ s12 /λ s23 = 1.5 (Figure 4a, b). Although the out-of-band reflection between the central and right edge channel is decreased, it still reaches 95%. Further increases in Δn up to the limiting case of Δn = 0.11, will result in a rapidly growing difference between channel spacings, λ s12 /λ s23 of up to 2.7 with a decrease of the  For significantly higher values of Δn > > 0.11 than those considered in this paper, the central cavity acts as a single cavity, independent of the edge cavities. The left-edge channel will be shifted out of the SB, and the right edge channel will be merged with the central channel, thus changing the resonator mode from triple mode to single mode. Obviously, the opposite manipulation of the refractive indices, i.e. increasing the n The tuning approach demonstrated here can be applied to the real devices (Table 2) as previously considered, allowing their actual characteristics to approach those in the original design (Table 1). In Figure 5, this is

Conclusion
To summarise, an approach based on the individual tunability of multi-channel filters has been proposed for the compensation of optical parameter deviations caused by structural fluctuations. Electro-optical tuning of individual central (or edge) channels was demonstrated by up to 25% of the channel spacings, which is sufficient for the optimization of devices with fabrication tolerances of up to 20%. The approach suggested here can be utilised for the optimization of multi-channel silicon devices over a wide infrared range.