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Scalable optical space switches compatible with high bit rates which can be reconfigured on-the-fly are needed to increase the flexibility of optical networks. We present the design of integrated Fabry-Perot filters working at oblique incidence, which can be used to build optical space switches. A comprehensive planar waveguide optimization was conducted to minimize radiation losses in the deep-etch features forming the filter mirrors. Four high order cavities were coupled to create a 200 GHz comb wavelength response with passbands larger than 50 GHz and extinction ratio greater than 20 dB over the entire C-band. Gaussian beam propagation analysis showed that the minimum beam waist required to avoid distortion increases rapidly with incident angle.
©2009 Optical Society of America
1. Introduction
Fabry-Perot (FP) interferometers have found application in many areas of science and engineering since their introduction in 1897 [1]. In optical telecommunications, they are commonly used as wavelength selective filters to isolate channels. In addition, by modulating the cavity refractive index, FP filters can operate as switchable reflectors that reflect or transmit a light beam, in a similar way to micro-electro-mechanical system (MEMS) pop-up mirrors [2]. By using tuneable FP filters at oblique incidence one could create optical space switches with layouts similar to two dimensional (2-D) MEMS switches but where the switching mechanism can be thermo-optic, electro-optic, or all-optical (see Fig. 1 ). This way, it is possible to avoid speed limitations inherent to moving parts while having a simple and scalable architecture. However, FP filters must be carefully engineered to provide a wavelength response that is broad and sharp in order to offer transparency to bit rate. They must also have a high extinction ratio, low insertion losses, and cause little chromatic dispersion. This can be achieved by coupling multiple cavities. Furthermore, the number of wavelength channels that a FP switch can transmit/reflect can be increased by creating a filter with a comb response, as demonstrated previously in a micro-ring resonator switch [3]. To reduce the volume over which the switching mechanism must operate, and hence decrease switching times, the FP filters may be implemented in a planar waveguide. Fast optical space switches are required to create agile photonic networks in which wavelength connections can be reconfigured on-the-fly [4]. The required switching speeds are impossible to achieve with the technologies used in the current generation of commercial reconfigurable optical add/drop multiplexers [5]. Furthermore, the other switch configurations able to provide the desired speeds that were reported are limited in their scalability or have high losses and/or crosstalk [6].
Fig. 1 Schematic of a 3x3 switch with tunable coupled cavity Fabry-Perot filters in a crossbar configuration.
In this work, we report on the development of integrated off-axis FP filters designed for optical space switching in planar waveguides. The transfer matrix method (TMM) and Eigenmode expansion were used to optimize various filter configurations to find the best theoretical performance and investigate the impact of fabrication limits. The modeling approach used to simulate integrated filters at oblique incidence that includes both guided-wave modes and free-space propagations was verified experimentally. Since deep etching is required to obtain the large refractive index contrast necessary to create broadband mirrors, the first section of this paper presents a study which for the first time evaluates simultaneously the impact of a waveguide layer structure and refractive index contrast on radiation losses. We conclude that the most efficient waveguide configuration varies as a function of etch depth and that index contrast and layer structure must be optimized concurrently to obtain the best result. In the next section, the optimized waveguide effective index is used with the TMM to design a filter that provides a clear bandwidth of at least 50 GHz and a free-spectral range (FSR) of 200 GHz. This comb response makes it possible to switch from reflection to transmission one out of every two channels on a 100 GHz grid over 40 nm with a refractive index change of only 0.053%. Thus, the switch can work on bands of “odd” or “even” channels. Investigation of the filter tolerances showed that mirrors formed of a single trench are less sensitive to fabrication errors than those made of two trenches. Lastly, in section four the propagation of Gaussian beams through integrated off-axis FP filters is analyzed. It was found that 99.99% of the incident beam angular spectrum must lie within the filter clear angular bandwidth to maintain a passband width greater than 50 GHz and that of the previously identified impairments for finite beams propagating in thin-film filters, only lateral beam shift is significant in the device presented here.
2. Planar waveguide optimization
Figure 2 illustrates the planar waveguide structure and defines the variables considered in the design optimization. Use of a GaAs/AlGaAs structure, in addition to being readily available, offers many degrees of freedom for planar waveguide optimization since all of the waveguide layer thicknesses and refractive indices can be adjusted. Deep etching is required to obtain a large index contrast to form broadband mirrors for the FP filters. Unfortunately, beam expansion occurs in the regions where the waveguide is removed, leading to radiation losses. This problem has been extensively studied in the context of photonic crystals [7–12] and of high reflectance Bragg mirrors [13–20]. It was shown that both the waveguide index contrast [7–9, 12] and its layer structure [11, 12] need to be optimized to minimize losses. The trench geometry also plays an important role [10, 15, 18, 20, 21]. However, the previous reports only covered one or two of these features, and to our knowledge, this study first to consider them simultaneously during waveguide optimization. Only the effects of conical or slated wall trenches were not included. The results presented here were obtained with the 2-D Eigenmode expansion algorithm described in [22] and at a wavelength of 1550.12 nm (193.4 GHz).
Although this algorithm cannot model oblique incidence on FP filters, it is able to rapidly simulate a device with multiple coupled cavities at normal incidence. However, during the optimization a single trench was simulated to avoid variation in radiation losses resulting from a change in the device frequency response. As waveguide parameters are scanned, the guided mode propagation constant changes and thus the device response changes from reflecting to transmitting or vice versa. Because deep-etched trenches themselves can act as FP cavities, transmitted wavelengths resonate within them, and the lack of confinement leads to radiation losses. On the other hand, light that is reflected from a trench quickly couples back into the waveguide and hence suffers less loss. In order to ensure that the simulations were always done with a trench maximizing transmission, and hence provided a worst case evaluation of radiation losses, the trench width was always equal to an integer number of half wavelengths. In their investigation on the effect of a waveguide index contrast on radiation losses, Bogaerts et. al. reported that there are two regimes where low losses are achievable [9]. One is in low index contrast waveguides for which light is slowly diverging in etched areas, and the other is only attainable in semi-infinite periodic structures with high index contrast where it is possible to couple to lossless Bloch modes. Since Bloch modes do not exist in FP filters, it is sufficient to consider a single trench and low index contrast to accurately optimize the waveguide structure.
An alternative to using a single trench would be to express the device lengths in terms of wavelength and to adjust its physical dimensions according to the guided mode propagation constant at the beginning of each iteration. However, doing so with modal expansion methods requires careful adjustment of the simulation parameters since in the regions without waveguides most of the propagating light couples to higher order modes that interact strongly with the simulation boundaries. As a result, the effective refractive index in the etched regions can be affected by the boundary conditions and the difference in refractive index that it creates can change significantly the device wavelength response.
The graphs in Fig. 3 show the radiation loss as a function of refractive index contrast and core thickness for a trench that is half a wavelength wide (at 1550.12 nm) and etched either 3 μm or 4 μm deep. For each index contrast and core thickness case, the top cladding thickness was varied between 0.5 μm and 3.0 μm and only the result from the most efficient configuration is reported in Fig. 3. The optimum waveguide designs vs. etch depth are given in Table 1 . As the etch depth increases, the waveguide configuration that minimize radiation losses requires a lower index contrast and a thicker core and top cladding. The optimum top cladding thickness as a function of index contrast and core thickness is shown in Fig. 4 for a 4 μm etch depth. As with the other etch depth, the top cladding thickness values for the best waveguides with 4 μm deep trenches are all well within the range of scanned values, which ensures that the optimization routine found the best solutions. Furthermore, single mode operation was verified for all waveguide configurations considered.
Fig. 3 Radiation loss vs. refractive index contrast and waveguide core thickness for (a) a 3 μm, and (b) 4 μm. The optimum top cladding thickness was found for each waveguide configuration.
Table 1. Optimum GaAs/AlGaAs waveguide configurations vs. trench depth
Fig. 4 Optimum top cladding thickness vs. refractive index contrast and waveguide core thickness for 4μm deep and half a wavelength long trench.
The results shown in Fig. 3 illustrate the complex interactions that must be considered to minimize radiation losses when designing a waveguide. The divergence of the beam exiting the waveguide should be minimized to avoid light from radiating out of the trench. This is achieved by having a wide and smooth mode profile. On the other hand, the beam needs to be isolated from the trench floor where substrate leakage might occur and this requires a compact mode profile. Thus, the optimum waveguide structure is the one that offers the least diverging mode profile that does not couple with the substrate. Furthermore, the fact that there is a unique optimum configuration of refractive index and waveguide layer thicknesses for each trench depth proves that both the waveguide mode width and its shape are important. For instance, Fig. 5(a) shows two modes of equal width (measured at the 1/e2 intensity points) from different waveguides. The first waveguide has the optimum profile for a 4 μm deep trench (see Table 1) whereas the second one has a 0.81% index contrast with a 0.35 μm top cladding and 0.75 μm core. The optimum waveguide has radiation losses of 3.3% whereas the other one suffers losses of 14.1% after propagating through a half wavelength trench that is 4 μm deep. This difference can be explained in terms of the mode angular spectrums. The spectrum of the mode with the higher losses is wider (see Fig. 5(b)) and hence it has energy propagating at larger angles than the other mode. Also, since the optimum index contrast varies with the layer structure, both must be considered simultaneously to find the best waveguide configuration.
Fig. 5 (a) Two intensity profiles with the same 1/e2 width and (b) their angular spectrum. The blue line shows the profile for the optimum waveguide configuration for a 4 μm trench described in Table 1. The red line is corresponds to a waveguide with an index contrast of 0.8%, a top cladding of 0.35 μm, and a core of 0.75 μm. Its radiation loss for a half-wavelength trench is 14%. The green line in (a) indicates the wafer surface and trench position with respect to the intensity profiles.
The reduction in radiation losses gained by etching deeper is shown in Fig. 6 and the effect of the cavity order is presented in Fig. 7 . In addition to minimizing radiation losses, the planar waveguide must also be tolerant to variations in the fabrication process. As can be seen in Fig. 3, the optimum configuration for a 3 μm etch is at the limit where the waveguide becomes multimode. Thus, it might be desirable to sacrifice some efficiency to improve fabrication tolerances. Moreover, Fig. 3 shows that tolerances increase for larger etch depths. If the maximum etch depth achievable is not exactly known before waveguide growth, it is preferable to choose a waveguide design that is optimum for an etch depth shallower than expected. In Fig. 8 the radiation losses vs. etch depth are shown for the waveguide configurations given in Table 1. If a waveguide designed for a 5 μm etch depth is used but only 4 μm deep trenches are obtained, radiation losses will be 6.1% higher than with a waveguide optimized for 4 μm trenches. Conversely, if an etch depth of 5 μm is achieved with a waveguide structure designed for 4 μm trenches, radiation losses will only be 1.1% higher than with the optimum waveguide for that depth. Furthermore, these losses are compounded when multiple trenches are used, such as in the case of coupled-cavity FP filters, and thus it is important to find the best realizable waveguide configuration.
Fig. 8 Radiation losses vs. etch depth for the optimum waveguide configurations described in Table 1.
Radiation losses can be decreased at the expense of a reduction in reflectivity by filling the trenches with a low index material. For instance, the loss of a 4 μm deep and 1.55 μm wide trench can be reduced from 6.9% to 2.2% by filling it with silicon dioxide even though that transforms a second order trench (when filled with air) into a third order one (with SiO2 filling). Radiation losses can even be completely eliminated if a waveguide made of low index materials is formed inside the trenches, provided that the modes supported by both the high and low index waveguide match [8].
3. Fabry-Perot filter design
To create efficient space switches, the filter response must have a flat and wide passband with sharp transitions between its regions of high transmission and reflection. This is achieved by coupling multiple FP cavities [23–25]. Furthermore, to make the switch practical, it must work over a broad wavelength range. Unfortunately, the refractive index modulation provided by fast phenomena, such as electro-optic or non-linear optical effects, is too small to shift a FP filter transmission peak over several channels. To extend the bandwidth over which the switch can operate, the filter free-spectral range (FSR) was designed to produce a comb response that transmit (or reflect) one out of every two channels on the desired frequency grid. This approach was previously used to extend the bandwidth of a micro-ring resonator switch [3]. One drawback of this technique is that FP filters loose their ability to work as demultiplexers because comb filters can only distinguish between two bands of wavelengths and not individual channels. It also leads to larger devices. To demonstrate the integrated FP filter switch concept, a filter was designed to meet the following objectives:
- − The clear bandwidth (i.e. −0.5 dB) should be flat and greater than 50 GHz;
- − The comb response should have a 200 GHz pitch so that the switch is compatible with the 100 GHz I.T.U. grid;
- − The on/off ratio should be greater than 20 dB;
- − The switch should work over the entire C-band.
The relationships between the mirror reflectivity and the cavity length defined in [25] were used to obtain a preliminary design and to constrain the number of variables during optimization. Initially, the outer mirrors were made of a single trench whereas the inner ones had two since, according to [25], the inner mirrors must have a higher reflectivity than the outer ones. This design was refined using a commercial software based on the TMM in which the planar waveguide was modeled by its effective refractive index. A similar technique was used to design deeply etched antireflective waveguide terminators [21]. It became apparent that a single trench would be sufficient for the inner mirrors since their width were not close to multiple of quarter wavelengths. This increased bandwidth by improving cavity coupling and reducing radiation losses. The double trench mirror design has a maximum bandwidth of 36 nm and transmission losses of 0.9 dB whereas with single trench mirrors, the switch can cover the entire C-band and has maximum losses of 0.2 dB. Furthermore, the tolerance to variations in trench width is higher for single trench mirrors, as shown in Fig. 9 . To model accurately what takes place during fabrication, the width that was added/removed to a trench was removed/added to the neighboring cavities. Variations larger than ± 20 nm destroyed the double trench design response. Changing the inner mirror design doubled the tolerance to variations in width, as shown in Fig. 9(b). Also, errors in trench width that are consistent across the entire filter will shift its wavelength response up (for larger trenches) or down (for smaller trenches).
Fig. 9 Filter normalized transmission response as a function of the mirrors trench width tolerance for (a) a design with double trench mirrors and (b) a design with single trench mirrors.
Four cavities were needed to obtain a clear bandwidth greater than 50 GHz. This is more than 10 times the clear bandwidth of a single cavity comb filter with a similar rejection ration of the adjacent channels. Because of the small FSR required to get the desired comb response, the cavities needed to be long, which led to large filters. Furthermore, as demonstrated in [23], the passband width decreases with increasing cavity order. Thus, more cavities are required in comb filters to obtain a 50 GHz passband than in single channel filters. However, small FSRs provide sharper responses for the same mirror reflectivity [26]. The increase in passband gained by having multiple cavities provides many advantages that outweigh the increase in size. First, it allows the switch to handle signals with higher modulation rates. It is more tolerant to wavelength drift in input signals and to temperature variations [27]. Also, it relaxes the control needed on the switching mechanism. Although chromatic dispersion cannot be avoided in autoregressive filters [28], with careful design its peaks can be moved towards the passband edges and a low dispersion region can be achieved in the passband center. Thus, a wider passband is useful to mitigate dispersion. Moreover, when designing FP filters, a tradeoff must be made between the rejection ratio and dispersion. This is done by adjusting the mirror reflectivity. Figure 10(a) shows the transmission spectrum of two filters with four coupled cavities but with different mirror reflectivity. The filter with the high reflectivity mirrors has a sharper response and a higher rejection of the adjacent channels but, as shown in Fig. 10(b), it suffers from stronger chromatic dispersion. Lastly, as explained below, a larger passband allows smaller beams or beams with a greater angle of incidence to travel through the filter undistorted.
Fig. 10 (a) Transmission response and (b) chromatic dispersion for filters with different mirror reflectivity calculated with the TMM. The filter with the low reflectivity is the 1st order design described in Table 2 below. At 1550.12 nm the mirror reflectivity is 10%, 51% and 66% for the 1st, 2nd and 3rd mirror, respectively. The 4th and 5th mirrors are identical to the 1st and 2nd. The high reflectivity filter has the same number of layers but the reflectivity of its 1st, 2nd and 3rd mirrors is 14%, 57% and 70% respectively.
In addition to meeting the performance requirement cited above, the filter design must also be compatible with the fabrication process. Table 2 shows the ideal dimensions for the four cavity design at normal incidence. Some trenches are very small and might not be realizable with a given fabrication process. One possible way to circumvent this issue is to increase the trench width by integer multiple of half wavelengths. However, this increases radiation losses, which in turn reduces the extinction ratio below 20 dB. It also increases the cavity coupling sensitivity to wavelength, which causes ripples in the passband that increase as the incident wavelength gets farther from the design value (see Fig. 11 ). Enlarging the trenches from first to second order increases radiation losses by 0.87 dB in transmission and 0.49 dB in reflection when the filter is implemented with 4 μm deep trenches. For 5 μm trenches, the loss increase is 0.54 dB in transmission and 0.31 dB in reflection. The impact on the bandwidth is much worse since the range over which of the on/off ratio exceeds 20 dB decreased from the whole C-band (40 nm) to only 12 channels (8.8 nm) for both etch depth. If 3 μm trenches are used, the first order filter also works over the entire C-band in that case, with losses of 0.65 dB and 0.12 dB in transmission and reflection, respectively. However, it is impossible to obtain 20 dB of extinction for any channels with the second order design. The maximum extinction ratio is 16.1 dB for transmitted channels. With the first order design, the bandwidth over which the extinction ratio is 20 dB starts to decrease for etch depths of less than 2 μm and disappears completely below 1.5 μm.
Fig. 11 Wavelength response in transmission and reflection for filters with (a) first and (b) second order mirrors.
Radiation losses not only reduce efficiency but also affect the wavelength response. The response for the filter with single trench mirrors described in Table 2 was calculated at normal incidence with the Eigenmode method (see Fig. 11). The radiation losses are slightly higher for the longer wavelength in the passband, and this creates a slope in the passband.
Table 2. Four cavity filters layer characteristics at normal incidence
4. Beam propagation in integrated off-axis filters
The filter wavelength response was designed assuming plane wave illumination at normal incidence, which is very different from the small beams used in integrated optics. To evaluate the impact of finite size beam on the filter response, Gaussian beams were propagated through the filter with the TMM by decomposing them into their angular components. Figure 12 shows the filter wavelength response around 1550.12 nm for beams with different fraction of their angular spectrum power included within the filter angular clear bandwidth. Even with 99.99% of the incident beam angular spectrum located within the filter angular clear bandwidth the plane wave response is not completely recovered but it is enough to obtain a passband greater than 50 GHz. Once this relationship between the filter and the incident beam angular spectrum is established, it is possible to relate the filter angular bandwidth with the minimum beam waist required to avoid degradation of the filter response and beam distortion:
Where wo is the 1/e Gaussian beam waist, and θ is the filter clear angular bandwidth in radian. The constant 1.945 is the beam waist multiplication factor within which 99.99% of the plane wave spectrum power in is contained for a one-dimensional Gaussian beam. Although the choice of 99.99% is probably a good rule of thumb for the ratio between a filter angular bandwidth and the angular spectrum of an incident Gaussian beam, the exact number required to recover the filter plane wave response is dependent on the response shape. The motivation to find a relationship between the filter and the beam angular spectrum is that the former changes with the angle of incidence. Hence, Eq. (1) provides a way to find the appropriate beam size as a function of incident angle.Fig. 12 Transmission efficiency vs. wavelength for input beams with different fraction of their angular spectrum within the filter angular clear bandwidth.
Although a filter wavelength response can be conserved as the angle of incidence is changed by scaling its normal incidence widths by the cosine of the new angle, the width of its angular response decreases rapidly as the incident angle increases. This is because the optical path difference caused by a given angular difference increases with the angle of incidence. The minimum beam waist required for undistorted propagation as a function of incident angle for the same filter is shown in Fig. 13 . Because the increase in beam width is very rapid, the incident angle of the filter must be small to keep the beam size compatible with integrated optics [29]. Adjusting the filter dimensions maintains the optical path difference between the interfering beams but it does not compensate for the changes in reflectivity at the interfaces, and thus this limits the angular range over which a given design can be used. That is why the curve in Fig. 13 approaches an asymptote for angles larger than 10 degrees. In that range the filter response widens because the filter phase is altered by the variations in reflectivity. Also, the maximum incident angle is limited by total internal reflection between guided and etched regions. Nevertheless, the filter angular bandwidth continues to decrease as the incident angle increases even if it is implemented with a lower index contrast, and thus, the constraint on the incident angle still exists for low index contrast filters.
Fig. 13 Minimum 1/e Gaussian beam width for undistorted propagation through a four cavity filter as a function of incident angle.
In previous studies of Gaussian beam propagation in thin-film filters, four impairments were identified [30–33]: lateral beam shift, waist magnification, angular shift, and focal shift. Lateral beam shift, which is the difference in the beam centroid position between that predicted by simple ray propagation and that created by interference, increases with the incident angle and must be taken into account when positioning the input and output ports of a device including off-axis FP filters. However, the waist magnification, angular shift, and focal shift caused by the FP filters described here are well below one percent of their original values. According to [32], nonspecular phenomena in multilayer dielectric stacks are significant in the regime where their responses vary rapidly. Therefore, the wide and flat passband provided by the coupled cavities attenuates these undesired effects. At oblique incidence, the filter response may vary with polarization. This is the case with the FP filter designed here and it was optimized to work with the planar waveguide TE polarization only. Furthermore, working off-axis can lead to polarization conversion. To evaluate whether this occurred in these integrated filters, their wavelength response was simulated with the TMM formulation developed by Hodgkinson and Wu [34], and it predicted that no polarization conversion would take place.
5. Comparison with experimental results
In order to verify the design and optimization approach developed here, a filter prototype was fabricated. It was implemented in a planar waveguide with a 0.6 μm Al0.06Ga0.94As top cladding, a 2.1 μm GaAs core, and a 5.5 μm Al0.06Ga0.94As bottom cladding. These layers were grown by metal-organic chemical vapor deposition on n + GaAs substrates. Although the expected etch depth for the mirror trenches was 4.0 μm, the above waveguide configuration was chosen instead of the one given in Table 1 because of its greater tolerance to top cladding thickness variations. Only 0.1 μm of additional top cladding is sufficient to make the optimum waveguide multimode whereas the configuration used in the prototype can have a 0.5 μm thicker cladding and still remain single mode. The final mirror etch depth was 4.9 μm. The filter pattern was defined by hard contact photolithography before being transferred in the planar waveguide by reactive ion etching with Cl2 and Ar using only the photoresist as mask. Two dimensional waveguides and parabolic collimating mirrors were also fabricated to input and collect the light from the filter, which had an incident angle of 1.2 degrees. The fabricated device dimensions were measured from scanning electron micrograph, and they were used with the Eigenmode method to simulate the prototype wavelength response (see Fig. 14 ). Over-etching shifted the prototype response to the L-band and reduced cavity coupling. The filter cavities had a thickness of 213 301.33 nm and 213 255.10 nm for the outer and inner cavities, respectively. These dimensions correspond to about the 898 order at 1595.49 nm. Further details on the prototype layout and the experimental set-up can be found in [35]. The shape of the simulated wavelength responses is similar to the experimental one, which indicates that the simulation approach was valid. Unfortunately, misalignments of the input and output waveguides caused significant insertion loss, and thus the radiation losses cannot be compared directly. The results in Fig. 14 were adjusted by subtracting the collimation system loss measured without the presence of a filter. Nevertheless, the average loss difference between the transmitted and reflected signals predicted by the Eigenmode method (0.9 dB) and measured experimentally (1.2 dB) agreed.
Fig. 14 Prototype wavelength response measured experimentally and simulated with the Eigenmode expansion method in (a) transmission and (b) reflection.
6. Conclusion
We have described the design and optimization of off-axis broadband integrated coupled cavity FP filters. We observed that the planar waveguide index contrast and its layer thicknesses must be considered simultaneously to minimize radiation losses. Furthermore, the depth of the etched features is also an important factor in the waveguide optimization. If at the time of epitaxial growth the maximum depth achievable is unknown, it is best to choose a waveguide configuration for a conservative estimation of the etch depth because the gain in efficiency provided by having the optimum waveguide for a deep etch are less than the loss suffered when the required depth is not obtained.
Once the effective refractive index that made up the filter had been determined, an initial design was conceived from previous work on multi-mirror FP interferometers and was later refined with a commercial thin film design software. This led to a device containing four coupled cavity filters with a 200 GHz comb response, a transmission passband larger than 50 GHz, and a theoretical on/off ratio greater than 20 dB over the entire C-band. The clear bandwidth obtained from with multiple cavities is over an order of magnitude larger than for a filter with a single cavity. Having a wide passband allows the filter to handle higher bit rates and makes it more tolerance to wavelength drift in the input signal. Although first order mirrors provided better cavity coupling and a larger bandwidth, larger second order mirrors were used for the prototype because of their compatibility with our fabrication process. However, having larger trenches reduced the bandwidth by 82% and increased the transmission and reflection losses by 0.6 dB and 0.3 dB, respectively, for 5 μm deep trenches.
Working off-axis requires that careful attention be paid to the incident beam size because the angular spectrum of FP filters decreases quickly with increasing incident angle. To avoid distortion, including waist magnification, focal shift, and angular shift, 99.99% of the beam angular spectrum must be included in the filter clear angular passband. However, lateral beam shift is always present and it must be considered in the switch layout.
We demonstrated that using the TMM for filter optimization and combining it with Eigenmode expansion to evaluate radiation losses is a valid approach to design integrated off-axis filters by comparing our simulation results with experimental data. If a small refractive index modulation of 0.053% is applied on the cavities, the FP filters presented here can be used as switching element in an optical space switch. They can be layed out in a crossbar or Benes configurations [29] since no waveguide interconnect is required between filters.
Acknowledgments
The authors thank the staff of McGill Nanotools Microfab for their help and financial support with the prototype fabrication. This work was supported in part by le Fonds québécois de la recherche sur la nature et les technologies and the Agile All-Photonic Networks' strategic research network of the Canadian National Science and Engineering Research Council.
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