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Abstract
Wavelength filters are among the most important building blocks required for integrated optical circuits. However, existing filter building blocks provide only basic functionality with limited options for controlling the filter function unless sophisticated filter design techniques are employed. Conversely, in the microwave regime, elegant and powerful filter synthesis techniques exist which use coupled resonators. As waveguide ring resonators have emerged, researchers in the optical domain have sought to translate these techniques but the multi-wavelength spacing required to couple optical ring resonator structures severely limits the types of filters that can be realized. In this paper we show how recently reported ridge resonance structures can be arranged as coupled resonators with very close spacing and thus can be harnessed to achieved many of the filter functionalities available in the field of microwave engineering. Our filter is comprised of multiple parallel ridges on a common silicon slab, with each resonator exhibiting a resonant frequency and quality factor which can be controlled through engineering the geometry of the ridge. It is thus possible to choose appropriate combinations of ridge geometries to satisfy the conditions required by filter synthesis prototype. We demonstrate through rigorous simulation how our approach can be used to achieve high order optical bandpass filters at 1.55 $\mu$m center wavelength with Butterworth or Chebychev responses and analyse the impact of non-ideal behaviours on filter performance.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Silicon photonics has been commonly accepted as a platform for large scale photonic integrated circuits (PICs). Sophisticated PICs with many components can now be realized using CMOS-compatible fabrication processes thanks to multi-project wafer (MPW) services offered by many CMOS foundries [1]. Optical filters are an important component in the PIC toolkit. Highly selective bandpass/bandstop filters with low insertion loss, low in-band ripple, steep band-edge roll-off and high out-of-band rejection are important for many applications, especially wavelength multiplexed data communications using lasers in uncooled operation [2]. A number of structures have been proposed to realized filters with such requirements.
Low loss, sharp roll-off filters can be realized by cascading multiple resonators such as coupled resonator optical waveguides (CROW’s) to achieve high order filtering functions [3]. By designing the properties of individual resonators and the coupling between resonators or their spacing, different filter characteristics can be realized. A number of approaches have been proposed to design such optical filters [4–8]. However, the procedures to design the properties of individual optical resonators or the coupling coefficients between resonators are often complicated or may require numerical optimization [9]. This is not the case in the field of microwave engineering, where simple but accurate techniques for synthesizing coupled resonator array filters are available. In microwave engineering the properties of individual resonators can be drawn from a filter synthesis table or using simple analytical equations [10,11]. The application of similar techniques to synthesize optical filters using coupled ring resonators has been reported [3,12]; however, the filter performance when applying such design rules has been quite limited [12,13]. These limitations stem from the large spacing between the optical resonators and it has been noted that these limitations could be overcome if the size of each resonator and the spacing between them could be made close to the operating wavelength [13]. Wavelength scale resonators and inter-resonator spacing is typically the case for filters in the microwave domain, but is difficult to achieve for common optical resonators posing a challenge for integrated photonic filter synthesis.
Recently, resonance behaviour based on bound states in the continuum has been proposed [14] and experimentally demonstrated using a silicon on insulator (SOI) ridge waveguide at 1.55 $\mu$m as shown in Figs. 1(a) and 1(b) [15]. This resonance results from the coupling between the continuum of TE slab modes and a discrete TM-like mode in a ridge via so called lateral leakage [16] when the ridge is excited with a obliquely incident broad beam of TE slab mode from the slab region on one side of the ridge. This resonant behaviour has been called ‘ridge resonance’. Close to the resonant wavelength, a ridge resonator exhibits a Lorentzian-like line-shape with the resonant bandwidth and resonant wavelength determined by the leakage loss and effective index of the TM-like mode in the ridge which can be controlled by the ridge geometries [17,18]. Resonators with an arbitrary bandwidth (or quality (Q) factor) can be realized by designing the leakage loss of the quasi-TM mode supported by the ridge. The width of the ridge resonator can be smaller than an optical wavelength in the lateral dimension. For example, in [15] at the free-space operating wavelength of 1.55 $\mu$m, ridge width is 650 nm, while an optical wavelength in the silicon slab is 775 nm. In addition, it has been shown that the continuum of TE slab modes can act as an optical bus allowing coupling between adjacent ridges [19,20]. Therefore, it should be possible to arrange multiple ridge resonators with a small spacing next to each other to realize high order compact integrated photonic filters [16] as illustrated in Fig. 1(c). It has been shown that a flat-top filter can be realized by using a coupled ridge resonator structure with multiple identical ridges [21]. However, coupled resonators with identical ridges and equal spacing can have very strong side-lobes as shown in Fig. 1(d). In [21], numerical optimization was used to optimize the spacing between resonators in order to minimize the side-lobes. Instead of numerical optimization, considering the small spacing between these resonators, it should possible to use filter synthesis techniques from microwave engineering to design these coupled resonator structures.
Fig. 1. (a) Optical microscope image of an SOI ridge resonator formed by a single ridge on a slab [15]. Parabolic mirrors were used to generate and collect broad TE beams of slab mode in the SOI slab. The arrows illustrate the beam propagation directions; (b) Measured transmission as a function of wavelength showing the resonance; (c) High-order coupled resonators formed by multiple parallel ridges; (d) Simulated reflection and transmission of a coupled resonator structure with 5 identical ridges and spacing equal to an odd multiple of a quarter of wavelength, comparing to the responses of a single ridge resonator.
In this paper, we propose an approach based on microwave filter synthesis to design high-order engineered photonic filters operating at 1.55 $\mu$m wavelength using ridge resonators as the fundamental elements. We show that common microwave filter types such as a Butterworth or Chebyshev filter can be synthesized with the desired response, using a simple filter synthesis technique. Factors that can affect the performance of the synthesized filters are discussed. The ability to synthesize filters using ridge resonators and a simple filter synthesis technique opens up a new approach to realize flat-top, sharp roll-off and high extinction ratio filters for on-chip wavelength filtering.
2. Filter synthesis technique
In electronics or microwave engineering, the filter synthesis process often starts with a normalized low-pass filter prototype such as an LC ladder network as illustrated in Fig. 2(a) [10]. Based on the specifications, such as bandwidth, pass-band ripple and stop-band attenuation, the filter type (e.g Butterworth, Chebyshev) and the filter order (i.e. the number of L and C elements in the low-pass filter prototype) are determined [22]. The values of all elements (g-values) can be obtained from filter design tables or calculated by simple formulae available in most microwave filter synthesis textbooks [10,11]. Afterwards, frequency scaling and element transformations are carried out to convert the normalized low-pass prototype filter to the required frequency and filter characteristic (low-pass, high-pass, bandpass or bandstop).
Fig. 2. (a) Normalized LC ladder low-pass filter prototype; (b) Filter implementation using resonators of different Q-factors connecting by quarter-wave transformers.
At microwave or optical frequencies, high-order bandpass or bandstop filters can be implemented using resonators with different Q-factors, which are connected by quarter-wave transformers as illustrated in Fig. 2(b). The Q-factor of each resonator is directly related to the g-value of the corresponding element of the low-pass filter prototype by:
where $\lambda _0$ is the resonant wavelength which is the same as the filter center wavelength, $B_k$ is the resonator bandwidth, $B$ is the desired filter bandwidth and $g_k$ is the g-value of the corresponding element in the low-pass filter prototype. The quarter-wave transformers act like impedance inverters to convert shunt elements in the low-pass filter prototype into series elements.The filter synthesis described above is generic for both microwave and photonic regimes. In photonics, there are different approaches to implement resonators such as ring resonators or Bragg gratings [12,13]. The impedance inverters are often implemented using short sections of transmission line. An ideal impedance inverter has a wavelength independent phase-shift of $(2m + 1)\pi /2$. However, impedance inverters based on transmission lines have the correct phase response only at a single wavelength. In order to maintain the correct phase shift over a wide wavelength range, the length of the transmission line must be kept to a minimum. In other words, the spacing between resonators should be very small, ideally equal to a quarter of wavelength at the operating frequency. This requirement can easily be achieved at microwave frequencies. However, optical resonators are often much larger than the operating wavelength, therefore, it is difficult to maintain the correct phase shift between resonators over a wide bandwidth, resulting in non-ideal filter response, especially for out-of-band frequencies [12].
The width of the recently reported ridge resonators [14,15] can be smaller than an optical wavelength. This opens the possibility of placing such ridge resonators very close to each other, connected with very short transmission lines. This compactness can reduce the phase errors of the impedance inverters for out-of-band frequencies, making such a filter configuration very attractive for the realization of higher order engineered optical filter as illustrated in Fig. 2(b).
3. Filter element synthesis - ridge resonator
As discussed in Section 2, resonators used for high-order filters can have different bandwidths or Q-factors, but must have the same resonant wavelength [see also Eq. (1)]. The Q-factor of a ridge resonator is mainly determined by the ridge width [15]. However, changing the ridge width also changes the effective index of the guided TM mode in the ridge, which as a consequence results in unwanted shift of the resonant wavelength [15]. In order to achieve ridge resonators with different Q factors but having the same resonant wavelength, another degree of freedom in the ridge resonator design is required. One possible approach to realize this new degree of freedom is to introduce some additional dielectric loading, such as poly-silicon stripes on either side of the ridge [23,24], as illustrated in Fig. 3(a). With such configuration, one might expect that by changing the ridge width as well as the loading width, the leakage loss of the TM guided modes can be controlled while keeping the mode effective index constant. If this can be achieved, then ridge resonators with different Q-factors but the same resonant wavelength should be feasible.
Fig. 3. (a) Cross-section of a ridge resonator with poly silicon loading on either side of the ridge. (b) illustration of the ridge resonator operation: blue arrows are obliquely incident, reflected and transmitted broad beams of vertically guided TE slab mode; the red arrow illustrates the generated guided TM-like mode in the ridge due to the coupling between the TE slab mode and guided TM-like mode in the ridge. (c) The simulated ridge resonance Q-factor (color map) and effective index of the guided TM mode in the ridge (contour lines) as functions of both the silicon core ridge width and the poly silicon loading width.
To investigate if the Q-factor and the resonant wavelength of a ridge resonator can be tuned independently of each other by introducing extra dielectric loading, we calculated the effective index and leakage loss of the TM-like mode for different ridge widths and loading widths using a full-vector eigenmode solver based on mode-matching method [25]. The Q-factor of the ridge resonance was calculated from the leakage loss [15]. The offset between the ridge and the loading was kept constant at 100 nm. All other dimensions are given in Fig. 3(a). Figure 3(c) shows the resonator Q-factor and the effective index of the TM-like mode at 1.55 $\mu$m wavelength in the ridge as functions of both the ridge width and the loading width. The solid lines in Fig. 3(c) are the lines of constant effective index, while the color contour-map indicates the Q-factor. One can see from Fig. 3(c) that the contour lines of constant TM-like effective index traverse regions of varying Q-factor. This shows that it is possible to pick ridge width and loading width combinations to realize ridge resonators with different Q-factors, while maintaining the same resonant wavelength.
4. High-order filter synthesis examples
In Section 2, we described a generic filter synthesis technique and discussed how implementation would require resonators with different Q-factors but the same resonant wavelength. In Section 3 we showed that these requirements can be realized using ridge resonators with a dielectric loading. Therefore, it should be possible to synthesize complex high-order filters using ridge resonators. Figure 4 illustrates the configuration of the proposed high-order filters. A filter is formed by multiple parallel ridge resonators, which are coupled by the common slab. The cross-section of individual ridge resonator is illustrated in Fig. 3(a). Similar to the operation of a ridge resonator [14,15], the filter is excited with a broad beam of vertically guided TE slab mode, obliquely incident on the ridge array. The incident beam and reflected beams can be generated and collected by, for example, on-chip lenses or parabolic mirror systems interfacing to single mode waveguides [15,26]. The ridge width and loading width of each resonator as well the spacing between adjacent resonators are found using the filter synthesis technique. In the following, examples of synthesizing high-order Butterworth and Chebyshev bandpass filters operating in reflection with such filter structures are investigated.
Fig. 4. A high-order filter composing of multiple parallel poly-silicon loading ridge resonators. The filter is excited by a laterally broad but vertically guided TE beam obliquely incident on the ridge array.
4.1 Butterworth bandpass filters
A Butterworth bandpass filter exhibits a maxima flat response in the passband and no ripple in the out-of-band response. The g-values of the normalized low-pass prototype Butterworth filter is given by [11]:
where $N$ is the filter order. For an odd-order filter, the g-values are symmetric, i.e. $g_k = g_{N-k+1}$. Therefore, the number of resonators with different Q-factors for an odd-order Butterworth filter is only $(N + 1)/2$.For a given filter bandwidth and filter order, the required Q-factors of individual resonators that make up a Butterwordth filter can be found using Eqs. (2) and (1). For example, the required Q-factors for a third-order and a fifth-order Butterworth filter with a 3-dB bandwidth of 5 nm centering at 1.55 $\mu$m wavelength are (620, 310, 620) and (1003, 383, 310, 383, 1003), respectively. After knowing the required Q-factors, the ridge width and loading width of each resonator were selected using the results presented in Fig. 3(c). This was done by following a contour line of constant effective index ($n_{eff} = 1.94$) of the guided TM-like mode and choosing different combinations of ridge width and loading width to achieve the required Q-factors. The resulting ridge configurations are shown as red crosses (for a third-order filter) and blue squares (for a fifth-order filter) in Fig. 3(c).
Using a full vector simulation model based on mode-matching method [15], the wavelength responses of all designed ridge resonators were rigorously simulated. All ridge resonators were excited using a broad TE beam from the cladding region with the same excitation angle, which is the phase-matching angle between the TE beam and guided TM-like mode in the ridge. Figure 5(a) shows the simulated reflection as a function of wavelength for three different resonators required for the fifth order filter when they are isolated. The wavelength responses of ideal resonators with the same Q-factors are also shown for comparison. It can be seen that all synthesized ridge resonators have the same resonant wavelength but different Q-factor. Close to the resonant wavelength, the response of each resonator closely reassembles that of the corresponding ideal resonator. The resonator reflection deviates from the ideal response for wavelengths further away from the resonant wavelength. This is due to the fact that apart from the reflection due to resonantly coupling between TE slab mode in the slab and TM-like mode in the ridge, there is also a background Fresnel reflection present due to the index difference at boundaries of the loading and ridge regions. The rigorously simulated reflection includes both the resonant reflection and non-resonant background reflection [15]. Away from the resonant wavelength, the resonant reflection and the background non-resonant reflection interfere constructively on one side of the resonant wavelength, while interfere destructively on the other side, resulting in a Fano-type resonance with asymmetric wavelength response.
Fig. 5. (a) Wavelength responses of the three synthesized ridge resonators for the fifth-order Butterworth filter, comparing with the ideal resonator responses; (b) Wavelength responses of the synthesized and ideal third-order and fifth-order Butterworth filters.
After designing the required resonators, the final step of the filter synthesis process is to find the separation between resonators so that the phase shift between adjacent resonators at the center wavelength is an odd multiple of $\pi /2$, taking into account the phase responses of the resonators. While the separation should be as small as possible to minimize the phase error over a wide bandwidth as discussion in Section 2, it also need to be sufficiently large to ensure that the evanescent coupling between guided TM-like modes in adjacent ridges is negligible. This requires that the spacing between adjacent ridges need to be larger than 1 $\mu$m.
Having all the required parameters, the performance of the synthesized filters can be evaluated. Using the mode-matching method, the reflection of the filters was simulated when they are excited with a broad TE beam. Figure 5(b) shows the rigorously simulated reflection and the ideal filter response as a function of wavelength for the third order and fifth order Butterworth filters. The in-band response and roll-off of the synthesized filters using ridge resonators match very well to the ideal filter responses. As expected for a Butterworth filter, there is no ripple in the in-band reflection. In addition, the fifth-order filter has steeper roll-off than the third-order filter. While an ideal Butterworth filter has a monotonically decrease in reflection in the out-of-band, the synthesized filters have out-of-band ripples. The level of maximum out-of-band reflection increases slightly for wavelengths far away from the center wavelength. The reasons for these out-of-band ripples will be discussed in more detail in Section 5.
Figure 6 shows the electric field distribution of the fifth-order filter when the filter is excited with TE slab mode at different wavelengths. Strong reflection and low transmission of the TE field component is evident when the filter is excited with a wavelength within the filter bandwidth. The guided TM modes in the ridges are also excited for these wavelengths. Right at the center wavelength, only the guided TM mode of the first ridge is excited. The field penetrates deeper into the resonator array when the wavelength moves closer to the band edge. Because adjacent ridges are kept well separated, there is no direct evanescent coupling between TM mode fields. Outside the pass-band, there is no reflection in the TE field and no TM mode field in any of the ridge resonators.
Fig. 6. Cross-section field distributions of the horizontal ($|E_x|$, TE) and vertical ($|E_y|$, TM) components of the electric field vector when the fifth-order Butterworth filter is excited with a broad TE beam at different wavelengths .
4.2 Chebyshev bandpass filters
In the second example, we use the same filter synthesis procedure to synthesize Chebyshev filters with different orders and out-of-band attenuation. By allowing some ripple in the out-of-band attenuation, a Chebyshev filter can have sharper roll-off than a Butterworth filter with the same order. For an odd-order Chebyshev filter, the g-values of the normalized low-pass filter prototype are given as [11]:
Similar to the Butterworth example, here we consider Chebyshev filters with the same center wavelength and 3 dB bandwidth specifications. Two cases of the out-of-band attenuation specification of 20 dB and 40 dB were investigated. Using these filter specifications, the g-values of the low-pass filter prototype were calculated using Eq. (3), then the Q-factors of the corresponding resonators of the bandpass filter were calculated using Eq. (1). The geometry dimensions of the ridges were found from Fig. 3(c), similar to before.
Figures 7(a) and 7(b) show the rigorously simulated responses and the ideal responses of the synthesized third order and fifth order Chebyshev filters, with an out-of-band attenuation of 20 dB and 40 dB, respectively. One can see that a fifth-order filter has sharper roll-off than a third-order filter as expected. In addition, a filter with lower out-of-band attenuation has a sharper roll-off. Similar to the case of Butterworth filters, the in-band responses of the synthesized filters match very well with the ideal filter responses. However, the synthesized filters have lower out-of-band suppression than the ideal filters.
Fig. 7. Wavelength responses of synthesized and ideal third-order and fifth-order Chebyshev filters with 20 dB (a) and 40 dB (b) out-of-band attenuation specifications.
5. Effects of non-ideal conditions
It has been shown in Sections 4.1 and 4.2 that although the in-band responses of the synthesized Butterworth and Chebyshev filters match very well to the ideal filter responses, they do not match as well for the out-of-band responses. In this section, the causes of this non-ideal out-of-band response are analyzed in more detail.
In an ideal high-order filter, the wavelength response of each resonator has a perfect Lorentzian lineshape. In addition, the phase shift of the quarter-wave transformers connecting adjacent resonators is always an odd multiple of 90 degrees regardless of the operating wavelength. These ideal conditions are not always fulfilled in filters using ridge resonators. In the following, we will investigate how these non-ideal conditions affect the filter performance.
In the synthesized filers using ridge resonators, quarter-wave transformers were realized using a short length of slab waveguide. The length of the waveguide was calculated to give an odd multiple of 90 degree phase-shift at the center wavelength of 1.55 $\mu$m. However, due to the finite length and dispersion of the slab waveguide mode, the phase-shift is not constant but varies as a function of the operating wavelength. The phase-shift of the quarter-wave transformer connecting two adjacent ridge resonators in the synthesized filters of Section 4 was simulated. The dispersion of the TE slab mode in the slab between ridge resonators were taken into account. Figure 8 shows the phase-shift between two adjacent resonators as a function of wavelength. Correct 90 degree phase-shift can only be obtained at the center wavelength. Therefore, the slab waveguide between two adjacent ridge resonators behaves like an ideal quarter wave transformer only at the center wavelength.
Fig. 8. Phase response of a quarter-wave transformer connecting two adjacent ridge resonators in the synthesized filters.
To investigate how the non-ideal phase shift due to finite spacing between resonators influences the filter response, the filter responses were simulated with all resonators having ideal Lorentzian responses, but they are connected with quarter-wave transformers with a phase response given by Fig. 8 to eliminate the effect of non-ideal resonator response. Figures 9(a) and 9(b) show the responses of the fifth-order Butterworth and Chebyshev filters, respectively. It can be observed that the out-of-band response is strongly influenced by the phase error of the quarter-wave transformers.
Fig. 9. Wavelength responses of a fifth-order Butterworth (a) and Chebyshev (b) filter when the ridge resonators are replaced by ideal resonators separated by slab waveguides as the synthesized filters.
The response of a ridge resonator is determined by the interaction between the resonant-assisted coupling process between TE slab mode and guided TM mode and the non-resonant background Fresnel reflection from the difference in refractive index at different waveguide regions [15]. This interaction can create a Fano-like resonance with an asymmetric line shape as shown in Fig. 5(a). Far away from the resonant wavelength, the responses of the resonators are strongly determined by the non-resonant background responses. To evaluate the impact of this non-ideal resonator response, the response of the synthesized filters were simulated but with the slab waveguides connecting adjacent resonators replaced by ideal quarter-wave transformers with constant 90 degree phase-shift. Figures 10(a) and 10(b) show the simulated response for a fifth-order Butterworth and Chebyshev filters, respectively. One can see that within the pass-band, the resonator response due to TE/TM resonant coupling is much stronger than the non-resonant background response. Hence, the impact of the non-ideal background resonator response on the filter response is minimal. One can also observe that the level of out-of-band attenuation is strongly influenced by the non-resonant background reflection. The level of out-of-band attenuation is approximately equal to the reflection due to non-resonant background reflection.
Fig. 10. Wavelength responses of a fifth-order Butterworth (a) and Chebyshev (b) filter when the the slabs connecting resonators are replaced by ideal quarter-wave transformers.
Comparing the analysis in Fig. 9 and Fig. 10, we can see that for both synthesized Butterworth and Chebyshev filters, the effect of non-ideal resonator response due to the non-resonant background reflection is much stronger than the effect of non-ideal quarter-wave transformer resulting from the finite slab length and TE slab mode dispersion. For the given poly silicon loaded SOI ridge structure investigated in this work, the background non-resonant Fresnel reflection level is approximately -30 dB. This determines the maximum out-of-band attenuation that can be achieved for this particular configuration. In order to achieve higher out-of-band suppression, the background reflection needs to be further reduced, for example by reducing the ridge etching depth or loading thickness or using another loading material with lower refractive index such as silicon nitrite to reduce the difference in refractive index between different waveguide regions. When the background reflection can be made negligible then the out-of-band attenuation will be dominated by the phase error in the quarter-wave transformers. This error can be reduced by minimising the spacing between adjacent ridge resonators. However, the ridge resonators cannot be too close. Otherwise, guided TM modes in the ridge start coupling through the evanescent field resulting in distorted in-band response. It is also possible to minimize the phase error by engineering the dispersion of TE slab mode.
6. Conclusions
In this paper, we have proposed an approach for synthesising high order engineered optical filters based on synthesis technique that is commonly used in the field of microwave engineering. The fundamental building block of the synthesized filters are the recently reported SOI ridge resonators. A dielectric loading of poly silicon was used in order to independently control the Q-factor and resonant wavelength of each ridge resonator. Using simple filter design formulae or tables that are readily available in many microwave filter design textbooks, the Q-factor of individual resonators and spacing between adjacent resonators can be easily found. Based on this simple filter synthesis technique, we have shown that different filter functions such as bandpass filters with Butterworth or Chebyshev responses operating at 1.55 $\mu$m wavelength can be synthesized, showing excellent agreement with the ideal filter passband responses. This is possible due to the small width of the ridge resonator, allowing small separations between adjacent resonators and therefore small phase errors. Rigorous simulations indicate that the in-band response should be close to ideal, but out of band suppression may be compromised. We believe that the primary limitation is due to Fano resonance interference between the desirable Lorenzian ridge resonance and the background Fresnel reflection due to the refractive index difference between the slab and the loading/ridge core regions of a ridge resonator. We believe that this background reflection can be improved through further optimisation of the ridge and loading geometries. It should be noted that the width of each individual ridge resonator as well as the spacing between resonators are small. However, the ridges must be excited with a broad TE beam, hence can be relatively long. Our previous work has experimentally demonstrated that it is possible to realise suitable beams in foundry compatible silicon photonics using on-chip waveguides and lenses. However, we are currently exploring techniques to reduce the real estate required.
Funding
Australian Research Council (DP150101336).
Disclosures
The authors declare no conflicts of interest.
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