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A major artifact of realistic photonic filters is the waveguide power loss. Its detrimental effect on the allpass structure is particularly alarming because the phase response is highly sensitive to perturbations. While the loss can be simply captured into a variation on the unit delay in signal processing analysis, its non-linearity makes it mathematically difficult to address. We present an allpass filter design algorithm that is able to provide filter coefficients that compensate for the waveguide power loss. By absorbing the loss parameter into the design cost function, the optimization problem becomes non-convex and NP hard. Our approach solves this problem by utilizing an iterative algorithm in conjunction with the branch and bound global optimization technique. The proposed algorithm is expected to improve the performance and increase the utilization of allpass filters for optical signal phase based applications such as distortion compensation and group delay equalization.
© 2013 Optical Society of America
1. Introduction
Allpass filters have long been used in digital communications and signal processing. These structures are excellent for applications that involve manipulation of the digital signal’s phase, such as phase equalization, Hilbert transform, fractional delay, and magnitude filtering using allpass substructures [1, 2]. It has been pointed out that allpass filters are also crucial in photonic systems. With a wide range of applications towards phase equalization in wavelength division multiplexing (WDM) [3, 4] and general optical filtering [5, 6], the digital allpass filter has demonstrated its solid connection to optical signal processing. A variety of fundamental photonic components naturally behave as allpass elements, such as ring resonators [7], microdisk resonators [8], and Bragg reflector pairs [9]. The fact that a complex allpass filters can be realized in lattice or cascade assemblies of first-ordered sections make these structures even more readily applicable for photonic operations.
In low loss systems such as fiber optics (loss coefficient ∼ 2 × 10−6 dB/cm [10]) the association of optical components with allpass elements is well justified. However, in the context of integrated optical systems such as dielectric waveguides (loss coefficient ∼ 5 dB/cm [11, 12]) and plasmonic waveguides (loss coefficient ∼ 2 × 103 dB/cm [13]) the association becomes difficult to justify, particularly when effects such as bending loss are considered [14]. The implications for optical filter design are significant. In particular, as the devices are integrated and miniaturized it will become necessary to confront the issue of optical loss. The allpass assumption is also violated in situations that require long phase delay lengths and therefore long sections of waveguide, narrowband filters being an example [6]. These realistic considerations limit the utilization and performance of allpass filters in optical systems, especially in its direct phase related applications such as dispersion compensation in WDM systems [15, 16].
In this paper, we consider the design procedure of a photonic allpass filter that is capable of compensating for the deviation from ideal behavior of these optical elements in the phase. From a signal processing perspective, the waveguide power loss can be readily modeled as a variation on the delay element z−1. It has been shown that the loss affects a transfer function by modifying the original response H(z) into H(γ−1z), where γ = e−αL/20 is the fraction of power retained by the filter sub-elements after losses are accounted for [3,5]. The scattering and absorption effects are captured by the parameter α, and L is the length of the device. Currently, there is no allpass design algorithm that considers the compensation for a variation on the delay element z−1. Although significant efforts have been put into allpass analysis to demonstrate robustness to coefficient perturbations [1, 17] in DSP filter design, its error resilience to this unique photonic filtering error has never been addressed.
Given these considerations, we present a minimax allpass filter design algorithm that is able to compensate for the delay element variation in terms of phase performance. Since the waveguide power loss can be considered as a predetermined effect [5], we capture the γ parameter into the filter design cost function. Without any manipulation, the design problem is mathematically challenging because it involves an optimization problem that is non-convex and NP hard. Instead, we solve a relaxed problem by employing an iterative approach in conjunction with a Branch and Bound global optimization technique. From the proposed technique, we are able to derive photonic parameters that allow the realization of allpass filters that can mitigate the performance degradation caused by the waveguide power loss.
The rest of the paper is outlined as follows: Section 2 provides the signal processing background on an ideal allpass filter behavior and operation, as well as the connection to photonic realizations. Section 3 discusses the waveguide loss issue from both photonic and signal processing perspectives. Section 4 sets up the mathematical problem and presents our algorithm on the waveguide loss compensating allpass design. Section 5 shows results and comparisons of our design, and section 6 concludes the paper.
2. Allpass filters
An ideal allpass filter describes a system that exhibits no attenuation in the magnitude for all frequencies, but alters the phase of the input signal according to a prescribed profile. The transfer function for an Nth-order allpass filter can be written as [2]
where di’s are the filter coefficients. The allpass filter exhibits a unique poles and zeros pattern in that given a zero at zi, A(z) must also contain a pole at . The precise relation between a pole and zero pair of the allpass filter allows the system A(z) to receive equal magnitude contributions from D(z) and , and thus results in a constant response.Since the allpass filter is expected to have a unity response in the magnitude for all frequencies, the design freedom lies only in the phase. The challenge in allpass design is therefore to find the optimal di’s so that the filter phase matches a given prescribed phase response Θpre(ω). It is then immediately evident that the allpass filter is excellent for photonic applications such as dispersion compensation and group delay equalization, where the operations required are only in the phase.
The allpass filter is highly suitable for physical realization using optical components because it can be easily modularized. A first-order allpass filter can then be readily realized using a variety of nanoscale photonic elements. To demonstrate the operation of a photonic allpass filter, we will consider the Bragg reflectors setup as shown in Fig. 1. The traversal time of the waveguide loop serves as the discrete delay element z−1, and is directly related to the optical path length L. The reflection coefficient ρ of the mirror and the phase delay s in the structure directly relate to the first-order allpass filter coefficient d1. Under ideal fabrication conditions, the transfer function of a first-order photonic allpass filter is
in which case d1 = −ρejs. The setup can be easily extended to higher order photonic allpass filter, which is simply a cascade or lattice of first order sections.
Fig. 1 First order allpass realization using a Bragg mirror topology. The optical path length is represented by L, s is the phase delay of the unit, and ρ is the reflection coefficient.
Note that in filter design, we are interested in finding the coefficient vector d = [1 d1 ··· dN]T of the Nth-order structure, yet the parameters of individual sections are needed for realization. The single stage parameters can be readily obtained post design through factorization in the cascade realization, and recursion in the lattice. Furthermore, in this paper we focus on the design of real coefficient allpass filters. In other words, the phase delay s is assumed to be an integer multiple of π. This requirement can be easily absorbed into the dimensions of the waveguides. Note that realisic tooling inaccuracies hinder such setup and result in a random phase error in the individual sections. While the effect of phase deviations in undeniably present [18], the average performance of the overall system often converges to the ideal design [6]. Its impact on the allpass performance is therefore far less drastic in comparison to waveguide loss. We will therefore assume ideal fabrication conditions and neglect the effect of random phase error in this paper.
3. Waveguide power loss
There are two mechanisms responsible for loss within an optical waveguide: scattering loss and material absorption. Scattering loss is a consequence of artifacts like roughness in the waveguide sidewalls [19] or excessive bending of the waveguide [20]. It is therefore amenable to reduction through improved fabrication precision. In contrast, absorption is an intrinsic material property and is inescapable without abandoning the material system. In the context of digital signal processing, all loss has the same effect regardless of provenance. As such, it is most pertinent to consider the aggregate absorption coefficient of the waveguide. This is also convenient in the sense that loss characterization methods generally determine the aggregate absorption coefficient [21].
A generalized allpass design algorithm that maximizes the allowable waveguide loss while retaining comparable performance is extremely valuable. It will allow relatively lossy waveguides that are inexpensive and simple to fabricate to replace low loss waveguides that are costly and complicated to fabricate [22, 23]. The eased fabrication constraints will also increase the compatibility of photonics with other platforms, such as CMOS. Furthermore, the method will mitigate the increased loss effect from miniaturizing photonic devices. Finally, it will eliminate the performance limitations of photonic allpass filters used in WDM systems for dispersion compensation [15, 16].
Since the power loss is the result of an optical signal traversing through a non-ideal waveguide, it can be considered to be coupled with the unit delay element z−1 [3]. The effect on an Nth-order allpass transfer function with real coefficients from γ is
where 0 ≤ γ ≤ 1 captures the remaining signal power after experiencing waveguide power loss. Notice that by changing the argument of A(z) from z to γ−1z, we are effectively scaling the poles and zeroes radii by γ. Such effect is particularly alarming in an allpass filter, because a precise relation between each pole and zero pair is needed to main a constant magnitude response. Figure 2 shows the effect of γ = 0.9 on the magnitude and phase for a 6th order allpass. As shown in the plots, because the relationship of having zero pole pairs at zi and is violated, the magnitude response can no longer be maintained at unity for all frequencies. The root scaling results in a notch effect at every pole and zero pair location, causing attenuations in the magnitude response. Also notice that the loss causes significant phase distortion near the transition band edges.Fig. 2 Effect of γ = 0.9 on the magnitude and phase responses of a 6th order allpass. Note that minor changes in the phase response can result in significant distortion. The solid lines are the ideal allpass magnitude and phase responses.
Note that the distortion in the magnitude response can be corrected by a follow-up finite impulse response (FIR) linear phase filter. Such structure can be easily fabricated using a tap delay line architecture with negligible loss [3]. Furthermore FIR linear phase filters presents no additional processing on the phase, thus preserving the output of the allpass filter. We will therefore focus our design efforts on providing a desirable phase behavior from the waveguide loss corrupted allpass filter, and assume that any magnitude distortion can be compensated by additional post filtering operations.
4. Loss compensating design
To create an allpass filter design algorithm with waveguide loss compensation, we first examine the transfer function incorporating the effect of γ. Since the effect can be captured as Aerr(z) = A(γ−1z), we can write the transfer function as
The goal of our algorithm is therefore to minimize the difference between the filter’s phase response Θ(ω) and a prescribed phase requirement Θpre(ω). The error function can be expressed as where and β (ω) ≜ Θpre(ω) + Nω. The details to the formulation of the objective function can be found in Appendix A. To derive filter coefficients that can result in optimal phase performance under waveguide loss, we seek to minimize ΔΘ(ω) in either the least squares sense or minimax with respect to d. In filter design, the minimax approach is generally more desirable because it ensures optimal performance at all frequencies [1]. We will therefore consider the minimization of the ℒ∞-norm cost function Notice that this optimization is challenging due to both the fraction form and non-semi definite matrices A−1P(ω)A and A−1Q(ω)A. The problem cannot be reduced to any standard convex formulation, and is highly non-linear.To solve the optimization problem, we propose a two part algorithm that consists of an initial guess step followed by refinement. We first consider a relaxed problem that can be efficiently solved using the global optimization technique Branch and Bound [24]. Note that the setup we consider for an initial solution is still non-linear, because the original optimization setup cannot be reduced to a convex form while still retaining its proper characterization of the loss effect. The resulting dopt is further refined using an iterative algorithm that brings the initial guess closer to a solution to the actual optimization problem in (7). The details of the derivation are provided in Appendix B. The overall algorithm is outlined in Algorithm 1.
| Require: filter order N, prescribed response to match Θpre(ω), waveguide power loss γ, number of samples L. |
| compute d0 using Branch and Bound on (24). |
| formulate A = diag {1 γ ··· γN}. |
| calculate P(ω), Q(ω) according to (19). |
| k=1; |
| repeat |
| formulate weight function |
| solve using Branch and Bound |
| k++; |
| until d converges |
Note that the refinement step can also be applied independently to existing allpass designs. We present a design algorithm that not only can be used to achieve loss compensating allpass filters, but also applied to improve existing implementations.
5. Example results
To demonstrate the performance of our algorithm, let us first consider the equalization of the phase response shown in Fig. 3. This phase profile exemplifies the response of a bandpass filter used in WDM that introduces nonlinear chromatic dispersion near the band edges. With the current phase response, only a narrow bandwidth at the center of the filter is usable because other frequency components within the passband will experience non-uniform group delay (negative derivative of the phase). The purpose of a followup allpass filter is then to equalize the filter’s phase to be linear, which subsequently increase the usable region of the passband [16]. We will therefore consider the design of a 4th-order allpass filter that is able to linearize the given phase response in the passband region of the bandpass filter from 0.15π to 0.6π. In this example we will examine the resulting allpass responses for a congregated loss effect of γ = 0.95. For a 1μm by 0.5μm SiNx strip waveguide cladded in SiO2 with α = 25dB/cm [14, 25, 26], this would correspond to a bending radius of ∼ 11μm, or equivalently ∼1% loss per 90 degree turn. Reducing the allowable bend radius is important for filter miniaturization, since the footprint occupied by a ring resonator unit cell is proportional to the square of the bend radius.
Fig. 3 Actual and desired phase responses for a bandpass structure with nonlinear chromatic dispersion near the pass band edges. The solid portions represent the frequency range of interest. The profile of the magnitude response is also shown for reference.
Figure 4 shows the desired response and those of the traditional and proposed methods. The branch and bound method in the proposed algorithm is implemented in MATLAB using the YALMIP [27] toolbox. To derive the response from a traditional approach, an ideal allpass filter is first designed using the minimax criterion and then subsequently corrupted according to (3). From the graph, it is immediately evident that since traditional method does not consider the effect of γ in the optimization, the phase profile under waveguide loss significantly digresses from the desired profile. Since we directly consider the effect in the optimization setup, the phase response from the proposed method is able to closely track the desired profile even under waveguide loss.
Fig. 4 Phase responses of 4th-order allpass filters that are designed using the traditional minimax approach and the proposed algorithm for γ = 0.95. Note that the region of interest are confined to within the passband of the example bandpass filter (0.15π to 0.6π).
Let us consider a second example where the prescribed phase response is of the following form
Here N is the filter order, ωp is a normalized frequency value that governs the passband, and ωs is the stopband. In comparison to the previous example, this prescribed response is more tolerating because it only contains a discontinuity. However, the effect of waveguide loss is still prominent, and can be easily identified when we directly examine the phase error. Figure 5 shows the phase error comparison results for N = 7, γ = 0.9, ωp = 0.55π, ωs = 0.6π. Notice that using the traditional design, the corrupted allpass response shows large errors near the band edges. The proposed design shows significant improvement in terms of maximum ΔΘ(ω).Fig. 5 Phase errors for the traditional minimax allpass and that of the proposed design. The ideal response of the minimax is also plotted for reference.
Because the loss effect is directly captured in the optimization setup, the result phase error from the proposed method is evenly distributed throughout the frequencies. The designed allpass will therefore not only have an minimal maximum error, but also reduced variations throughout. To demonstrate the overall effectiveness and robustness of our method, Table 1 summarizes the comparison results for various design parameters of the form shown in (10). Notice that the proposed method does not display an improvement when the effect of γ is negligible under highly tolerant requirements and low loss. In all other conditions, the proposed design is able to outperform because traditional approaches do not consider the effect of waveguide power loss. In general, the proposed algorithm’s superiority is most obvious for 0.8 < γ < 0.99. The variance measurements demonstrate that our method is not only able to reduce the maximum phase error, but also improve the phase matching across all frequencies. As shown by the tables, even minor changes in the waveguide power with γ = 0.93 causes significant phase differences. The proposed design and the refinement technique are both aimed to minimize this effect.
Table 1. Normalized maximum phase differences for N = 7 allpass filters under various waveguide loss. Note that γ = 0.8 would correspond to a photonic allpass filter whose individual sections are composed of Bragg reflectors and 1μm × 0.5μm SiNx strip waveguides cladded in SiO2 that are 485 μm long.
Table 2. Variance measurements across the frequencies for N = 7 allpass filters under various waveguide loss. Units are in 1 × 10−4
The fact that the proposed method is able to reduce the phase error variation after waveguide loss is also ideal for operations outside of the typical phase based applications. It has been demonstrated that allpass filters are excellent for photonic system realization of arbitrary magnitude filter [5]. Under ideal fabrication, traditional frequency selective filter (Butterworth, Chebyshev, Elliptic) can be decomposed into allpass substructures without losing optimality [28]. Specifically, a general bandpass filter can be obtained by placing two allpass filters in a parallel setup. We can further require one of the allpass filters to be a simple delay for reduced design and fabrication complexity
where N is the order of the allpass filter Aerr(z). Ideally, the system operates by requiring the phase response of Aerr(z) to be −(N − 1)ω in the passband regions to creating complex addition, and −(N − 1)ω − π in the stopband regions for subtraction. Under waveguide power loss, the phase response of Aerr(z) is distorted, and traditional designs can no longer provide optimal performance. Figure 6 shows the comparison result of lowpass filters created using the traditional minimax approach and the proposed method for γ = 0.9. Note that the phase requirement for the allpass filter takes the form in (10). Without accounting for the loss effect, the magnitude response displays large ripples due to the increased peak phase error and large variations. By reducing the phase error of the loss corrupted allpass, the proposed design is able to provide smoother magnitude responses in the passband and stopband. While the attenuations in the passband and stopband have been affected by the loss, the effect can be minimized by post processing gain adjustments.Fig. 6 Linear magnitude responses of lowpass filters generated using waveguide loss corrupted minimax and the proposed compensating design. The ideal magnitude response from a minimax design is also shown for reference.
6. Conclusion
This paper presents an allpass design method that can mitigate the effect of waveguide power loss. We propose a two part optimization algorithm that can be used individually, or applied to an existing design to improve the results in the phase. By directly considering the waveguide loss in the algorithm, we are able to produce allpass filter coefficients that can provide optimal response given a prescribed phase requirement even under non-ideal fabrications. Comparison results with the minimax approach demonstrate that the proposed algorithm is able to provide performances gains for both high and low loss devices. The design method will be able to lift the performance limitation of allpass filters used in WDM applications such as distortion compensation and group delay equalization, as well as general photonic filtering.
7. Appendix A. Cost function
The phase response of the allpass filter under error is
or in vector notations where Our goal is then to have Θerr(ω) best match a prescribed phase response Θpre(ω). In other words, we would like to minimize For notation simplicity, let us define β (ω) ≜ Θpre(ω) + Nω, then we have With trigonometric manipulation, we can arrive at where the n, m element in matrices P(ω) and Q(ω) are We can further simplify by first considering Taylor series expansion on arctan and discard all the non-linear terms for small approximation errors, which yields Notice that if the denominator is fixed, then we can simply consider the following optimization with quadratic constraints While the objective is still not convex, it is much smoother and there are relaxation techniques and global optimization methods that can be applied.B. Algorithm details
While introducing a constraint on the denominator reduces the irregularities in the original cost function, (22) remains a nonconvex quadratic programming problem and NP hard. A direct approach to solving this non-linear optimization problem to utilize a global optimization technique such as branch and bound [24]. In branch and bound, the original feasible set to the optimization problem is branched into smaller subsets. Within each subset, the maximum and minimum costs are then computed. A branch is discarded when its minimum is greater than the maximum of another branch. This process continues until a termination criterion is met, or when a single solution is found. In application for nonlinear programming, the original optimization problem is partitioned into subsets, and a convex envelope is found within each subset. The algorithm continues to branch and terminates when the upper and lower bounds meet. We will use this approach to solve the following optimization problem
Instead of the ℒ∞-norm, we consider the equivalent problem of enforcing L constraints for computational savings. The final step in our algorithm is to combine (23) with the constraint on the denominator. The complete optimization problem is therefore Note that we have introduced a new optimization variable ε to relax the original equality constraint. The proposed algorithm is able to compensate for waveguide power loss in photonic filters because the effect is directly captured in the optimization problem by the variable A.B.1. Refinement
While (24) can already provide loss compensating designs, we can further improve the algorithm by introducing an iterative refinement step. In (24), the optimal result is obtained when we examine a further confined feasible set. However, the quantity dTA−1Q(ω)Ad does not necessarily have to be 1 to achieve minimal cost in the original problem. We can therefore refine the result by searching in a small neighborhood near the solution from (24).
To implement the local search, we consider an iterative approach to remove the denominator in the original optimization problem (7) through
where k is the iteration variable. The procedure is analogous to treating the numerator and denominator as two separate optimization problems. Note that when the denominator is a fixed quantity, we can simply modify (24) to solve the numerator optimization problem.Since the denominator is solved using only a naive update, we must control the step size to search only within a small neighborhood from the original guess d0. To ensure the denominator does not differ significantly from iteration k − 1 to iteration k, we introduce the following constraint
where λ is the step size control, and can be either manually chosen or introduced as an optimization variable.Acknowledgments
The authors would like to thank Professor Philip Gill and Professor Jiawang Nie of University of California San Diego for critical discussions on solving the optimization problem.
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