Allpass filter design with waveguide loss compensation

Allpass filter design with waveguide losscompensation

Yujia Wang,∗ Andrew Grieco, Truong NguyenDepartment of Electrical and Computer Engineering, University of California, San Diego

9500 Gilman Dr., La Jolla, CA 92093, USA∗[email protected]

Abstract: A major artifact of realistic photonic filters is the waveguidepower loss. Its detrimental effect on the allpass structure is particularlyalarming because the phase response is highly sensitive to perturbations.While the loss can be simply captured into a variation on the unit delayin signal processing analysis, its non-linearity makes it mathematicallydifficult to address. We present an allpass filter design algorithm that isable to provide filter coefficients that compensate for the waveguide powerloss. By absorbing the loss parameter into the design cost function, theoptimization problem becomes non-convex and NP hard. Our approachsolves this problem by utilizing an iterative algorithm in conjunction withthe branch and bound global optimization technique. The proposed algo-rithm is expected to improve the performance and increase the utilization ofallpass filters for optical signal phase based applications such as distortioncompensation and group delay equalization.

© 2013 Optical Society of America

OCIS codes: (070.2025) Discrete optical signal processing; (070.2615) Frequency filtering;(130.7408) Wavelength filtering devices.

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#199848 - $15.00 USD Received 28 Oct 2013; revised 5 Dec 2013; accepted 10 Dec 2013; published 17 Dec 2013(C) 2013 OSA 30 December 2013 | Vol. 21, No. 26 | DOI:10.1364/OE.21.032040 | OPTICS EXPRESS 32040

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1. Introduction

Allpass filters have long been used in digital communications and signal processing. Thesestructures are excellent for applications that involve manipulation of the digital signal’s phase,such as phase equalization, Hilbert transform, fractional delay, and magnitude filtering usingallpass substructures [1, 2]. It has been pointed out that allpass filters are also crucial in pho-tonic systems. With a wide range of applications towards phase equalization in wavelengthdivision multiplexing (WDM) [3, 4] and general optical filtering [5, 6], the digital allpass filterhas demonstrated its solid connection to optical signal processing. A variety of fundamentalphotonic components naturally behave as allpass elements, such as ring resonators [7], mi-crodisk resonators [8], and Bragg reflector pairs [9]. The fact that a complex allpass filters canbe realized in lattice or cascade assemblies of first-ordered sections make these structures evenmore readily applicable for photonic operations.

In low loss systems such as fiber optics (loss coefficient ∼ 2×10−6 dB/cm [10]) the associ-ation of optical components with allpass elements is well justified. However, in the context ofintegrated 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 becomesdifficult to justify, particularly when effects such as bending loss are considered [14]. The im-plications for optical filter design are significant. In particular, as the devices are integratedand miniaturized it will become necessary to confront the issue of optical loss. The allpass as-sumption is also violated in situations that require long phase delay lengths and therefore longsections of waveguide, narrowband filters being an example [6]. These realistic considerationslimit the utilization and performance of allpass filters in optical systems, especially in its directphase 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 capableof 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 avariation on the delay element z−1. It has been shown that the loss affects a transfer functionby modifying the original response H(z) into H(γ−1z), where γ = e−αL/20 is the fraction ofpower retained by the filter sub-elements after losses are accounted for [3,5]. The scattering andabsorption 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 delayelement z−1. Although significant efforts have been put into allpass analysis to demonstraterobustness to coefficient perturbations [1, 17] in DSP filter design, its error resilience to thisunique photonic filtering error has never been addressed.

Given these considerations, we present a minimax allpass filter design algorithm that is ableto compensate for the delay element variation in terms of phase performance. Since the waveg-uide power loss can be considered as a predetermined effect [5], we capture the γ parameterinto the filter design cost function. Without any manipulation, the design problem is mathemat-ically 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 aBranch and Bound global optimization technique. From the proposed technique, we are ableto derive photonic parameters that allow the realization of allpass filters that can mitigate theperformance degradation caused by the waveguide power loss.

The rest of the paper is outlined as follows: Section 2 provides the signal processing back-ground on an ideal allpass filter behavior and operation, as well as the connection to photonicrealizations. Section 3 discusses the waveguide loss issue from both photonic and signal pro-cessing perspectives. Section 4 sets up the mathematical problem and presents our algorithmon the waveguide loss compensating allpass design. Section 5 shows results and comparisonsof 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 forall frequencies, but alters the phase of the input signal according to a prescribed profile. Thetransfer function for an Nth-order allpass filter can be written as [2]

A(z) =d∗N +d∗N−1z−1 + · · ·+d∗1z−(N−1)+1z−N

1+d1z−1 + · · ·dN−1z−(N−1)+dNz−N= z−N D∗(1/z∗)

D(z)(1)

where di’s are the filter coefficients. The allpass filter exhibits a unique poles and zeros patternin that given a zero at zi, A(z) must also contain a pole at pi =

1z∗i

. The precise relation betweena pole and zero pair of the allpass filter allows the system A(z) to receive equal magnitudecontributions from D(z) and D∗(1/z∗), and thus results in a constant response.

Since the allpass filter is expected to have a unity response in the magnitude for all frequen-cies, the design freedom lies only in the phase. The challenge in allpass design is therefore tofind 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 suchas dispersion compensation and group delay equalization, where the operations required areonly in the phase.

The allpass filter is highly suitable for physical realization using optical components becauseit can be easily modularized. A first-order allpass filter can then be readily realized using avariety of nanoscale photonic elements. To demonstrate the operation of a photonic allpassfilter, we will consider the Bragg reflectors setup as shown in Fig. 1. The traversal time of the


Fig. 1. First order allpass realization using a Bragg mirror topology. The optical path lengthis represented by L, s is the phase delay of the unit, and ρ is the reflection coefficient.

waveguide loop serves as the discrete delay element z−1, and is directly related to the opticalpath length L. The reflection coefficient ρ of the mirror and the phase delay s in the structuredirectly relate to the first-order allpass filter coefficient d1. Under ideal fabrication conditions,the transfer function of a first-order photonic allpass filter is

A(1)(z) =−e js−ρe− js + z−1

1−ρe jsz−1 (2)

in which case d1 = −ρe js. The setup can be easily extended to higher order photonic allpassfilter, which is simply a cascade or lattice of first order sections.

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 areneeded for realization. The single stage parameters can be readily obtained post design throughfactorization in the cascade realization, and recursion in the lattice. Furthermore, in this paperwe focus on the design of real coefficient allpass filters. In other words, the phase delay s isassumed to be an integer multiple of π . This requirement can be easily absorbed into the di-mensions of the waveguides. Note that realisic tooling inaccuracies hinder such setup and resultin a random phase error in the individual sections. While the effect of phase deviations in unde-niably present [18], the average performance of the overall system often converges to the idealdesign [6]. Its impact on the allpass performance is therefore far less drastic in comparison towaveguide loss. We will therefore assume ideal fabrication conditions and neglect the effect ofrandom phase error in this paper.

3. Waveguide power loss

There are two mechanisms responsible for loss within an optical waveguide: scattering loss andmaterial absorption. Scattering loss is a consequence of artifacts like roughness in the waveg-uide sidewalls [19] or excessive bending of the waveguide [20]. It is therefore amenable toreduction through improved fabrication precision. In contrast, absorption is an intrinsic ma-terial property and is inescapable without abandoning the material system. In the context ofdigital signal processing, all loss has the same effect regardless of provenance. As such, it ismost pertinent to consider the aggregate absorption coefficient of the waveguide. This is alsoconvenient in the sense that loss characterization methods generally determine the aggregateabsorption coefficient [21].

A generalized allpass design algorithm that maximizes the allowable waveguide loss whileretaining comparable performance is extremely valuable. It will allow relatively lossy waveg-uides that are inexpensive and simple to fabricate to replace low loss waveguides that are costlyand complicated to fabricate [22, 23]. The eased fabrication constraints will also increase thecompatibility of photonics with other platforms, such as CMOS. Furthermore, the method willmitigate the increased loss effect from miniaturizing photonic devices. Finally, it will eliminatethe 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 waveg-

uide, it can be considered to be coupled with the unit delay element z−1 [3]. The effect on anNth-order allpass transfer function with real coefficients from γ is

Aerr(z) =dN + γdN−1z−1 + · · ·+ γNz−N

1+ γd1z−1 + · · ·+ γNdNz−N = A(γ−1z) (3)

where 0 ≤ γ ≤ 1 captures the remaining signal power after experiencing waveguide powerloss. Notice that by changing the argument of A(z) from z to γ−1z, we are effectively scalingthe poles and zeroes radii by γ . Such effect is particularly alarming in an allpass filter, becausea precise relation between each pole and zero pair is needed to main a constant magnituderesponse. Figure 2 shows the effect of γ = 0.9 on the magnitude and phase for a 6th orderallpass. As shown in the plots, because the relationship of having zero pole pairs at zi and 1

z∗iis

Fig. 2. Effect of γ = 0.9 on the magnitude and phase responses of a 6th order allpass. Notethat minor changes in the phase response can result in significant distortion. The solid linesare the ideal allpass magnitude and phase responses.

violated, the magnitude response can no longer be maintained at unity for all frequencies. Theroot scaling results in a notch effect at every pole and zero pair location, causing attenuationsin the magnitude response. Also notice that the loss causes significant phase distortion near thetransition band edges.

Note that the distortion in the magnitude response can be corrected by a follow-up finiteimpulse response (FIR) linear phase filter. Such structure can be easily fabricated using a tapdelay line architecture with negligible loss [3]. Furthermore FIR linear phase filters presentsno additional processing on the phase, thus preserving the output of the allpass filter. We willtherefore focus our design efforts on providing a desirable phase behavior from the waveguideloss corrupted allpass filter, and assume that any magnitude distortion can be compensated byadditional post filtering operations.

4. Loss compensating design

To create an allpass filter design algorithm with waveguide loss compensation, we first examinethe 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

Aerr(z) = γNz−N D(γz−1)

D(γ−1z)(4)

The goal of our algorithm is therefore to minimize the difference between the filter’s phase re-sponse Θ(ω) and a prescribed phase requirement Θpre(ω). The error function can be expressed


as

∆Θ(ω)≈ dTA−1P(ω)AddTA−1Q(ω)Ad

(5)

where

d =[1 d1 · · · dN

]TA = diag

{1 γ · · · γN}

Pn,m(ω) = sin(β (ω)−nω−mω)

Qn,m(ω) = cos(β (ω)−nω−mω) (6)

and β (ω) , Θpre(ω) +Nω . The details to the formulation of the objective function can befound in Appendix A. To derive filter coefficients that can result in optimal phase performanceunder waveguide loss, we seek to minimize ∆Θ(ω) in either the least squares sense or minimaxwith respect to d. In filter design, the minimax approach is generally more desirable because itensures optimal performance at all frequencies [1]. We will therefore consider the minimizationof the L∞-norm cost function

minimized

maxω

W (ω)

∣∣∣∣ dTA−1P(ω)AddTA−1Q(ω)Ad

∣∣∣∣ (7)

Notice that this optimization is challenging due to both the fraction form and non-semi definitematrices A−1P(ω)A and A−1Q(ω)A. The problem cannot be reduced to any standard convexformulation, and is highly non-linear.

To solve the optimization problem, we propose a two part algorithm that consists of an initialguess step followed by refinement. We first consider a relaxed problem that can be efficientlysolved using the global optimization technique Branch and Bound [24]. Note that the setup weconsider for an initial solution is still non-linear, because the original optimization setup cannotbe 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 guesscloser to a solution to the actual optimization problem in (7). The details of the derivation areprovided in Appendix B. The overall algorithm is outlined in Algorithm 1.

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 allpassfilters, but also applied to improve existing implementations.

5. Example results

To demonstrate the performance of our algorithm, let us first consider the equalization of thephase response shown in Fig. 3. This phase profile exemplifies the response of a bandpass filterused in WDM that introduces nonlinear chromatic dispersion near the band edges. With the cur-rent phase response, only a narrow bandwidth at the center of the filter is usable because otherfrequency components within the passband will experience non-uniform group delay (negativederivative of the phase). The purpose of a followup allpass filter is then to equalize the filter’sphase to be linear, which subsequently increase the usable region of the passband [16]. We willtherefore consider the design of a 4th-order allpass filter that is able to linearize the given phaseresponse in the passband region of the bandpass filter from 0.15π to 0.6π . In this examplewe will examine the resulting allpass responses for a congregated loss effect of γ = 0.95. Fora 1µm by 0.5µm SiNx strip waveguide cladded in SiO2 with α = 25dB/cm [14, 25, 26], thiswould 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


Algorithm 1 Allpass loss refinementRequire: filter order N, prescribed response to match Θpre(ω), waveguide power loss γ , num-

ber 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

Wk(ω) =1∣∣dT

k−1A−1Q(ω)Adk−1∣∣ (8)

solve using Branch and Bound

minimizedi,δ ,γ

δ +λ

subject to −δ ≤Wk(ω)dTi A−1P(ω)Adk ≤ δ

−λ ≤ dTk A−1Q(ω)Adk−dT

k−1A−1Q(ω)Adk−1 ≤ λ

(9)

k++;until d converges

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 chro-matic dispersion near the pass band edges. The solid portions represent the frequency rangeof 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. Thebranch and bound method in the proposed algorithm is implemented in MATLAB using theYALMIP [27] toolbox. To derive the response from a traditional approach, an ideal allpass filteris 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 theeffect of γ in the optimization, the phase profile under waveguide loss significantly digressesfrom the desired profile. Since we directly consider the effect in the optimization setup, the


Fig. 4. Phase responses of 4th-order allpass filters that are designed using the traditionalminimax approach and the proposed algorithm for γ = 0.95. Note that the region of interestare confined to within the passband of the example bandpass filter (0.15π to 0.6π).

Fig. 5. Phase errors for the traditional minimax allpass and that of the proposed design. Theideal response of the minimax is also plotted for reference.

phase response from the proposed method is able to closely track the desired profile even underwaveguide loss.

Let us consider a second example where the prescribed phase response is of the followingform

Θpre(ω) =

{−(N−1)ω 0≤ ω ≤ ωp

−(N−1)ω−π ωs ≤ ω ≤ π(10)

Here N is the filter order, ωp is a normalized frequency value that governs the passband, andωs is the stopband. In comparison to the previous example, this prescribed response is moretolerating because it only contains a discontinuity. However, the effect of waveguide loss isstill prominent, and can be easily identified when we directly examine the phase error. Figure5 shows the phase error comparison results for N = 7, γ = 0.9, ωp = 0.55π , ωs = 0.6π . Noticethat using the traditional design, the corrupted allpass response shows large errors near the bandedges. The proposed design shows significant improvement in terms of maximum ∆Θ(ω).

Because the loss effect is directly captured in the optimization setup, the result phase er-ror from the proposed method is evenly distributed throughout the frequencies. The designedallpass will therefore not only have an minimal maximum error, but also reduced variationsthroughout. To demonstrate the overall effectiveness and robustness of our method, Table 1summarizes 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 isnegligible under highly tolerant requirements and low loss. In all other conditions, the pro-


posed design is able to outperform because traditional approaches do not consider the effectof waveguide power loss. In general, the proposed algorithm’s superiority is most obvious for0.8 < γ < 0.99. The variance measurements demonstrate that our method is not only able toreduce 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 sig-nificant phase differences. The proposed design and the refinement technique are both aimed tominimize this effect.

Table 1. Normalized maximum phase differences for N = 7 allpass filters under variouswaveguide loss. Note that γ = 0.8 would correspond to a photonic allpass filter whose indi-vidual sections are composed of Bragg reflectors and 1µm× 0.5µm SiNx strip waveguidescladded in SiO2 that are 485 µm long.

Method loss passband, stopband0.4, 0.6 0.5, 0.6 0.55, 0.6 0.575, 0.6

corrupted minimax γ = 0.93 0.0119 0.0406 0.0902 0.1975γ = 0.9 0.0163 0.0561 0.1542 0.2598γ = 0.8 0.0626 0.1758 0.2599 0.2955

compensating design γ = 0.93 0.0230 0.0582 0.1322 0.2232(no refinement) γ = 0.9 0.0231 0.0494 0.1075 0.2010

γ = 0.8 0.0252 0.0298 0.0209 0.0701

compensating design γ = 0.93 0.0142 0.0266 0.0583 0.0804(with refinement) γ = 0.9 0.0127 0.0220 0.0378 0.0487

γ = 0.8 0.0129 0.0147 0.0162 0.0313minimax refined γ = 0.93 0.0186 0.0412 0.0732 0.1120

γ = 0.9 0.0184 0.0355 0.0578 0.0919γ = 0.8 0.0135 0.0097 0.0172 0.0560

Table 2. Variance measurements across the frequencies for N = 7 allpass filters under vari-ous waveguide loss. Units are in 1 ×10−4

Method loss passband, stopband0.4, 0.6 0.5, 0.6 0.55, 0.6 0.575, 0.6

corrupted minimax γ = 0.93 0.148 1.469 7.098 20.11γ = 0.9 0.237 2.371 11.94 28.27γ = 0.8 1.672 12.10 32.72 54.48

compensating design γ = 0.93 0.148 0.611 2.550 8.864(no refinement) γ = 0.9 0.158 0.422 1.645 6.562

γ = 0.8 0.498 0.508 0.268 0.639

compensating design γ = 0.93 0.228 0.746 2.605 4.858(with refinement) γ = 0.9 0.214 0.476 1.288 2.310

γ = 0.8 0.223 0.252 0.315 0.632

minimax refined γ = 0.93 0.330 1.496 5.102 7.523γ = 0.9 0.264 1.254 2.937 5.403γ = 0.8 0.222 0.073 0.230 2.262

The fact that the proposed method is able to reduce the phase error variation after waveg-uide loss is also ideal for operations outside of the typical phase based applications. It hasbeen demonstrated that allpass filters are excellent for photonic system realization of arbitrarymagnitude filter [5]. Under ideal fabrication, traditional frequency selective filter (Butterworth,


Fig. 6. Linear magnitude responses of lowpass filters generated using waveguide loss cor-rupted minimax and the proposed compensating design. The ideal magnitude response froma minimax design is also shown for reference.

Chebyshev, Elliptic) can be decomposed into allpass substructures without losing optimal-ity [28]. Specifically, a general bandpass filter can be obtained by placing two allpass filtersin a parallel setup. We can further require one of the allpass filters to be a simple delay forreduced design and fabrication complexity

H(z) =12

(z−(N−1)+Aerr(z)

)(11)

where N is the order of the allpass filter Aerr(z). Ideally, the system operates by requiringthe phase response of Aerr(z) to be −(N− 1)ω in the passband regions to creating complexaddition, and −(N−1)ω−π in the stopband regions for subtraction. Under waveguide powerloss, the phase response of Aerr(z) is distorted, and traditional designs can no longer provideoptimal performance. Figure 6 shows the comparison result of lowpass filters created usingthe traditional minimax approach and the proposed method for γ = 0.9. Note that the phaserequirement 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 largevariations. By reducing the phase error of the loss corrupted allpass, the proposed design is ableto provide smoother magnitude responses in the passband and stopband. While the attenuationsin the passband and stopband have been affected by the loss, the effect can be minimized bypost processing gain adjustments.

6. Conclusion

This paper presents an allpass design method that can mitigate the effect of waveguide powerloss. We propose a two part optimization algorithm that can be used individually, or applied toan existing design to improve the results in the phase. By directly considering the waveguideloss in the algorithm, we are able to produce allpass filter coefficients that can provide optimalresponse given a prescribed phase requirement even under non-ideal fabrications. Comparisonresults with the minimax approach demonstrate that the proposed algorithm is able to provideperformances gains for both high and low loss devices. The design method will be able tolift the performance limitation of allpass filters used in WDM applications such as distortioncompensation 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

Θerr(ω) =−Nω + arctan∑

Nk=0 dkγ−k sinkω

∑Nk=0 dkγ−k coskω

+ arctan∑

Nk=0 dkγk sinkω

∑Nk=0 dkγk coskω

(12)

or in vector notations

Θerr(ω) =−Nω + arctandTA−1s(ω)

dTA−1c(ω)+ arctan

dTAs(ω)

dTAc(ω)(13)

where

d =[1 d1 · · · dN

]Ts(ω) =

[0 sinω · · · sinNω

]Tc(ω) =

[1 cosω · · · cosNω

]TA = diag

{1 γ · · · γN} (14)

Our goal is then to have Θerr(ω) best match a prescribed phase response Θpre(ω). In otherwords, we would like to minimize

∆Θ(ω) = Θpre(ω)−Θerr(ω) (15)

For notation simplicity, let us define β (ω), Θpre(ω)+Nω , then we have

∆Θ(ω) = β (ω)− arctandTA−1s(ω)

dTA−1c(ω)− arctan

dTAs(ω)

dTAc(ω)(16)

With trigonometric manipulation, we can arrive at

∆Θ(ω) = arctandTA−1P(ω)AddTA−1Q(ω)Ad

(17)

where the n,m element in matrices P(ω) and Q(ω) are

Pn,m(ω) = sin(β (ω)−nω−mω) (18)Qn,m(ω) = cos(β (ω)−nω−mω) (19)

We can further simplify by first considering Taylor series expansion on arctan

arctanx = x− x3

3+

x5

5− x7

7+ · · · (20)

and discard all the non-linear terms for small approximation errors, which yields

∆Θ(ω)≈ dTA−1P(ω)AddTA−1Q(ω)Ad

(21)


Notice that if the denominator is fixed, then we can simply consider the following optimizationwith quadratic constraints

minimized

maxω

W (ω)∣∣dTA−1P(ω)Ad

∣∣subject to dTA−1Q(ω)Ad = 1

(22)

While the objective is still not convex, it is much smoother and there are relaxation techniquesand global optimization methods that can be applied.

B. Algorithm details

While introducing a constraint on the denominator reduces the irregularities in the originalcost function, (22) remains a nonconvex quadratic programming problem and NP hard. A di-rect approach to solving this non-linear optimization problem to utilize a global optimizationtechnique such as branch and bound [24]. In branch and bound, the original feasible set to theoptimization problem is branched into smaller subsets. Within each subset, the maximum andminimum costs are then computed. A branch is discarded when its minimum is greater thanthe maximum of another branch. This process continues until a termination criterion is met, orwhen a single solution is found. In application for nonlinear programming, the original opti-mization 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. Wewill use this approach to solve the following optimization problem

minimizeδ ,d

δ

subject to −δ <W (ω)dTA−1P(ω)Ad < δ

(23)

Instead of the L∞-norm, we consider the equivalent problem of enforcing L constraints forcomputational savings. The final step in our algorithm is to combine (23) with the constraint onthe denominator. The complete optimization problem is therefore

minimized,δ ,ε

δ + ε

subject to −δ ≤W (ω)dTA−1P(ω)Ad≤ δ

− ε ≤ dTA−1Q(ω)Ad−1≤ ε

(24)

Note that we have introduced a new optimization variable ε to relax the original equality con-straint. The proposed algorithm is able to compensate for waveguide power loss in photonicfilters 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 algo-rithm by introducing an iterative refinement step. In (24), the optimal result is obtained whenwe examine a further confined feasible set. However, the quantity dTA−1Q(ω)Ad does notnecessarily have to be 1 to achieve minimal cost in the original problem. We can thereforerefine 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

minimized

maxω

W (ω)

∣∣∣∣∣ dTk A−1P(ω)Adk

dTk−1A−1Q(ω)Adk−1

∣∣∣∣∣ (25)

where k is the iteration variable. The procedure is analogous to treating the numerator anddenominator as two separate optimization problems. Note that when the denominator is a fixedquantity, 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 tosearch only within a small neighborhood from the original guess d0. To ensure the denomina-tor does not differ significantly from iteration k− 1 to iteration k, we introduce the followingconstraint

|dTk A−1Q(ω)Adk−dT

k−1A−1Q(ω)Adk−1|< λ (26)

where λ is the step size control, and can be either manually chosen or introduced as an opti-mization variable.

Acknowledgment

The authors would like to thank Professor Philip Gill and Professor Jiawang Nie of Universityof California San Diego for critical discussions on solving the optimization problem.


Allpass filter design with waveguide loss compensation

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