Efficient wavelength multiplexers based on asymmetric response filters

We propose integrated photonic wavelength multiplexers based on serially cascaded channel add-drop filters with an asymmetric frequency response. By utilizing the through-port rejection of the previous channel to advantage, the asymmetric response provides optimal rejection of the adjacent channels at each wavelength channel. We show theoretically the basic requirements to realize an asymmetric filter response, and propose and evaluate the possible implementations using coupled resonators. For one implementation, we provide detailed design formulas based on a coupledmode theory model, and more generally we provide broad guidelines that enumerate all structures that can provide asymmetric passbands in the context of a pole-zero design approach to engineering the device response. Using second-order microring resonator filter stages as an example, we show that the asymmetric multiplexer can provide 2.4 times higher channel packing (bandwidth) density than a multiplexer using the same order stages (number of resonators) using conventional all-pole maximally-flat designs. We also address the sensitivities and constraints of various implementations of our proposed approach, as it affects their applicability to CMOS photonic interconnects. © 2013 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (130.7408) Wavelength filtering devices; (250.5300) Photonic integrated circuits. References and links 1. X. Zheng, J. Lexau, Y. Luo, H. Thacker, T. Pinguet, A. Mekis, G. Li, J. Shi, P. Amberg, N. Pinckney, K. Raj, R. Ho, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultra-low-energy all-CMOS modulator integrated with driver,” Opt. Express 18, 3059–3070 (2010). 2. G. Li, X. Zheng, J. Yao, H. Thacker, I. Shubin, Y. Luo, K. Raj, J. E. 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Introduction
The integration of photonics and CMOS electronics has been an active area of research in recent years [1][2][3][4][5][6][7].With cloud based computing driving the production of larger and more bandwidth intensive data centers as well as the increasing number of processors in multi-core CPUs, the need for more energy efficient and higher bandwidth density communication links between CPUs and RAM has motivated research into photonic CPU-memory communication links [8,9] and the first generation of monolithic electronics-photonics integration [5][6][7].Photonic communication links, using wavelength division multiplexing (WDM), have the potential to greatly increase the bandwidth density and energy efficiency compared to electrical links [10].
At the heart of many photonic communication links and network implementations are wavelength (de)multiplexers typically comprising serially cascaded microring filter stages.Microring filters have a free spectral range (FSR) that is determined by the ring circumference and the guided mode group index (i.e.dispersion).In a WDM communication link, the FSR, ad-jacent channel rejection, and required filter bandwidth with a certain maximum insertion loss determine how many WDM channels can fit in one FSR of the microring-based filters.The total bandwidth (and bandwidth utilization, Gbps data/GHz optical bandwidth) increases with an increasing number of WDM channels in a given optical wavelength range; for this reason, a designer would like to use higher order filter responses to permit denser channel spacing.Higher order filters, however, require a larger number of microring resonators which requires additional thermal tuning to compensate for fabrication variations and align to a WDM grid.Thermal tuning has a substantial energy cost and significantly impacts the energy efficiency of a proposed photonic link design.Therefore, it is of interest to investigate methods to achieve maximal bandwidth efficiency with a given number of resonator elements.
In this work, we propose a type of multiplexer/demultiplexer that we will call a "pole-zero" (de)multiplexer.It relies on the cascade of a number of stages of a novel filter design that enables asymmetric response shapes, which we will refer to as a "pole-zero" filter.A pole-zero (de)multiplexer enables very dense wavelength channel packing using low-order filters, denser by a factor of 2.4 than conventional Butterworth designs of the same order when using secondorder stages.We choose a Butterworth response for comparison since it is a commonly used passband shape.
To design a pole-zero filter, we directly control the placement of the resonant frequencies (poles) and transmission zeros of the filter response (S-matrix element of interest) in the complex-frequency plane.Utilizing poles and zeros in the complex frequency plane is familiar in electrical and RF/microwave circuits, and has been previously explored in photonic devices [11][12][13][14].In this paper, we focus on the simplest way to achieve an asymmetric response by the introduction of a single transmission zero on one side of the passband.We present a coupling of modes in time (CMT) model and use it to design an asymmetric filter response that lends itself to design of WDM demultiplexers with very densely packed channels.The advantage of this approach is that it provides a physically intuitive design technique that naturally leads to efficient implementations based on various criteria and constraints, as described later.The resulting asymmetric response is equivalent to that of previously studied asymmetricresponse RF filters based on standing wave cavities [15].Based on some general guidelines for designing pole-zero filters [13], we also consider all possible topologies of a second-order design.These alternative designs have advantages and disadvantages, e.g. in terms of sensitivity to various fabrication parameters, number of degrees of freedom that need to be controlled, etc.

Coupling of modes in time model
A CMT model [16] is used to design the filter response, namely, the shape of the passband and the location of the transmission zero.First, we show what is required in a general photonic system to achieve one finite-detuning transmission zero in the drop port, and we present a physical implementation that can achieve this.We then consider an approximate solution to the CMT equations for an N th -order system.The specific device we are interested in is a 2 ndorder implementation for which we rigorously derive the CMT model and solve the full design equations.
Note that in the CMT model we consider a single resonance per resonant cavity; accordingly, when referring to a single pole or some number of zeros, this refers in a real cavity to a certain number per mode, i.e. per FSR of the system.We assume a narrowband approximation, i.e. that the passband is much smaller than the FSR, so that the adjacent azimuthal modes do not contribute to the same passband.However, these constraints are artificial and the same approach can be applied if a single cavity is used to supply multiple resonances that contribute to the passband, for example.

Designing a response with one transmission zero at finite detuning from the passband center
Consider an abstract photonic circuit representing a filter with one input port and two output ports as shown in Fig. 1(a).The two outputs are the familiar through and drop ports.Since the system has two resonances, all of the ports share the same two poles in the complex-frequency plane.The ports can have 0 to N finite-detuning transmission zeros.Assuming a lossless system, we choose to constrain the system to have two real zeros in the through port to ensure 100% transmission at those frequencies in the drop-port passband by power conservation, with a second-order rolloff.We put one real zero in the drop port transfer function.The zero is placed on one side of the passband to make the response function asymmetric, for reasons that will become clear in Section 3. Next, one must determine a physical implementation of a photonic circuit that can achieve the desired response.A simple rule can be used to determine the number of finite-detuning transmission zeros in the response function from the input to a given output port, i.e. in each s-parameter, S j,input , j ∈ {thru,drop}.In general, the number of finite transmission zeros in each s-parameter is equal to N, the resonant order of the system, minus the minimum number of resonators that must be traversed from input to output [13].Using this rule as a guide, the circuit shown in Fig. 1(b) can achieve the desired transmission response.Specifically, the resonant order of the system is N = 2, and the minimum number of resonators that the light must pass through is one to the drop port, and zero to the through port.
In the drop port response, the light can take the blue path shown in Fig. 1(b) and bypass the second resonator.Using our general rule, this results in: (2 poles)−(1 minimum resonator traversed to drop) = 1 transmission zero in the drop port response.Similarly, we find two zeros in the through port, which create the familiar rejection band in the same way as for regular serially-coupled ring filters [17,18].

Approximate design equations for an N th -order system
Because the transmission zero is placed off resonance, to enhance the drop-port response rejection band, it is possible to find a simple model for the position of the transmission zero, by assuming off-resonant excitation of the resonances in the system.
The time evolution of the mode energy amplitudes in a lossless N th order resonant system of Fig. 2. Higher-order filters with a drop-port transmission zero: (a) a device architecture that can produce one drop-port zero in arbitrarily high order filters; (b) example response of a 2 nd -order filter using Eqs.6,7 and a zero placed at δ ω zd = 10r i .
serially coupled resonators can be written as N first order differential equations, where a k is the energy amplitude in the k th ring, ω k is the resonant frequency of the k th ring, r i,d are the decay rates to the input bus and drop bus, respectively, μ kl is the energy coupling rate to the k th ring from the l th ring, s i is the amplitude of the input wave, and s d is the amplitude of the drop-port output wave.If the resonant frequencies of all rings are set equal, as is the case for typical square passband responses, the desired filter shape is synthesized through choice of the ring-ring couplings, μ kl , and the input and drop port decay rates, r i and r d .When a monochromatic input wave is sufficiently far detuned in wavelength from the passband center wavelength, the coupling in each equation is dominated by the forward coupling from one ring to another (i.e.μ (N)(N+1) a N+1 / μ (N)(N−1) a N−1 << 1).This is because off- resonance the rings do not like to exchange energy (i.e. when the detuning is much larger than the coupling rate [19]), so coupling from ring 1 to ring 2 is weak, and back from ring 2 back to ring 1 is weaker still because it is a second-order effect in the detuning-induced suppression of coupling.Hence, in the coupling equations we can assume dominant coupling from the ring energy amplitude that is closer to the input bus.This simplifies Eq. 1 by completely decoupling the equations, and should perfectly recover the response in the off-resonant wings of the passband (only).Our goal is to design a circuit to achieve one finite transmission zero in the drop port of the device.Figure 2(a) shows an extension of Fig. 1(b) to achieve this for increasing order microring filters.In all of these filters, bypassing the N th ring with a tap at the (N − 1) th ring coupled directly to the drop port enables the asymmetric response by ensuring a single drop-port response function zero.The weaker the tap coupling, the further detuned the transmission zero is from the passband.In the limit of zero tap coupling to the (N − 1) th ring, the standard symmetric response is recovered.
Equations 1, with one modification, can be solved for the asymmetric response of the circuits shown in Fig. 2(a).The modification is an additional term that describes the direct coupling of the drop port to the (N − 1) th ring.After further making the foregoing off-resonant approxi-mation, i.e. that the energy amplitude a k is excited primarily by the previous energy amplitude a k−1 , Eqs. 1 can be simplified to where r t is the decay rate to the tap port, φ is the propagation phase accumulated in the interference arm, and s d [see Fig. 3(a)] is given by The output wave, s d , can be then be found from Letting d/dt → jω to solve for the steady state frequency response of the system, Eqs.2-4 can be solved for the transfer function, S d,i (ω) ≡ s d /s i (valid off resonance) (5) The root of the numerator in Eq. 5 gives the frequency position of the transmission zero which, since it is off resonant, can be found from this approximate model.Setting the imaginary part of the root to zero to place the transmission zero on the real frequency axis and introducing δ ω zd as the desired detuning from the passband (resonant) frequency to the transmission zero, two simple design equations can be derived that give the phase delay needed in the interference arm [see Fig. 1(b)] as well as the decay rate to the tap port: The remaining decay rates and ring-ring couplings can be taken from the standard all-pole design synthesis techniques [18,20,21].Figure 2(b) shows the transmission for a 2 nd -order pole-zero filter using Eqs.6 and 7 with a design zero location of δ ω/r i = 10.The specific reason for our choice of this implementation is because it converges, in the limit of zero coupling at the tap, to a standard all-pole design.This means that any fabrication uncertainties introduced in the additional tap interferometer will affect only the position of the zero to first order, and will require weak coupling for substantially detuned zeros, making the design fairly insensitive to variations (or at least not considerably more so than a standard all-pole design).We will consider alternative geometries in Section 4.

Rigorous solution of the 2 nd -order filter synthesis problem
In the previous section, approximate design equations were derived, and it was shown that a finite transmission zero in the drop port can be achieved with the proper choice of the tap decay rate and the interference phase.This approximate model, however, is not applicable when it is desirable to place the zero close to the resonant frequency (since the approximate model is valid far from resonance).To derive the full design equations, we begin with a 3 × 3 system whose tap port is not connected to the drop port as shown in Fig. 3(a).
The CMT equations for the 3 × 3 system can be written in state-variable form [13,22]: where I is the identity matrix, and H, M i , M o , a, s + , and s − are defined as follows: The state variables a 1,2 are the energy amplitudes in rings 1 and 2; s i , s a , and s a are input port wave amplitudes; and s t , s d , and s d are output port wave amplitudes.s a and s d are temporary ports which will be later connected together.The ring resonant frequencies are ω 1 and ω 2 , the ring decay rates due to coupling to waveguides are r i , r t , and r d , and the related couplings are given by √ 2r i , √ 2r t , and √ 2r d [18, 19, 22].The coupling phases, φ 1,2,3 , can be set to zero without loss of generality.
With d/dt → jω, a transmission matrix T 3×3 , where s − = T 3×3 • s + , can be derived, The coupling arm can now be connected.That is, port s d is connected to s a after a finite propagation distance.For our 3-port model, this eliminates one input and output port, and reduces the model to a 2 × 2 T-matrix, described by where where φ (≡ β L) is the phase accumulated in the coupling arm.The reduced transmission matrix is We are interested in the through port response, T 11 , and the drop port response, T 21 , given by where δ ω is the relative detuning of the two rings such that their resonances are at ω 1 = ω 0 + δ ω and ω 2 = ω 0 − δ ω .
We can now require that the through-port zeros be placed on the real frequency axis to ensure 100% power transfer to the drop port (in the lossless approximation).To first find the zeros, the numerator of T 11 is set to 0 and solved for δ ω, giving This equation produces two constraints that must be satisfied in order to have real zeros in the through port: The constraint in Eq. 19 simplifies to which leads to a design equation that determines δ ω : We next consider the drop port response T 21 (δ ω), given by Eq. 16.Its transmission zero is given by δ Since φ is not yet determined, this equation for the zero location can be interpreted graphically as a root locus in the complex-δ ω plane.For various φ , the zero is located on a circle with radius μ r d /r t centered at −(δ ω , r d ), as shown in Fig. 3(b).This interpretation leads to a simple design equation for φ , the phase accumulated in the interference arm, to place the zero on the real frequency axis: as well as for the tap coupling, r t , At this point, we have fixed all degrees of freedom of the model.The total list of parameters of the device relevant in our synthesis includes r i , μ, δ ω zd , r t , r d , δ ω and φ .The first three (r i , μ, δ ω zd ) are chosen to be inputs to the model.The choice of r i and μ largely determines the passband shape (maximally flat, equiripple, bandwidth, etc.), and these exist in all-pole (serially-coupled) ring filters.Detuning δ ω zd is the desired location of the drop port zero.Without loss of generality, here, we choose r i and μ to be those of an all-pole 2 nd -order Butterworth filter.This is given by r i = μ [18,20].After the three input parameters are chosen, Eqs.18,21,23, and 24 are solved for r t , r d , δ ω and φ using the derived expressions.shows the transmission of a representative pole-zero filter, and a 2 nd -order Butterworth filter of equal 3 dB bandwidth for comparison.
In general, when the transmission zero is very close the passband, the r i and μ no longer give exactly the bandwidth and passband ripple that a Butterworth or Chebyshev design sets for them, but they are close enough for all practical designs that they can either be used as is or adjusted slightly to get the exact desired parameters.The fixing of the zeros ensures that the passband is not distorted and has (in principle) complete dropping of wavelengths in the passband.

Design of a serial demultiplexer based on asymmetric response stages
In this section, we show the advantages that can be obtained from using filters with the asymmetric response shown in Fig. 4 to design an efficient serial wavelength demultiplexer.In the drop port response, the transmission rolls off more slowly than a standard 2 nd -order Butterworth response on the left side of the passband, and it rolls off much faster on the right side between the center frequency and the zero location.To the right of the zero, there is again an increase in transmission.If this increase in transmission is detrimental in a particular application, e.g. as a crosstalk level, the designer must be able to set bounds on the maximum tolerable out-of-band transmission.In general, there is a tradeoff between how close a zero is to the passband (allowing a sharper rolloff) and the worst-case off-resonant rejection out of band.For the purposes of a serial demultiplexer, this will affect the adjacent channel rejection.The zero location in Fig. 4 was chosen to ensure a minimum 20 dB adjacent channel rejection.
Using the asymmetric-response filter as a building block, a serial demultiplexer can be designed to achieve a symmetrized response in the drop port that has fast rolloff on both sides of the center frequency.Figure 5  Figure 6(a) shows the drop port responses from a 4 channel serial demultiplexer where each successive channel has a resonant frequency that is detuned from the previous channel's center wavelength by the zero detuning.To make clear the advantage gained from using the polezero filters in comparison to conventional all-pole Butterworth filters, it is helpful to convert the normalized plots shown thus into an actual example implementation.For a filter bank that has a passband of 20 GHz defined at a 0.05dB ripple and at least 20 dB adjacent channel rejection, the multiplexer based on pole-zero filters can achieve a channel spacing of 44 GHz, or 45% bandwidth utilization.The all-pole 2 nd -order Butterworth filter achieves a channel spacing of 106GHz, i.e. bandwidth utilization under 19%.The pole-zero filter bank gives a 2.4 times denser channel packing, i.e. higher bandwidth density, with no increase in filter order.Figures 6(a) and 6(b) show the responses of the example pole-zero and Butterworth demultiplexers based on second-order filter stages for comparison.It should be noted that although the pole-zero filters were derived from the Butterworth design, the transmission zero causes there to be a slight ripple in the passband.Comparing the pole-zero filter to a Chebyshev filter with approximately the same ripple, the channel packing is about 1.8 times denser using a pole-zero filter bank compared to a Chebyshev filter bank.If the interference arm path length is chosen to give the correct accumulated phase and the two rings only need to be slightly tuned to line up their resonance frequencies, then the filter can be tuned with only 2 heaters on the ring resonators and still achieve reasonable extinction at the zero location.Since a benefit of this design is in part in the reduction of the number of thermal tuning elements (thermal power) compared to using a higher order filter, it is of interest to investigate both the sensitivity of the zero placing mechanism (the interferometer arm and tap coupling) to fabrication variations, and to investigate the possible geometries that can realize the proposed asymmetric response in search of the simplest, and most symmetric, possible implementation.

Alternative topologies for a filter response with two poles and one zero
The topology shown in Fig. 1 is not the only physical implementation that supports one transmission zero in the drop-port response with a finite detuning from the passband.Using the rule discussed in Section 2.1 for the number of finite-detuning zeros in a given S-matrix transfer function (i.e. in transmission to a given port), in a simple system such as this, it is straightforward to consider all device topologies that can result in one transmission zero.Figure 7(a) shows all of the degrees of freedom for a photonic circuit based on 2 traveling-wave resonators, that can produce a 2-pole, 1-zero response.This calls for two waveguides (2 pairs of input and output ports), with each waveguide coupled to each resonator.The designer has access to 5 couplings and 2 phases.Not all of the degrees of freedom are needed to design a response with one finite-detuning transmission zero.
Figures 7(b)-7(d) show alternate configurations with Fig. 7(b) being the configuration that was analyzed in detail in this paper.The configurations are grouped by which couplings they use.All of them share the characteristic that a minimum of one ring must be traversed from the input port to the drop port.Each configuration has strengths and weaknesses.Figure 7(b) can be viewed as a perturbation to the standard all-pole filters, so intuition used for the all-pole filter mostly applies.The drop port adjacent channel rejection is robust because a serially cascaded ring array naturally rejects signal off resonance (there is no requirement for a carefully tuned phase relationship between the energy amplitudes in ring 1 and ring 2, as is the case in some geometries like parallel-coupled rings [23]).Figure 7(c) has a robust through port rejection since the light is forced to pass through both of the rings, but the 2 nd -order portion of the drop port rolloff is sensitive due to the necessity of the proper phase relationship between rings 1 and 2. This filter geometry has been used previously for filtering and for its high-Q states, but the responses realized either had no transmission zero [23,24], or one in the center of the passband not usable for bandpass filtering [24,25].In this work, we propose that this geometry can be used to design asymmetric bandpass filters, subject to appropriate choice of the phases between the two resonators.The second configuration in Fig. 7(c) uses the same coupling as the first configuration in part (c).Design of the phases here requires unequal waveguide lengths and hence some adjustments to the waveguide geometry.This crossed configuration lends itself to a convenient network topology.Filter designs like this geometry, with waveguides crossed orthogonally (only), have been demonstrated [26], but they have again been used exclusively for all-pole filter design.Our work shows that in principle, pole-zero filters are realizable with different phase shifts.Figure 7(d) is the simplest implementation requiring only 3 coupling points which may make it more tolerant to fabrication uncertainties.The degenerate case of this implementation has also been previously studied [27].Although Fig. 7(d) can achieve a transmission zero, it is forced to be between the split resonances of the supermodes of the cavity.Hence, it is not useful for the type of spectral response we are pursuing in this paper.All of these configurations have a first-order asymptotic rolloff far from the center frequency, and second order rolloff when detuned closer to the passband than the transmission zero.
It is outside the scope of this paper to consider in detail specific implementations and fabrication processes and conditions, but future work will investigate the best designs in terms of robustness to various common types of error in realistic fabrication processes such as those seen in monolithic integration efforts [5,6].

Conclusion
We have proposed pole-zero filter designs with asymmetric filter responses, based on coupled resonators on chip.We have developed a physical model and design approach that gives insight into passband design and sensitivity.We have also proposed, and through simulations demonstrated, the benefit of a pole-zero based design in increasing the bandwidth density of serial wavelength demultiplexers, without an increase in filter order.A presented example design shows a factor of 2.4 reduction in the channel spacing, i.e. increase in bandwidth density, needed when using a pole-zero filter in comparison to an all-pole Butterworth filter of the same order.We have also shown that there is a finite number of possible implementations, at an abstract level (independent of resonator type), that can produce responses with two poles and one finite detuning transmission zero in the drop port.These designs can be applied to many physical geometries and implementations and can be extended to apply to standing-wave cavities in a straightforward way.Some of the investigated geometries enforce restrictions on where the zeros can be placed, to the designer's advantage or disadvantage.
The demand for very dense WDM communication links will push designers to using higher order filters; asymmetric response filters and (de)multiplexers can provide more densely packed channels using fewer resonators compared to all-pole designs.Because of the importance of energy considerations, these designs could make an impact in on-chip systems and interconnects, if simple and robust implementations can be demonstrated that are competitive in sensitivity to standard all-pole (and higher order) designs.

Fig. 1 .
Fig. 1.Resonant systems capable of a 2-pole, 1-zero response: (a) abstract representation of a 2-pole, 1-zero photonic circuit; (b) a physical implementation that uses a weak tap coupler to give rise to interference that produces the transmission zero.

Fig. 3 .
Fig. 3. Abstract photonic circuit used to derive the T-matrix of the tapped-filter: (a) schematic of a 2-ring filter with 3 input and 3 output ports; (b) graphical representation of the drop-port zero location in the complex-δ ω plane.

#Fig. 4 .
Fig.4.Comparison of a pole-zero filter with an asymmetric response, and an all-pole Butterworth filter.In the pole-zero filter, the zero is placed at δ ω/r i = 2.3.

Fig. 5 .
Fig. 5. Constructing a serial demultiplexer with symmetrized, densely packed passbands by using asymmetric response filters: (a) illustration of a two-channel demultiplexer; (b) Channel 1 drop port response, |T 31 | 2 ; (c) Channel 1 through port response, |T 21 | 2 ; (d) Channel 2 drop port response, |T 61 | 2 , showing a highly selective response due to the transmission zero on the right, and through-port extinction of the previous stage on the left.

Fig. 6 .
Fig.6.Design example demonstrating higher bandwidth density and denser channel packing in a serial demultiplexer based on asymmetric second-order filter stages: (a) example demultiplexer design using pole-zero filters shows 20 GHz passbands with 44 GHz channel spacing; (b) example design using conventional, all-pole Butterworth filters (same passband shape and bandwidth) is limited to 106 GHz channel spacing.

Fig. 7 .
Fig. 7. Proposed device topologies that support a 2-pole, 1-zero drop-port response: (a) general, abstract design with all possible (non-trivial) degrees of freedom; (b) tap-coupler implementation (analyzed in detail in this paper), (c) phased parallel-coupled-ring implementation, (d) 2-poles, 1-zero with minimal degrees of freedom.(b-d) are limiting cases of the general geometry in (a).