Efficient wavelength multiplexers based on asymmetric ... · Efficient wavelength multiplexers based on asymmetric response filters Mark T. Wade∗ and Miloˇs A. Popovi c´ Department

Efficient wavelength multiplexers basedon asymmetric response filters

Mark T. Wade∗ and Milos A. PopovicDepartment of Electrical, Computer, and Energy Engineering, University of Colorado,

Boulder, Colorado 80309, USA∗[email protected]

http://plab.colorado.edu

Abstract: We propose integrated photonic wavelength multiplexers basedon serially cascaded channel add-drop filters with an asymmetric frequencyresponse. By utilizing the through-port rejection of the previous channelto advantage, the asymmetric response provides optimal rejection of theadjacent channels at each wavelength channel. We show theoretically thebasic requirements to realize an asymmetric filter response, and propose andevaluate the possible implementations using coupled resonators. For oneimplementation, we provide detailed design formulas based on a coupled-mode theory model, and more generally we provide broad guidelines thatenumerate all structures that can provide asymmetric passbands in thecontext of a pole-zero design approach to engineering the device response.Using second-order microring resonator filter stages as an example, weshow that the asymmetric multiplexer can provide 2.4 times higher channelpacking (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 implementationsof our proposed approach, as it affects their applicability to CMOS photonicinterconnects.

© 2013 Optical Society of America

OCIS codes: (130.3120) Integrated optics devices; (130.7408) Wavelength filtering devices;(250.5300) Photonic integrated circuits.

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#186079 - $15.00 USD Received 27 Feb 2013; revised 20 Apr 2013; accepted 22 Apr 2013; published 26 Apr 2013(C) 2013 OSA 6 May 2013 | Vol. 21, No. 9 | DOI:10.1364/OE.21.010903 | OPTICS EXPRESS 10903

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1. Introduction

The integration of photonics and CMOS electronics has been an active area of research in re-cent years [1–7]. With cloud based computing driving the production of larger and more band-width 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 betweenCPUs and RAM has motivated research into photonic CPU-memory communication links [8,9]and the first generation of monolithic electronics-photonics integration [5–7]. Photonic com-munication links, using wavelength division multiplexing (WDM), have the potential to greatlyincrease the bandwidth density and energy efficiency compared to electrical links [10].

At the heart of many photonic communication links and network implementations are wave-length (de)multiplexers typically comprising serially cascaded microring filter stages. Micror-ing filters have a free spectral range (FSR) that is determined by the ring circumference andthe 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 lossdetermine how many WDM channels can fit in one FSR of the microring-based filters. Thetotal bandwidth (and bandwidth utilization, Gbps data/GHz optical bandwidth) increases withan 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 requiresadditional 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 ofa proposed photonic link design. Therefore, it is of interest to investigate methods to achievemaximal 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 thatenables 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, denserby a factor of 2.4 than conventional Butterworth designs of the same order when using second-order stages. We choose a Butterworth response for comparison since it is a commonly usedpassband shape.

To design a pole-zero filter, we directly control the placement of the resonant frequen-cies (poles) and transmission zeros of the filter response (S-matrix element of interest) in thecomplex-frequency plane. Utilizing poles and zeros in the complex frequency plane is famil-iar in electrical and RF/microwave circuits, and has been previously explored in photonic de-vices [11–14]. In this paper, we focus on the simplest way to achieve an asymmetric responseby the introduction of a single transmission zero on one side of the passband. We present acoupling of modes in time (CMT) model and use it to design an asymmetric filter responsethat lends itself to design of WDM demultiplexers with very densely packed channels. Theadvantage of this approach is that it provides a physically intuitive design technique that natu-rally leads to efficient implementations based on various criteria and constraints, as describedlater. The resulting asymmetric response is equivalent to that of previously studied asymmetric-response RF filters based on standing wave cavities [15]. Based on some general guidelines fordesigning pole-zero filters [13], we also consider all possible topologies of a second-order de-sign. These alternative designs have advantages and disadvantages, e.g. in terms of sensitivityto various fabrication parameters, number of degrees of freedom that need to be controlled, etc.

2. Coupling of modes in time model

A CMT model [16] is used to design the filter response, namely, the shape of the passbandand the location of the transmission zero. First, we show what is required in a general photonicsystem to achieve one finite-detuning transmission zero in the drop port, and we present aphysical implementation that can achieve this. We then consider an approximate solution tothe CMT equations for an Nth-order system. The specific device we are interested in is a 2nd-order implementation for which we rigorously derive the CMT model and solve the full designequations.

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 certainnumber 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 notcontribute to the same passband. However, these constraints are artificial and the same approachcan be applied if a single cavity is used to supply multiple resonances that contribute to thepassband, for example.


2nd-orderresonant

system

(2 poles, 2 zeros)

(2 poles, 1 zero)

THRUINPUT

DROP

(a) (b)

IN THRU

DROP

si st

sa

a1

a2

rt

rd

μ

sd

ri

s′d

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

2.1. Designing a response with one transmission zero at finite detuning from the passbandcenter

Consider an abstract photonic circuit representing a filter with one input port and two outputports as shown in Fig. 1(a). The two outputs are the familiar through and drop ports. Since thesystem has two resonances, all of the ports share the same two poles in the complex-frequencyplane. The ports can have 0 to N finite-detuning transmission zeros. Assuming a lossless sys-tem, we choose to constrain the system to have two real zeros in the through port to ensure100% transmission at those frequencies in the drop-port passband by power conservation, witha second-order rolloff. We put one real zero in the drop port transfer function. The zero isplaced on one side of the passband to make the response function asymmetric, for reasons thatwill become clear in Section 3.

Next, one must determine a physical implementation of a photonic circuit that can achievethe desired response. A simple rule can be used to determine the number of finite-detuningtransmission zeros in the response function from the input to a given output port, i.e. in eachs-parameter, S j,input, j ∈ {thru,drop}. In general, the number of finite transmission zeros ineach s-parameter is equal to N, the resonant order of the system, minus the minimum number ofresonators that must be traversed from input to output [13]. Using this rule as a guide, the circuitshown in Fig. 1(b) can achieve the desired transmission response. Specifically, the resonantorder of the system is N = 2, and the minimum number of resonators that the light must passthrough 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 bypassthe second resonator. Using our general rule, this results in: (2 poles)−(1 minimum resonatortraversed to drop) = 1 transmission zero in the drop port response. Similarly, we find two zerosin the through port, which create the familiar rejection band in the same way as for regularserially-coupled ring filters [17, 18].

2.2. Approximate design equations for an Nth-order system

Because the transmission zero is placed off resonance, to enhance the drop-port response re-jection band, it is possible to find a simple model for the position of the transmission zero, byassuming off-resonant excitation of the resonances in the system.

The time evolution of the mode energy amplitudes in a lossless Nth order resonant system of


IN INTHRU THRU IN THRU

DROP

DROP

DROP

(a) (b)

-40 -20 0 20 40-100

-80

-60

-40

-20

0

Detuning [ ]δω/r i

Tran

smis

sion

[dB

]

Fig. 2. Higher-order filters with a drop-port transmission zero: (a) a device architecture thatcan produce one drop-port zero in arbitrarily high order filters; (b) example response of a2nd-order filter using Eqs. 6,7 and a zero placed at δωzd = 10ri.

serially coupled resonators can be written as N first order differential equations,

ddt

a1 = ( jω1 − ri)a1 − jμ12a2 − j√

2ri si

ddt

a2 = jω2a2 − jμ21a1 − jμ23a3

...

ddt

aN = ( jωN − rd)aN − jμ(N)(N−1)aN−1 − j√

2rd sd (1)

where ak is the energy amplitude in the kth ring, ωk is the resonant frequency of the kth ring, ri,d

are the decay rates to the input bus and drop bus, respectively, μkl is the energy coupling rateto the kth ring from the lth ring, si is the amplitude of the input wave, and sd is the amplitude ofthe drop-port output wave. If the resonant frequencies of all rings are set equal, as is the casefor typical square passband responses, the desired filter shape is synthesized through choice ofthe ring-ring couplings, μkl , and the input and drop port decay rates, ri and rd .

When a monochromatic input wave is sufficiently far detuned in wavelength from the pass-band center wavelength, the coupling in each equation is dominated by the forward couplingfrom one ring to another (i.e.

∣∣μ(N)(N+1)aN+1∣∣/∣∣μ(N)(N−1)aN−1

∣∣ << 1). This is because off-resonance the rings do not like to exchange energy (i.e. when the detuning is much larger thanthe coupling rate [19]), so coupling from ring 1 to ring 2 is weak, and back from ring 2 backto ring 1 is weaker still because it is a second-order effect in the detuning-induced suppres-sion of coupling. Hence, in the coupling equations we can assume dominant coupling fromthe ring energy amplitude that is closer to the input bus. This simplifies Eq. 1 by completelydecoupling the equations, and should perfectly recover the response in the off-resonant wingsof the passband (only). Our goal is to design a circuit to achieve one finite transmission zeroin the drop port of the device. Figure 2(a) shows an extension of Fig. 1(b) to achieve this forincreasing order microring filters. In all of these filters, bypassing the Nth ring with a tap at the(N −1)th ring coupled directly to the drop port enables the asymmetric response by ensuring asingle drop-port response function zero. The weaker the tap coupling, the further detuned thetransmission 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 circuitsshown in Fig. 2(a). The modification is an additional term that describes the direct coupling ofthe drop port to the (N − 1)th ring. After further making the foregoing off-resonant approxi-


mation, i.e. that the energy amplitude ak is excited primarily by the previous energy amplitudeak−1, Eqs. 1 can be simplified to

ddt

a1 = j (ω1 + jri)a1 − j√

2risi

ddt

a2 = jω2a2 − jμ21a1

...

ddt

aN−1 = jωN−1aN−1 − jμ(N−1)(N−2)aN−2 − rtaN−1

ddt

aN = j (ωN−1 + jrd)aN − jμ(N)(N−1)aN−1 − j√

2rd s′d e− jφ (2)

where rt is the decay rate to the tap port, φ is the propagation phase accumulated in the inter-ference arm, and s′d [see Fig. 3(a)] is given by

s′d =− j√

2rt aN−1. (3)

The output wave, sd , can be then be found from

sd = s′de− jφ − j√

2rd aN . (4)

Letting d/dt → jω to solve for the steady state frequency response of the system, Eqs. 2–4 canbe solved for the transfer function, Sd,i(ω)≡ sd/si (valid off resonance)

sd

si=

μN−2

jδωN−1 + rt

(N−3

∏k=1

μk

jδωk+1

)− j

√2ri

jδω1 + ri

(√2rd(

jμN−1 +2√

rdrte− jφ)

jδωN + rd−√

2rte− jφ

)

.

(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 partof the root to zero to place the transmission zero on the real frequency axis and introducingδωzd 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 interferencearm [see Fig. 1(b)] as well as the decay rate to the tap port:

cosφ =δωzd√

δω2zd + r2

d

(6)

rt =rdμ2

N−1

r2d +δω2

zd

. (7)

The remaining decay rates and ring-ring couplings can be taken from the standard all-poledesign synthesis techniques [18, 20, 21]. Figure 2(b) shows the transmission for a 2nd-orderpole-zero filter using Eqs. 6 and 7 with a design zero location of δω/ri = 10.

The specific reason for our choice of this implementation is because it converges, in thelimit of zero coupling at the tap, to a standard all-pole design. This means that any fabricationuncertainties introduced in the additional tap interferometer will affect only the position ofthe zero to first order, and will require weak coupling for substantially detuned zeros, makingthe design fairly insensitive to variations (or at least not considerably more so than a standardall-pole design). We will consider alternative geometries in Section 4.


(a) (b)

Im{δω}

Re{δω}√rd

rtμ

−δω′,−rd

)δω +

rd

δωzd

δωzd

φ

si st

s′d sa

s′a

a1

a2

rt

rd

μ

sd

ri

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 representationof the drop-port zero location in the complex-δω plane.

2.3. Rigorous solution of the 2nd-order filter synthesis problem

In the previous section, approximate design equations were derived, and it was shown that afinite transmission zero in the drop port can be achieved with the proper choice of the tap decayrate and the interference phase. This approximate model, however, is not applicable when it isdesirable to place the zero close to the resonant frequency (since the approximate model is validfar from resonance). To derive the full design equations, we begin with a 3× 3 system whosetap 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]:

ddt�a = jH ·�a− jMi ·�s+ (8)

�s− =− jMo ·�a+ I ·�s+ (9)

where I is the identity matrix, and H, Mi, Mo,�a,�s+, and�s− are defined as follows:

H =

[ω1 + j (ri + rt) −μ

−μ ω2 + jrd

]

Mi =

[√2rie jφ1 0

√2rte jφ3

0√

2rde jφ2 0

]

Mo =

⎡

⎣

√2rie− jφ1 0

0√

2rde− jφ2√2rte− jφ3 0

⎤

⎦

�a =

[a1

a2

]

�s+ =

⎡

⎣si

s′asa

⎤

⎦

�s− =

⎡

⎣st

sd

s′d

⎤

⎦ .

(10)


The state variables a1,2 are the energy amplitudes in rings 1 and 2; si, sa, and s′a are input portwave amplitudes; and st , s′d , and sd are output port wave amplitudes. s′a and s′d are temporaryports which will be later connected together. The ring resonant frequencies are ω1 and ω2,the ring decay rates due to coupling to waveguides are ri,rt , and rd , and the related couplingsare given by

√2ri,

√2rt , and

√2rd [18, 19, 22]. The coupling phases, φ1,2,3, can be set to zero

without loss of generality.With d/dt → jω , a transmission matrix T3×3, where�s− = T3×3 ·�s+, can be derived,

T3×3 = I+ jMo ·(

δωI−H)−1 ·Mi. (11)

The coupling arm can now be connected. That is, port s′d is connected to s′a after a finite prop-agation distance. For our 3-port model, this eliminates one input and output port, and reducesthe model to a 2×2 T-matrix, described by

T = P ·(

I−T3×3 ·B)−1 ·T3×3 ·A (12)

where

P =

[1 0 00 1 0

], B =

⎡

⎣0 0 00 0 e− jφ

0 0 0

⎤

⎦ , A =

⎡

⎣1 00 00 1

⎤

⎦ (13)

where φ (≡ βL) is the phase accumulated in the coupling arm. The reduced transmission matrixis [

st

sd

]=

[T11 T12

T21 T22

]·[

si

sa

]. (14)

We are interested in the through port response, T11, and the drop port response, T21, given by

T11(δω) =− j2

√rdrt μ + e jφ [( jrt − jri −δω +δω ′)(− jrd +δω +δω ′)+μ2

]

− j2√

rdrt μ + e jφ [(rt + ri + jδω − jδω ′)(rd + jδω + jδω ′)+μ2](15)

T21(δω) =2√

ri[√

rt (rd − jδω − jδω ′)+ je jφ√rdμ]

− j2√

rdrt μ + e jφ [(rt + ri + jδω − jδω ′)(rd + jδω + jδω ′)+μ2](16)

where δω ′ is the relative detuning of the two rings such that their resonances are at ω1 =ω0 +δω ′ and ω2 = ω0 −δω ′.

We can now require that the through-port zeros be placed on the real frequency axis to ensure100% power transfer to the drop port (in the lossless approximation). To first find the zeros, thenumerator of T11 is set to 0 and solved for δω , giving

δωz,thru =12

[j (rd − ri + rt)±

√− j8e− jφ√rdrt μ − (rd + ri − rt + j2δω ′)2 −4μ2

](17)

This equation produces two constraints that must be satisfied in order to have real zeros in thethrough port:

rd − ri + rt = 0 (18)

Im

{√− j8e− jφ√rdrt μ − (rd + ri − rt + j2δω ′)2 −4μ2

}= 0 (19)

The constraint in Eq. 19 simplifies to

(rd + ri − rt)δω ′+2√

rdrt μ cosφ = 0 (20)


-0

-15 -10 -5 0 5 10 15-40

-30

-20

-10

Tran

smis

sion

[dB

]

THRUDROPBUTTERSPEC

Detuning [ ]δω/r i

DROP

THRUIN

Fig. 4. Comparison of a pole-zero filter with an asymmetric response, and an all-pole But-terworth filter. In the pole-zero filter, the zero is placed at δω/ri = 2.3.

which leads to a design equation that determines δω ′:

δω ′ =−2√

rdrt μ cosφrd + ri − rt

. (21)

We next consider the drop port response T21(δω), given by Eq. 16. Its transmission zero isgiven by

δω =−(δω ′+ jrd)+ e jφ μ√

rd

rt≡ δωzd . (22)

Since φ is not yet determined, this equation for the zero location can be interpreted graphicallyas a root locus in the complex-δω plane. For various φ , the zero is located on a circle withradius μ

√rd/rt centered at −(δω ′,rd), 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 zeroon the real frequency axis:

cosφ =δω ′+δωzd√

(δω ′+δωzd)2 + r2

d

. (23)

as well as for the tap coupling, rt ,

rt =μ2rd

(δω ′+δωzd)2 + r2

d

. (24)

At this point, we have fixed all degrees of freedom of the model. The total list of parame-ters of the device relevant in our synthesis includes ri, μ , δωzd , rt , rd , δω ′ and φ . The firstthree (ri, μ , δωzd) are chosen to be inputs to the model. The choice of ri and μ largely de-termines the passband shape (maximally flat, equiripple, bandwidth, etc.), and these exist inall-pole (serially-coupled) ring filters. Detuning δωzd is the desired location of the drop portzero. Without loss of generality, here, we choose ri and μ to be those of an all-pole 2nd-orderButterworth filter. This is given by ri = μ [18,20]. After the three input parameters are chosen,Eqs. 18,21,23, and 24 are solved for rt , rd , δω ′ and φ using the derived expressions. Figure 4


1 2 4

63

5Channel 1 Channel 2

(a)

(b) (c) (d)

Tran

smis

sion

[dB

]

|T61(δω)|2|T21(δω)|2|T31(δω)|2

Detuning [ ]δω/r iDetuning [ ]δω/r iDetuning [ ]δω/r i

Fig. 5. Constructing a serial demultiplexer with symmetrized, densely packed passbands byusing asymmetric response filters: (a) illustration of a two-channel demultiplexer; (b) Chan-nel 1 drop port response, |T31|2; (c) Channel 1 through port response, |T21|2; (d) Channel2 drop port response, |T61|2, showing a highly selective response due to the transmissionzero on the right, and through-port extinction of the previous stage on the left.

shows the transmission of a representative pole-zero filter, and a 2nd-order Butterworth filter ofequal 3 dB bandwidth for comparison.

In general, when the transmission zero is very close the passband, the ri and μ no longergive exactly the bandwidth and passband ripple that a Butterworth or Chebyshev design setsfor them, but they are close enough for all practical designs that they can either be used as isor adjusted slightly to get the exact desired parameters. The fixing of the zeros ensures thatthe passband is not distorted and has (in principle) complete dropping of wavelengths in thepassband.

3. 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 asym-metric response shown in Fig. 4 to design an efficient serial wavelength demultiplexer. In thedrop port response, the transmission rolls off more slowly than a standard 2nd-order Butterworthresponse on the left side of the passband, and it rolls off much faster on the right side betweenthe center frequency and the zero location. To the right of the zero, there is again an increase intransmission. If this increase in transmission is detrimental in a particular application, e.g. as acrosstalk level, the designer must be able to set bounds on the maximum tolerable out-of-bandtransmission. In general, there is a tradeoff between how close a zero is to the passband (allow-ing a sharper rolloff) and the worst-case off-resonant rejection out of band. For the purposes ofa serial demultiplexer, this will affect the adjacent channel rejection. The zero location in Fig. 4was chosen to ensure a minimum 20 dB adjacent channel rejection.

Using the asymmetric-response filter as a building block, a serial demultiplexer can be de-signed to achieve a symmetrized response in the drop port that has fast rolloff on both sidesof the center frequency. Figure 5 shows a 2-channel serial demultiplexer and the transferfunction for Channel 1 through and drop port, |T21|2 and |T31|2, and Channel 2 drop port,


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

|T61|2 = |T64T21|2, where |T64|2 (and |T31|2) has the response shown in Fig. 5(b). The throughport response of the Channel 1 filter shapes the left side of the drop port response at the Channel2 filter. This outcome is achieved when the channel spacing is set equal to the detuning of thezero from the passband center.

Figure 6(a) shows the drop port responses from a 4 channel serial demultiplexer where eachsuccessive channel has a resonant frequency that is detuned from the previous channel’s centerwavelength by the zero detuning. To make clear the advantage gained from using the pole-zero filters in comparison to conventional all-pole Butterworth filters, it is helpful to convertthe normalized plots shown thus into an actual example implementation. For a filter bank thathas a passband of 20 GHz defined at a 0.05dB ripple and at least 20 dB adjacent channel re-jection, the multiplexer based on pole-zero filters can achieve a channel spacing of 44 GHz, or45% bandwidth utilization. The all-pole 2nd-order Butterworth filter achieves a channel spac-ing of 106GHz, i.e. bandwidth utilization under 19%. The pole-zero filter bank gives a 2.4times 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 demulti-plexers based on second-order filter stages for comparison. It should be noted that although thepole-zero filters were derived from the Butterworth design, the transmission zero causes thereto be a slight ripple in the passband. Comparing the pole-zero filter to a Chebyshev filter withapproximately the same ripple, the channel packing is about 1.8 times denser using a pole-zerofilter bank compared to a Chebyshev filter bank.


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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-couplerimplementation (analyzed in detail in this paper), (c) phased parallel-coupled-ring imple-mentation, (d) 2-poles, 1-zero with minimal degrees of freedom. (b-d) are limiting cases ofthe general geometry in (a).

If the interference arm path length is chosen to give the correct accumulated phase and thetwo rings only need to be slightly tuned to line up their resonance frequencies, then the filtercan be tuned with only 2 heaters on the ring resonators and still achieve reasonable extinctionat the zero location. Since a benefit of this design is in part in the reduction of the numberof thermal tuning elements (thermal power) compared to using a higher order filter, it is ofinterest to investigate both the sensitivity of the zero placing mechanism (the interferometerarm and tap coupling) to fabrication variations, and to investigate the possible geometries thatcan realize the proposed asymmetric response in search of the simplest, and most symmetric,possible implementation.

4. 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 trans-mission zero in the drop-port response with a finite detuning from the passband. Using the rulediscussed in Section 2.1 for the number of finite-detuning zeros in a given S-matrix transferfunction (i.e. in transmission to a given port), in a simple system such as this, it is straight-forward 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 andoutput ports), with each waveguide coupled to each resonator. The designer has access to 5couplings and 2 phases. Not all of the degrees of freedom are needed to design a response withone finite-detuning transmission zero.

Figures 7(b)-7(d) show alternate configurations with Fig. 7(b) being the configuration thatwas analyzed in detail in this paper. The configurations are grouped by which couplings theyuse. All of them share the characteristic that a minimum of one ring must be traversed from theinput port to the drop port. Each configuration has strengths and weaknesses. Figure 7(b) can beviewed 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 cascadedring array naturally rejects signal off resonance (there is no requirement for a carefully tunedphase relationship between the energy amplitudes in ring 1 and ring 2, as is the case in somegeometries like parallel-coupled rings [23]). Figure 7(c) has a robust through port rejectionsince the light is forced to pass through both of the rings, but the 2nd-order portion of the dropport rolloff is sensitive due to the necessity of the proper phase relationship between rings 1and 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 thepassband not usable for bandpass filtering [24,25]. In this work, we propose that this geometrycan be used to design asymmetric bandpass filters, subject to appropriate choice of the phasesbetween the two resonators. The second configuration in Fig. 7(c) uses the same coupling asthe first configuration in part (c). Design of the phases here requires unequal waveguide lengthsand hence some adjustments to the waveguide geometry. This crossed configuration lends itselfto a convenient network topology. Filter designs like this geometry, with waveguides crossedorthogonally (only), have been demonstrated [26], but they have again been used exclusivelyfor all-pole filter design. Our work shows that in principle, pole-zero filters are realizable withdifferent phase shifts. Figure 7(d) is the simplest implementation requiring only 3 couplingpoints which may make it more tolerant to fabrication uncertainties. The degenerate case ofthis implementation has also been previously studied [27]. Although Fig. 7(d) can achieve atransmission zero, it is forced to be between the split resonances of the supermodes of thecavity. Hence, it is not useful for the type of spectral response we are pursuing in this paper. Allof these configurations have a first-order asymptotic rolloff far from the center frequency, andsecond 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 fab-rication processes and conditions, but future work will investigate the best designs in terms ofrobustness to various common types of error in realistic fabrication processes such as thoseseen in monolithic integration efforts [5, 6].

5. Conclusion

We have proposed pole-zero filter designs with asymmetric filter responses, based on coupledresonators on chip. We have developed a physical model and design approach that gives in-sight into passband design and sensitivity. We have also proposed, and through simulationsdemonstrated, the benefit of a pole-zero based design in increasing the bandwidth density ofserial wavelength demultiplexers, without an increase in filter order. A presented example de-sign 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 sameorder. We have also shown that there is a finite number of possible implementations, at an ab-stract level (independent of resonator type), that can produce responses with two poles and onefinite detuning transmission zero in the drop port. These designs can be applied to many physi-cal geometries and implementations and can be extended to apply to standing-wave cavities ina straightforward way. Some of the investigated geometries enforce restrictions on where thezeros can be placed, to the designer’s advantage or disadvantage.

The demand for very dense WDM communication links will push designers to using higherorder filters; asymmetric response filters and (de)multiplexers can provide more densely packedchannels using fewer resonators compared to all-pole designs. Because of the importance of en-ergy 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 tostandard all-pole (and higher order) designs.


Acknowledgments

This work was supported in part by DARPA POEM program award W911NF-10-1-0412; a Uni-versity of Colorado Boulder/NIST Measurement Science and Engineering Fellowship awardedto M. Wade; and University of Colorado College of Engineering startup funding. We thank J.M.Shainline and C.M. Gentry for helpful discussions.


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Efficient wavelength multiplexers based on asymmetric ... · Efficient wavelength multiplexers based on asymmetric response filters Mark T. Wade∗ and Miloˇs A. Popovi c´ Department