optimization of ultrafast all-optical resonator switching

Optimization of ultrafast all-opticalresonator switching

Stephan GuldeIBM Research, Zurich Research Laboratory, Saumerstrasse 4, 8803 Ruschlikon, Switzerland

[email protected]

Asma JebaliCommunication Photonics Group, IFH, Swiss Federal Institute of Technology, ETH-Zentrum,

Gloriastrasse 35, 8092 Zurich, Switzerland

IBM Research, Zurich Research Laboratory, Saumerstrasse 4, 8803 Ruschlikon, Switzerland

Nikolaj MollIBM Research, Zurich Research Laboratory, Saumerstrasse 4, 8803 Ruschlikon, Switzerland

[email protected]

Abstract: We present general optimization arguments for resonator-basedall-optical switching. Several generic resonator geometries, namely Fabry-Perot resonators, circular gratings as well as micro-ring resonators, arediscussed and their particular features highlighted. We establish analyticalmodels which allow a direct comparison of the different all-optical switchgeometries. For the parameter range investigated, we find a clear advantageof photonic band-gap resonators (based on Bragg-type reflection) overmicro-ring resonators (based on total internal reflection).

© 2005 Optical Society of America

OCIS codes: (190.4360) Nonlinear optics, devices; (190.3270) Kerr effect; (230.1150) All-optical devices; (230.5750) Resonators; (320.70800) Ultrafast devices.

References and links1. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley Interscience, New York, 2002).2. R. W. Boyd, Nonlinear Optics(Academic Press, San Diego, 2003).3. G. I. Stegeman and A. Miller, “Physics of All-Optical Switching Devices,” in Photonics in Switching, Vol. 1,

J. E. Midwinter, ed. (Academic Press, San Diego, 1993).4. U. Peschel, T. Peschel, and F. Lederer, “Optimization of bistable planar resonators operated near half the band

gap,” IEEE J. Quantum Electronics 30, 1241-1249 (1994).5. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt.

Lett. 24, 847-849 (1999).6. G. Priem, I. Notebaert, P. Bienstman, G. Morthier, and R. Baets, “Resonator-based all-optical Kerr-nonlinear

phase shifting: Design and limitations,” J. Appl. Phys. 97, 023104-1 (2005).7. M. F. Yanik, S. Fan, M. Soljacic and J. D. Joannopoulos, “All-optical transistor action with bistable switching in

a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506-2508 (2003).8. B. Luther-Davies and M. Samoc, “Third-order nonlinear optical organic materials for photonic switching,” Curr.

Opin. Solid State Mater. Sci. 2, 213-219 (1997).9. D. Vujic and S. John, “Pulse reshaping in photonic crystal waveguides and microcavities with Kerr nonlinearity:

Critical issues for all-optical switching,” Phys. Rev. A 72, 013807-1 (2005).10. D. I. Babic and S. W. Corzine, “Analytic Expressions for the Reflection Delay, Pnetration Depth, and Absorptance

of Quarter-Wave Dielectric Mirrors,” IEEE J. Quantum Electronics 28, 514-524 (1992).11. The reflection phase φ used in this paper is related to the reflection phase θ used in Ref. [10] by φ = −θ .12. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386,

143-149 (1997).

(C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9502#8746 - $15.00 USD Received 15 September 2005; revised 31 October 2005; accepted 7 November 2005

mailto:[email protected]

mailto:[email protected]

13. A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, G.-L. Bona, W. Bachtold, “Lasing in organic circular gratingstructures,” J. Appl. Phys. 96, 3043-3049 (2004).

14. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, “InGaAsP Annular Bragg Lasers: Theory, Applications,and Modal Properties,” IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005).

15. G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel,“Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers,” J.Appl. Phys. 98, 023105-1 (2005).

16. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, “Lasing from a circular Bragg nanocavity with anultrasmall modal volume,” Appl. Phys. Lett. 86, 251101-1 (2005).

17. D. Ochoa, R. Houdre, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, “Diffraction of cylindrical Braggreflectors surrounding an in-plane semiconductor microcavity,” Phys. Rev. B 61, 4806-4812 (2000).

18. J. A. Stratton, Electromagnetic Theory(McGraw-Hill, New York, 1941).19. C. Manolatou and H. A. Haus, Passive Components for Dense Optical Integration(Kluwer Academic Publishers,

Boston, 2002).20. E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” The Bell System Technical Journal, September issue,

2103-2132 (1969).

1. Introduction

Although light propagation in waveguides is a well understood tool for data communication,the all-optical era has yet not been reached for data networks. In optical networks, severalkey components are still lacking for which to date only the electronic counterpart is available.The demand for higher transmission capacity and faster communication networks is growing atan incredible rate. Photonics offers much greater bandwidth than traditional copper networksand can carry multiple signals simultaneously without interference. Therefore, in the futureall-optical switches might play a key role in the transition to optics for interconnects in datatransmission.

In an all-optical switch, an optical signal is directly controlled by another optical signal. Thisrequires coupling between two light pulses. This coupling is mediated by a nonlinear opticalmaterial in the physical device. There is a variety of all-optical switch device types [1, 2, 3]. Oneexample are interferometer-based switches such as a Mach-Zehnder interferometer in whichone arm contains the nonlinear material. In this paper we focus on optical resonator-basedswitches [4, 5, 6, 7]. These have the advantage that they are physically compact and thereforeallow a very high integration density. Another important advantage of resonator-based switchesis the enhancement of optical power inside the optical resonator. This increases the desiredeffect in the nonlinear material which is placed inside the resonator. We focus on the opticalKerr effect, which is a change of refractive index with the applied light intensity. More precisely,the change of the refractive index n is proportional to the optical intensity I and the nonlinearKerr coefficient n2 of the material:

n(I) = n+n2I . (1)

Using a Kerr material inside an optical resonator will lead to an intensity-dependent resonancefrequency. This causes an optical bistability [1] which enables all-optical switching. We do notdeal with further material issues here, i.e. we assume that the material figures of merit [8] interms of low linear and nonlinear absorption are favorable. We rather focus on the physics ofdifferent optical resonator types and their optimization for all-optical switching.

The paper is structured as follows. After a general analysis, which applies to any opticalresonator (Sec. 2), we discuss the particular aspects of several generic devices. We start withan ideal one-dimensional optical resonator, i.e. a Fabry-Perot (FP) resonator with ideal mirrorswhich is filled with the nonlinear material (Sec. 3). In Sec. 4 we consider the more realistic caseof a FP resonator with dielectric quarter wave stack mirrors. Employing dielectric mirrors for aFP resonator significantly influences the all-optical switch performance of the device. In prac-tice, a two-dimensional waveguide-based optical network using integratedoptical resonators as


rm

dIinput

Ioutput

(d)

rc

(c)

IoutputIinput

dHdL

Iinput Ioutput

(b) L

Iinput Ioutput

(a) L

Fig. 1. Four generic optical resonator geometries: (a) an ideal FP resonator where the mirrorplanes have zero thickness; (b) a FP resonator with dielectric mirrors, i.e. quarter wavestacks; (c) a circular grating resonator consisting of concentric rings of alternating refractiveindex, and (d) a micro-ring resonator made of single-mode waveguides. The Kerr nonlinearmaterial is indicated in red in all cases.

switches is of interest. As a generic example we will consider here circular grating resonators(Sec. 5), which can be regarded as the two-dimensional analogue of the dielectric mirror FP(DMFP) resonators. We use transfer-matrix calculations to show that circular gratings behavevery similarly to the DMFP resonators. Next, the particular features of a micro-ring resonatorswitch where the ring is made of a nonlinear waveguide material are highlighted (Sec. 6). InSec. 7 we put together the results of all previous sections for a general performance comparisonof all resonator switch geometries investigated. Finally, in Sec. 8, we conclude our investiga-tions.

2. Optical Resonators for All-Optical Switching

The resonance frequency of a resonator depends on its materials and its geometry. The fourgeometries discussed in this paper are shown in Fig. 1.

Independent of the exact physical design, there are two important parameters for an all-optical switch: the maximum transmission T, which is the transmission on resonance, and thequality factor Q of the device. For a high-quality switch, T should be close to 1. This corre-sponds to a low insertion loss. A transmission smaller than 1 accounts for either losses or anasymmetry in the coupling between the input/output channels and the resonator.

The second parameter, the quality factor Q, is defined as the resonance frequency divided byits full width at half height maximum (FWHM) δν:

Q =νres

δν. (2)

An optical resonator can be used as a switch by shifting its resonance frequency ν0 in and out


of resonance with the frequency of an incoming signal light pulse at frequency νl . In this way,the transmitted signal can be switched on and off. The fastest time scale at which this switchingcan happen is determined by the intrinsic time scale of the optical resonator (related to its decaytime) and reads

τswitch =1

δν=

Qνres

=Qλres

c. (3)

More precisely, a binary logic signal pulse to be switched cannot be shorter than τswitch. Other-wise it would become spectrally too broad with respect to the resonance width δν and conse-quently could only be partially transmitted. Moreover, the transmitted pulse would be spectrallyconsiderably distorted [9]. Eq. (3) is derived under the assumption that the signal pulse has aduration of τswitch and a transform-limited spectral FWHM of δνpulse. Furthermore we haveassumed that the pulse is spectrally more narrow than the cavity resonance by a factor of 2, i.e.δν = 2δνpulse. It is precisely this condition which yields Eq. (3). Note that this result is inde-pendent of the physical design, i.e. the exact geometry of the resonator. We would like to pointout that the above factor of 2 is just a representative choice. In a practical application, wherehigh transmission and low distortion are essential, a higher factor between δν and δνpulse mightbe chosen.

3. Ideal Fabry-Perot Resonator

The resonance frequencies of an ideal FP resonator are given by

νN =c

2nLN , (4)

where the peak order N is a positive integer, n is the refractive index of the medium containedin the resonator, and L is the length of the resonator, i.e. the distance between the two planarmirrors.

Changing the refractive index n by the intensity of the light will change the resonance fre-quencies. This mechanism can be employed to perform the resonance shifting required forswitching. If there is only one light channel this leads to self-switching. Switching one lightpulse with another one in the case of a FP resonator can, for example, be achieved by using twolight beams under a shallow angle and symmetrical incidence with respect to the normal vectorof the FP plane.

A good optimization parameter for any resonator switch is the input channel intensity re-quired for switching, which should be as small as possible. We define Iinput as the steady stateinput channel intensity which will shift a resonance by its width δν, assuming a transmissionof one. This is an approximation, i.e. Iinput is different from the true (average) intensity Iswitch

required for a pulse of duration τswitch to actually achieve switching. In order to determine theexact value of Iswitch the quite complicated temporal dynamics of such bistable switches [1, 9]has to be investigated. In general, Iinput underestimates the true switch intensity, i.e. Iinput will besmaller than Iswitch. Iinput will be close to Iswitch provided that the input pulse is spectrally muchnarrower than the cavity resonance. Even if the input pulse is only spectrally narrower by afactor of 2 [as assumed in Eq. (3)], Iinput is still a good relativemeasure for comparing differentresonators which have the same Q. This is precisely the focus of our paper. We compare opticalresonators of different geometry but the same Q, and look for the geometry with the lowestIinput.

The change of resonance frequency with refractive index is given by the derivative of Eq.(4):

dνN = −νN

ndn. (5)


The change of the refractive index of the material in the cavity by the Kerr effect is

∆n = Icn2 = 2pIinputn2 , (6)

where Ic is the intra-cavity intensity (on resonance) and p is the intensity enhancement factorof the resonator. The factor of 2 reflects the fact that in the resonator a forward and a backwardpropagating wave exist which add up to a standing wave. Precisely speaking, Eq. (6) includes aspacial averaging of Ic ·n2 over one optical period of the standing wave. We will assume high-Qresonators with Q� 1. Then only small relative changes of n are required for switching, and wecan insert Eq. (6) into Eq. (5) using dn= ∆n. Setting the modulus of the resonance frequencyshift |dνN| equal to the resonance width δν, one finds

Iinput =n

2p|n2|δννN

=n

2pQ|n2| . (7)

Note that we assume here that the resonance of order N is used for switching and that Q denotesthe quality factor of exactly this resonance.

Regarding the enhancement p, two regimes can be distinguished: The regime Q/N � 1 isthe high-finesse regime in which [1]

Q/N = F ≈ π1−R

� 1 , (8)

with the finesse F and the reflectivity R of the mirrors. In this regime and on resonance,

p =1

1−R≈ Q

πN� 1 . (9)

In the other regime, where Q/N ≤ 1, p asymptotically approaches 1. We will assume that Q isfixed, its value being given by switch-speed (i.e. bandwidth) requirements [Eq. (3)]. Then wesee from Eq. (7) that in order to reduce Iinput one wants to operate in the high-finesse regime,where p� 1, and then [Eq. (9) into Eq. (7)]:

Iinput = I0N , (10)

withI0 =

πn2Q2|n2| . (11)

Based on this result optimization of the switch is straightforward. With given Q and nonlin-ear material (n and n2), I0 is fixed. Then, a device with the lowest possible N yields the bestperformance, i.e. the lowest Iinput. The lowest possible N is 1, corresponding to a λ /2 resonator.

As explained, the high-finesse regime is advantageous and according to Eq. (8) requires highreflectivities R, say R > 0.95. An ideal FP resonator with such high reflectivities, however,cannot be fabricated because the ideal mirrors are not available: A single refractive index jumpbetween two dielectric materials yields R≈ 0.3 at most. Metallic mirrors with R> 0.95 havesignificant losses in transmission, thus the overall maximum device transmission is then T � 1.

Dielectric mirrors solve this problem as they can be fabricated with very high reflectivitiesand low losses. Using dielectric mirrors for a FP resonator modifies the device behavior, andthis will be discussed in the following section.

4. Fabry-Perot Resonator with Dielectric Mirrors

A dielectric mirror stack consists of alternating layers of two transparent materials that havedifferent refractive indices nH and nL, with nH > nL. Such a stack is optimized for a certain


center wavelength λ0 or center frequency ν0 = c/λ0, where its reflectivity is maximum. Thisis done by choosing the optical thickness of each layer to be λ0/4. In a FP resonator withdielectric mirrors, the distance L between the mirrors will typically be adjusted such that theresonance of interest is near λ0, and we will focus on this case here.

Compared with an ideal mirror, dielectric mirrors have the additional feature that the light-wave penetrates into the dielectric stack at the reflection event. This penetration of the lightwaveinto the dielectric mirror stack can also be expressed in terms of the phase shift upon reflectionφ, which is related to the reflection delay time τ by [10]

τ =∂φ∂ω

, (12)

with ω = 2πν. In the optical frequency range around ν0, the reflection delay time τ is constantto a very good approximation. Furthermore, for high-finesse dielectric mirrors, Babic et al.[10]find an analytical expression for τ in that frequency range:

τ =πω

D , (13)

with a constant D that depends only on the refractive indices of the materials involved:

D =nLI

nHI

nH

nH −nL. (14)

Here, nLI and nHI are the lower and higher refractive indices, respectively, at the interface be-tween the incident medium and the first dielectric mirror layer. The absolute value of φ at themirror center frequency ω0 is either φ0 = 0 or φ0 = π, depending on the sign of the refractiveindex contrast between the incident medium and the first dielectric mirror layer. Altogether weget [11]

φ = φ0 +(ω−ω0)τ . (15)

For a FP resonator with dielectric mirrors, this reflection phase φ adds to the round-tripphaseϕ of light inside the resonator:

ϕ = 2

(nLω

c+φ

)= 2

(nLω

c+(ω−ω0)τ

), (16)

Note that here we dropped a term 2φ0 as it is either 0 or 2π. The resonance condition readsϕ = 2πN:

nLωc

+(ω−ω0)τ = πN . (17)

At ω = ω0 this resonance condition is the same as for a FP resonator with ideal mirrors [Eq.(4)]. However, the gradientof ϕ is now different:

dϕdω

= 2

(nLc

+ τ)

. (18)

Assuming that the resonance of interest is tuned to ω0, we can insert Eq. (4) and get

dϕdω

= 2

(Nπω0

+ τ)

. (19)

Compared with the case of an ideal mirror FP resonator, which corresponds to τ = 0, thisgradient of ϕ has changed by a factor

κ = 1+τωNπ

= 1+DN

. (20)


Here, Eq. (13) has been inserted. The consequence of this change of the round-trip phase gradi-ent is essentially that the entire (transmission) spectrum of the DMFP resonator is compressedin ω by a factor of κ towards ω0 (where it is unchanged). Another consequence is that thechange of the resonance frequency under a change of the refractive index n of the cavity mate-rial is decreased by the factor κ with respect to an ideal mirror cavity:

dνN = − dnνN

n1κ

. (21)

This result can obtained by determining dω/dn from Eq. (17) and evaluating it for ω0 ≈ ω.Similarly, the Q-value for a certain mirror reflectivity R (and in the high-finesse regime) isincreased by a factor of κ :

Q =Nπ

1−Rκ . (22)

The intensity enhancement factor on the other hand is still given by p= (1−R)−1, and therefore

p =Q

πN1κ

. (23)

If we now calculate our optimization parameter Iinput for a DMFP cavity the factor κ enterstwice, once via Eq. (21) and once via Eq. (23), and we get

Iinput = I0 f (N) , (24)

with the “reference intensity” I0 as defined before [Eq. (11)] and the dimensionless function

f (N) = Nκ 2 =(N+D)2

N. (25)

The input intensity Iinput increases by a factor of κ 2 compared with the ideal FP resonator. Notethat a D-value of 0 formally reproduces the ideal FP resonator case.

As done for the ideal mirror FP cavity, we look for the minimum of Iinput [Eq. (24)] assumingthat Q is fixed according to switch speed requirements. Note that under this prerequisite I0contains only material parameters while all geometric parameters are contained in f . Hence, thegoal is to minimize the function f (N), which is depicted in Fig. 2 for several typical dielectricmaterial parameters. In most of the following discussion we will take N as a continuous realnumber greater than or equal to 1, but keep in mind that in reality N is a positive integer number.Fig. 2 illustrates that for an ultra-high refractive index contrast, where D < 1, the minimum off (N) is at N = 1 as in the ideal FP resonator case (D = 0). In this range, i.e. for D < 1 andNopt = 1, the optimized switching intensity is therefore

Ioptinput = I0 × (1+D)2 . (26)

For D > 1, which, in practice, is the more frequent situation, f (N) has a minimum at N = D.Inserting this into Eq. (24), one finds an optimized switching intensity of

Ioptinput = I0 ×4D . (27)

The latter two equations show that the switching intensity Ioptinput depends crucially on D. Con-

sequently, a small D, which is achieved by maximum refractive index contrast at all interfaces[Eq. (14)], is largely beneficial for the switching performance.

Note that according to Eqs. (26) and (27) Ioptinput ∝ (Q2|n2|)−1, which can also be written as

Ioptinput ∝ (τ 2

switch|n2|)−1. This means that an increase of all-optical switching speed by a factor oftwo requires a four times larger |n2| of the Kerr material. This explains why ultra-fast all-opticalswitching to date is limited by the lack of highly nonlinear materials.


D=0 D=0

D=0.8D=0.8

D=2.3D=2.3

D=3.1D=3.1

Order N

f

Fig. 2. The function f = Iinput/I0 for several types of mirrors which are characterized bytheir D-values. The D = 0 curve corresponds to the ideal FP resonator. The other curvesrepresent FP resonators with dielectric mirrors. In all cases, the refractive index of the non-linear material filling the cavity is taken to be n = nLI = 1.7 (representing e.g. a nonlinearpolymer) and the low-index material of the dielectric stack has nL = 1.45 (e.g. quartz). Thedifference between the curves lies in the high-index material of the dielectric stack: D = 0.8corresponds to nH = 3.5 (silicon in the IR), D = 2.3 corresponds to nH = 2.2 (a very highrefractive index dielectric in the NIR), D = 3.1 corresponds to nH = 2.0 (a high refractiveindex dielectric in the VIS/NIR).

5. Circular Gratings

In a FP resonator there is no intrinsic lateral confinement of the optical mode. The actual switch-ing poweris inversely proportional to the lateral confinement, therefore maximum lateral con-finement should be aimed for. This can be achieved in an integrated device where a 2D micro-resonator is coupled to single-mode waveguides for in- and output. Examples of integratedmicro-resonators are photonic crystal cavities [12] and circular grating resonators [Fig. 1(c)].In this paper we will not deal with the problem of coupling, i.e. mode matching between waveg-uide and resonator modes, but assume that this problem can be solved. Moreover, we will focuson circular grating resonators [13, 14, 15, 16]. However, we believe that any photonic crystalcavity will, for the same refractive index contrasts, behave very similarly.

There are different ways to tailor the resonances of a circular grating structure [13, 14]. Thecircular grating geometry that we focus on is illustrated in Fig. 1(c). It consists of a centraldisk of radius rc made of Kerr-nonlinear material of refractive index n (red), surrounded byconcentric rings of alternating refractive indices nH > n (black) and nL = n (blue). All low-index rings (nL) have the same width dL, and all high-index rings (nH) the same width dH.

To model the circular grating resonator we use a 2D transfer-matrix method described inRefs. [13, 17]. We select the TM polarization defined by the electric field vector being perpen-dicular to the object plane. The circular grating is designed to be in the first order, i.e. the opticalthickness of the high- and low-index rings is near λ /4 (see below). From the transfer-matrixmodelling we calculate the “power ratio” (the function Rm of Ref. [13]) describing the ratioof energy density in the inner circle and in the outermost ring as a function of frequency (Fig.3). This allows us to determine, for a given circular geometry, the width ∆ω and the center fre-quency ω0 of the band-gap, as well as the frequency ωRes and order of resonances located inside


Fig. 3. “Power ratio” of a circular grating for the resonance orders m = 0 and m = 1 ofTM polarization. The order m describes the number of azimuthal nodal lines of the mode,i.e. its rotational symmetry. The grating parameters are: nH = 1.95, n = nL = 1.6, q = qopt,rc = 0.81a. It shows that for these example parameters there is a m= 0 resonance but nom= 1 resonance inside the band-gap.

the band-gap. The spectral shape, i.e. the spectral width of these resonances, however, does notyield their Q-values correctly. Therefore, we calculate Q separately and from first principles,namely as [18]

Q =ωResW

P. (28)

Here, W is the energy stored in the resonator and P the energy leakage rate out of the resonator.We can calculate W and P from the transfer-matrix parameters. A detailed and more extensivedescription of the method will be presented elsewhere.

Our goal is to compare circular grating resonators and DMFP resonators for the same refrac-tive indices of the materials involved. In a first step of the circular grating modelling, the dutycycle

q =dH

dL(29)

of the grating was optimized by maximizing the relative band-gap width ∆ω/ω0. In addition,for the optimum duty cycle q the center frequency of the band-gap was determined. All thiswas done for several refractive index combinations (nL, nH) with indices ranging from 1 to 3.5.In all cases we find the same values as for a quarter-wave dielectric mirror stack, namely

qopt =nL

nH, ω0 =

2πca

nH +nL

4 ·nH ·nL, (30)

where a is the full period of the grating, a = dL +dH. This result could be expected and servedprimarily as a confirmation of our model.

Next, we selected one exemplary refractive index combination, nH = 1.95 and nL = 1.6, anddetermined the Q-values of the lowest-order resonance Q0 (Fig. 4). The optimum duty cycle qopt

was chosen. The overall geometry is then fully determined by the number of high-index rings,Z, and the central radius rc [cf. Fig. 1(c), where Z = 4]. The value of rc was chosen such thatthe lowest-order resonance (Fig. 4) was located in the center of the band-gap. This turned out tobe the case for rc = 0.830a. Now the number of rings Z was varied, and Q0 was determined for


Fig. 4. Electric field amplitude in arbitrary units of the lowest-order (m= 0) TM resonanceof a circular grating resonator (see text for geometric parameters).

Table 1. Q-values of corresponding circular grating resonators (Q0) and DMFP resonators[Q(1)]. Z denotes the number periods of the grating or the dielectric mirrors, respectively.

Z Q0 Q(1) Q(1)/Q0

16 1823 2465 1.3520 8849 11960 1.3524 43050 58210 1.3532 1019000 1379000 1.35

each Z-value. The same was done for the corresponding DMFP cavity. For DMFP resonatorsQ(N) can be determined from Eq. (22). The dielectric mirror reflectivities R can be calculatedfrom standard optics formulae [10]. Again the lowest-order resonance (N = 1) was chosen andlocated in the center of the band-gap. The results are displayed in Table 1. It shows that theQ-values of the 1D DMFP resonator and those of the 2D circular grating resonator are similar.More precisely, for all Z the DMFP resonator Q-values are larger by a factor of about 1.35.

Next, in analogy to Eq. (21), we definethe parameter

κν = −(

dνdn

nν

)−1

(31)

for an arbitrary nonlinear optical resonator. Using this definition we determine κν for a circulargrating resonator. Again we focus on the lowest-order resonance and take the refractive indicesnH = 1.95, nL = 1.6. With the resonance in the center of the band-gap for the starting value ofn = 1.6, we varied n by ±0.1. We found that the resonance frequency ν varied linearly withn, and the gradient of this variation allowed us to determine κν according to Eq. (31). Thevalue obtained is κν = 4.1. This can be compared with the κν = κ of the corresponding DMFPresonator (same nH and nL, resonance order N = 1) calculated from Eqs. (20) and (14). Wefind κ = 5.6. This is similar to the κν of the corresponding circular grating resonator. Moreprecisely, the DMFP resonator value is larger by a factor of 1.37. This is very similar to thefactor of 1.35 found above for the difference in Q-values. This suggests that in all relevantrespects the circular grating resonator behaves like the corresponding DMFP resonator with κ


reduced by a factor pκ ≈ 1.36. In Sec. 4 we found that Iinput [Eq. (24)] increases quadraticallywith κ . Therefore we conclude that in terms of switching performance the circular grating isbetter by a factor of p2

κ ≈ 1.8 than the corresponding DMFP resonator .

6. Micro-Ring Resonator

A losslessmicro-ring resonator [Fig. 1(d)] behaves like the idealFP resonator described above,including the optimization arguments. In the corresponding Eqs. (4) to (10), the length L is tobe replaced by the half circumference of the ring and the refractive index n by the effectiveindex neff of the ring waveguide mode. For Eq. (4) this gives

νN =c

2πneffrmN , (32)

where rm is the mean ring radius. A difference to standing wave FP resonators is, however, thatthe factor 2 of Eq. (6) has to be dropped because a ring is a travelling wave resonator. Equation(7) therefore turns into

Iinput =neff

pQ|n2| . (33)

Moreover, the switch optimization then leads to low-order resonances. However, for low ordersN, i.e. small ring radii rm, bending lossesset in as an additional characteristic compared withthe ideal FP resonator. Hence, a compromise has to be made regarding rm.

Bending losses are a particular feature of curved waveguides and increase exponentially withdecreasing radius of curvature. They lead to a reduced transmission T of the device and limitthe Q-value of the micro-ring resonator. The total Q of a ring resonator device is given by

1Q

=1

Q0+

1Qe

. (34)

Q0 is the quality factor of the uncoupled ring resonator, which is essentially equal to the Q-valueof the freestanding ring (limited by bending losses). Qe is the contribution related to couplingwith the propagating modes of the input and output strip waveguides. The transmission of aring resonator which is side-coupled to two waveguides in a symmetric way can be written as afunction of Q0 and Qe [19]:

T =

∣∣∣∣∣1

1+ QeQo

∣∣∣∣∣2

, (35)

and with Eq. (34) one obtains

T =∣∣∣∣1− Q

Q0

∣∣∣∣2

. (36)

Again we assume that Q is fixed by the required switching speed [see Eq. (3)]. For high trans-mission and for the total Q not to be bending-loss-limited, one wants

Q0 � Q. (37)

Regarding the enhancement factor p, in the high-transmission regime (T ≈ 1), the losses in thering can be neglected and Eq. (9) remains valid:

p≈ QπN

. (38)

Inserting this into Eq. (33), we get

Iinput =πneff

Q2|n2| N . (39)


This equation is very similar to the Iinput we obtained above for the FP resonator devices. More-over, if we assume that the same materials are used we get neff ≈ n, and therefore [cf. Eq.(11)]

Ioptinput = I0 ×2N(Q,T,ni) . (40)

N(Q,T,ni) is the smallest possible N (corresponding to the smallest possible ring radius)achievable for certain refractive indices ni and given Q and device transmission T. More pre-cisely, for given Q and T, the Q0 of the freestanding ring can be determined using Eq. (36). Forgiven refractive indices ni this Q0 corresponds to one particular ring radius rm, which in turncorresponds to a certain order

N(Q,T,ni) =2πneffrm

λ(41)

of the ring resonance. A low N(Q,T,ni) is achieved by a high index contrast between ringmaterial and the surrounding media.

7. Comparison of Geometries

In the previous sections we have investigated various all-optical resonator switch geometrieswhich we will compare in this section. In Sec. 5 we analyzed circular grating cavities. Wedid not derive analytical formulae but argued that circular grating resonators, at least for thelowest resonance order, behave similar to, in fact even better than a DMFP resonator in termsof all-optical switching performance.

We will now compare the performance of the FP resonator switches with micro-ring res-onator switches. For the optimized versions of all these geometries we found the quality pa-rameter, the switching input intensity Iinput, to be of the form

Ioptinput = I0 ×Γ , (42)

with the same I0 [Eq. (11)] and a dimensionless parameter Γ, which is different for each geom-etry. Therefore, our comparison of geometries is simplified to a comparison of their Γ-values.For an FP resonator with ideal mirrors we found Γ = 1. The ideal-mirror FP resonator switch isan instructive example. Yet, the ideal mirrors cannot be fabricated with high finesse as desired.A practically realizable device is the DMFP. For this geometry we found

Γ ={

(1+D)2 for D ≤ 14D for D ≥ 1

. (43)

For a micro-ring resonator we found

Γ = 2N(Q,T,ni) . (44)

The Γ of the micro-ring resonator switch depends on Q and T, and therefore we have to selectsome Q and T in order to make a comparison. We will assume ultra-fast switching at a rateof 100 GHz, thus having a logic pulse duration of τswitch = 10 ps, which, using Eq. (3) andassuming the Datacom wavelength of λ = 850 nm, yields Q = 3500. We will furthermore re-quire the micro-ring cavity switch to have a transmission of T = 95%. Equation (35) then givesQ0 = 140 000. We consider a 2D micro-ring and use the results of Marcatili [20] to estimatethe ring radius rm that corresponds to Q0 = 140 000 for different refractive index combina-tions. More precisely, we keep the refractive index of the material inside and outside the ring atnL = 1.6 and vary the index nH of the ring material in nH = {2.4, 2.13, 2.0, 1.92}. At each nH

we select the width of the ring waveguide to be the maximum width at which the waveguide issingle-moded. As done for the circular grating resonators, we select the TM polarization of the


DMFP

micro-ring

Fig. 5. The value of Γ = Ioptinput/I0 of different resonator geometries as a function of the

refractive index contrast D. Γ describes the switch intensity of an optimized geometry inunits of the geometry-independent intensity I0. Squares represent a selection of micro-rings(see text); the dotted line is a guide to the eye. The solid line represents DMFP cavities ofthe same refractive index contrasts [Eq. (43)]. Note that the point {D = 0, Γ = 1}, markedby a green circle, corresponds to the ideal FP resonator.

electromagnetic field. For each nH we calculate rm/λ using Marcatili’s results. From that weget N(Q,T,ni) using Eq. (41) and assuming the effective index of the waveguide mode to beneff ≈ 0.95 ·nH. The results are shown in Fig. 5 along with the Γ-values of the correspondingDMFP switches. The DMFPs were chosen with n = nL and thus

D =nL

nH −nL. (45)

The Γ-values in Fig. 5 are plotted as a function of D. The four nH values chosen for the micro-ring calculations correspond to D = {2, 3, 4, 5}. The Γ-values of the DMFP switch are lowerthan those of the micro-ring resonator switch by a factor of roughly 14 in the examples studied.In other words, the DMFP although its performance is reduced by lower index contrast stilloutperforms the micro-ring resonator switch in our example. Note that only at higher switchspeed (> 100 GHz), i.e. lower Q, the micro-ring performance will improve with respect to thatof the DMFP. At lower switch speed (and with the same T of the micro-ring device), the DMFPperformance will be better by even more than the factor of 14 found here.

8. Conclusions

We have derived analytical formulae that describe the all-optical switching performance ofideal-mirror FP resonators, DMFP resonators and micro-ring resonators. It was found for thelatter two that a high refractive index contrast is beneficial for the switching performance.

Furthermore we have investigated circular gratings as a 2D integrated analogue of DMFPresonators. We have shown that for the parameter set investigated the circular grating resonatorbehaves qualitatively the same as the corresponding DMFP resonator. Quantitatively, i.e. interms of switch performance, the circular grating resonator performs even better than the DMFPresonator. Any integrated optical resonator device based on Bragg-type reflection, includingphotonic crystal or waveguide Bragg-grating-based cavities, will qualitatively behave the sameas a DMFP resonator. This means that (1) a low refractive index contrast is detrimental to


all-optical switching performance, and (2) for extremely high refractive index contrasts theoptimum switch performance is achieved with the lowest spatial resonance order, i.e. a smallsize of the defect that defines the cavity. For decreasing index contrast, the optimum resonanceorder increases.

The simple form of our formulae describing switch performance allows a direct comparisonof the geometries, in particular DMFP versus micro-ring resonators. We found that for switchspeeds of up to 100 GHz the DMFP device clearly outperforms the micro-ring device. Sincewe found before that the example circular grating resonator performed even better than thecorresponding DMFP resonator we expect circular grating resonators to outperform micro-ring resonators even more clearly than DMFP resonators do. Further above 100 GHz, we donot expect a cross-over but rather an asymptotic approach of the performance values betweenphotonic band-gap-based and ring-based devices.

Acknowledgments

We are grateful to Selim Jochim and Rainer Mahrt for many fruitful discussions. We thankthe members of the Photonics Group at IBM’s Zurich Research Laboratory and Daniel Ernifor useful conversations. We gratefully acknowledge funding by the EU PHOENIX projectNr. IST-2001-38919 and the Swiss National Center for Competence in Research - “QuantumPhotonics”.


optimization of ultrafast all-optical resonator switching

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