Optimization of ultrafast all-optical resonator switching - OSA Publishing

Optimization of ultrafast all-optical resonator switching Stephan Gulde IBM Research, Zurich Research Laboratory, Säumerstrasse 4, 8803 Rüschlikon, Switzerland [email protected]

Asma Jebali Communication Photonics Group, IFH, Swiss Federal Institute of Technology, ETH-Zentrum, Gloriastrasse 35, 8092 Zurich, Switzerland IBM Research, Zurich Research Laboratory, Säumerstrasse 4, 8803 Rüschlikon, Switzerland

Nikolaj Moll IBM Research, Zurich Research Laboratory, Säumerstrasse 4, 8803 Rüschlikon, Switzerland [email protected]

Abstract: We present general optimization arguments for resonator-based all-optical switching. Several generic resonator geometries, namely FabryPerot resonators, circular gratings as well as micro-ring resonators, are discussed and their particular features highlighted. We establish analytical models which allow a direct comparison of the different all-optical switch geometries. For the parameter range investigated, we find a clear advantage of photonic band-gap resonators (based on Bragg-type reflection) over micro-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) Alloptical devices; (230.5750) Resonators; (320.70800) Ultrafast devices.

References and links 1. 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).

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Received 15 September 2005; revised 31 October 2005; accepted 7 November 2005

14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9502

13. A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, G.-L. Bona, W. Bächtold, “Lasing in organic circular grating structures,” 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 an ultrasmall 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 Bragg reflectors 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, several key 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 at an incredible rate. Photonics offers much greater bandwidth than traditional copper networks and can carry multiple signals simultaneously without interference. Therefore, in the future all-optical switches might play a key role in the transition to optics for interconnects in data transmission. In an all-optical switch, an optical signal is directly controlled by another optical signal. This requires coupling between two light pulses. This coupling is mediated by a nonlinear optical material in the physical device. There is a variety of all-optical switch device types [1, 2, 3]. One example are interferometer-based switches such as a Mach-Zehnder interferometer in which one arm contains the nonlinear material. In this paper we focus on optical resonator-based switches [4, 5, 6, 7]. These have the advantage that they are physically compact and therefore allow a very high integration density. Another important advantage of resonator-based switches is the enhancement of optical power inside the optical resonator. This increases the desired effect in the nonlinear material which is placed inside the resonator. We focus on the optical Kerr 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 nonlinear Kerr coefficient n2 of the material: (1) n(I) = n + n2 I . Using a Kerr material inside an optical resonator will lead to an intensity-dependent resonance frequency. This causes an optical bistability [1] which enables all-optical switching. We do not deal with further material issues here, i.e. we assume that the material figures of merit [8] in terms of low linear and nonlinear absorption are favorable. We rather focus on the physics of different optical resonator types and their optimization for all-optical switching. The paper is structured as follows. After a general analysis, which applies to any optical resonator (Sec. 2), we discuss the particular aspects of several generic devices. We start with an ideal one-dimensional optical resonator, i.e. a Fabry-Perot (FP) resonator with ideal mirrors which is filled with the nonlinear material (Sec. 3). In Sec. 4 we consider the more realistic case of a FP resonator with dielectric quarter wave stack mirrors. Employing dielectric mirrors for a FP resonator significantly influences the all-optical switch performance of the device. In practice, a two-dimensional waveguide-based optical network using integrated optical resonators as #8746 - $15.00 USD

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(a)

(b)

L

Iinput

Iinput

Ioutput

(c)

L

dH

Ioutput

(d)

dL

Ioutput Iinput

rc

rm

Ioutput Iinput

d

Fig. 1. Four generic optical resonator geometries: (a) an ideal FP resonator where the mirror planes have zero thickness; (b) a FP resonator with dielectric mirrors, i.e. quarter wave stacks; (c) a circular grating resonator consisting of concentric rings of alternating refractive index, and (d) a micro-ring resonator made of single-mode waveguides. The Kerr nonlinear material 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 behave very similarly to the DMFP resonators. Next, the particular features of a micro-ring resonator switch where the ring is made of a nonlinear waveguide material are highlighted (Sec. 6). In Sec. 7 we put together the results of all previous sections for a general performance comparison of all resonator switch geometries investigated. Finally, in Sec. 8, we conclude our investigations. 2.

Optical Resonators for All-Optical Switching

The resonance frequency of a resonator depends on its materials and its geometry. The four geometries discussed in this paper are shown in Fig. 1. Independent of the exact physical design, there are two important parameters for an alloptical switch: the maximum transmission T , which is the transmission on resonance, and the quality factor Q of the device. For a high-quality switch, T should be close to 1. This corresponds to a low insertion loss. A transmission smaller than 1 accounts for either losses or an asymmetry 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 by its full width at half height maximum (FWHM) δ ν : Q=

νres . δν

(2)

An optical resonator can be used as a switch by shifting its resonance frequency ν0 in and out #8746 - $15.00 USD

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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 switching can happen is determined by the intrinsic time scale of the optical resonator (related to its decay time) and reads 1 Q Qλres . (3) τswitch = = = δν νres c More precisely, a binary logic signal pulse to be switched cannot be shorter than τswitch . Otherwise it would become spectrally too broad with respect to the resonance width δ ν and consequently could only be partially transmitted. Moreover, the transmitted pulse would be spectrally considerably distorted [9]. Eq. (3) is derived under the assumption that the signal pulse has a duration of τswitch and a transform-limited spectral FWHM of δ νpulse . Furthermore we have assumed that the pulse is spectrally more narrow than the cavity resonance by a factor of 2, i.e. δ ν = 2δ νpulse . It is precisely this condition which yields Eq. (3). Note that this result is independent of the physical design, i.e. the exact geometry of the resonator. We would like to point out that the above factor of 2 is just a representative choice. In a practical application, where high transmission and low distortion are essential, a higher factor between δ ν and δ νpulse might be chosen. 3.

Ideal Fabry-Perot Resonator

The resonance frequencies of an ideal FP resonator are given by

νN =

c N, 2nL

(4)

where the peak order N is a positive integer, n is the refractive index of the medium contained in the resonator, and L is the length of the resonator, i.e. the distance between the two planar mirrors. Changing the refractive index n by the intensity of the light will change the resonance frequencies. This mechanism can be employed to perform the resonance shifting required for switching. If there is only one light channel this leads to self-switching. Switching one light pulse with another one in the case of a FP resonator can, for example, be achieved by using two light beams under a shallow angle and symmetrical incidence with respect to the normal vector of the FP plane. A good optimization parameter for any resonator switch is the input channel intensity required for switching, which should be as small as possible. We define Iinput as the steady state input channel intensity which will shift a resonance by its width δ ν , assuming a transmission of 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 the exact 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 be smaller than Iswitch . Iinput will be close to Iswitch provided that the input pulse is spectrally much narrower than the cavity resonance. Even if the input pulse is only spectrally narrower by a factor of 2 [as assumed in Eq. (3)], Iinput is still a good relative measure for comparing different resonators which have the same Q. This is precisely the focus of our paper. We compare optical resonators of different geometry but the same Q, and look for the geometry with the lowest Iinput . The change of resonance frequency with refractive index is given by the derivative of Eq. (4): νN dn . (5) d νN = − n #8746 - $15.00 USD

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The change of the refractive index of the material in the cavity by the Kerr effect is ∆n = Ic n2 = 2pIinput n2 ,

(6)

where Ic is the intra-cavity intensity (on resonance) and p is the intensity enhancement factor of the resonator. The factor of 2 reflects the fact that in the resonator a forward and a backward propagating wave exist which add up to a standing wave. Precisely speaking, Eq. (6) includes a spacial averaging of Ic · n2 over one optical period of the standing wave. We will assume high-Q resonators with Q 1. Then only small relative changes of n are required for switching, and we can insert Eq. (6) into Eq. (5) using dn = ∆n. Setting the modulus of the resonance frequency shift |d νN | equal to the resonance width δ ν , one finds Iinput =

δν n n . = 2p|n2 | νN 2pQ|n2 |

(7)

Note that we assume here that the resonance of order N is used for switching and that Q denotes the quality factor of exactly this resonance. Regarding the enhancement p, two regimes can be distinguished: The regime Q/N 1 is the high-finesse regime in which [1] Q/N = F ≈

π 1, 1−R

(8)

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

1 Q ≈ 1. 1 − R πN

(9)

In the other regime, where Q/N ≤ 1, p asymptotically approaches 1. We will assume that Q is fixed, its value being given by switch-speed (i.e. bandwidth) requirements [Eq. (3)]. Then we see 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 = I0 N , with I0 =

πn . 2Q2 |n2 |

(10) (11)

Based on this result optimization of the switch is straightforward. With given Q and nonlinear material (n and n2 ), I0 is fixed. Then, a device with the lowest possible N yields the best performance, 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 high reflectivities 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 jump between two dielectric materials yields R ≈ 0.3 at most. Metallic mirrors with R > 0.95 have significant 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 reflectivities and low losses. Using dielectric mirrors for a FP resonator modifies the device behavior, and this 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 have different refractive indices nH and nL , with nH > nL . Such a stack is optimized for a certain #8746 - $15.00 USD

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center wavelength λ0 or center frequency ν0 = c/λ0 , where its reflectivity is maximum. This is done by choosing the optical thickness of each layer to be λ0 /4. In a FP resonator with dielectric mirrors, the distance L between the mirrors will typically be adjusted such that the resonance 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 lightwave penetrates into the dielectric stack at the reflection event. This penetration of the lightwave into the dielectric mirror stack can also be expressed in terms of the phase shift upon reflection φ , 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 constant to 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=

π D, ω

(13)

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

nH nLI . nHI nH − nL

(14)

Here, nLI and nHI are the lower and higher refractive indices, respectively, at the interface between the incident medium and the first dielectric mirror layer. The absolute value of φ at the mirror center frequency ω0 is either φ0 = 0 or φ0 = π , depending on the sign of the refractive index contrast between the incident medium and the first dielectric mirror layer. Altogether we get [11] φ = φ0 + (ω − ω0 )τ . (15) For a FP resonator with dielectric mirrors, this reflection phase φ adds to the round-trip phase ϕ of light inside the resonator: nLω nLω +φ = 2 + (ω − ω0 )τ , ϕ =2 (16) c c Note that here we dropped a term 2φ0 as it is either 0 or 2π . The resonance condition reads ϕ = 2π N: nLω + (ω − ω0 )τ = π N . (17) c At ω = ω0 this resonance condition is the same as for a FP resonator with ideal mirrors [Eq. (4)]. However, the gradient of ϕ is now different: nL dϕ +τ . =2 (18) dω c Assuming that the resonance of interest is tuned to ω0 , we can insert Eq. (4) and get Nπ dϕ =2 +τ . dω ω0

(19)

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

κ = 1+ #8746 - $15.00 USD

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τω D = 1+ . Nπ N

(20)



Here, Eq. (13) has been inserted. The consequence of this change of the round-trip phase gradient is essentially that the entire (transmission) spectrum of the DMFP resonator is compressed in ω by a factor of κ towards ω0 (where it is unchanged). Another consequence is that the change of the resonance frequency under a change of the refractive index n of the cavity material is decreased by the factor κ with respect to an ideal mirror cavity: d νN = − dn

νN 1 . n κ

(21)

This result can obtained by determining d ω /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) is increased by a factor of κ : Nπ Q= κ. (22) 1−R The intensity enhancement factor on the other hand is still given by p = (1−R)−1 , and therefore Q 1 . (23) πN κ If we now calculate our optimization parameter Iinput for a DMFP cavity the factor κ enters twice, once via Eq. (21) and once via Eq. (23), and we get p=

Iinput = I0 f (N) ,

(24)

with the “reference intensity” I0 as defined before [Eq. (11)] and the dimensionless function f (N) = N κ 2 =

(N + D)2 . N

(25)

The input intensity Iinput increases by a factor of κ 2 compared with the ideal FP resonator. Note that 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)] assuming that Q is fixed according to switch speed requirements. Note that under this prerequisite I0 contains only material parameters while all geometric parameters are contained in f . Hence, the goal is to minimize the function f (N), which is depicted in Fig. 2 for several typical dielectric material parameters. In most of the following discussion we will take N as a continuous real number 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 of f (N) is at N = 1 as in the ideal FP resonator case (D = 0). In this range, i.e. for D < 1 and N opt = 1, the optimized switching intensity is therefore opt

Iinput = 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 opt

Iinput = I0 × 4D .

(27) opt

The latter two equations show that the switching intensity Iinput depends crucially on D. Consequently, a small D, which is achieved by maximum refractive index contrast at all interfaces [Eq. (14)], is largely beneficial for the switching performance. opt Note that according to Eqs. (26) and (27) Iinput ∝ (Q2 |n2 |)−1 , which can also be written as 2 |n2 |)−1 . This means that an increase of all-optical switching speed by a factor of Iinput ∝ (τswitch two requires a four times larger |n2 | of the Kerr material. This explains why ultra-fast all-optical switching to date is limited by the lack of highly nonlinear materials. opt

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D=3.1 D=2.3 D=0.8

f

D=0

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

5.

Circular Gratings

In a FP resonator there is no intrinsic lateral confinement of the optical mode. The actual switching power is inversely proportional to the lateral confinement, therefore maximum lateral confinement should be aimed for. This can be achieved in an integrated device where a 2D microresonator is coupled to single-mode waveguides for in- and output. Examples of integrated micro-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 waveguide and resonator modes, but assume that this problem can be solved. Moreover, we will focus on circular grating resonators [13, 14, 15, 16]. However, we believe that any photonic crystal cavity 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]. The circular grating geometry that we focus on is illustrated in Fig. 1(c). It consists of a central disk of radius rc made of Kerr-nonlinear material of refractive index n (red), surrounded by concentric rings of alternating refractive indices nH > n (black) and nL = n (blue). All lowindex 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 in Refs. [13, 17]. We select the TM polarization defined by the electric field vector being perpendicular to the object plane. The circular grating is designed to be in the first order, i.e. the optical thickness of the high- and low-index rings is near λ /4 (see below). From the transfer-matrix modelling we calculate the “power ratio” (the function Rm of Ref. [13]) describing the ratio of 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 frequency ω0 of the band-gap, as well as the frequency ωRes and order of resonances located inside

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Fig. 3. “Power ratio” of a circular grating for the resonance orders m = 0 and m = 1 of TM 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 no m = 1 resonance inside the band-gap.

the band-gap. The spectral shape, i.e. the spectral width of these resonances, however, does not yield their Q-values correctly. Therefore, we calculate Q separately and from first principles, namely as [18] ωResW . (28) Q= P 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 extensive description of the method will be presented elsewhere. Our goal is to compare circular grating resonators and DMFP resonators for the same refractive indices of the materials involved. In a first step of the circular grating modelling, the duty cycle dH (29) q= dL 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 this was 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 2π c nH + nL , ω0 = , nH a 4 · nH · nL

(30)

where a is the full period of the grating, a = dL + dH . This result could be expected and served primarily as a confirmation of our model. Next, we selected one exemplary refractive index combination, nH = 1.95 and nL = 1.6, and determined 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 that the lowest-order resonance (Fig. 4) was located in the center of the band-gap. This turned out to be the case for rc = 0.830a. Now the number of rings Z was varied, and Q0 was determined for #8746 - $15.00 USD

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Fig. 4. Electric field amplitude in arbitrary units of the lowest-order (m = 0) TM resonance of 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 16 20 24 32

Q0 1823 8849 43050 1019000

Q(1) 2465 11960 58210 1379000

Q(1)/Q0 1.35 1.35 1.35 1.35

each Z-value. The same was done for the corresponding DMFP cavity. For DMFP resonators Q(N) can be determined from Eq. (22). The dielectric mirror reflectivities R can be calculated from standard optics formulae [10]. Again the lowest-order resonance (N = 1) was chosen and located in the center of the band-gap. The results are displayed in Table 1. It shows that the Q-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 define the parameter d ν n −1 κν = − (31) dn ν for an arbitrary nonlinear optical resonator. Using this definition we determine κν for a circular grating resonator. Again we focus on the lowest-order resonance and take the refractive indices nH = 1.95, nL = 1.6. With the resonance in the center of the band-gap for the starting value of n = 1.6, we varied n by ±0.1. We found that the resonance frequency ν varied linearly with n, and the gradient of this variation allowed us to determine κν according to Eq. (31). The value obtained is κν = 4.1. This can be compared with the κν = κ of the corresponding DMFP resonator (same nH and nL , resonance order N = 1) calculated from Eqs. (20) and (14). We find κ = 5.6. This is similar to the κν of the corresponding circular grating resonator. More precisely, the DMFP resonator value is larger by a factor of 1.37. This is very similar to the factor of 1.35 found above for the difference in Q-values. This suggests that in all relevant respects the circular grating resonator behaves like the corresponding DMFP resonator with κ #8746 - $15.00 USD

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reduced by a factor pκ ≈ 1.36. In Sec. 4 we found that Iinput [Eq. (24)] increases quadratically with κ . Therefore we conclude that in terms of switching performance the circular grating is better by a factor of p2κ ≈ 1.8 than the corresponding DMFP resonator . 6.

Micro-Ring Resonator

A lossless micro-ring resonator [Fig. 1(d)] behaves like the ideal FP resonator described above, including the optimization arguments. In the corresponding Eqs. (4) to (10), the length L is to be replaced by the half circumference of the ring and the refractive index n by the effective index neff of the ring waveguide mode. For Eq. (4) this gives c νN = N, (32) 2π neff rm where rm is the mean ring radius. A difference to standing wave FP resonators is, however, that the factor 2 of Eq. (6) has to be dropped because a ring is a travelling wave resonator. Equation (7) therefore turns into neff . (33) Iinput = pQ|n2 | Moreover, the switch optimization then leads to low-order resonances. However, for low orders N, i.e. small ring radii rm , bending losses set in as an additional characteristic compared with the ideal FP resonator. Hence, a compromise has to be made regarding rm . Bending losses are a particular feature of curved waveguides and increase exponentially with decreasing radius of curvature. They lead to a reduced transmission T of the device and limit the Q-value of the micro-ring resonator. The total Q of a ring resonator device is given by 1 1 1 = + . Q Q0 Qe

(34)

Q0 is the quality factor of the uncoupled ring resonator, which is essentially equal to the Q-value of the freestanding ring (limited by bending losses). Qe is the contribution related to coupling with the propagating modes of the input and output strip waveguides. The transmission of a ring resonator which is side-coupled to two waveguides in a symmetric way can be written as a function of Q0 and Qe [19]: 1 2 (35) T = , 1 + Qe Qo

and with Eq. (34) one obtains

Q 2 . T = 1 − Q0

(36)

Again we assume that Q is fixed by the required switching speed [see Eq. (3)]. For high transmission 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 the ring can be neglected and Eq. (9) remains valid: p≈

Q . πN

(38)

Inserting this into Eq. (33), we get Iinput = #8746 - $15.00 USD

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π neff N. Q2 |n2 |

(39)



This equation is very similar to the Iinput we obtained above for the FP resonator devices. Moreover, if we assume that the same materials are used we get neff ≈ n, and therefore [cf. Eq. (11)] opt (40) Iinput = I0 × 2N(Q, T, ni ) . 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 precisely, for given Q and T , the Q0 of the freestanding ring can be determined using Eq. (36). For given refractive indices ni this Q0 corresponds to one particular ring radius rm , which in turn corresponds to a certain order 2π neff rm N(Q, T, ni ) = (41) λ of the ring resonance. A low N(Q, T, ni ) is achieved by a high index contrast between ring material and the surrounding media. 7.

Comparison of Geometries

In the previous sections we have investigated various all-optical resonator switch geometries which we will compare in this section. In Sec. 5 we analyzed circular grating cavities. We did not derive analytical formulae but argued that circular grating resonators, at least for the lowest resonance order, behave similar to, in fact even better than a DMFP resonator in terms of all-optical switching performance. We will now compare the performance of the FP resonator switches with micro-ring resonator switches. For the optimized versions of all these geometries we found the quality parameter, the switching input intensity Iinput , to be of the form Iinput = I0 × Γ , opt

(42)

with the same I0 [Eq. (11)] and a dimensionless parameter Γ, which is different for each geometry. 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 is an 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 (1 + D)2 for D ≤ 1 . (43) Γ= 4D for D ≥ 1 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 select some Q and T in order to make a comparison. We will assume ultra-fast switching at a rate of 100 GHz, thus having a logic pulse duration of τswitch = 10 ps, which, using Eq. (3) and assuming the Datacom wavelength of λ = 850 nm, yields Q = 3500. We will furthermore require the micro-ring cavity switch to have a transmission of T = 95%. Equation (35) then gives Q0 = 140 000. We consider a 2D micro-ring and use the results of Marcatili [20] to estimate the ring radius rm that corresponds to Q0 = 140 000 for different refractive index combinations. More precisely, we keep the refractive index of the material inside and outside the ring at nL = 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 is single-moded. As done for the circular grating resonators, we select the TM polarization of the #8746 - $15.00 USD

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micro-ring

DMFP

opt

Fig. 5. The value of Γ = Iinput /I0 of different resonator geometries as a function of the refractive index contrast D. Γ describes the switch intensity of an optimized geometry in units 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 of the same refractive index contrasts [Eq. (43)]. Note that the point {D = 0, Γ = 1}, marked by a green circle, corresponds to the ideal FP resonator.

electromagnetic field. For each nH we calculate rm /λ using Marcatili’s results. From that we get N(Q, T, ni ) using Eq. (41) and assuming the effective index of the waveguide mode to be neff ≈ 0.95 · nH . The results are shown in Fig. 5 along with the Γ-values of the corresponding DMFP 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 microring calculations correspond to D = {2, 3, 4, 5}. The Γ-values of the DMFP switch are lower than 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 still outperforms the micro-ring resonator switch in our example. Note that only at higher switch speed (> 100 GHz), i.e. lower Q, the micro-ring performance will improve with respect to that of the DMFP. At lower switch speed (and with the same T of the micro-ring device), the DMFP performance 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 of ideal-mirror FP resonators, DMFP resonators and micro-ring resonators. It was found for the latter 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 DMFP resonators. We have shown that for the parameter set investigated the circular grating resonator behaves qualitatively the same as the corresponding DMFP resonator. Quantitatively, i.e. in terms of switch performance, the circular grating resonator performs even better than the DMFP resonator. Any integrated optical resonator device based on Bragg-type reflection, including photonic crystal or waveguide Bragg-grating-based cavities, will qualitatively behave the same as a DMFP resonator. This means that (1) a low refractive index contrast is detrimental to #8746 - $15.00 USD

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all-optical switching performance, and (2) for extremely high refractive index contrasts the optimum switch performance is achieved with the lowest spatial resonance order, i.e. a small size of the defect that defines the cavity. For decreasing index contrast, the optimum resonance order increases. The simple form of our formulae describing switch performance allows a direct comparison of the geometries, in particular DMFP versus micro-ring resonators. We found that for switch speeds of up to 100 GHz the DMFP device clearly outperforms the micro-ring device. Since we found before that the example circular grating resonator performed even better than the corresponding DMFP resonator we expect circular grating resonators to outperform microring resonators even more clearly than DMFP resonators do. Further above 100 GHz, we do not expect a cross-over but rather an asymptotic approach of the performance values between photonic band-gap-based and ring-based devices. Acknowledgments We are grateful to Selim Jochim and Rainer Mahrt for many fruitful discussions. We thank the members of the Photonics Group at IBM’s Zurich Research Laboratory and Daniel Erni for useful conversations. We gratefully acknowledge funding by the EU PHOENIX project Nr. IST-2001-38919 and the Swiss National Center for Competence in Research - “Quantum Photonics”.

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