ACHIEVING on-chip optical signal isolation is a fundamental

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1 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 2, MARCH/APRIL Integrated Nonmagneti Optial Isolators Based on Photoni Transitions Zongfu Yu and Shanhui Fan, Senior Member, IEEE (Invited Paper) Abstrat Based on the effets of photoni transitions, here we show that a linear, broadband, and nonreiproal isolator an be aomplished by spatial temporal refrative index modulations that simultaneously impart frequeny and wave vetor shifts during the photoni transition proess. This paper demonstrates that on-hip isolation an be aomplished with dynami photoni strutures in standard material systems that are widely used for integrated optoeletroni appliations. Index Terms Integrated optial isolator, nonreiproal optis, photoni transition, refrative index modulation. I. INTRODUCTION ACHIEVING on-hip optial signal isolation is a fundamental diffiulty in integrated photonis [1]. The need to overome this diffiulty, moreover, is beoming inreasingly urgent, espeially with the emergene of silion nanophotonis [2] [4], whih promises to reate on-hip optial systems at an unpreedented sale of integration. To reate omplete optial signal isolation requires simultaneous breaking of both the time reversal and the spatial inversion symmetry. In bulk optis, this is ahieved using materials exhibiting magneto-optial effets [5] [10]. However, on-hip integration of magneto-optial materials, espeially in silion in a CMOS-ompatible fashion, remains a hallenge sine magneto-optial materials are not the standard material system used in the CMOS proess. Alternatively, optial isolation has also been observed using nonlinear optial proesses [11], [12], or in eletro-opti modulators [13]. In either ase, however, optial isolation ours only at speifi power ranges, or with either assoiated modulation sidebands or substantial intrinsi loss [13]. In addition, there have been works aiming to ahieve partial optial isolation in reiproal strutures that have no inversion symmetry (e.g., hiral strutures) [14]. In these systems, the apparent isolation ours by restriting the allowed photon states in the bakward diretion, and would not work for This paper should have appeared in Seleted Topis in Quantum Eletronis, Speial Issue on Silion Photonis, January/February 2010, Vol. 16, No. 1. It was inadvertently omitted from the issue. Manusript reeived May 5, 2009; revised June 7, First published November 6, 2009; urrent version published April 7, This work was supported by the National Siene Foundation under Grant ECS Z. Yu is with the Department of Applied Physis, Stanford University, Stanford, CA USA ( zfyu@stanford.edu). S. Fan is with the Department of Eletrial Engineering, Stanford University, Stanford, CA USA ( shanhui@stanford.edu). Color versions of one or more of the figures in this paper are available online at Digital Objet Identifier /JSTQE arbitrary bakward inoming states. None of the aforementioned nonmagneti shemes an provide omplete optial isolation. In this paper, we review and expand our reent works [15], [16] on reating omplete and linear optial isolation using dynami photoni strutures, where the refrative index of the struture is modulated as a funtion of time and spae. In these works, the temporal profile of the modulation is hosen to break the time-reversal symmetry, while the spatial profile of the modulation is hosen to break the spatial inversion and the mirror symmetry. While the existene of nonreiproity in eletro-opti and aousti modulation has long been noted before [17], the possibility of reating omplete optial isolation based on suh modulation was not expliitly studied and is the subjet of this paper. The underlying physial mehanism of our isolator strutures relies upon the photoni transition effet as indued by refrative index modulation. It was shown theoretially [18] that when photoni strutures are subjeted to temporal refrative index modulation, photon states an go through interband transitions, in a diret analogy to eletroni transitions in semiondutors. Suh photoni transitions have been reently demonstrated experimentally in silion miroring resonators [19]. In our reent work [15], [16], we showed that with appropriately designed band struture, and by hoosing a spatially and temporally varying modulation format that simultaneously imparts frequeny and momentum shifts of photon states during the transition proess, the transmission behavior of a photoni struture an beome nonreiproal. As seen in the finite-differene timedomain simulations [20], when a silion waveguide is under a modulation, the details of whih will be desribed in the following setions of the paper, light of frequeny ω 1 in forward diretion is onverted to a higher frequeny mode ω 2 by the modulation [see Fig. 1(a)]. At the same time, light of frequenies ω 1 or ω 2 in the bakward diretion are not affeted by the modulation [Fig. 1(b) and ()]. Combined with an absorption filter entered at ω 2, this struture an absorb all light inident from one diretion at ω 1, while passing those in the opposite diretion, and thus, reates a omplete isolator behavior. It was also shown that the finite-differene time-domain simulations an also be well reprodued by oupled mode theory [15]. In this paper, we will use the oupled mode theory to disuss the performane and design onsiderations for dynami isolator shemes. The paper is organized as follows. In Setion II, we review the physial mehanism of using interband transition for optial isolation. In Setion III, we analyze the performane of the isolator inluding the bandwidth and devie size. Finally, in X/$ IEEE

2 460 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 2, MARCH/APRIL 2010 Fig. 1. Finite-differene time-domain simulation of an isolator based on photoni transitions. The box indiates the regions where the refrative index is modulated. Blue/red show the amplitude of eletri fields. Arrows indiate propagation diretions. Setion IV, we demonstrate two alternative isolator strutures to suit different performane requirements. II. THEORY OF THE NONRECIPROCAL MODE CONVERSION IN INTERBAND PHOTONIC TRANSITIONS We start by onsidering the photon transition proess in a silion waveguide. The waveguide (assumed to be 2-D for simpliity) is represented by a dieletri distribution ε s (x), whih is time-independent and uniform along the z-diretion [Fig. 2(b)]. Suh a waveguide possesses a band struture, as shown in Fig. 2(a), with symmetri and antisymmetri modes loated in the first and seond band, respetively. An interband transition between two modes with frequenies and wave vetors (ω 1,k 1 ), (ω 2,k 2 ) loated in these two bands, an be indued by modulating the waveguide with an additional dieletri perturbation ε (x, z, t) =δ(x) os(ωt qz) (1) where δ(x) is the modulation amplitude distribution along the diretion transverse to the waveguide and Ω=ω 2 ω 1 is the modulation frequeny. Fig. 2() shows the profile of the modulation. Suh a transition with k 1 k 2 is referred to as an indiret photoni transition, in analogy with indiret eletroni transitions in semiondutors. We assume that the wave vetor q approximately satisfies the phase mathing ondition, i.e., k = k 2 k 1 q 0. Inthe modulated waveguide, the eletri field beomes E(x, z, t)=a 1 (z)e 1 (x)e i( k 1 z+ω 1 t) +a 2 (z)e 2 (x)e i( k 2 z+ω 2 t) (2) where E 1,2 (x) are the modal profiles, satisfying the orthogonal ondition (For simpliity, we have assumed the TE modes where the eletri field has omponents only along the y-diretion.) v gi ε(x)ei E j = δ ij. (3) 2ω i In (3), the normalization is hosen suh that a n 2 is the photon number flux arried by the nth mode. By substituting (2) into Fig. 2. (a) Band struture of a slab waveguide. (b) Struture of the silion (ε s =12.25) waveguide. Modulation is applied to the dark region. () Modulation profile at two sequential time steps. the Maxwell s equations, and using slowly varying envelope approximation, we an derive the oupled mode equation d dz ( a1 where a 2 ) 0 i π exp( i kz) = 2l i π exp(i kz) 0 2l l = 4π ε 0 δ(x)e 1(x)E 2 (x)dx ( a1 is the oherene length. With an initial ondition a 1 (0) = 1 and a 2 (0) = 0, the solution to (4) is [ ( ) a 1 (z) =e iz k/2 z os π2 +(l k) 2l 2 ( )] l k + i π2 +(l k) sin z π2 +(l k) 2 2l 2 ( ] a 2 (z) =ie iz k/2 π π2 +(l k) sin z π2 +(l k) 2 2l 2. a 2 ) (4) (5) (6)

3 YU AND FAN: INTEGRATED NONMAGNETIC OPTICAL ISOLATORS BASED ON PHOTONIC TRANSITIONS 461 Fig. 3. (a) Spatial evolution of the photon number flux of two modes (dashed line: mode 1 and solid line: mode 2) when a phase mathing modulation is applied to the waveguide. (b) Maximum photon flux in mode 2 for nonzero phase mismath. In the ase of perfet phase mathing, i.e., k =0, a photon initially in mode 1 will make a omplete transition to mode 2 after propagating over a distane of oherene length l [see Fig. 3(a)]. In ontrast, in the ase of strong phase mismath, i.e., l k 1, the transition amplitude is negligible [Fig. 3(b)]. The system desribed earlier exhibits strong nonreiproal behavior: the modulation in (1) does not phase math the mode at (ω 1, k 1 ) with any other mode of the system [Fig. 2(a)]. Thus, while the mode at (ω 1,k 1 ) undergoes a omplete photoni transition, its time-reversed ounterpart at (ω 1, k 1 ) is not affeted at all. Suh nonreiproity arises from the breaking of both time reversal and spatial inversion symmetries in the dynamis. The modulation in (1) is not invariant with either t t or z z. As a speifi example, we onsider a silion (ε =12.25) waveguide of 0.27 µm wide, hosen suh that the first and seond bands of the waveguide have the same group veloity around wavelength 1.55 µm (or a frequeny of 193 THz). The modulation has a strength δ max /ε s =5 10 4, a frequeny Ω/2π =20GHz, and a spatial period 2π/ q =0.886 µm (all these parameters should be ahievable in experiments). The modulation is applied to half of the waveguide width so that the even and odd modes an ouple effiiently. The modulation length L is hosen as the oherene length l 0 =2.19 mm [Fig. 2(b)] for operation frequeny ω 0 at 1.55 µm wavelength. Fig. 4(a) shows the transmission for forward and bakward diretions. The bandwidth is 5 nm with ontrast ratio above 30 db. For the loss indued by refrative index modulation shemes, e.g., arrier injetion modulation, the ontrast ratio remains approximately the same as the lossless ase, sine the modulation loss applies to transmission in both diretions. Thus, the isolation effet is not affeted. As an example, the modulation Fig. 4. Forward and bakward transmission spetra without (a) and with (b) modulation loss. strength used here δ/ε s = results in a propagating loss of 1.5m 1 in silion [21]. This auses an insertion loss of about 3.5 db while the bandwidth remains approximately unhanged [Fig. 4(b)]. In general, similar nonreiproal effets an also be observed in intraband transitions. However, sine typially Ω ω 1, and the dispersion relation of a single band an typially be approximated as linear in the viinity of ω 1, a asaded proess [19], whih generates frequenies at ω 1 + nω with n > 1, is unavoidable, and it ompliates the devie performane. In ontrast, the interband transition here eliminates the asaded proesses. We would like to emphasize that the modulation frequeny an be far smaller than the bandwidth of the signal. This is, in fat, one of the key advantages of using interband transition. The transition ours from a fundamental even mode to a seondorder odd mode. The generated odd mode an be removed with the use of mode filters that operate based on modal profiles. Examples of suh mode filters an be found in [22] and [23]. It is important to point out that suh mode filters are purely passive and reiproal, and an be readily implemented on hip in a very ompat fashion. Moreover, in Setion IV, we will disuss the implementation of an isolator without the use of modal filters. In order to ahieve the required modulation profile, partiularly the spatial periodi modulation, one an use three uniformly modulated regions for eah spatial period. Eah region is modulated by an osillating signal of frequeny Ω. In addition, the neighboring region has a phase differene of 2π/3. Suh sheme provides a simplified implementation of the modulation profile in (1) with negligible performane differene, as verified by oupled mode alulation. III. DEVICE PERFORMANCE The isolation effet and results of the oupled mode theory have been verified by numerial simulation using the finitedifferene time-domain method [15], whih solves Maxwell

4 462 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 2, MARCH/APRIL 2010 equation without any approximation. Based on the oupled mode theory, we analyze in detail various aspets regarding the performane of the proposed isolator, inluding, in partiular, its operational bandwidth and devie size, later in this setion. A. Bandwidth The dynami isolator struture reates ontrast between forward and bakward propagations by ahieving omplete frequeny onversion only in the forward diretion. As disussed earlier, the modulation is hosen suh that it indues a phasemathed transition from an even mode at the frequeny ω 0 to an odd mode at the frequeny of ω 0 +Ω. The length of the waveguide is hosen to be the oherene length l (ω 0 ) for this transition, suh that omplete onversion ours at this frequeny ω 0 for the inident light. In order to ahieve a broadband operation, one would need to ahieve near-omplete onversion for all inident light having frequenies ω in the viinity of ω 0 as well. From (6), broadband operation, therefore, requires that k(ω) =0 (7) l (ω) =L = l (ω 0 ). The first ondition in (7) implies that the phase mathing ondition needs to be ahieved over a broad range of frequenies, and the seond ondition implies that the oherene length should not vary as a funtion of frequeny. Deviations from these onditions result in a finite operational bandwidth. We onsider the phase mathing ondition first. In the viinity of the design frequeny ω 0, the wave vetor mismath an be approximated by ( ) 1 k = k 1 (ω) k 2 (ω +Ω) q v g1 (ω) 1 ω v g2 (ω +Ω) + 1 ( d 2 ) k 1 (ω) 2 dω 2 d2 k 2 (ω +Ω) ω dω 2 ω 2. =ω0 Thus, to minimize the phase mismath, it is neessary, first of all, that the two bands have the same group veloities, i.e., the two bands are parallel to eah other. Moreover, it is desirable that the group veloity dispersion of the two bands mathes with one another. As a quantitative estimate, assuming that l (ω) L for all frequenies, Fig. 5(a) shows the forward transmission as a funtion of L k. For a transmission below 30 db, this requires a phase mismath of L k <0.1. As a onrete example for omparison purposes, Fig. 6(a) shows the phase mismath L k as a funtion of wavelength for the struture simulated in Fig. 4. Notie that L k <0.1 over a bandwidth of 5 nm due to the mismath of group veloity dispersion in the two guided mode bands. For the seond ondition in (7), we note that in most waveguide strutures, sine the oherene length is determined by the modal profile, it generally varies slowly as a funtion of frequeny. For example, for a waveguide with parameters hosen in Setion II, the oherene length varies less than 2% over 20 nm bandwidth around 1.55 µm wavelength [Fig. 6(b)]. As a simple estimate of how oherene length variation impats devie performane, assuming k(ω) =0overabroad frequeny Fig. 5. (a) Forward transmission as a funtion of phase mismath. (b) Coherene length variation. Fig. 6. (a) Phase mismath and (b) oherene length as a funtion of wavelength for the devie simulated in Fig. 4. range, we alulate the forward transmission as a funtion of oherene length given the modulation length L = l (ω = ω 0 ) [Fig. 5(b)]. For 2% variation of the oherene length, the forward transmission remains below 30 db. Comparing Fig. 6(a) and (b), therefore, we onlude that for the struture simulated in Fig. 4, the 5 nm bandwidth is primarily limited by group veloity dispersion of the two waveguide bands. Sine the struture used in Fig. 4 is rather simple, we believe that substantial further enhanement of operating bandwidth is ahievable by optimization of waveguide geometry.

5 YU AND FAN: INTEGRATED NONMAGNETIC OPTICAL ISOLATORS BASED ON PHOTONIC TRANSITIONS 463 Fig. 7. (a) Transition diagram for low-frequeny modulation. (b) Spatial evolution of photon flux in the bakward diretion for an even mode at 1.55 µm wavelength (dashed line) and an odd mode (solid line) that is 20 GHz lower in frequeny. The struture has the same parameters as desribed in Setion II. B. Devie Size The size of the isolator is determined by the oherene length l. Starting from (5), and taking into aount the normalization of E field [see (3)], the oherene length an be written as 4π l = ε 0 δ(x)e 1(x)E 2 (x) dx = 2π vg1 v g2 γ ω 1 ω 2 λ 0 1 γ v g where γ = δ(x)e 1(x)E 2 (x) dx ε(x) E 1 2 dx ε(x) E 2 2 dx haraterizes the effet of modulation. In deriving (8), we assume that ω 1 ω 2 2π/λ 0, where λ 0 is the wavelength in vauum, sine the modulation frequeny is typially far smaller than the optial frequeny. Moreover, the two bands are assumed to be parallel to eah other, i.e., v g1 v g2 v g. Equation (8) indiates that the devie size is proportional to the group veloity and is inversely proportional to the modulation strength. For a rough estimate, with a modulation strength γ (δ/ε) 10 4, operating at a wavelength of λ 1.5 µm and v g /3, the oherene length l 5mm. To redue the size, one an use stronger modulation strength and/or slow light waveguides. C. Near-Phase-Mathed Transition in the Bakward Diretion In general, due to energy onservation onstraint, a mode with a frequeny of ω 1 an only make a transition to modes at ω 1 ± Ω. In our design, the modulation is hosen to reate a phase-mathed transition in the forward diretion. However, for most eletro-opti or aousti opti modulation shemes, the (8) modulation frequeny Ω 100 GHz is muh smaller than the optial frequeny. Consequently, as an be seen from Fig. 7(a), in the bakward diretion, the transition to the mode in the seond band with a frequeny ω 3 = ω 1 Ω beomes nearly phase-mathed. The wave vetor mismath of this transition is k b = k 2 (ω 1 Ω) + k 1 (ω 1 )+q 2Ω. (9) v g Suh a transition results in loss in the bakward diretion, and thus, a redution of ontrast between the forward and bakward diretions. To alulate suh transmission loss in the bakward diretion, we replae k in (6) with k b. In general, in order to suppress suh bakward transmission loss, one needs to have k b L 1. (10) Combining with (8), the ondition of (10) is then transformed to 2λ 0 Ω 1. (11) γ Remarkably, we note from (11) that for eletro-opti or aousti opti modulation shemes, the effets of weak refrative index modulation γ and low modulation frequeny Ω anel eah other out. The use of weak refrative index modulation results in a long oherene length, whih helps in suppressing the transition proesses that are not phase-mathed. It is preisely suh a anellation that enables the onstrution of dynami isolators with pratial modulation mehanisms. For the example shown in Fig. 4, the near-phase-mathed transition in the bakward diretion has a k b =2π/2.06 mm, and thus, k b L =6.7, whih results in a loss of 0.22 db for the bakward transmission [Fig. 7(b)]. IV. DESIGN FLEXIBILITY In the previous setions, we have shown that by using interband transition, one an reate nonreiproal mode onversion in a waveguide. Suh a waveguide works as an isolator when ombined with a modal filter. The performane of suh a devie an be analyzed and optimized using oupled mode theory. In this setion, we present two examples to show that suh nonreiproal photon transition an be exploited in a wide range of strutures to form nonreiproal optial devies that satisfy diverse performane requirements. In the first example, we design a four-port isolator/irulator using nonreiproal phaseshift in the interband transitions. In the seond example, we use a nonreiproal ring resonator to demonstrate a ompat design for optial isolation. A. Four-Port Cirulator Fig. 8(a) shows the design of a four-port irulator [16]. The struture onsists of a Mah Zehdner interferometer, in whih one waveguide arm is subjet to the dynami modulation desribed previously. In ontrast to the design in Setion III, however, here the length of the modulation region is hosen to be twie the oherene length L =2l. Thus, light passing through the modulated waveguide in the forward diretion

6 464 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 2, MARCH/APRIL 2010 Fig. 8. (a) Shemati and (b) transmission spetrum of a four-port irulator. The dynami index modulation is applied to the waveguide in the dashed red box. will return to the inident frequeny [Fig. 3(a)]. However, suh light experienes a nonreiproal phaseshift due to the photoni transition effet. The use of a Mah Zehnder interferometer onfiguration then allows one to onstrut a irulator. Here, no filter is required, whih signifiantly redues the devie omplexity. For onreteness, we assume that the interferometer has two arms with equal length, and uses two 50/50 waveguide ouplers. For suh an interferometer, the transmission is desribed by [24] ( ) ( )( ) bu = 1 1 i T exp(iϕp ) 0 b l 2 i 1 0 exp(iϕ OUT p ) ( )( ) 1 i bu. (12) i 1 Here, the subsript IN and OUT label the input or output, b u/l are the input or output amplitudes in the upper/lower arm, and ϕ p is the phase aquired due to propagation in the absene of modulation. In (12), the transmission oeffiient through the upper arm has an additional ontribution from the photon transition [ ( ) z T = e iz k/2 os π2 +(l k) 2l 2 l k + i π2 +(l k) 2 b l IN ( ) z sin ] π2 +(l k) 2l 2 (13) whih influenes both the transmission amplitude and the phase as the wave passes through the upper arm. In our design, we assume a phase mathing modulation with k =0for the forward diretion, and use a modulated region with L =2l. Equation (13) shows T = 1. In ontrast, for the light in the bakward diretion in the upper arm, in general, the phase mathing ondition is not satisfied. Hene, T 1. Thus, in this design, the modulation does not reate any frequeny onversion. Instead, Fig. 9. (a) Shemati of ring resonator designed for nonreiproal frequeny onversion. The dark regions are modulated. (b) Shemati of the modes in the ring waveguide oupling region. its sole effet is to indue a nonreiproal π-phaseshift in the upper arm. The interferometer in Fig. 8 exploits suh nonreiproal phase to reate a irulator. We have used the oupled mode theory developed in Setion III to simulate this struture, assuming the same waveguide parameters as in Fig. 4. The results, shown in Fig. 8, indiate that lights injeted into port 1 ompletely output through port 3, while in the time-reversed ase, lights injeted into port 3 end up in port 2. Therefore, this devie has exatly the same response funtion of a four-port irulator [25]. Unlike onventional design, however, no magneti omponents are used inside the struture. Alternatively, the devie an also funtion as a two-port isolator. Fig. 8(b) shows the transmission spetra in both diretions between ports 1 and 4: lights inident from port 4 transmit to port 1, while the reverse transmission is ompletely suppressed. The ontrast ratio for the two diretions is above 30 db for a bandwidth of 5 nm [see Fig. 8(b)]. B. Nonreiproal Ring Resonator As disussed before, the devie size is determined by the oherene length, whih, typially, is above millimeters, unless slow light waveguides are used. Substantial redution of the devie footprint an be aomplished using resonator struture at the expense of a smaller operating bandwidth [15]. As an example, we onsider a ring resonator [Fig. 9(a)] that supports two antilokwise rotating resonanes, at frequenies ω 1 and ω 2, respetively. Eah resonane is further haraterized by its wave vetors k 1 and k 2 in the waveguide that forms the ring. These two resonanes are oupled by applying a dieletri onstant modulation along the ring with a profile δ(x) os[(ω 1 ω 2 )t (k 1 k 2 )z], where z measures the propagation distane on the irumferene of the ring in ounterlokwise diretion. To desribe the ation of this struture, we note that upon ompleting one round trip, the irulating amplitudes a 1,2 and b 1,2 of these two modes [Fig. 9(b)] are related by ( ) ( )( ) a1 T11 T 12 b1 = (14) a 2 T 21 T 22 b 2

7 YU AND FAN: INTEGRATED NONMAGNETIC OPTICAL ISOLATORS BASED ON PHOTONIC TRANSITIONS 465 and width of the waveguide.) A phase mathing modulation is applied to the ring with a oherene length l =2.37 mm. At the design wavelength 1.55 µm, the forward transmission is ompletely suppressed (see Fig. 10). Here, omplete isolation is ahieved with a devie size muh smaller than the oherene length. Fig. 10. Transmission spetra of a ring resonator isolator. ω 0 orresponds to 1.55 µmwavelength. The waveguide ring transmit oeffiient is assumed to be r 1,2 =0.95. where the matrix elements are related to the transition amplitudes for a single round trip, and an be alulated using (4). Eah of these modes is also oupled to an external waveguide as desribed by b 1 r 1 jt a 1 B 1 b 2 = jt 1 r A r 2 jt 2 a 2. (15) B jt 2 r 2 A 2 The external waveguide is also assumed to support two modes with opposite symmetry at the frequenies ω 1 and ω 2, respetively. Here, the subsripts label the two frequenies, and A 1,2 and a 1,2 (B 1,2 and b 1,2 ) are the photon flux amplitudes in the external and ring waveguides before (after) the oupler. The oeffiients r and t are taken to be real [26] and r1,2 2 + t 2 1,2 =1. With inident light in mode 1 (i.e., A 1 =1, A 2 =0)ofthe external waveguide, ombining (14) and (15), we have B 1 = r 1 T 11 r 1 r 2 T 22 + r 2 Det[T ] (16) 1 r 1 T 11 r 2 T 22 + r 1 r 2 Det[T ] where Det stands for determinant. Thus, the ondition for omplete frequeny onversion (i.e., B 1 =0)is r 1 T 11 r 1 r 2 T 22 + r 2 Det[T ]=0. (17) In the ase that ring is lossless, Det[T] = 1 and T 11 = T 22 = os( π L 2 l ), where l is the oherene length and L is the irumferene of the ring. Complete onversion between the two modes an be ahieved when the length of the ring is hosen to be ( ) π L os = r 1 + r 2. (18) 2 l 1+r 1 r 2 With r 1,2 1, L/l 0, the devie, therefore, an provide omplete frequeny onversion even when its length is far smaller than the oherene length. As an example, now we use the same waveguide disussed in Fig. 4 to form a ring with a radius r =12.3 µm. Suh a ring supports two resonant modes: a first band resonant mode at 1.55 µm and a seond band mode that is 50 GHz higher in frequeny. (This is always ahievable by fine-tuning the radius V. CONCLUSION In this paper, we have provided some of the detailed theoretial onsiderations for the dynami isolator strutures that we have reently proposed. In ontrast to previously onsidered isolators based on material nonlinearity [11], [12], where isolation is only ahievable for a range of inident power, the photoni transition effet studied here is linear with respet to the inident light: the effet does not depend upon the amplitude and phase of the inident light. Having a linear proess is ruial beause the devie operation needs to be independent of the format, the timing, and the intensity of the pulses used in the system. In onlusion, the struture proposed here shows that on-hip isolation an be aomplished with dynami modulation in standard material systems that are widely used for integrated optoeletroni appliations. REFERENCES [1] M. Soljai and J. D. Joannopoulos, Enhanement of nonlinear effets using photoni rystals, Nat. Mater., vol. 3, pp , [2] L. Pavesi and D. J. Lokwood, Silion Photonis. Berlin, Germany: Springer-Verlag, [3] V. R. Almeida, C. A. Barrios, P. R. Panepui, and M. Lipson, All-optial ontrol of light on a silion hip, Nature, vol. 431, pp , [4] D. A. B. Miller, Optial interonnets to silion, IEEE J. Sel. Topis Quantum Eletron., vol. 6, no. 6, pp , Nov./De [5] R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood, Jr., and H. Dötsh, Magneto-optial nonreiproal phase shift in garnet/silion-on-insulator waveguides, Opt. 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8 466 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 2, MARCH/APRIL 2010 [17] S. Harris and R. Wallae, Aousto-opti tunable filter, J. Opt. So. Amer., vol. 59, pp , [18] J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, Interband transitions in photoni rystals, Phys.Rev.B,vol.59,pp ,1999. [19] P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, Induing photoni transitions between disrete modes in a silion optial miroavity, Phys. Rev. Lett., vol. 100, pp , [20] A. Taflove and S. C. Hagness, Computational Eletrodynamis: The Finite-Differene Time-Domain Method, 2nd ed. Boston, MA: Arteh House, [21] R. Soref and B. Bennett, Eletrooptial effets in silion, IEEE J. Quantum Eletron., vol. 23, no. 1, pp , Jan [22] Y. Jiao, S. Fan, and D. A. B. Miller, Demonstrations of systemati photoni rystal design and optimization by low rank adjustment: An extremely ompat mode separator, Opt. Lett., vol. 30,pp ,2005. [23] B. T. Lee and S. Y. Shin, Mode-order onverter in a multimode waveguide, Opt. Lett., vol. 28, pp , [24] B. E. A. Saleh and M. C. Teih, Fundamentals of Photonis, 2nd ed. Hoboken, NJ: Wiley, [25] Y. Okamura, T. Negami, and S. Yamamoto, Integrated optial isolator and irulator using nonreiproal phase shifters: A proposal, Appl. Opt., vol. 23, pp , [26] H. A. Haus, Wave and Fields in Optoeletronis. Englewood Cliffs, NJ: Prentie-Hall, Shanhui Fan (M 05 SM 06) reeived the Ph.D. degree in theoretial ondensed matter physis from Massahusetts Institute of Tehnology (MIT), Cambridge, in He was a Researh Sientist at the Researh Laboratory of Eletronis at MIT prior to his appointment at Stanford. He is urrently an Assoiate Professor of eletrial engineering at Stanford University, Stanford, CA. He has authored or oauthored more than 160 refereed journal artiles, has given over 120 invited talks, and holds 34 US patents. His urrent researh interests inlude omputational and theoretial studies of solid state and photoni strutures and devies, espeially photoni rystals, miroavities, and nanophotoni iruits and elements. Dr. Fan is a Fellow of the Optial Soiety of Ameria (OSA), a Fellow of the Amerian Physial Soiety (APS), and a member of The International Soiety for Optial Engineers (SPIE). He reeived a National Siene Foundation Career Award in 2002, a David and Luile Pakard Fellowship in Siene and Engineering in 2003, the National Aademy of Sienes Award for Initiative in Researh in 2007, and the Adolph Lomb medal from the OSA in Zongfu Yu reeived the B.S. degree from the University of Siene and Tehnology of China, Hefei, China, in He is urrently working toward the Ph.D. degree in applied physis at Stanford University, Stanford, CA. He has authored or oauthored more than ten refereed journal artiles. His urrent researh interests inlude silion photonis, dynami photoni strutures, plasmoni devies, and omputational eletromagnetis.