Linearized electro-optic racetrack modulator based on double injection method in silicon

Transcription

1 Linearized electro-optic racetrack modulator based on double injection method in silicon Roei Aviram Cohen, * Ofer Amrani, and Shlomo Ruschin School of Electrical Engineering, Faculty of Engineering, Tel-Aviv University, Tel Aviv, 69978, Israel * roeicohe@tau.ac.il Abstract: Racetrack-based modulator of increased linearity for optical links is presented and analyzed. The modulator is referred to as FLAME - Finer Linearity Amplitude Modulation Element. Linearity is improved via the introduction of a Double Injection approach. Large spurious-freedynamic-range (SFDR) of 32dB Hz 4/5 can thus be theoretically obtained. The FLAME is studied for silicon platform and requires small footprint size (00 50µm 2 ) and low operation voltage, 2.5V. This makes the FLAME an appealing candidate for large scale integration in RF photonics. 205 Optical Society of America OCIS codes: ( ) Silicon; (30.030) Integrated optics; (30.40) Modulators; ( ) Resonators; ( ) Electro-optical devices; ( ) Waveguide modulators. References and links. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, Micrometre-scale silicon electro-optic modulator, Nature 435(7040), (2005). 2. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor, Nature 427(6975), (2004). 3. T. Tanabe, K. Nishiguchi, E. Kuramochi, and M. Notomi, Low power and fast electro-optic silicon modulator with lateral p-i-n embedded photonic crystal nanocavity, Opt. Express 7(25), (2009). 4. J. Ding, S. Member, R. Ji, L. Zhang, and L. Yang, Electro-Optical Response Analysis of a 40 Gb / s Silicon Mach-Zehnder Optical Modulator, J. Lightwave Technol. 3, (203). 5. R. Soref, The past, present, and future of silicon photonics, IEEE J. Sel. Top. Quantum Electron.2(6), (2006). 6. M. Streshinsky, A. Ayazi, Z. Xuan, A. E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, Highly linear silicon traveling wave Mach-Zehnder carrier depletion modulator based on differential drive, Opt. Express 2(3), (203). 7. T. Ismail and C. Liu, High-dynamic-range wireless-over-fiber link using feedforward linearization, IEEE Technol. J. 25, (2007). 8. C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, Limits on the performance of RF-over-fiber links and their impact on device design, IEEE T. Microw. Theory 54(2), (2006). 9. J. C. Fan, C. L. Lu, and L. G. Kazovsky, Dynamic range requirements for microcellular personal communication systems using analog fiber-optic links, IEEE T. Microw. Theory 45(8), (997). 0. E. Ackerman and A. Daryoush, Broad-band external modulation fiber-optic links for antenna-remoting applications, IEEE T. Microw. Theory 45(8), (997).. A.-R. H and K. S, Radio over Fiber Technologies for Mobile Comm (Artech House, 2002). 2. M. Sauer, A. Kobyakov, and J. George, Radio over fiber for picocellular network architectures, J. Lightwave Technol. 25(), (2007). 3. S. Dubovitsky, W. H. Steier, S. Yegnanarayanan, and B. Jalali, Analysis and improvement of Mach-Zehnder modulator linearity performance for chirped and tunable optical carriers, J. Lightwave Technol. 20(5), (2002). 4. W. Bridges and J. Schaffner, Distortion in linearized electrooptic modulators, IEEE T. Microw. Theory 43(9), (995). 5. F. Vacondio, M. Mirshafiei, J. Basak, Ansheng Liu, Ling Liao, M. Paniccia, and L. A. Rusch, A silicon modulator enabling RF over fiber for 802. OFDM signals, IEEE J. Sel. Top. Quantum Electron. 6(), 4 48 (200). 6. A. Karim and J. Devenport, Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links, IEEE Photon. Technol. Lett. 9(5), (2007). 7. H. Tazawa and W. Steier, Bandwidth of linearized ring resonator assisted Mach-Zehnder modulator, IEEE Photonics Technol. Lett. 7(9), (2005). 8. B. Dingel, N. Madamopoulos, A. Prescod, and R. Madabhushi, Analytical model, analysis and parameter optimization of a super linear electro-optic modulator (SFDR>30dB), Opt. Commun. 284(24), (20). 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2252

2 9. P. Yue, X. Yi, Q.-N. Li, T. Wang, and Z.-J. Liu, MMI-based ultra linear electro-optic modulator with high output RF gain, Int. J. Light Electron Opt. 24(7), (203). 20. A. Gutierrez and J. Galan, High linear ring-assisted MZI electro-optic silicon modulators suitable for radioover-fiber applications, in International Conference on Group IV Photonics (GFP) (202), Vol. 4, pp D. M. Gill, S. S. Patel, M. Rasras, A. E. White, A. Pomerene, D. Carothers, R. L. Kamocsai, C. M. Hill, and J. Beattie, CMOS-Compatible Si-Ring-Assisted Mach Zehnder Interferometer With Internal Bandwidth Equalization, IEEE J. Sel. Top. Quantum Electron. 6(), (200). 22. J. Cardenas, P. A. Morton, J. B. Khurgin, A. Griffith, C. B. Poitras, K. Preston, and M. Lipson, Linearized silicon modulator based on a ring assisted Mach Zehnder inteferometer, Opt. Express 2(9), (203). 23. M. Song, L. Zhang, S. Member, R. G. Beausoleil, S. Member, and A. E. Willner, Nonlinear Distortion in a Silicon Microring-Based Electro-Optic Modulator for Analog Optical Links, IEEE J. Sel. Top. Quantum Electron. 6, 85 9 (200). 24. A. Ayazi, T. Baehr-Jones, Y. Liu, A. E.-J. Lim, and M. Hochberg, Linearity of silicon ring modulators for analog optical links, Opt. Express 20(2), (202). 25. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, R. Cohen, N. Izhaky, J. Basak, and M. J. Paniccia, Recent advances in high speed silicon optical modulator, Proc. SPIE 6477, (2007). 26. R. Soref and B. Bennett, Electrooptical effects in silicon, IEEE J. Quantum Electron. 23(), (987). 27. V. Passaro and F. Dell Olio, Scaling and optimization of MOS optical modulators in nanometer SOI waveguides, IEEE Trans. NanoTechnol. 7(4), (2008). 28. L. Zhang, S. Member, Y. Li, J. Yang, M. Song, R. G. Beausoleil, and A. E. Willner, Silicon-Based Microring Resonator Modulators for, IEEE J. Sel. Top. Quantum Electron. 6(), (200). 29. C. Barrios and M. Lipson, Modeling and analysis of high-speed electro-optic modulation in high confinement silicon waveguides using metal-oxide-semiconductor configuration, J. Appl. Phys. 96(), 6008 (2004). 30. F. Gardes, G. Reed, N. Emerson, and C. Png, A sub-micron depletion-type photonic modulator in Silicon On Insulator, Opt. Express 3(22), (2005). 3. M. Streshinsky, R. Ding, Y. Liu, A. Novack, Y. Yang, Y. Ma, X. Tu, E. K. S. Chee, A. E.-J. Lim, P. G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, Low power 50 Gb/s silicon traveling wave Mach-Zehnder modulator near 300 nm, Opt. Express 2(25), (203). 32. C. Barrios, Electrooptic modulation of multisilicon-on-insulator photonic wires, J. Lightwave Technol. 24(5), (2006). 33. D. Samara-Rubio and M. Paniccia, Scaling the modulation bandwidth and phase efficiency of a silicon optical modulator, IEEE J. Sel. Top. Quantum Electron. (2), (2005). 34. H. Yamada and T. Chu, Optical directional coupler based on Si-wire waveguides, IEEE Photon. Technol. Lett. 7(3), (2005). 35. A. Yariv, Universal relations for coupling of optical power between microresonators and dielectric waveguides, Electron. Lett. 36(4), 32 (2000). 36. K. Ho and J. Kahn, Optical frequency comb generator using phase modulation in amplified circulating loop, Photonics Technol. Lett. IEEE 5, (993). 37. A. Yariv and P. Yeh, Optical Waves in Crystals : Propagation and Control of Laser Radiation (984), Vol. 2, p. xi, 589 p. 38. F. Xia, L. Sekaric, and Y. A. Vlasov, Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators, Opt. Express 4(9), (2006).. Introduction Many electro-optical (EO) devices for telecommunication in Silicon-on-Insulator (SOI) platform have been proposed and demonstrated in recent years by both academy and industry [ 4]. SOI-based EO devices are attractive primarily due to the ability of integrating them with silicon-based circuits manufactured in traditional electronic fabrication facilities [5,6]. EO communications can be divided into two typical transmissions architectures: analog and digital transmission. For analog transmission, the linearity of the power at the modulator output is a significant performance factor in the optical link. A common measure for linearity of devices is the spurious-free-dynamic-range (SFDR), which quantifies the maximum dynamic range the device can achieve [7 0]. A compact linear EO modulator with high SFDR performance can be employed in various analog applications including Radio-over- Fiber (RoF) communication, phased array antenna, true-time-delay systems, cable television (CATV) and satellite communication [ 3]. A classical architecture for modulating light via electrical control is by employing the Mach-Zehnder Interferometer (MZI) modulator [4,5]. However, since the inherent output of MZI is a non-linear function of the input power, the linear range of the transmitted output is limited (SFDR = 09.3dB Hz 2/3 ). When operated with bias-shifted level and increased input power, an SFDR of 2dB Hz 2/3 was reported using a Lithium Niobate 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2253

3 (LiNbO 3 ) MZI modulator [6]. However, using low bias level results with high 2nd order harmonic distortion, and higher input power is required. Furthermore, LiNbO 3 -based modulators are typically of large dimensions. For improving linearity, several MZI with ring-assisted schemes (RAMZI) [7 22] have been demonstrated both theoretically and experimentally. The ring resonators, while nonlinear in their own right, are employed for extending dynamic range of the MZI by suppressing the intermodulation term (IM3) via the ring resonance effect. Compared to the FLAME device, these devices have larger footprint, possess higher fabrication complexity for the electrical design, and require the use of multiple voltage supplies. Regular ring resonator schemes have also been analyzed and measured [23,24] using differently chosen parameters, such as the coupling coefficients. A ring resonator, by nature, possesses a more complex interference effect than does a MZI, and hence, it provides a nonlinear transmittance of even smaller linear range. Notably, while there can be found in the literature reports of devices that may theoretically achieve similar SFDR to FLAME (highest is 33.5dB Hz 4/5 RAMZI of [9]), it is the associated electro-optical design and the fabrication complexity of a device that will strongly dominate its SFDR performance in practice. While ring-assisted MZI devices require the optical fabrication of MZI, a coupled ring as well as 2 to 3 electrodes in different sizes and shapes, the FLAME device requires but a single ring and an electrode. The relative simplicity of the FLAME's configuration increases the probability of realizing high SFDR value in practice. 2. FLAME modulator introduction and analysis The FLAME design, depicted in Fig., incorporates a micron-size racetrack Add-Drop Filter (ADF) and a basic optical splitter. The splitter is required to split a single-mode light wave into two equal-power single-mode waves. Here we employ a traditional Y-coupler (3dB coupler). A x2 multi-mode interference (MMI) coupler, or a directional coupler (DC), could also be employed. While these couplers typically enjoy smaller footprint and lower losses, they do impose bandwidth limitation and demonstrate lower fabrication tolerance [25]. Furthermore, an MMI coupler may generate reflections that may degrade the linearity of the device The two waves travel through the waveguides towards the resonator, where they are injected into the racetrack in an opposite directions (Double Injection method). Because the injected waves are of the same wavelength, they interfere with each other inside the racetrack and thus increase the diversity of independent parameters that may, in turn, be employed for obtaining a variety of interesting transmission functions. For a FLAME design, 0 parameters participate in shaping the transmission function: the injected fields amplitude and their phase difference (E i, E i2, Φ i ), the coupling coefficients amplitude (τ, τ 2 ), the coupling coefficients phase (φ τ, φ τ2, φ κ, φ κ2 ), and the loss coefficient (α). In this paper we try to optimize these parameters with the aim of achieving an improved linear transmission. The two waveguides exiting the Y-coupler are curved so as to deliver equal amplitudes and phases to the racetrack. The signal of interest comes out of the throughput port, i.e. E t. The light continuing to the drop port, i.e. E t2, must be attenuated in order to suppress recoupling to the racetrack via reflections. The attenuation, schematically portrayed by the purple rectangular in Fig., can be realized by manipulating the waveguide geometry to yield high losses, or by coupling the mode to the radiation modes. It is also possible to deposit metals or other materials with high absorption quality. 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2254

4 Fig.. Schematic illustration of the FLAME modulator (not to scale). The modulator consists of a Y-coupler, racetrack ADF resonator and is driven by a MOS capacitor electrode. In silicon, the refractive index can be altered by means of the plasma dispersion effect [26]. As such, the refractive index of the racetrack's waveguide can be altered by an electrical structure subjected to applied voltage, consequently changing the phase accumulated by the light travelling through the racetrack. Among the common electrical controls are the PN/PIN diode and the MOS capacitor. Diodes are inherently nonlinear devices, whereas a MOS capacitor, operating in accumulation mode, exhibits a linear response in voltage carrier relation [27,28]. A MOS capacitor and a PN diode reach modulation frequencies in order of tens of gigahertz [29,30]. In most designs, however, the capacitor suffers from higher optical losses due to the short gap (<200nm) between the waveguide core and the gate. A PIN diode operating in injection mode (forward biased), can inflict the largest change in refractive index, and hence it may yield a smaller device. On the downside, it is limited in modulation speed due to minority carrier mechanism. In order to achieve the best linearity possible we chose a MOS capacitor design under accumulation mode. To support the MOS electrical control we consider a rib type waveguide, designed for single mode operation. A detailed description of the capacitor model we employed is given in [27]. Due to the linear relation between the applied voltage and the effective index of the optical mode, the linearity of the modulator strongly depends on the optical layout of the device. 2. Device dimensions The dimensions of the FLAME depend mostly on the electrical control segment of the device in-charge of modulation. A tradeoff exists between the length of the electrode surrounding the racetrack and the applied voltage required for utilizing the maximum possible SFDR. Lower operation voltage requires longer electrode and thus a longer racetrack, consequently increasing the footprint of the device. In silicon technology and for the sake of CMOS compatibility, low operation voltage is of major importance and much research have been carried in recent years [3] in an attempt to lower it. For our MOS capacitor model, a spatial carrier density with its steady-state value linearly proportional to the peak-to-peak value of the driving voltage is used so that the effective index change is calibrated to be at 2.5V (.2V bias). Similar performance can be achieved using several other proposed electrode designs [32,33]. The oxide thickness is 0.5nm and the rib waveguide cross-section is nm with a slab height of 230nm. A switching rate in excess of 40G signals per second is achievable. The figure of merit is V π L c = 0.692[V cm], where V π is the half-wave voltage and L c, is the electrode length. 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2255

5 The model devised for the FLAME modulator suggests that the maximum SFDR is obtained when the modulating signal is applied at the center-point of optical power transmission function (0.5V π ). This is commonly obtained by electrically biasing the modulator or by designing the center point of the transmission function to fall on the optical wavelength. We define V SFDR-Max as the maximum voltage that can be applied to the modulator whilst at its center point before IM3 distortion appears above the noise floor. The model shows that the SFDR is constant at 32dB Hz 4/5 and V SFDR-Max /V π = 6.0%, respectively, for any V π and any electrode length. Consequently, the minimum electrode length required to utilize the maximum possible SFDR is given by VSFDR Max λ0 Lc (min) =, () V 2Δn where n c is the induced change in effective-index obtained by the electrical control and λ 0 is the vacuum wavelength. For our modulator, operating under V pp = 2.5V and optical wavelength of 550nm, the minimum electrode length would be 66µm. Note that by driving the capacitor at V pp = 5V, the electrode length reduces in half. The racetrack resonator comprises of two directional couplers. The coupling coefficients modeling these couplers depend on the gap, waveguide cross-section and interaction length of each coupler unit. For similar waveguide cross-sections and a gap of 200nm, a complete power transfer occurs between the straight waveguides after a distance of 5µm [34]. In order to form a racetrack shape, both coupling units need to be of the same length. The electrical electrode is required not to interact with the coupling units; hence, an addition of 0µm has been added to the racetrack perimeter. Given the minimum required electrode length, common dimensions for the DC and Y couplers, the total footprint of the FLAME modulator can be found to be as small as 00 50µm 2. Due to the resonator design, these footprint are about one order of magnitude smaller than MZI-based linear modulators. 2.2 Basic model Basic mathematical model for the FLAME device can be based on Yariv's [35] analytical model for a micro-ring resonator. In steady state, E(ω), the electromagnetic field dependence on the frequency for an ADF can be shown to be * iθ * iθ2 ( τ τ αe ) κκα e π 2 iφ 2 2 i iφi2 t = * * i iθ * * iθ i2 ττα 2 e ττα 2 e E E e E e where τ and κ are the transmitted and coupled lumped coefficients of the directional coupler involving the bus waveguide and the racetrack structure, respectively. α represents the loss mechanism that the light experiences while traveling through the ring, and θ(ω) is the phase accumulated by the light per unit round. E i,e i2 and Φ i,φ i2 are the amplitudes and phases of the fields arrived from the Y-coupler respectively. The subscript 2 denotes half a ring. The power at the throughput port follows directly from Eq. (2): where P E ( ) E E η η t = t = τ + τ2α 2ττ 2α cos θ ϕτ ϕτ2 i + κκ2 α i2 ( θ τ ) ( θ Σ τ Σ) 2 κκ 2 ατ cos ϕ + ϕ τα 2 cos ϕ 2 ϕ E 2, 2 2 i Ei η c ( ) τ τ2 η = + ττ α 2ττ α cos θ ϕ ϕ ϕ = ϕ ϕ Φ +Φ. Σ κ κ2 i i2, (2) (3) (4) 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2256

6 The top line in Eq. (3) represents constant power contribution emanating from the completion of throughput and drop ports (energy conservation). The bottom line represents the power carried by the interference between the two injected inputs, which is due to the counterpart Double Injection method. Figure 2(a) demonstrates the optical transfer function of the FLAME plotted as a function of the racetrack phase with τ = 0.650, τ 2 = and E i = E i2 = (0.5) /2, while the phases φ τ, φ τ2 and φ Σ are set to zero. These parameters were optimized via simulations with the aim of linearizing the transmission of the device by trying to extinct the 3rd order intermodulation. The loss coefficient α is dependent on the length, shape and type of electrical control applied to a modulator. It is hence common in the literature to set α = (lossless modulator) in order to be able to compare different modulators [7 9, 23]. The FLAME performance is hence analyzed for the case α =, as well as for a lossy case where the loss is assumed to be linearly dependent on the applied voltage, as described in Sec. 3.. The device Free-Spectral Range, Finesse, Full-Width-Half-Maximum, Q-Factor and photon lifetime are 0.5nm,.75, 6nm, 258 and 0.2ps, respectively. 2.3 Double Injection method In order to identify the contribution of the various terms in Eq. (3) on the transfer function of the FLAME, we can plot each term of P t separately as ( A B ) t t i i2 C i Ei2 P = E = E + E E (5) Fig. 2. (a) Optical transfer function of the FLAME modulator as a function of the phase. (b) Transmission versus phase plot of the optical base term and the interference term which is present due to the Double Injection method. Figure 2(b) clearly demonstrates that the shape of the transfer function is dominated by the counter directional Double Injection term, C E i E i Low frequency modulation The accumulated phase per unit round, θ, appearing in Eq. (2) and the applied voltage in steady state, or under low frequency modulation, are interrelated as follows: V θ = π + θ (6) V π where θ 0 = 2πn eff L/λ 0 with n eff and L being the effective index and perimeter of the racetrack waveguide, respectively. Figure 3 shows the steady-state intensity transfer function of the FLAME modulator compared with a standard MZI modulator. The improvement in linearity of the FLAME modulator is readily observed from the figure. 0, 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2257

7 Transfer Function FLAME MZI Voltage Fig. 3. Low frequency transfer function of the FLAME and MZI modulators in normalized units. 2.5 High frequency modulation (Cavity Dynamics) For calculating the SFDR of the FLAME while properly considering effects associated with high frequency modulation, it is necessary to model the dynamic response of the resonator. We analyze the transient time of the resonator by employing the multiple-rounds approach [36]. The transient time, associated with the device cavity dynamics is related to the photon lifetime in the racetrack. The analysis is aimed at accounting for the mismatch between the duration it takes the light to reach steady state and the rate of change of the applied electrical modulation signal. Using the multiple-rounds approach, the equation describing the output field can be shown to be θ () t 2 n n i n κ n * * iθn () t 2 * * * iθn () t Et() t = τ * ( ττα 2 e ) Ei() t e κκ 2 α ( ττα 2 e ) Ei2() t, τ n= n= (7) with n being the number of rounds the light traversing the ring. Note that by using common series formulas, Eq. (7) can be reduced to the steady state equation, Eq. (2). Simulations reveal that after 0 rounds the output field practically converges to the asymptotic state (n ). In case of a lumped electrode, the phase velocity of the electrical and optical waves may not be in sync, and hence the electro-optic effect is lessened. The model we employed for dealing with the phase velocity mismatch is based on Yariv [37]. For the ring resonator, it is imperative to account for the multiple rounds the light traveling through the ring. The terms involving the phase in the field expression, Eq. (7), can then be written in the following form V sin( ( n ) ψm) sin( ( n ) ψm2) n n i n ( n ) θ0 π sin( ωmt ( n ) ψm) sin( ωm2t ( n ) ψm2) + + * * iθ () * * n V n t π ψm ψ m2 ( τταe ) = ( ττα) e 2 2 n= n= (8) e sin ψm sin ψm2 θ0 V 2 2 ψ i sin m ψ + π ω 2 sin m m t ω 2 () 2 2 m t + θ t V 2 π ψm ψm2 i 2 e. = (9) Here, a two-tone sinusoid signal of frequencies ω m and ω m2 modulate the optical signal; ψ m = ω m n Si L/(2c) is the phase velocity factor, c is the speed of light and n Si is the refractive index of the silicon layer. Note that the effective electrode length is reduced by a factor of sin(ψ m )/ψ m depending on the modulation frequency. 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2258

8 3. FLAME linearity (SFDR) In order to quantify the linearity of the FLAME we shell study its dynamic range behavior and obtain knowledge about its inter-modulation distortions. Table details the optical link parameters typically used to evaluate the SFDR [7 9,22,23]. Table. Optical Link Parameters Used to Evaluate SFDR Parameter Symbol Value Unit Laser Power P laser 00 mw Laser Noise RIN 60 dbm/hz Total Optical Loss IL 0 db Modulator Impedance R m 50 Ω Modulator Responsivity r m.0 A/W Detector Impedance R D 50 Ω Detector Responsivity r D 0.7 A/W Noise Bandwidth BW Hz Half Voltage V π 0 V Two tones electrical modulation under low frequency, which is commonly set at f m = 0.9Hz, f m2 =.0Hz, are applied to the model described in Sec. 2.5 via the phase term, θ(t). A dominant factor affecting modulator performance is the 3rd order intermodulation (IM3) distortion carried by the frequencies 2f m f m2 and 2f m2 f m. By substituting Eq. (8)-(9) into Eq. (7) and using Bessel function identities, we derive the transmitted RF signal-power of the FLAME for any modulation frequency: n 2 κ * Psignal Ei J0( δ *, n ) J( δ2, n ) + Ei 2κκ2 αj0( δ, n ) J0( δ2, n ) J0( δ2, n ) J2( δ, n ) γn n= τ (0) Similarly, the IM3 power is given by n 2 κ * PIM 3 Ei J ( δ * 2, n ) J2 ( δ, n ) + Ei 2κκ 2 αj0 ( δ, n ) J0 ( δ2, n ) J ( δ, n ) J2 ( δ2, n ) γn, n= τ () where δ V sin (( n ) ψ ) and γ cos((2n 3) ψ). This derivation is significantly more involved compared to other modulator models iθ known in the art [7,23]. The reason for this is because of the interference term e iθ/2 Σe which is present due to the Double Injection method. Note that Eq. (0)-() show only some terms of the complete equations. Figure 4 can provide the SFDR behavior for the FLAME and MZI modulators using the parameters specified in Table. 2 2 Fig. 4. SFDR performance for FLAME (blue) and MZI (green) modulators under low frequency modulation. Solid lines represent the signal power and the dotted lines the intermodulation (IM3) power. 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2259

9 The SFDR obtained for the FLAME modulator is 32dB Hz 4/5 yielding additional 22.7dB compared to the MZI. For higher modulation frequencies, the transient time (cavity dynamics) and the phase velocities mismatch of the modulator can affect the behavior of the SFDR. The SFDR dependence on the modulation frequency for various driving voltage is shown in Fig. 5. The SFDR of the FLAME decreases as the frequency increases. Nevertheless, the FLAME still offers improved behavior as compared to the MZI for modulation frequencies in excess of 00GHz. It is possible to optimize the SFDR for a narrow bandwidth of modulation frequencies by properly adjusting the length of the electrical electrode and the driving voltage. Note that the while the MOS capacitor is typically limited in bandwidth to 40GHz, in Fig. 5 we wish to demonstrate the potential of the optical structure to perform under the high frequency effects described above. The phase velocity factor can be expressed proportionally to the ring's FSR, ψ ω m / FSR. When looking at normalized frequencies (normalized by the FSR), we obtain that the SFDR of the FLAME (~0dB) approaches that of the MZI (~09dB) at 60% of the FSR. The resulting bandwidth is 4 times better compared to RAMZI modulators, which can reach top frequency of only 5% of the FSR [7]. Note that the SFDR of the MZI is independent of the modulation frequency under the push pull configuration. Fig. 5. SFDR dependence on the modulation frequency for the FLAME and MZI modulators. Cavity dynamics and phase velocities mismatch effects are included. The Double Injection approach was aimed at improving linearity, yet at the same time, it yields a device of relatively low Q-factor. In this case, however, low Q factor may be viewed as an advantage since it comes with an increased operation bandwidth. Thus, the FLAME has improved bandwidth as compared to a high Q ring modulator whose speed is inherently limited by operation. 3. Dynamic and static losses The loss coefficient of the modulator, α, is influenced mostly by the induced loss of the electrical electrode and curved waveguide sections comprising the racetrack geometry (0.009dB for all curves). Note that the DC units can also produce losses; however, such couplers with gap larger than 20nm yield negligible loss [38]. For the electrical electrode, the dynamic optical loss induced due to a large modulating signal of 2.5V was calculated to be [27]: loss MOS db = V. cm (2) 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 2260

10 These losses reduce α from constant unity to the range of 0.80<α<0.82 depending on the applied voltage, and consequently degrade the SFDR performance. We incorporated the dynamic and static losses in our model and optimized the optical parameters to obtain the highest SFDR possible. For τ = 0.592, τ 2 = 0.09 and E i = E i2 = (0.5) /2, while the phases φ τ, φ τ2 and φ Σ are set to zero, we obtain an SFDR of 27.3dB Hz 4/5 (under low frequency modulation). It is important to note that all other modulators suggested in the art will suffer dynamic range degradation when introducing such losses. Due to the FLAME relative simple configuration (one ring and electrode), it is reasonable to expect better performance from the FLAME compared to other modulators in the literature. 4. Conclusion We have presented and analyzed an EO modulator with improved linearity characteristics. The modulator, termed FLAME, consists of a racetrack resonator and a single MOS electrode. It offers an SFDR of 32dB Hz 4/5 under driving voltage of 2.5V. The modulator has been analyzed in SOI platform and has a footprint of 00 50µm 2 making it an appealing candidate for large scale integration in RF analog applications. High frequency modulation analysis has been carried out revealing superior SFDR behavior compared to MZI and RAMZI modulators. Optical and electrical losses were also analyzed revealing smaller than 5dB degradation in the SFDR. These advantages are achieved thanks to the newly introduced Double Injection design which improves linearity and provides larger bandwidth. 205 OSA 9 Feb 205 Vol. 23, No. 3 DOI:0.364/OE OPTICS EXPRESS 226