Frontiers in Guided Wave Optics and Optoelectronics Part 8 pdf

All-Optical Wavelength-Selective Switch by Intensity Control in Cascaded Interferometers 265 3.2 Switch B The switching operation with switch B is also verified by FD-BPM simulation. The model used in the simulation is shown in Fig. 9. The total length of the switch is L=8.85 mm. Raman amp. α I A I B O A O B 3dB coupler L c L s L p L D d W z x Fig. 9. Two-dimensional model of switch B for FD-BPM simulation. -20 1 10log|E| 2 A in B out A out [dB] (a) 1 α = - 10 15. 484 A in B out A out 10log|E| 2 [dB] (b) 3.383 α = Fig. 10. Distribution of optical fields with two different switching conditions in swith B. The switching operation at λ=1550 nm is confirmed as shown in Fig. 10. Although the output intensities from output port O A and O B are different, switching is successfully simulated. The wavelength dependence is shown in Fig. 11. At the designed wavelength λ=1550nm, the switching extinction ratio is larger than 25 dB. The wavelength range to achieve the extinction ratio larger than 20 dB is approximately 30 nm, though the 10-dB extinction ration is obtained over 80 nm. Frontiers in Guided Wave Optics and Optoelectronics 266 -30 -25 -20 -15 -10 -5 0 5 1500 1520 1540 1560 1580 1600 Wavelength (nm) Relative output intensity (dB) A out B out (a) 1 α = -20 -15 -10 -5 0 5 10 15 20 25 1500 1520 1540 1560 1580 1600 Wavelength (nm) Relative output intensity (dB) A out B out (b) 3.383 α = Fig. 11. Wavelength dependence of switched outputs for the switch B designed at wavelength λ=1550nm. 4. Improvement of wavelength dependency Waveguide-type Raman amplifiers do not depend on wavelength bands to be used because stimulated Raman scattering which is the base effect of Raman amplification can occur at any wavelength bands. Meanwhile, 3dB couplers have wavelength dependency in general, that is, the function of dividing an incident optical wave into two waves at the rate of 50:50 is available at some particular wavelength bands. The main cause of the wavelength dependency is the wavelength dependence of the coupling coefficient κ in eq.(1). For improving the characteristics of wavelength dependency of the switch and utilizing it at any All-Optical Wavelength-Selective Switch by Intensity Control in Cascaded Interferometers 267 wavelength bands, wavelength-independent (or wavelength-flattened) optical couplers should be employed. Fiber-type wavelength-independent couplers, that can be used for 50:50 of the dividing rate at wavelength bands such as 1550 nm ± 40 nm and 1310 nm ± 40 nm, have already been on the market. However, waveguide-type wavelength-independent couplers have advantage from the viewpoint of integrating the switch elements. An alternative for improving wavelength dependence is to replace the directional couplers by asymmetric X-junction couplers (Izutsu et al., 1982; Burns & Milton, 1980; Hiura et al., 2007). The asymmetric X-junction coupler has basically no dependence on wavelength and helps to improve the wavelength dependency of the proposed switch (Kishikawa et al., 2009a; Kishikawa et al., 2009b). 5. Another issue in implementation Phase shift of the signal pulse experienced in the waveguide-type Raman amplifiers should be discussed because it can impact the operation of the switch. The phase shift is induced from refractive index change caused by self-phase modulation (SPM), cross-phase modulation (XPM), free carriers generated from two-photon absorption (TPA) (Roy et al., 2009), and temperature change. Although the structure of the switch becomes more complex, the effect of SPM and TPA-induced free carriers can be cancelled by installing the same nonlinear waveguides as those of the waveguide-type Raman amplifiers into counter arms of the Mach-Zehnder interferometers of the switch. The influence of XPM and temperature change involved with high power pump injection can also be suppressed by injecting pump waves, having the same power and different wavelengths that do not amplify the signal pulse, into the counterpart nonlinear waveguides. 6. Conclusion We proposed a novel all-optical wavelength-selective switching having potential of a few tens of picosecond or faster operating speed. We discussed the theory and the simulation results of the switching operation and the characteristics. Moreover, the dynamic range over 25dB was also obtained from the simulation results of the switch. This characteristics can be wavelength- selective switching operation. More detailed analysis and simulation taking the nonlinearity of Raman amplifiers into account are required to confirm the operation with actual devices. Although the principle and the fundamental verification were performed with the switches consisting of directional couplers, the idea can be similarly applied to switches consisting of other components such as asymmetric X-junction couplers to increase the wavelength range. 8. References Doran, N. J. & Wood, D. (1988). Nonlinear-Optic Loop Mirror, Optics Lett., vol.13, no.1, pp.56-58, Jan. 1988. Burns, W. K. & Milton, A. F. (1980). An Analytic Solution for Mode Coupling in Optical Waveguide Branches, IEEE J. Quantum Electron., vol.QE-16, no.4, pp.446-454, Apr. 1980. Goh, T., Kitoh, T., Kohtoku, M., Ishii, M., Mizuno, T. & Kaneko, A. (2008). Port Scalable PLC- Based Wavelength Selective Switch with Low Extinction Loss for Multi-Degree ROADM/WXC, The Optical Fiber Communication Conference and the National Fiber Optic Engineers Conference (OFC/NFOEC2008), San Diego, OWC6, Mar. 2008. Frontiers in Guided Wave Optics and Optoelectronics 268 Goto, N & Miyazaki, Y. (1990). Integrated Optical Multi-/Demultiplexer Using Acoustooptic Effect for Multiwavelength Optical Communications, IEEE J. on Selected Areas in Commun., vol.8, no.6, pp.1160-1168, Aug. 1990. Hadley, G. R. (1992). Wide-Angle Beam Propagation Using Pade Approximant Operators, Opt. Lett., vol.17, no.20, pp.1426-1428, Oct. 1992. Hiura, H., Narita, J. & Goto, N. (2007). Optical Label Recognition Using Tree-Structure Self- Routing Circuits Consisting of Asymmetric X-junction, IEICE Trans. Commun., vol.E90-C, no.12, pp.2270-2277, Dec. 2007. Izutsu, M., Enokihara, A. & Sueta, T. (1982). Optical-Waveguide Hybrid Coupler, Opt. Lett., vol.7, no.11, pp.549-551, Nov. 1982. Kishikawa, H. & Goto, N. (2005). Proposal of All-Optical Wavelength-Selective Switching Using Waveguide-Type Raman Amplifiers and 3dB Couplers, J. Lightwave Technol., vol.23, no.4, pp.1631-1636, Apr. 2005. Kishikawa, H. & Goto, N. (2006). Switching Characteristics of All-Optical Wavelength- Selective Switch Using Waveguide-Type Raman Amplifiers and 3-dB Couplers, IEICE Trans. Electron., vol.E89-C, no.7, pp.1108-1111, July 2006. Kishikawa, H. & Goto, N. (2007a). Optical Switch by Light Intensity Control in Cascaded Coupled Waveguides, IEICE Trans. Electron., vol.E90-C, no.2, pp.492-498, Feb. 2007. Kishikawa, H. & Goto, N. (2007b). Designing of Optical Switch Controlled by Light Intensity in Cascaded Optical Couplers, Optical Engineering, vol.46, no.4, pp.044602-1-10, Apr. 2007. Kishikawa, H., Kimiya, K., Goto, N. & Yanagiya, S. (2009a). All-Optical Wavelength-Selective Switch Controlled by Raman Amplification for Wide Wavelength Range, Optoelectronics and Communications Conf., OECC2009, Hong Kong, TuG3, July 2009. Kishikawa, H., Kimiya, K., Goto, N. & Yanagiya, S. (2009b). All-Optical Wavelength-Selective Switch by Amplitude Control with a Single Control Light for Wide Wavelength Range", Int. Conf. on Photonics in Switching, PS2009, Pisa, PT-12, Sept. 2009. Kitagawa, Y., Ozaki, N., Takata, Y., Ikeda, N., Watanabe, Y., Sugimoto, Y. & Asakawa, K. (2009). Sequential Operations of Quantum Dot/Photonic Crystal All-Optical Switch With High Repetitive Frequency Pumping, J. Lightwave Technol., vol.27, no.10, pp.1241-1247, May 2009. Nakamura, S., Ueno, Y., Tajima, K., Sasaki, J., Sugimoto, T., Kato, T., Shimoda, T., Itoh, M., Hatakeyama, H., Tamanuki, T. & Sasaki, T. (2000). Demultiplexing of 168-Gb/s Data Pulses with a Hybrid-Integrated Symmetric Mach-Zehnder All-Optical Switch, IEEE Photon. Tech. Lett., vol.12, no.4, pp.425-427, Apr. 2000. Raghunathan, V., Boyraz, O & Jalali, B. (2005). 20dB On-Off Raman Amplifiation in Silicon Waveguides, Conf. Lasers and Electro-Optics (CLEO2005), Baltimore, CMU1, May 2005. Rong, H., Liu, A., Nicolaescu, R., Paniccia, M., Cohen, O. & Hak, D. (2004). Raman Gain and Nonlinear Optical Absorption Measurements in a Low-Loss Silicon Waveguide, Appl. Phys. Lett., vol.85, no.12, pp.2196-2198, Sept. 2004. Roy, S., Bhadra, S. K. & Agrawal, G. P. (2009). Raman Amplification of Optical Pulses in Silicon Waveguides: Effects of Finite Gain Bandwidth, Pulse Width, and Chirp, J. Opt. Soc. Am. B, vol. 26, no. 1, Jan. 2009. Suto, K., Saito, T., Kimura, T., Nishizawa, J. & Tanabe, T. (2002). Semiconductor Raman Amplifier for Terahertz Bandwidth Optical Communication, J. Lightwave Technol., vol.20, no.4, pp.705-711, Apr. 2002. Suzuki, S., Himeno, A. & Ishii, M. (1998). Integrated Multichannel Optical Wavelength Selective Switches Incorporating an Arrayed-Waveguide Grating Multiplexer and Thermooptic Switches, J. Lightwave Technol., vol.16, no.4, pp.650-655, Apr. 1998. 14 Nonlinear Optics in Doped Silica Glass Integrated Waveguide Structures David Duchesne 1 , Marcello Ferrera 1 , Luca Razzari 1 , Roberto Morandotti 1 , Brent Little 2 , Sai T. Chu 2 and David J. Moss 3 1 INRS-EMT, 2 Infinera Corporation, 3 IPOS/CUDOS, School of Physics, University of Sydney, 1 Canada 2 USA 3 Australia 1. Introduction Integrated photonic technologies are rapidly becoming an important and fundamental milestone for wideband optical telecommunications. Future optical networks have several critical requirements, including low energy consumption, high efficiency, greater bandwidth and flexibility, which must be addressed in a compact form factor (Eggleton et al., 2008; Alduino & Paniccia, 2007; Lifante, 2003). In particular, it has become well accepted that devices must possess a CMOS compatible fabrication procedure in order to exploit the large existing silicon technology in electronics (Izhaky et al., 2006; Tsybeskov et al., 2009). This would primarily serve to reduce costs by developing hybrid electro-optic technologies on-chip for ultrafast signal processing. There is still however, a growing demand to implement all-optical technologies on these chips for frequency conversion (Turner et al., 2008; Venugopal Rao et al., 2004), all-optical regeneration (Salem et al., 2008; Ta’eed et al., 2005), multiplexing and demultiplexing (Lee et al., 2008; Bergano, 2005; Ibrahim et al., 2002), as well as for routing and switching (Lee et al., 2008; Ibrahim et al., 2002). The motivation for optical technologies is primarily based on the ultrahigh bandwidth of the optical fiber and the extremely low attenuation coefficient. Coupled with minimal pulse distortion properties, such as dispersion and nonlinearities, optical fibers are the ideal transmission medium to carry information over long distances and to connect optical networks. Unfortunately, the adherence of the standard optical fiber to pulse distortions is also what renders it less than perfectly suited for most signal processing applications required in telecommunications. Bending losses become extremely high in fibers for chip-scale size devices, limiting its integrability in networks. Moreover, its weak nonlinearity limits the practical realization (i.e. low power values and short propagation lengths) of some fundamental operations requiring nonlinear optical phenomena, such as frequency conversion schemes and switching (Agrawal, 2006). Several alternative material platforms have been developed for photonic integrated circuits (Eggleton et al., 2008; Alduino & Panicia, 2007; Koch & Koren, 1991; Little & Chu, 2000), including semiconductors such as AlGaAs and silicon-on-insulator (SOI) (Lifante, 2003; Frontiers in Guided Wave Optics and Optoelectronics 270 Koch and Koren, 1991; Tsybeskov et al., 2009; Jalali & Fathpour, 2006), as well as nonlinear glasses such as chalcogenides, silicon oxynitride and bismuth oxides (Ta’eed et al., 2007; Eggleton et al., 2008; Lee et al., 2005). In addition, exotic and novel manufacturing processes have led to new and promising structures in these materials and in regular silica fibers. Photonic crystal fibers (Russell, 2003), 3D photonic bandgap structures (Yablonovitch et al., 1991), and nanowires (Foster et al., 2008) make use of the tight light confinement to enhance nonlinearities, greatly reduce bending radii, which allows for submillimeter photonic chips. Despite the abundance of alternative fabrication technologies and materials, there is no clear victor for future all-optical nonlinear devices. Indeed, many nonlinear platforms require power levels that largely exceed the requirements for feasible applications, whereas others have negative side effects such as saturation and multi-photon absorption. Moreover, there is still a fabrication challenge to reduce linear attenuation and to achieve CMOS compatibility for many of these tentative photonic platforms and devices. In response to these demands, a new high-index doped silica glass platform was developed in 2003 (Little, 2003), which combines the best of both the qualities of single mode fibers, namely low propagation losses and robust fabrication technology, and those of semiconductor materials, such as the small quasi-lossless bending radii and the high nonlinearity. This book chapter primarily describes this new material platform, through the characterization of its linear and nonlinear properties, and shows its application for all-optical frequency conversion for future photonic integrated circuits. In section 2 we present an overview of concurrent recent alternative material platforms and photonic structures, discussing advantages and limitations. We then review in section 3 the fundamental equations for nonlinear optical interactions, followed by an experimental characterization of the linear and nonlinear properties of a novel high-index glass. In section 4 we introduce resonant structures and make use of them to obtain a highly efficient all-optical frequency converter by means of pumping continuous wave light. 2. Material platforms and photonic structures for nonlinear effects 2.1 Semiconductors Optical telecommunications is rendered possible by carrying information through waveguiding structures, where a higher index core material (n c ) is surrounded by a cladding region of lower index material (n s ). Nonlinear effects, where the polarization of media depends nonlinearly on the applied electric field, are generally observed in waveguides as the optical power is increased. Important information about the nonlinear properties of a waveguide can be obtained from the knowledge of the index contrast (Δn = n c -n s ) and the index of the core material, n c . The strength of nonlinear optical interactions is predominantly determined through the magnitude of the material nonlinear optical susceptibilities (χ (2) and χ (3) for second order and third order nonlinear processes where the permittivity depends on the square and the cube of the applied electromagnetic field, respectively), and scales with the inverse of the effective area of the supported waveguide mode. Through Miller’s rule (Boyd, 2008) the nonlinear susceptibilities can be shown to depend almost uniquely on the refractive index of the material, whereas the index contrast can easily be used to estimate the area of the waveguide mode, where a large index contrast leads to a more confined (and thus a smaller area) mode. It thus comes to no surprise that the most commonly investigated materials for nonlinear effects are III-V semiconductors, such as silicon and AlGaAs, which possess a large index of refraction at the telecommunications wavelength (λ = 1.55 μm) and Nonlinear Optics in Doped Silica Glass Integrated Waveguide Structures 271 where waveguides with a large index contrast can be formed. For third order nonlinear phenomena such as the Kerr effect 1 , the strength of the nonlinear interactions can be estimated through the nonlinear parameter γ = n 2 ω/cA (Agrawal, 2006), where n 2 is the nonlinear index coefficient determined solely from material properties, ω is the angular frequency of the light, c is the speed of light and A the effective area of the mode, which will be more clearly defined later. The total cumulative nonlinear effects induced by a waveguide sample can be roughly estimated as being proportional to the peak power, length of the waveguide and the nonlinear parameter (Agrawal, 2006). In order to minimize the energetic requirements, it is thus necessary either to have long structures and/or large nonlinear parameters. Focusing on the moment on the nonlinear parameter, in typical semiconductors, the core index n c > 3 (~3.5 for Si and ~3.3 GaAs) leads to values of n 2 ~10 -18 – 10 -17 m 2 /W, to be compared with fused silica (n c = 1.45) where n 2 ~2.6 x 10 -20 m 2 /W. Moreover, etching through the waveguide core allows for a large index contrast with air, permitting photonic wire geometries with effective areas below 1 um 2 , see Fig. 1. This leads to extremely high values of γ ~ 200,000W -1 km -1 (Salem et al., 2008; Foster et al., 2008) (to be compared with single mode fibers which have γ ~ 1W -1 km -1 (Agrawal, 2006)). This large nonlinearity has been used to demonstrate several nonlinear applications for telecommunications, including all-optical regeneration at 10 Gb/s using four-wave mixing and self-phase modulation in SOI (Salem et al., 2008; Salem et al., 2007), frequency conversion (Turner et al., 2008; Venugopal Rao et al., 2004; Absil et al., 2000), and Raman amplifications (Rong et al., 2008; Espinola et al., 2004). Fig. 1. (left) Silicon-on-insulator nano-waveguide (taken from (Foster et al., 2008)) and inverted nano-taper (80nm in width) of an AlGaAs waveguide (right). Both images show the very advanced fabrication processes of semiconductors. There are however major limitations that still prevent their implementation in future optical networks. Semiconductor materials typically have a high material dispersion (a result of being near the bandgap of the structure), which prevents the fabrication of long structures. To overcome this problem, small nano-size wire structures, where the waveguide dispersion dominates, allows one to tailor the total induced dispersion. The very advanced fabrication technology for both Si and AlGaAs allows for this type of control, thus a precise waveguide 1 We will neglect second order nonlinear phenomena, which are not possible in centrosymmetric media such as glasses. See (Boyd, 2008) and (Venugopal Rao et al., 2004; Wise et al., 2002) for recent advances in exploiting χ (2) media for optical telecommunications. 80nm Frontiers in Guided Wave Optics and Optoelectronics 272 geometry can be fabricated to have near zero dispersion in the spectral regions of interest. Unfortunately, the small size of the mode also implies a relatively large field along the waveguide etched sidewalls (see Fig. 1). This leads to unwanted scattering centers and surface state absorptions where initial losses have been higher than 10dB/cm for AlGaAs (Siviloglou et al., 2006; Borselli et al., 2006; Jouad & Aimez, 2006), and ~ 3 dB/cm for SOI (Turner et al.,2008). Another limitation comes from multiphoton absorption (displayed pictorially in Fig. 2 for the simplest case, i.e. two-photon absorption) and involves the successive absorption of photons (via virtual states) that promotes an electron from the semiconductor valence band to the conduction band. This leads to a saturation of the transmitted power and, consequently, of the nonlinear effects. For SOI this has been especially true, where losses are not only due to two-photon absorption, but also to the free carriers induced by the process (Foster et al., 2008; Dulkeith et al., 2006). Moreover, the nonlinear figure of merit (= n 2 /α 2 λ, where α 2 is the two photon absorption coefficient), which determines the feasibility of nonlinear interactions and switching, is particular low in silicon (Tsang & Liu, 2008). Lastly, although reducing the modal area enhances the nonlinear properties of the waveguide, it also impedes coupling from the single mode fiber into the device; for comparison the modal diameter of a fiber is ~10μm whereas for a nanowire structure it is typically 20 times smaller. This leads to high insertion losses through the device, necessitating either expensive amplifiers at the output, or of complicated tapers often requiring mature fabrication technologies and sometimes multi-step etching processes (Moerman et al., 1997) (SOI waveguides make use of state-of-the-art inverse tapers which limits the insertion losses to approximately 5dB (Almeida et al., 2003; Turner et al., 2008)). Fig. 2. Schematic of two-photon absorption in semiconductors. In the most general case of the multiphoton absorption process, electrons pass from the valence band to the conduction band via the successive absorption of multiple photons, mediated via virtual states, such that the total absorbed energy surpasses the bandgap energy. 2.2 High index glasses In addition to semiconductors, a number of high index glass systems have been investigated as a platform for future photonic integrated networks, including chalcogenides (Eggleton et K E hf<E g < 2hf h f E g h f N o al. , ( W n o w h H o Fa b R u si g al. , p h al. , n o re q A Li t an ch e an T h pr o se c pr o Fe r H y m a (L i Fi g u p A s m o g e n pr o o nlinear Optics in D o , 2008; Ta’eed et W orhoff et al., 200 o nlinear paramet e h ich has been u o wever, all of t b rication proces s u an et al., 2004) a g nificantl y better , 2006). Photose n h otonic structure s , 2006). Wherea s o nlinear absorpti o q uired to reduce p hi g h-index, dop e t tle Optics in 20 0 d the nonlinear e mical vapour d e d reactive ion et c h e wave g uides ar o cess CMOS co m c tion is 1.45 x 1. o pa g ation losses r rera et al., 2008 ) y dex wave g uide s a terial platform h i ttle et al., 2004), a g . 3. Scannin g ele p per SiO 2 deposit i s will be show n o derate nonline a n erate si g nifican o duce the nece s o ped Silica Glass I n al., 2007), silico n 2). Chalco g enid e e rs approachin g u sed to demonst r t hese platforms s es for chalco g e n a nd while the y g than silicon, for n sitivit y and ph o s , can sometimes s other hi g h-ind o n (virtuall y in f p ropa g ation loss e e d silica g lass m 0 3 as a compro m properties of se m e position. Subse q c hin g , producin g e then buried in m patible and re q 5 μm 2 as show n have been sho w ) . In addition, fib s , with couplin g l h as alread y been a s well as the op t ctron microscop y i on), and electro m n in the subseq u a rit y , and coupl e t nonlinear effec t s sar y equations n tegrated Wavegui d n nitride (Gonda r e s in particular h γ ~ 100,000W -1 k r ate demultiple x suffer from s h n ide g lasses are s g enerall y possess example - it can o to-darkenin g , w place limits on t h ex g lasses, suc h f inite fi g ure of m e s, makin g the e n m aterial called H y m ise between the m iconductors. F i q uentl y , wave g u i wave g uide side w standard fused s i q uirin g no furth e n in Fi g . 3. The l w n to be as low er pi g tails have b l osses on the ord e exploited to ach t ical sensin g of bi y picture of the h i m a g netic field di s u ent sections bel e d with lon g o r t s with low pow g overnin g li g h d e Structures r enko et al., 2009 ) ave been shown k m -1 in nanotap e x in g at 160 Gb/ h ortcomin g s of o s till under devel o a ver y hi g h no n be an issue for s o w hile powerful t o h e material stabi l h as silicon ox yn m erit), hi g h te m n tire process no n - y dex ® (Little, 20 0 attractive linear i lms are first de p i des are formed u w alls with excep t i lica g lass, maki n e r anneal. The t y l inear index at λ as 0.06 dB/cm b een desi g ned fo e r of 1.5dB. The l ieve filters with omolecules (Yal c ig h-index g lass w s tribution of the f ow, this materi a r resonant struc er requirements. t propa g ation i n ) and silicon ox yn to have extreme l e rs (Yeom et al., s (Pelusi et al., o ne form or a n o pment (Li et al. n linear fi g ure of m o me g lasses (La m o ols for creatin g l it y (Shokooh-Sa r n itride, have ne g m perature annea l - CMOS compati b 0 3), was develo p features of silic a p osited usin g st a u sin g photolitho g t ionall y low rou g ng the entire fabr i y pical wave g uid e = 1.55 μm is 1. (Duchesne et al. , r couplin g to an d l inear properties >80dB extinctio n c in et al., 2006). w ave g uide (prior f undamental mo d a l platform also tures, can be u s In the next sect i n a nonlinear m 273 n itride ly hi g h 2008), 2007). n other. , 2005; m erit - m ont et novel r emi et g li g ible l in g is b le. p ed b y a g lass a ndard g raph y g hness. i cation e cross 7, and , 2009; d from of this n ratios to d e. has a s ed to i on we m edia, Frontiers in Guided Wave Optics and Optoelectronics 274 followed by a characterization method for the nonlinearity, and explain the possible applications achievable by exploiting resonant and long structures. 3. Light dynamics in nonlinear media In order to completely characterize the nonlinear optical properties of materials, it is worthwhile to review some fundamental equations relating to pulse propagation in nonlinear media. In general, this is modelled directly from Maxwell’s equations, and for piecewise homogenous media one can arrive at the optical nonlinear Schrodinger equation (Agrawal, 2006; Afshar & Monro, 2009): 2 22 212 1 2 222 i HOD i HOL zt A t ψψβψ α α βψγψψψψ ∂∂ ∂ ++ ++= − − ∂∂ ∂ (1) Where ψ is the slowly-varying envelope of the electric field, given by: ( ) 00 '( , ) ( , )expEztFx y izit ψβω =−, where ψ’ has been normalized such that 2 ψ represents the optical power. ω 0 is the central angular frequency of the pulse, β 0 the propagation constant, β 1 is the inverse of the group velocity, β 2 the group velocity dispersion, α 1 the linear loss coefficient, α 2 the two-photon absorption coefficient, γ (= n 2 ω 0 /cA) the nonlinear parameter, t is time and z is the propagation direction. Here F(x,y) is the modal electric field profile, which can be found by solving the dispersion relation: 22 22 2 n FFF c ω β ∇+ = (2) The eigenvalue solution to the dispersion relation can be obtained by numerical methods such as vectorial finite element method (e.g. Comsol Multiphysics). From this the dispersion parameters can be calculated via a Taylor expansion: ()()() 23 3 2 01 0 0 0 26 β β ββ βωω ωω ωω = +−+ −+ −+ (3) The effective area can also be evaluated: 2 2 4 F dxd y A F dxd y ∞ ∞ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ = ∫∫ ∫∫ (4) In arriving to eq. (1), we neglected higher order nonlinear contributions, non-instantaneous responses (Raman) and non-phase matched terms; we also assumed an isotropic cubic medium, as is the case for glasses. These approximations are valid for moderate power values and pulse durations down to ~100fs for a pulse centered at 1.55 μm (Agrawal, 2006). The terms HOL and HOD refer to higher order losses and higher order dispersion terms, which may be important in certain circumstances (Foster et al., 2008; Siviloglou et al., 2006). Whereas eq. (1) also works as a first order model for semiconductors, a more general and exact formulation can be found in (Afshar & Monro, 2009). Given the material dispersion [...]... nonlinear interactions (for a fixed input power): 1) increasing the nonlinear parameter, or 2) increasing the propagation length To increase the former, one can reduce the modal size by having high-index contrast waveguides, and/ or using a high index material with a high value of n2 Thus, for nonlinear applications, the advantage for doped silica glass waveguides lies in exploiting its low loss and. .. dB with only 8. 8mW of input power Moreover, a cascade of four -wave mixing processes can be seen whereby the pump and 1st idler mix to generate a 3rd idler (the numbers refer to Fig 18) Nonlinear Optics in Doped Silica Glass Integrated Waveguide Structures 289 Fig 18 Four wave mixing across several resonances in the high Q resonator A conversion efficiency of -36db is obtained with only 8. 8mW of external... zero-GVD points are found to be at 1594.7nm and 1560.5nm for TE and TM, respectively  286 Frontiers in Guided Wave Optics and Optoelectronics In addition, the dispersion data can be used to predict the bandwidth over which four -wave mixing can be observed In resonators, the linear phase matching condition for the propagation constants is automatically satisfied as the resonator modes are related linearly... (Lin et al., 20 08) Fig 14 Phase matching diagram associated to four -wave mixing in the high Q micro-ring resonator (interpolated) The regions in black are areas where four wave- mixing is not possible, whereas the coloured regions denote possible four -wave mixing with the colour indicating the degree of frequency mismatch (blue implies perfect phase matching; colour scale is Δω in MHz) Nonlinear Optics. .. Ferrera et al., 20 08) Fig 10 Schematic of the vertically coupled high-index glass micro-ring resonator  284 Frontiers in Guided Wave Optics and Optoelectronics Two ring resonators will be discussed in this section, one with a radius of 47.5 μm, a Q factor ~65,000, and a bandwidth matching that for 2.5Gb/s signal processing applications, as well as a high Q ring of ~1,200,000, with a ring radius of 135μm... returning the material to thermal equilibrium This, however, may not leave the material in its’ original state and this is where femtosecond micromachining comes in with the aim of manipulating the modification to create useful devices in a highly controlled manner 2 98 Frontiers in Guided Wave Optics and Optoelectronics To look at multiphoton absorption in a little more depth let us consider the inscription... Moss, D J (20 08) Low power continuous -wave nonlinear optics in doped silica glass integrated waveguide structures Nat Photonics, Vol 2, 737-740 Nonlinear Optics in Doped Silica Glass Integrated Waveguide Structures 291 Ferrera, M.; Duchesne, D.; Razzari, L.; Peccianti, M.; Morandotti, R.; Cheben, P.; Janz, S.; Xu, D.-X.; Little, B E.; Chu, S & Moss, D J (2009) Low power four wave mixing in an integrated,... narrow linewidth, multi-wavelength sources, or correlated photon pair generation (Kolchin et al., 2006; Kippenberg et al., 2004; Giordmaine & Miller, 1965) In both cases the bus waveguides and the ring waveguide have the same cross section and fabrication process as previously described in Section 2.2 and 3 (see Fig 3) The 4-port ring resonator is depicted in Fig 10, and light is injected into the ring... Coupling coefficients and schematic of a typical 4-port ring resonator  281 Nonlinear Optics in Doped Silica Glass Integrated Waveguide Structures Fig 8 Typical Fabry-Perot resonance transmission at the drop port of a resonator (input port excited) Here a FSR of 500GHz and a Q of 25 were used 4.2 Four -wave mixing Section 3 discussed third order nonlinear effects following the propagation of a single... situation in which the 288 Frontiers in Guided Wave Optics and Optoelectronics pump power was varied for a fixed signal power also demonstrated the expected quadratic dependence, validating the approximations leading to Eq (16) Lastly, by tuning the signal wavelength slightly off-resonance and measuring the conversion efficiency, it was experimentally shown, Fig 17, that these results were in quasi-perfect . α − ⎡ ⎤ =−− ⎣ ⎦ (9) Frontiers in Guided Wave Optics and Optoelectronics 2 78 The nonlinear term introduces a nonlinear chirp in the temporal phase, which in the frequency domain corresponds to. (OFC/NFOEC20 08) , San Diego, OWC6, Mar. 20 08. Frontiers in Guided Wave Optics and Optoelectronics 2 68 Goto, N & Miyazaki, Y. (1990). Integrated Optical Multi-/Demultiplexer Using Acoustooptic. though the 10-dB extinction ration is obtained over 80 nm. Frontiers in Guided Wave Optics and Optoelectronics 266 -30 -25 -20 -15 -10 -5 0 5 1500 1520 1540 1560 1 580 1600 Wavelength (nm) Relative