Analysis of As 2 S 3 -Ti: LiNbO 3 Taper Couplers Using Supermode Theory

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1 Optis and Photonis Journal, 2012, 2, Published Online Deember 2012 ( Analysis of As 2 S 3 -Ti: LiNbO 3 Taper Couplers Using Supermode Theory Xin Xia, Yifeng Zhou, Christi K. Madsen Department of Eletrial and Computer Engineering, Texas A&M University, College Station, USA madsen@tamu.edu Reeived Otober 16, 2012; revised November 14, 2012; aepted November 24, 2012 ABSTRACT In this work, we develop a simulation method based on supermode theory and transfer matrix formalism, and then apply it to the analysis and design of taper ouplers for vertially integrated As 2 S 3 and Ti: LiNbO 3 hybrid waveguides. Test strutures based on taper ouplers are fabriated and haraterized. The experimental results onfirm the validity of the modeling method, whih in turn, is used to analyze the fabriated ouplers. Keywords: Optial Waveguides; Couplers; Coupled Mode Analysis 1. Introdution As study on integrated Optis proeeds, several shemes with regard to materials and strutures were developed, suh as silion-on-insulator, halogenide glass waveguides, III-V semiondutor waveguides and titanium diffused waveguides. While different shemes have their own merits and shortomings, reiproal benefits an be obtained from integration of them, namely, the hybrid waveguides. For example, preliminary result was reported on As 2 S 3 -on-ti: LiNbO 3 hybrid waveguide devies [1,2], whih benefit from the high index ontrast of As 2 S 3 and easy onnetion with ommerial single mode fibers. For integration of different waveguides, light oupling is the key. A diretional oupler is the simplest funtional devie to ouple light by transferring energy between two waveguides. However, in pratie its oupling effiieny an be fairly low due to the phase mismath and small tolerane to fabriation errors. Alternatively grating and taper ouplers are used, and taper ouplers are generally preferred owing to its simpliity in design and fabriation. Despite diverse forms, the general taper oupler is omposed of two parallel waveguides plaed in lose proximity: one is uniform whereas at one end of the other one, the width is gradually varied. Two ends of the taper math the wave guiding properties of two waveguides, so the mode is transformed gradually from one into another during propagation in the taper. Although the priniple is intuitively quite simple, the design in most ases is onservative beause of the lak of preise modeling guidelines and aurate modeling tools [3]. A lot of theoretial study was arried out to investigate them, and different approahes were devel- oped. Lee et al. proposed an equivalent waveguide onept employing a onformal mapping method, whih was ombined with the Beam Propagation Method (BPM) to ondut analysis [4]. In [3], tapered waveguides were analyzed by onsidering the whole taper as a suession of short linear taper fragments and modeling eah of them using a two-dimensional BPM that solves diretly the Helmholtz equation. However, most of the early work foused on orreting simulation methods to improve the auray, and the underlying physial mehanism governing the power transfer was not desribed [5]. Therefore, few guidelines an be found for designers. Thus more and more researhers began to look into taper ouplers from the angle of supermodes, i.e. loal modes. In [5], Xia et al. defined and distinguished between the resonant oupling and adiabati oupling from the view of supermodes [5]. Resonant ouplers are ompat and simple but highly sensitive to unavoidable variations during fabriation [5]. Adiabati ouplers, on the ontrary, don t require exat ontrol of taper length and gap, but need longer lengths [6]. Sun et al. onduted a series of studies on the behavior of supermodes in adiabati ouplers [6,7] and derived a mathematial expression of the shortest adiabati tapers [6]. As suh theoretial work ontributed a lot to our understanding of taper ouplers, the study on issues of pratial appliation and modeling is still laking. In pratie, we often need to balane the taper length and the oupling effiieny, sine we may not have suffiient spae to fulfill the adiabati ondition, and we may want ertain oupling effiieny that is not neessarily 100%. Mah-Zehnder interferene filters, for example, typially use 3 db ouplers. Moreover, the mate-

2 X. XIA ET AL. 345 rials and strutures used may limit the oupling. Thus, there are a lot of effiient but non-adiabati taper ouplers desired in pratie. In published papers most simulations were onduted based on beam propagation method (BPM) [8]. BPM alulates the eletromagneti fields during light propagation proess and gives distributions of eletri and magneti fields. It is highly aurate as long as ertain assumptions are met. However, limited knowledge of underlying mehanism an be obtained from the simulation proess, so it is widely used to as a means of examining the designed taper oupler instead of guiding the design at the first plae. Alternatively, the modeling of taper ouplers an be based on the onept of modes using the oupled mode theory, whih an provide insights to the mode evolvement in the oupler and thus provide immediate guidelines for design. 2. Modeling Methods A taper oupler, whih onsists of two adjaent waveguides, an be regarded as a modified diretional oupler. In eah waveguide, only one mode is allowed to propagate. The oupled mode theory analyzes the oupled waveguides by taking one waveguide as the subjet and studying the influene of the perturbation imposed by the presene of the other one. The supermode theory, however, views the oupled waveguides as a whole system, i.e. a omposite two-waveguide struture, and studies the normalized loal modes of the system, whih are alled supermodes. Nevertheless, both theories desribe mode oupling for senarios that oupled waveguides are invariable along the propagation diretion. But the taper oupler is a varying struture where the width of one of the waveguides is onstantly hanging along the propagation diretion. However, the oupler an be divided into a suession of infinitely short setions. The length of eah setion is so small that the width an be regarded as invariant. So the simulation of a taper oupler an be divided into two steps: modeling of individual divisions and a asade of individual models. For eah division, as the width is deemed onstant, it is atually a simple diretional oupler, in whih there are fundamental supermode and first order supermode, named as even mode (E e ) and odd mode (E o ) respetively aording to the symmetry of their field distributions. The total field is a linear ombination of the even and odd mode. If the propagation onstants of modes in individual waveguides are the same, namely, they are phase mathed, two lobes of even and odd mode have the same size. If two propagation onstants are different, that is, the phases are mismathed, the symmetry of lobes of E e and E o is broken, and their shapes are different. When phase mismath is large, two waveguides are effetively deoupled: a wave propagating in either one is virtually unaffeted by the existene of the other, and the supermodes of the omposite struture just beome those of the individual waveguides [9]. δ is defined as the differene of the propagation onstants of two individual modes while β is for two supermodes in a similar way in (1): 2 1 e o and (1) 2 2 As shown in Figure 1, if δ is muh smaller than 0, most energy of the even mode is loated in waveguide 1 while if it is muh larger than 0, most energy is loated in waveguide 2. The opposite is true for the odd mode. So, the essene of taper oupling is to spatially transfer the energy of a supermode (even mode) from one waveguide to the other by designing the tapered waveguide so that δ sweeps from a negative value to a positive value while suppressing the oupling to the other supermode (odd mode) [6]. The larger sope δ overs, the more thorough the energy transfer is. Ideally, δ hanges from negative infinity to positive infinity, whereas in pratie, the sope is determined by the materials and strutures. Solving the oupled mode equations by substituting the general supermode solutions into them, we an obtain the expressions of supermodes and the relationship between the phase mismath of supermodes (β ) and that of individual modes (δ) [7] (2) As δ and β are known, the oupling strength κ [9] an be alulated. Then we have a omplete mathematial desription of the model with parameter δ, κ and β. Following the same method, models of all the divisions in the taper oupler an be built. Subsequently, transfer matrix formalism is derived to asade all the models based on oupled mode equations. In the matrix form, the solution to oupled mode equations is (3). z E z j z z E j iz j iz sin ze sin ze 1 i z os sin e 2 j z z 0 0 i z 1 os sin e E2 E where E 1 0 and E 2 0 are the input eletri fields in waveguide 1 and 2 respetively. Let z z0 and re-form the equation to obtain the expression of vetor, let z z0 z to re-write (3), substitute the vetor expression into it, and we arrive at the transfer matrix formalism relating the model at z z to the model at 0 (3)

3 346 X. XIA ET AL. z 0 in (4). E1z0z E2z0z M z E z z z Mz z z ; 0 0 ; 0 E2 z0 j iz osz sin ze (4) j iz0zz0 sin ze j iz0zz0 sin ze j iz os z sin ze Then by multiplying the matries in order, the models are asaded. As a result, the eletri field at ertain point z an be obtained from the known input E 1 0 and E2 0. The algorithm is summarized in Table 1. In step 2, due to the omplexity of the waveguide struture, omputer software FIMMWAVE (Photon Design Ltd.) is used to model eah setion, i.e., to ompute mode parameters. The film mode mathing method is applied as the mode solver. It is good for strutures onsisting of large uniform areas, suh as As 2 S 3 retangular waveguides. The resolution and the size of simulation window are tested to prevent artifiial errors. Simulation starts with unoupled waveguides, and their eigen-modes are omputed individually without the presene of the other one. The propagation onstants of the Ti waveguide mode and the As 2 S 3 waveguide mode are found to be β 1 and β 2 respetively. Then the model for the oupled system is built, and the even mode (β e ) and odd mode (β o ) are found, as Figure 2 shows Figure 1. The supermodes of a taper oupler. Table 1. Algorithm of modeling the taper oupler. Disretize the taper oupler into a sequene of suffiiently small divisions; Regard eah setion as a diretional oupler and model it to obtain mode propagation onstants β 1, β 2, β e and β o, and ompute δ, β and κ; Calulate individual transfer matrix of eah division based on parameter δ, β and κ; Casade all the divisions together by multiplying matries in order; 5 Calulate the oupling effiieny. Figure 2. The fundamental mode of the Ti waveguide (a) and the As 2 S 3 waveguide (b) and the odd () and even mode (d) of oupled waveguides.

4 X. XIA ET AL. 347 The approximation of a width-varying waveguide with a sequene of width-onstant waveguides is mathematially equivalent to the approximation of a ontinuous integral with a disrete summation, whih indues error inevitably. As the matries asade, the previous error passes on, and ombines with the error of the present one. Consequently, suh aumulation of the errors will manifest at the end of the taper, even if very small error exists in intermediate models. Simulation experiments show that disretization spaing z is ritial to the numerial error: the larger an error exists, the smaller the spaing needs to be, and the heavier the omputation load is required. In order to redue the error at the first plae, the trapezoidal approximation algorithm X i X i1 z is adopted to substitute left Rie- 2 mann sum X i z in (4) ( X, and ). 3. Simulation Results The struture of an As 2 S 3 -Ti: LiNbO 3 oupler is illustrated in Figure 3. A titanium diffused waveguide is formed in lithium niobate substrate (Ti: LiNbO 3 ). On substrate surfae is a piee of tapered As 2 S 3 retangular waveguide, whih is separated from the titanium diffused waveguide by a few mirons. Both waveguides work in single mode ondition. In Ti: LiNbO 3 fabriation proess, the LiNbO 3 material under Ti pattern rises up from the substrate surfae during titanium diffusion, resulting in a 0.1 μm high bump. In order to avoid the sattering loss aused by the rough surfae of the bump, As 2 S 3 waveguide is plaed to the side of the bump (side oupling) instead of on the top. For simpliity, air ladding is used. The height of As 2 S 3 waveguide is 470 nm. The final width of As 2 S 3 waveguide is determined to be 3.5 μm, in order to have a good mode onfinement in the As 2 S 3 waveguide. As the width of As 2 S 3 taper varies, the mode propaga- tion onstants in eah setion are plotted against the average width of that setion in Figure 4. We see that the propagation onstant of As 2 S 3 mode inreases gradually as its width beomes larger whereas the Ti mode remains onstant due to the invariable Ti waveguide width. The propagation onstant of the even mode oinides with that of the Ti mode first and then gradually follows the trend of the As 2 S 3 mode. On the ontrary, for odd mode, the propagation onstant goes from the As 2 S 3 mode to the Ti mode. During this proess, there is a point that the propagation onstants of the As 2 S 3 mode and the Ti mode are equal, orresponding to the point that the phase mismath δ equals to 0. From the graph, it is the point where the β-as 2 S 3 and β-ti urves ross, orresponding to the width of 1.47 μm, alled as ritial width. It is the ritial point where two waveguides are phase mathed, and the energy is equally distributed in two waveguides for both even and odd mode. In other words, it an be regarded as the mid-point of mode oupling proess from Ti waveguide to As 2 S 3 waveguide. As the width of the As 2 S 3 waveguide inreases, the inreasing rate of propagation onstant β 2 gets smaller. That means the phase mismath δ, the differene between the propagation onstants of two waveguides, will eventually ease to grow. The normalized phase mismath γ [6] is introdued to haraterize suh variation [6], as shown in (5) and plotted in Figure 5. (5) Among various types of taper geometries, the linear taper is most straightforward and provides insights into Figure 3. Configuration of an As 2 S 3 -Ti: LiNbO 3 taper oupler (two-stage taper design). The inset piture shows a top view. Figure 4. The propagation onstants of four modes. The inset piture shows them in a larger sale (from 0.6 μm to 4 μm).

5 348 X. XIA ET AL. the general taper design. Figure 6 shows the oupling effiieny of linear tapers of different lengths, with width varying from 1.0 μm to 3.5 μm. The squares stand for the oupling effiieny and the bars represent the magnitude of osillation. There is an optimum point that the maximum oupling effiieny reahes 96% when the length is 5 mm. The inset urve shows the perentage of energy oupled as light propagates through a 5 mm long linear taper. We an see that it onsists of a monotonially asending part and a subsequent osillation part. The oupling is mostly ontributed by the former part while the latter is due to resonane effets. For the even mode, the larger γ is, the more energy is loated in As 2 S 3 waveguide and the less in Ti waveguide, while it is vie versa for the odd mode. Sine the even mode is the mode to ouple, the energy remaining in Ti Figure 5. γ of the As 2 S 3 -Ti: LiNbO 3 oupler. Figure 6. Coupling effiieny for tapers of different length, with the inset figure showing the oupling proess of a 5 mm long taper, i.e., the oupling effiieny versus the loation along the taper. waveguide imposes an ultimate limit to the oupling effiieny. From the urve of γ in Figure 5, we learn that at the end of the taper, γ is Beause γ is not large enough, there is still a oupling between two waveguides. Suh oupling deteriorates the oupling effiieny and auses it to osillate. The behavior of the oupler in this region is similar to that of a resonant oupler. As a result, a ertain amount of energy flows bak and forth between the two waveguide modes. From the view of supermode theory, the osillation is a result of beating between the even and odd modes. Although the even mode is desired, the oupling of the odd mode is not ompletely suppressed, for example, if the length of the taper is not long suffiiently aording to the adiabati riterion in [6]. When the odd mode propagates in the taper, there is oupling between the even and odd modes and a small amount of energy flows bak and forth onstantly. Sine at the end of taper, the majority of the energy of the even mode is in As 2 S 3 waveguide and that of the odd mode is in Ti waveguide, there is a onstant energy flow between two waveguides, and onsequently the oupling effiieny osillates. In the presene of mode beating, it is not neessarily the longer taper, the better oupling. There exists an optimum length for a taper with fixed width variation: if it is shorter than that, the mode is under-oupled sine it is far away from the adiabati riterion for 100% oupling; if onsiderably longer than that, the oupling effiieny is degraded by the resonant effet, as Figure 6 shows. In order to redue the problem of mode beating, we must enlarge γ, either by inreasing the phase mismath δ or by dereasing the oupling strength κ. δ is limited by the property of the materials whereas κ an be ontrolled by the struture. For example, κ an be redued by introduing a gap between As 2 S 3 waveguide and Ti waveguide. Although the oupling effiieny an be as high as 96%, it takes quite a few millimeters to get a deent oupling effiieny for linear tapers, whih is not aeptable for ultra-ompat design. Aording to the above analysis, effiient oupling takes plae in the first part of taper where As 2 S 3 waveguide expands aross the ritial width and orrespondingly the phase mismath δ hanges from a negative value to a positive one. That ontributes to effiient oupling and we want it to be suffiiently long. One most of energy has entered As 2 S 3 waveguide, the rest of the taper an be shortened. As a onsequene, we have arrived at a two-stage taper (Figure 3). Furthermore, sine the end width of the first stage (transition width) an now be a muh smaller value, the rate of width hange is redued largely. Simulation shows that for the first part of a two-stage taper, if the width varies from 1.0 μm to 1.6 μm (have some leeway for fabriation deviations) in the length of 2 mm, the width inreasing rate is

6 X. XIA ET AL , whih is equivalent to an 8.3 mm long linear taper. Along with a 1 mm long seond part, with width varying from 1.6 μm to 3.5 μm, the total length is 3 mm. The oupling effiieny an still reah above 90%, whereas the total length is redued by 64%. 4. Experiments To test As 2 S 3 -Ti: LiNbO 3 taper oupler design, S-shaped strutures are fabriated and tested on a near IR measurement setup. As shown in Figure 7, it is omposed of two taper ouplers and an S-shaped As 2 S 3 waveguide to onnet them. The taper ouplers follow the two-stage taper oupler design. The devie is fabriated using photolithography and dry-eth tehnology. The substrate LiNbO 3 is a birefringene rystal with refrative index n o = and n e = (λ = 1531 nm), plaed in x-ut, y-propagation manner. The titanium diffused waveguide is fabriated through sputtering of a 95 nm thik titanium layer, patterning into 7 μm wide strip with photolithography and reative ion ething (RIE), diffusion for 9 hours at 1025 C and optial polishing on end-faets. For As 2 S 3 waveguide fabriation, a layer of 0.47 μm thik As 2 S 3 film is deposited on the titanium waveguide sample using an RF sputtering system, along with a protetive layer of SiO 2 and Ti, whih protets the As 2 S 3 from being dissolved by ommerial alkaline-based developers. Then the projetion photolithography is arried out, and the 1.0 μm wide taper tip an be produed, nevertheless the subsequent hardbake auses an expansion to ertain degree. After that, the Ti-SiO 2 -As 2 S 3 stak is ethed through to the substrate by RIE. And Ti-SiO 2 is removed in diluted hydrofluori solution at last. The hardbake time is prolonged in order to obtain smother sidewalls by the resist reflow proess, whih, however, auses an expansion of As 2 S 3 waveguide to ertain degree, up to 0.5 μm. The average tip width (i.e. the initial width) of tapered As 2 S 3 waveguide after fabriation is 1.3 μm. Depending on the proess onditions suh as exposure and development, it an be smaller or larger than that. Simulation study on the influene of the tip width variation for two-stage tapers is shown in Figure 8, along with the oupling urve of a two-stage taper oupler. Measurement results onfirm the funtion of the taper oupler following the design in setion III (Table 2). Generally, the ross port aounts for 50% to 90% of the total output power. Negleting the exess loss aused by propagation in the low-loss As 2 S 3 and Ti waveguides, the average oupling effiieny is 73.2%. However, prior to extrating the preise oupling effiieny, the propagation loss and bending loss in As 2 S 3 waveguide have to be alibrated first. Many experiments need to be done for that, and the work is still ongoing. Instead of working at a single wavelength, these pratial taper ouplers are designed to work for a wavelength range. Aordingly, their oupling behaviors in frequeny domain are studied. The measured spetrum at the ross port is presumably to have the same trends of the oupling spetrum, with an offset from the exat values. That offers the information of taper ouplers in the frequeny domain and an be used as another means to test our simulation method. The typial measured spetrum, along with simulation results is shown in Figure 9. In simulation the wavelength is sanned orrespondingly from 1520 nm to 1600 nm, at the interval of 2 nm. The results show that, though the taper oupler exhibits ertain degree of wavelength dependeny, it has high oupling effiieny over a broad bandwidth. From the urve, we an see that the period of osillation is less than 10 nm, and longer wavelengths have a (a) Figure 8. Influene of tip width variation (a) and the oupling proess of a two-stage taper (b). Table 2. Measurement results. Sample Cross (db) Through (db) Cross in Total (%) (b) Figure 7. S bend strutures for testing taper ouplers

7 350 X. XIA ET AL. larger osillation period than shorter wavelengths: both are aptured by the simulation. The osillation of the oupling urve is a strong indiation of mode beating while the phenomenon that longer wavelengths have a slightly larger osillation period possibly omes from waveguide dispersion: the wavelength-dependent propagation onstant. Simulation shows that when the wavelength varies from 1530 nm to 1540 nm, the onfinement of the mode in As 2 S 3 waveguide hanges from to and the effetive index hanges from to Consequently, the propagation onstant hanges from to , dereasing by 0.7%. From the plot of γ in Figure 10, we an learn that different wavelengths have different ritial widths, whih shifts to a larger value as the wavelength inreases. Suh hange Figure 9. Measured (a) and simulated (b) oupling spetra of taper oupler with tip width = 1.3 μm. makes the mode at different wavelengths see the taper oupler slightly different, and the energy transfer does not take plae at the same loation: the mode of shorter wavelength ouples before that of a longer wavelength does. From the inset plot of γ in Figure 10, we also see that as wavelength inreases, the rate of shift inreases, onfirming the presene of dispersion. Beause of a 0.3 μm expansion during fabriation, the average tip width of tapered As 2 S 3 waveguide is 1.3 μm, and aordingly the transition width is 1.9 μm. Whether it is smaller or larger than that is dependent on the proess onditions, whih is hard to ontrol and manifested in the measured oupling spetra, as shown in Figures 11(a) and 12(a). Models are built to analyze them, in Figures 11(b) and 12(b). In Figure 11, there is a drop in oupling effiieny in long wavelength region, while the model shows if the tip width is redued to 1.2 μm, orrespondingly the end width of the first stage is 1.8 μm, suh a oupling spetrum will be resulted. The phenomenon an be understood from the plot of γ in Figure 10: at the wavelength of 1600 nm, the ritial width is read to be 1.87 μm, whih is larger than the atual transition width (1.8 μm). Hene the transfer of the energy has not ompleted yet at the end of the first stage, and resumes at the seond stage where the width varies very fast, and onsiderable energy is oupled to odd mode. Consequently, the oupling effiieny drops. Similarly, the model explains the drop of oupling effiieny in the short wavelength region for Figure 12. Provided that the tip width is larger, e.g. 1.4 μm, for short wavelengths suh as 1525 nm, the ritial width is 1.32 μm, whih is smaller than the initial tip width. As a result, the odd mode is exited at the input of the taper oupler, and the oupling effiieny in this wavelength region is degraded, as shown in the urve. Figure 10. γ of the taper ouplers at different wavelengths. Figure 11. Measured (a) and simulated (b) oupling spetra of taper oupler with tip width = 1.2 μm.

8 X. XIA ET AL Aknowledgements The authors would like to thank William Tim Snider and Travis E. James for the help in fabriation. This publiation was supported by the Pennsylvania State University Materials Researh Institute Nanofabriation Lab and National Siene Foundation Cooperative Agreement No , National Nanotehnology Infrastruture Network, with Cornell University. Figure 12. Measured (a) and simulated (b) oupling spetra of taper oupler with tip width = 1.4 μm. From the above analysis, we see a tradeoff between the appliable wavelength range and the design parameters, whih, on the other side, provides a way of ontrolling the frequeny domain behavior of taper ouplers: by adjusting the transition width between two stages we an ut off longer wavelengths and by hanging the tip width we an suppress the oupling of shorter wavelengths. 5. Conlusion A modeling method for taper ouplers is developed and applied to the study of As 2 S 3 -Ti: LiNbO 3 taper ouplers, whih are generally not adiabati but highly effiient in terms of pratial use. Simulations show that for those pratial tapers, both adiabati oupling and resonant oupling play an important role. There exists an optimum taper design with respet to the tip width, end width and length. A two-stage taper design an largely redue the total length of the taper by 64% while keeping high oupling effiieny above 90%. Following the guidelines, test strutures are fabriated. The measurement results agree with the simulation results well, suggesting a good oupling effiieny. Frequeny domain analysis shows that the taper ouplers work for a range of wavelengths, whih an be ontrolled by adjusting the transition width and the tip width. REFERENCES [1] M. E. Solmaz, et al., Compat Bends for Ahieving Higher Integration Densities for LiNbO 3 Waveguides, IEEE Photonis Tehnology Letters, Vol. 21, No. 9, 2009, pp doi: /lpt [2] M. E. Solmaz, et al., First Demonstration of an As 2 S 3 - on-linbo 3 Ring Resonator, Conferene on Optial Fiber Communiation Inudes Post Deadline Papers, San Diego, Marh 2009, pp [3] J. Haes, et al., Design of Adiabati Tapers for High- Contrast Step Index Waveguides, Linear and Nonlinear Integrated Optis, Lindau, April 1994, pp [4] C.-T. Lee, et al., Design and Analysis of Completely Adiabati Tapered Waveguides by Conformal Mapping, Journal of Lightwave Tehnology, Vol. 15, No. 2, 1997, pp [5] F. Xia, et al., Photoni Integration Using Asymmetri Twin-Waveguide (ATG) Tehnology: Part I-Conepts and Theory, IEEE Journal of Seleted Topis in Quantum Eletronis, Vol. 11, No. 1, 2005, pp doi: /jstqe [6] X. Sun, et al., Adiabatiity Criterion and the Shortest Adiabati Mode Transformer in a Coupled-Waveguide System, Optis Letters, Vol. 34, No. 3, 2009, pp doi: /ol [7] A. Yariv and X. Sun, Supermode Si/III-V Hybrid Lasers, Optial Amplifiers and Modulators: A Proposal and Analysis, Optis Express, Vol. 15, No. 15, 2007, pp doi: /oe [8] J. Haes, et al., A omparison between Different Propagative Shemes for the Simulation of Tapered Step Index Slab Waveguides, Journal of Lightwave Tehnology, Vol. 14, No. 6, 1996, pp [9] J.-M. Liu, Coupling of Waves and Modes, In: J.-M. Liu, Ed., Photoni Devies, Cambridge University Press, Cambridge, 2005, p