Mode conversion in tapered submicron silicon ridge optical waveguides

The mode conversion in tapered submicron silicon ridge optical waveguides is investigated theoretically and experimentally. Two types of optical waveguide tapers are considered in this paper. One is a regular lateral taper for which the waveguide width varies while the etching depth is kept the same. The other is a so-called “bi-level” taper, which includes two layers of lateral tapers. Mode conversion between the TM fundamental mode and higher-order TE modes is observed in tapered submicron siliconon-insulator ridge optical waveguides due to the mode hybridization resulting from the asymmetry of the cross section. Such a mode conversion could have a very high efficiency (close to 100%) when the taper is designed appropriately. This enables some applications e.g. polarizer, polarization splitting/rotation, etc. It is also shown that this kind of mode conversion could be depressed by carefully choosing the taper parameters (like the taper width, the etching depth, etc), which is important for the applications when low-loss propagation for the TM fundamental mode is needed. ©2012 Optical Society of America OCIS codes: (130.0130) Integrated optics; (230.5440) Polarization-selective devices. References and links 1. Y. Shani, C. Henry, R. Kistler, K. Orlowsky, and D. Ackerman, “Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon,” Appl. Phys. Lett. 55(23), 2389–2391 (1989). 2. R. Smith, C. Sullivan, G. Vawter, G. Hadley, J. Wendt, M. Snipes, and J. Klem, “Reduced coupling loss using a tapered-rib adiabatic-following fiber coupler,” IEEE Photon. Technol. Lett. 5, 1053–1056 (1993). 3. R. Zengerle, H. Bruckner, H. Olzhausen, and A. 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Introduction
Optical waveguide taper is a fundamental element for photonic integrated circuits (PIC's).It is often used to change the light spot size in order to have better coupling efficiency between two sections with different cross sections (e.g., a planar optical waveguide and a singlemode or lens fiber) [1][2][3][4][5][6][7][8].In order to achieve a low-loss taper, one usually makes the taper long enough to be adiabatic so that higher-order modes are not excited [9][10][11][12].This design rule works well usually especially for low index-contrast (∆) optical waveguides (e.g., SiO 2 -on-Si buried waveguides).However, the situation becomes complicated for small-sized high-∆ optical waveguides, e.g., submicron silicon-on-insulator (SOI) waveguides, which have been used widely for ultra-compact CMOS-compatible PIC's in the recent years [13][14][15][16][17][18][19][20][21][22][23][24].In a high-∆ optical waveguide, the mode hybridization is significant at some special waveguide widths [25][26][27][28][29][30] and consequently mode conversion may happen in a tapered structure [29,30].In Ref [29,30], the authors give a discussion on the mode conversion in a tapered SOI strip nanowire.When a SOI nanowire has a SiO 2 under-cladding and an air upper-cladding, which makes the SOI nanowire asymmetrical in the vertical direction, the mode conversion between the TM fundamental mode and the first-order TE mode is observed when light propagates along a taper structure.Such a mode conversion is not desired usually because it introduces some serious excess loss as well as crosstalk due to the excited higher order modes, e.g., in AWG (arrayed-waveguide grating) demultiplexer [31].Such undesired mode-conversion could be minimized by using several kinds of modified tapered structures suggested in Ref [29].A simple and easy way to depress such a mode conversion in a SOI-nanowire taper is to introduce a SiO 2 upper-cladding (instead of air) to make the SOI nanowire symmetrical in the vertical direction [30].On the other hand, such a kind of mode conversion could be very useful.For example, in our previous paper a SOI-nanowire taper was designed to have an almost 100% mode conversion efficiency from the TM fundamental (TM 0 ) mode to the first higher-order TE (TE 1 ) mode so that polarization splitter-rotators could be realized with a very simple design and easy fabrication process [30].
In this paper, we focus on the mode conversion in submicron SOI rib waveguides (other than SOI nanowires), which is also very popular for silicon-based integrated optoelectronics [19][20][21][22][23][24].One should note that there is a significant difference between an SOI rib waveguide and a SOI strip nanowire.A SOI strip nanowire could be symmetrical or asymmetrical in the vertical direction by simply choosing an appropriate material for the upper-cladding so that the mode conversion could be eliminated or enhanced accordingly [30].In contrast, for a SOI rib waveguide, it is still asymmetrical in the vertical direction even when having the same material for the upper-cladding and the under-cladding.Therefore, when it is desired to have a mode conversion between the TM 0 mode and the higher-order TE mode for the case of using a SOI rib waveguide, it is not necessary to choosing different materials for the uppercladding and the under-cladding.On the other hand, such an asymmetry also makes that one cannot yet avoid the mode conversion in a taper section due to the mode hybridization by simply choosing the same material for the upper-cladding and the under-cladding.Such a mode conversion will introduce a significant excess insertion loss as well as some channel crosstalk due to the excited higher-order modes (e.g., in AWG demultiplexers [31]).
In the following section, we give a detailed analysis for light propagation in SOI rib waveguide tapers and present the mode conversion numerically.The experimental observation for the mode conversion has also been presented.Two types of taper structures are considered here.One is a regular lateral taper for which the waveguide width varies while the etching depth is kept the same.Since the structure and the fabrication are very simple, a regular lateral taper is very popular for modifying the waveguide mode size in the lateral direction.The other is a so-called "bi-level" taper, which includes two layers of lateral tapers and consequently a double-etching process is needed for the fabrication.A bi-level taper is often used to connect two sections with different etching depths, e.g., from a shallowly-etched rib waveguide to a deeply-etched rib waveguide [7][8][32][33].For example, bi-level taper are very useful for the case when a singlemode rib waveguide is needed at the input/output ends of a chip while a strong confinement is desired for e.g., sharp bending.Our experimental and theoretical results show that one should be very careful when designing an adiabatic lateral taper or bi-level taper with a small-sized high-∆ optical waveguide, e.g., submicron SOI ridge waveguides considered in this paper.

Structure and analysis
In this paper, we consider tapered submicron SOI rib waveguides, which has been used very widely for silicon optoelectronics [19][20][21][22][23][24].Two types of taper structures are analyzed here.The first one is a regular lateral taper, and the other is the so-called bi-level taper [7][8][32][33]].In the present example, the SOI wafer has a 400nm-thick top Si layer and the refractive indices of Si and SiO 2 are n Si = 3.455, and n SiO2 = 1.445, respectively.A finite-difference method (FDM) mode-solver (from Fimmwave) is used to calculate the mode field profiles and the effective indices for all eigenmodes.show the 3D-view for the regular lateral taper and the cross section for the SOI rib waveguide.In this taper section, the waveguide width varies while the etching depth is kept the same.Such a taper is often used when it is needed to modify the mode size, e.g., at the input/output ends of a silicon photonic integrated chip in order to enhance the coupling efficiency between fibers and the chip.Regarding that the spot size of a commercialized lens fiber is usually around 3µm, in our calculation we give a modal analysis for a SOI rib waveguide whose core width varies from 3µm to 0.5µm in order to characterize the mode conversion in a waveguide taper to match the lens fiber [6].Figure 2(a)-2(c) show the effective indices for SOI rib waveguides with different etching depths h et as the core width w co increases from 0.5µm to 3µm.Here the etching depth is chosen as h et = 0.4H, 0.5H, and 0.6H, respectively.Particularly, for the case of h et = 0.4H, one should note that the TM 0 mode becomes leaky and is to be cutoff in the range of w co <0.95µm and thus the curve for the TM 0 mode in Fig. 2(a) stops at w 0 = 0.95µm.Here h et is given with a ratio in respect to H just to understand that the case considered here is with an etching depth around half of the total height (which is used very often).
Since a SOI rib waveguide is asymmetrical in the vertical direction, mode hybridization is observed in some special ranges of the rib width, e.g., around w co0 = 1µm, and 2.45µm, as shown by the circles labeled in Fig. 2(a)-2(c).Due to the mode hybridization around w co = w co0 , mode conversion between the two hybridized modes will happen when the light propagates along an "adiabatic" (long) taper structure whose end-widths (w 1 , and w 2 ) satisfy the condition: w 1 <w co0 <w 2 .In Fig. 2(a)-2(c), the arrowed curves indicate that the mode conversions between the TM 0 mode and the higher-order TE mode as the core width varies.Such a mode conversion is harmful when one expects to have a low-loss adiabatic taper [29].In order to avoid the undesired mode conversion, one can choose the taper widths (w 1 , w 2 ) so that there is no mode hybridization in the width range of w 1 <w<w 2 .In this way, there will not be mode conversion when light propagates along a long taper.From Fig. 2(a)-2(c), it can be seen that the mode hybridization region shifts when choosing different etching depths h et .One has a smaller w co0 (where the mode hybridization region locates) when the optical waveguide is etched less.This indicates that the mode conversion due to the mode hybridization could be modified by slightly adjusting the etching depth h et , which makes the design flexible.For example, when reducing the etching depth from 0.6H to h et = 0.4H, the first and second mode hybridization regions shift from around w co0 = 1.1µm and 2.55µm to around w co0 = 0.9µm and 2.35µm, respectively, as shown in Fig. 2(a).Then one can choose the taper end-widths in the range of 0.90µm<(w 1 , w 2 )<2.35µm so that no mode conversion happens in the designed taper for the case of h et = 0.4H.Particularly, regarding that the TM 0 mode becomes leaky when w co <0.95µm, one should choose the taper end-widths (w 1 , and w 2 ) to be larger than 0.95µm, i.e., (w 1 , w 2 )>0.95µm.Finally the end-widths of the low-loss taper should be 0.95µm<(w 1 , w 2 )<2.35µm.On the other hand, it is also possible to utilize such kind of mode conversion to obtain a polarization rotation, which is similar to the case of tapered SOI nanowires [30].In order to show the mode hybridization which causes the mode conversion in a tapered SOI rib waveguide, we consider the case of h et = 0.5H as an example.From Fig. 2(b), it can be seen that there are two regions (i.e., w co0 = 2.45µm, and 1.0µm) where mode hybridization happens.In the region around w co = 2.45µm, the mode hybridization happens between the TM 0 and the third-order TE (TE 3 ) mode.The mode profiles for these two modes are shown in Fig. 3(a)-3(b), respectively.It can be seen that the minor-component (E x or E y ) is comparable to the corresponding major-component (E y or E x ).In this case, it is hard to distinguish these two modes.When w co = 1.0µm, the mode hybridization is similar while it happens between the TM 0 mode and the first-order TE mode (TE 1 ), as shown in Fig. 4 As mentioned above, there are two regions (around w co0 = 1.0µm, and 2.45µm) where mode conversions happen when the core width is tapered from 3µm to 0.5µm.Therefore, here we examine two types of tapers.For the first one, the taper end-width is chosen as w 1 = 2µm, and w 2 = 2.7µm (w 1 <2.45µm<w 2 ).The second one has taper end-widths w 1 = 0.8µm, and w 2 = 1.5µm (w 1 <1µm<w 2 ).A commercial software (FIMMPROP, Photon Design, UK) employing an eigenmode expansion and matching method [34] is then used to simulate the light propagation in the defined taper structure.Figure 5 shows the mode conversion efficiencies coupled to the TM 0 mode and the TE 3 mode after the launched TM 0 mode propagates along the linear lateral taper with w 1 = 2.7µm, and w 2 = 2.0µm.From this figure, it can be seen that one could realize a very high efficiency (>90%) from the TM 0 mode to the TE 3 mode when choosing the taper length L tp appropriately.For example, here we choose L tp = 1500µm.When the launched field is chosen as the TE 0 or TM 0 modal field, the simulated light propagation along the designed taper are shown in Fig. 6(a) and 6(b), respectively.It can be seen that the mode conversion happens as predicted when the input filed is the TM 0 mode, while there is no mode conversion for the case with the TE 0 -mode input.When one choose a shorter taper (e.g., 350µm), the launched TM 0 mode is then converted to the TE 3 mode partially.Consequently two-mode interference happens, which introduces some undesired ripples when measuring the wavelength dependence of the output power.5, one can also see that the mode conversion could be very slight by choosing a very short non-adiabatic taper.For example, for a 10µm -long taper, the mode conversion from TM 0 to TE 3 is about 5% only and such a low loss is acceptable for some applications.Figure 7(a) and 7(b) show the simulation results for light propagating along a 10µm-long taper for the case with the TE 0 and TM 0 modes launched, respectively.From these two figures, it can be seen that there are some small ripples due to the multimode-interference effect.For the case when the TE 0 mode is launched, the TE 2 mode is excited slightly because the taper is not adiabatic.In this case, the dominant mode is the TE 0 mode which has a power ratio of 99.66% while the excited TE 2 mode has a low power ratio of about 0.26%.Figure 8 shows the mode conversion efficiencies to the TM 0 and the TE 1 mode after the launched TM 0 mode propagates along a linear lateral taper with w 1 = 1.5µm, and w 2 = 0.8µm, between which there is a mode hybridization region round w co0 = 1µm (see Fig. 2 (b)).From this figure, it can be seen that the mode conversion efficiency from the TM 0 mode to the TE 1 mode is close 100% when choosing the taper length L tp appropriately.
For example, here we choose L tp = 215µm.Figure 9(a) and 9(b) show the simulation results for light propagating along the designed taper when the launched field is chosen as the TE 0 and TM 0 modal field, respectively.It can be seen that the mode conversion happens for the input TM 0 modal field, while there is no mode conversion for the input TE 0 modal field, as predicted.When one chooses a shorter taper, the launched TM 0 mode is converted to the TE 1 mode partially.For example, when L tp = 22.4µm, the mode conversion from the TM 0 mode to the TE 1 mode is about 50%, and a significant multimode-interference effect is observed as shown in Fig. 10(a)-10(b).Particularly, when choosing L tp = 0 (i.e., no taper), the mode conversion from the TM 0 mode to the TE 1 mode is about 25% and more power (~75%) is preserved to the TM 0 mode as shown in Fig. 8.One could design a discontinuous taper structure by optimizing the widths w 1 and w 2 further to improve the preservation efficiency for the TM 0 mode, in a way similar to that suggested for the design of SOI strip nanowire tapers in Ref [29].In a short summary, for a regular lateral taper whose widths ranges from w 1 to w 2 (w 1 <w 2 ), the mode conversion between the TM 0 mode and higher-order TE modes (e. happens when there is a mode hybridization region in the range of w 1 <w<w 2 according to the simulation given above.Such a mode conversion could be very efficient (close to 100%) when the taper length is long enough, which is very useful some applications, e.g., polarization rotation.On the other hand, it is also possible to design a taper to minimize the mode conversion by choosing the etching depth (h et ) or the range for the taper end-widths (w 1 , and w 2 ) when it desired to achieve a low-loss waveguide taper.For example, one can choose the taper end-widths (w 1 , and w 2 ) so that there is no a mode hybridization region in the range of w 1 <w<w 2 .In this way, there will not be mode conversion when light propagates along a long taper.Generally speaking, the mode evolution in a gradually-varying taper structure is insensitive to the variation of the taper dimension (e.g., the height, the width) when the taper is long enough.However, when there are mode-hybridization regions as discussed here, one should choose the taper end-widths carefully to be tolerant to the taper dimension variation because the mode hybridization region shifts slightly as the taper dimension changes.For example, one can choose the taper end-widths not to be close to the mode hybridization regions (around w co0 ), which can make the taper tolerant to the dimension variation.
Bi-level taper is another type of taper structure used often to connect two sections with different etching depth [7-8, 32-33, 35].Usually it is assumed that no higher-order mode is excited when light propagates along a long bi-level taper, so that one achieves a low-loss smooth transition between the fundamental modes of the SOI ridge waveguide and the silicon strip waveguide.However, our simulation shows that higher-order modes might be generated even in a long bi-level taper for submicron SOI ridge waveguides.It is very essential to understand this issue when designing the waveguide taper.In the following part, we give a detailed analysis for the mode conversion in bi-level taper.Figure 12(a)-12(c) show the effective indices for an SOI double-rib waveguide with h et = 0.5H as the side-rib width w side decreases from 3µm to 0 when the central-rib width is chosen as w co = 0.85, 1, and 1.2µm, respectively, so that the SOI rib waveguide is quasi-singlemode.Since a SOI double-rib waveguide is not symmetrical in the vertical direction, mode hybridization might happen.The mode hybridization in a SOI double-rib waveguide depends a lot on the central rib width w co .For the present case (h et = 0.5H), it is found that mode hybridization and conversion happen as the side-rib width w side varies from 3µm to 0 when the central rib width w co = 1.0µm according to Fig. 12(b) and the mode profiles e.g.shown in Fig. 13(a)-(b) below.In contrast, when choosing w co = 0.85µm, and 1.2µm, there is no mode hybridization and conversion according to Fig. 12(a)-(c), and Fig. 14    (C) 2012 OSA a rectangular waveguide, which is symmetrical in the vertical direction and consequently no mode hybridization is observed.In contrast, for the case of w side = 0.5µm, it can be seen that the eigen-modes have significant major as well as minor components (E x and E y ) due to the hybridization as shown in Fig. 13(a)-13(b).This mode hybridization between the TM 0 mode and the TE 1 mode makes a mode conversion between them when light propagates along an adiabatic taper.In order to check the mode hybridization for the cases of w co = 0.85µm and 1.2µm, we also consider the waveguide with w side = 0.5µm and show the filed profiles (E x and E y ) for mode #1 and #2 in Fig. 14(a)-14(b), respectively.Note that the color scale are different for E x and E y .According to the field profiles, it can be seen that both mode #1 and #2 have a major components (E x or E y ), which indicates the mode hybridization is not significant.In order to show the mode conversion in a tapered SOI double-rib waveguide, the case of h et = 0.5H is considered as an example.We use a commercial software (FIMMPROP, Photon Design, UK) [34] to simulate the light propagation in the present structure.Figure 15 shows the mode conversion efficiencies to the TM 0 mode and the TE 1 mode after the launched TM 0 mode propagates along the bi-level taper.Here the side-rib width at the input end of the bilevel taper is chosen as w side1 = 3.0µm (see the inset in Fig. 15).From this figure, it can be seen that the mode conversion efficiency from the TM 0 mode to the TE 1 mode is close 100% when choosing the taper length L tp appropriately (e.g., >300µm).For example, here we choose L tp = 300µm.Figure 16(a)-16(b) show the simulation results for light propagating along the designed taper when the launched field is chosen as the TE 0 , and TM 0 modal field, respectively.It can be seen that there is a very efficient mode conversion observed between the TM 0 mode and the TE 1 mode as predicted, while there is no mode conversion when the (C) 2012 OSA TE 0 modal field is launched.The efficient mode conversion between the TM 0 mode and the TE 1 mode enables the realization of a polarization splitter-rotator with the assistance of an asymmetrical directional coupler as proposed in Ref [30].Note that the taper length could be shortened greatly by choosing a smaller side-rib width w side for the input end of the bi-level taper (e.g., w side1 = 1.0µm) since the mode conversion happens at the region very close to the output end of the bi-level taper (see Fig. 16(b)).In order to give a comparison, we also calculate the mode conversion efficiencies for the cases of w co = 0.85, and 1.2µm, as shown in Fig. 17(a) and Fig. 17 (b), respectively.From these figure, it can be seen that the mode conversion could be eliminated almost by choosing the taper length appropriately when the rib width is chosen as 0.85 or 1.2µm.As an example, Fig. 18(a)-(b) show the simulated light propagating in the taper with w co = 1.2µm when the launched field is the TE 0 , and TM 0 modal fields, respectively.Here the taper length is L tp = 100µm.It can be seen that there is no mode conversion observed between the TM 0 mode and the TE 1 mode as predicted.Therefore, one has to be careful when design a bi-level taper for TM polarization.It also indicates that the mode conversion between the TM 0 mode and the higher-order TE mode could be cancelled or enhanced by choosing the rib width or etching depth appropriately.

Experimental observation
We fabricated some straight waveguides with tapered structures, as shown in Fig. 19(a), by using a simple fabrication process including UV lithography, and ICP (inductively coupled plasma) etching.The etching depth of the fabricated SOI rib waveguide is h rib = 0.5H, and H = 400nm.The taper structure has a 1µm-wide straight waveguide in the middle while there are tapers at both ends, which have been used often to obtain better coupling to singlemode fibers.All the parameters for the widths of the tapers are: w 1 = 1µm, w 2 = 1.5µm, and w 3 = 3µm, as shown in Fig. 19(a).The taper length L tp = 100µm.Our simulation in Fig. 5 indicates that the mode conversion in the section tapering from w 2 = 3µm to w 3 = 1.5µm is not significant since the taper length is not long.Furthermore, since the TE 3 mode is going to be cutoff (see Fig. 2(b)), it is reasonable to assume that there is only TM 0 mode left in the 1.5µm-wide section after light goes through the taper section at the input end.In contrast, as shown in Fig. 8, the mode conversion between the TM 0 mode and the TE 1 mode is expected to be quite significant in the second taper from w 2 = 1.5µm to w 1 = 1µm.Therefore, two-mode interference happens between the TM 0 mode and the TE 1 mode in the 1µm-wide straight section, which has been observed theoretically (see Fig. 10(a)-10(b)).
Figures 19(b) and 19(c) shows the measured light transmissions of a series of taper structures when the TM 0 and TE 0 modes are launched respectively.The length L 1 of the 1µmwide straight section in the middle varies from 0 to 1020µm.From these figures, it can be seen that the spectral response is quasi-periodical due to the two-mode interference as expected when the launched field is the TM 0 mode.In contrast, there is no quasi-periodical responses when the TE 0 modal field is launched, which is also consistent with our prediction.When the TM 0 mode is launched, the beat length L π of the two-mode interference in the 1µm-wide straight section is then given by L π = π/[(n eff_TM0 -n eff_TE1 )k 0 ], where n eff_TM0 and n eff_TE1 are the effective indices of the TM 0 mode and the TE 1 mode, respectively, k 0 is the wavenumber in vacuum.Correspondingly, the free-spectral range (FSR) of the spectral response of the taper structure is given by 2 where n g_TM0 and n g_TE1 are the group indices for the TM 0 mode and the TE 1 mode, respectively and they are given by n g_TM0 = n eff_TM0 -λ(∂n eff_TM0 /∂λ), and n g_TE1 = n eff_TE1λ(∂n eff_TE1 /∂λ), respectively.With this formula, the calculated quasi-FSR of the spectral response for the taper structure as the length L 1 varies, is shown in Fig. 19(c).The quasi-FSR extracted from the measured spectral responses in Fig. 19(b) is also shown in order to give a comparison.From Fig. 19(c), it can be seen that the theoretical and experimental results agree with each other very well.
From the measurement results shown here, one should realize that the mode conversion in a taper structure might be very serious and influence the performances of optical waveguides and devices.It is necessary to design the taper very carefully to avoid the undesired mode conversions.
For bi-level tapers, the mode conversion could be removed by choosing a relatively deep rib, e.g., h et = 0.6H, as indicated in Fig. 12(d).Figure 20(a) and 20(b) show the measured light transmissions of straight waveguides with bi-level taper structures at the input/output ends when the TM 0 and TE 0 mode are launched, respectively.Here h et = 0.6H.The central-rib width is w co = 1µm.From these figures, it can be seen that the spectral response is quite smooth, which indicates that no mode conversion is observed for the case with a TM 0 launched field, as the theoretical calculation predicted.Such a design has been used in a MZIbased PBS successfully in our previous paper [35].

Conclusions
In this paper, the mode conversion in tapered submicron SOI rib optical waveguides has been studied.We have considered two typical optical waveguide tapers (i.e., regular lateral tapers, and bi-level tapers) for submicron SOI rib optical waveguides.For a SOI rib waveguide, it is still asymmetrical in the vertical direction even when choosing the same material for the upper-cladding and the under-cladding.Therefore, in a tapered SOI rib optical waveguide, mode conversion between the TM 0 mode and higher-order TE modes might happens due to the mode hybridization in some waveguide width ranges, which has been observed for both types of waveguide tapers in our simulation.Our experimental results have also been demonstrated to give the evidence of mode conversions.Such a mode conversion is not desired usually because some excess loss and crosstalk is introduced in some photonic integrated circuits.It has also been shown that such harmful mode conversion effect can be removed almost for both types of waveguide tapers by carefully designing the taper parameters (e.g., the width and length of the taper, the etching depth, etc).On the other hand, our simulation results have also shown that a very high mode-conversion efficiency (close to 100%) could be achieved in both SOI rib optical waveguide tapers.Such an efficient mode conversion could be useful for some applications, e.g., polarization rotation [30].

#
Figure1(a) and 1(b) show the 3D-view for the regular lateral taper and the cross section for the SOI rib waveguide.In this taper section, the waveguide width varies while the etching depth is kept the same.Such a taper is often used when it is needed to modify the mode size, e.g., at the input/output ends of a silicon photonic integrated chip in order to enhance the coupling efficiency between fibers and the chip.Regarding that the spot size of a commercialized lens fiber is usually around 3µm, in our calculation we give a modal analysis for a SOI rib waveguide whose core width varies from 3µm to 0.5µm in order to characterize the mode conversion in a waveguide taper to match the lens fiber[6].

Fig. 1 .
Fig. 1.(a) The schematic configuration of a regular lateral taper; (b) the cross section for a SOI rib waveguide.

Fig. 3 .
Fig. 3.The field profiles (Ex and Ey) for modes #1 and #2 of a SOI ridge waveguide with wco = 2.45µm, (a) mode #1; (b) mode #2.The total height of the Si core layer is H = 400nm, and the etching depth het = 0.5H.Here modes #1 and #2 are the two hybridization modes in the region around w = 2.45µm.

Fig. 4 .
Fig. 4. The field profiles (Ex and Ey) for modes #1 and #2 of a SOI ridge waveguide with wco = 1.0µm,(a) mode #1; (b) mode #2.The total height of the Si core layer is H = 400nm, and het = 0.5H.Here modes #1 and #2 are the two hybridization modes in the region around w = 1.0µm.

Figure 11 (
Figure 11(a) and 11(b) show the 3D-view for the bi-level taper and the cross section for the SOI double-rib waveguide.Such a taper is often used to connect two sections with different etching depths previously [7-8, 32-33].

Fig. 11 .
Fig. 11.(a) The schematic configuration of a bi-level lateral taper; (b) the cross section for an SOI double-rib waveguide in the taper section.
Figure12(a)-12(c) show the effective indices for an SOI double-rib waveguide with h et = 0.5H as the side-rib width w side decreases from 3µm to 0 when the central-rib width is chosen as w co = 0.85, 1, and 1.2µm, respectively, so that the SOI rib waveguide is quasi-singlemode.Since a SOI double-rib waveguide is not symmetrical in the vertical direction, mode hybridization might happen.The mode hybridization in a SOI double-rib waveguide depends a lot on the central rib width w co .For the present case (h et = 0.5H), it is found that mode hybridization and conversion happen as the side-rib width w side varies from 3µm to 0 when the central rib width w co = 1.0µm according to Fig.12(b) and the mode profiles e.g.shown in Fig.13(a)-(b) below.In contrast, when choosing w co = 0.85µm, and 1.2µm, there is no mode hybridization and conversion according to Fig.12(a)-(c), and Fig. 14(a)-(b) below.The mode hybridization and conversion can be also avoided by choosing a deeper etching depth h et .For example, when choosing h et = 0.6H (see Fig.12(d)), there is no mode conversion observed.

Figure 13 (
Figure13(a)-13(b) shows the field profiles (E x and E y ) for mode #1 and #2 in the case of w co = 1.0µm when w side = 0.5µm and 0µm, respectively.Here modes #1 and #2 are the two lowest order modes except the TE 0 mode.When w side = 0, the double-rib waveguide becomes

Fig. 19 .
Fig. 19.(a) The structure of the lateral taper in our experiments; (b) the measured spectral responses for taper structures when the TM0 modal field is launched; (c) the measured spectral responses for taper structures when the input TE0 modal field is launched; (d) the measured and calculated quasi-FSR.H = 400nm, and het = 0.5H.

Fig. 20 .
Fig. 20.(a) The measured spectral responses for bi-level taper structures when the TE0 modal field is launched; (b) the measured spectral responses for taper structures when the TM0 modal field is launched; The parameters are: H = 400nm, wco = 1µm, H = 400nm, and het = 0.6H.