1. Introduction

Silicon photonics has made high density integrated optical components on one chip possible [1–4]. The high refractive index contrast in silicon photonic technology leads to modes with very tight confinement. For some applications, such as hybrid integration [4,5], it is desirable to have strongly evanescent fields. In such cases, thin, shallow ridge silicon-on-insulator (SOI) waveguides have been proposed as a solution [6–8]. It has, however, been demonstrated that the TM-like mode of a thin, shallow ridge waveguide will leak into a radiating TE mode unless the waveguide width is maintained at a resonant, so called ‘magic’, width [6,9]. The guided TM and radiating TE modes are non-orthogonal due to the strong longitudinal field component of the TM mode and are also inherently phased matched. Thus, no surface or side-wall roughness is required for this coupling [6,9–11]. The phase matching of the guided TM and radiating TE slab modes means that radiation occurs only at a very specific angle to the propagation axis and in equal amounts on both sides of the waveguide [9].

This lateral leakage can be likened to a leaky waveguide antenna [12]. Drawing on this analogy with antennas, applications could be considered where this leakage is harnessed to transmit an optical signal between well separated waveguides through use of the unguided TE radiation. Before considering such a scenario, it is important that the characteristics of the radiation can be controlled and directed. For example, if we wished to transmit light efficiently between two adjacent waveguides, the radiation from the transmitting waveguide should emit from one side only.

In this paper, we present a novel waveguide structure based on lateral leakage in thin, shallow ridge SOI waveguides. The structure utilizes two parallel leaky waveguides that are excited with quadrature phase and separated by a quarter wavelength. The radiation from these two waveguides interfere constructively on one side and destructively on the other. We analyze this structure using a fully vectorial mode matching technique. It is shown that if the waveguides are excited at quadrature phase, it is possible to adjust the separation between the waveguides such that radiation on one side of the structure is suppressed. It is also shown that the waveguides remain coupled even for significant separations and thus care must be taken to balance both the radiation loss and phase velocity of the two supermodes involved.

2. Single side radiation device concept

Our aim is to conceive a waveguide configuration that radiates only from one side. To achieve this goal, we propose that two radiating waveguides can be excited with quadrature phase and placed in parallel with an appropriate separation such that the radiation on one side is coherently cancelled. Figure 1(a) presents the cross section and plan view of a single SOI shallow ridge waveguide. A ray diagram illustrates the path of the guided TM-like mode and coupling to the radiating TE mode at the side walls of the ridge [6]. Since the radiation is due to inherent phase matching and is not due to any scattering resulting from random perturbations, the TE-radiation propagates at a very specific angle to the waveguide axis and is symmetric on both sides of the waveguide.

 

Fig. 1 (a) X-Y cross section of a single SOI shallow ridge waveguide and plan view of the structure showing TM to TE mode coupling at the side wall of the ridge, (b) X-Y cross section of two identical and parallel SOI shallow ridge waveguides and plan view of the structure showing the mechanism of the summation of the radiation of the two waveguides in both sides of the structure. Waveguide dimensions are shown.

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Figure 1(b) shows the cross section and plan view of two parallel identical SOI shallow ridge waveguides. At this stage, we assume that evanescent coupling between the waveguides can be neglected. The fundamental TM mode of each of the two waveguides is excited with the same amplitude (A) but the excitation of the second waveguide has its phase shifted by Φ relative to the first waveguide. Figure 1 (b) illustrates the two TE waves radiating from each waveguide. Both TE waves propagate in the same direction but will have a relative phase that is dependent on the relative phase of excitation as well as the separation of the waveguides. The phase difference of the two equal components of TE radiation emanating from the right and left side of the structure can be written as:

where k₀ = 2π/λ is the wavenumber of free space, n_TE is the effective index of the TE slab mode and d is the distance between the phase fronts of the TE waves exiting the two waveguides as illustrated in Fig. 1(b). To suppress the radiation on one side, the phase difference should be Δφ = (2m + 1)π, where m is an integer. If the fundamental modes of the two waveguides are excited with phase difference of Φ = π/2 and the separation of the two waveguides (S) is chosen carefully such that k₀n_TEd = 2mπ + π/2, then, according to Eq. (1) and (2) the phase difference between the TE waves radiating to the right will be Δφ_R = (2m + 1)π, while the phase difference between the TE waves radiating to the left will be Δφ_L = 2mπ. If equal power radiates from each waveguide, then the radiation on the right side should coherently cancel while the radiation on the left side should coherently sum. Thus, in principle, the configuration of Fig. 1(b) should achieve single side radiation.

3. Simulation using superposition

Having conceived a structure that should provide single side radiation, we now wish to investigate theoretically whether it will truly suppress radiation on one side. To rigorously model thin-ridge SOI waveguides with lateral leakage, a fully vectorial mode matching technique [13] was employed. Since the computational window in mode matching technique is completely open in lateral directions, the lateral leakage can be simulated very accurately [9,11,14]. To avoid treatment of the continuum of the radiation modes vertically, the ridge waveguide was placed between two perfectly conducting planes above and below the structure. These conducing planes were placed far enough from the waveguide, so that their effect on the waveguide is negligible.

The mode matching model was used to simulate the guided modes of the waveguide structure of Fig. 1(a). The width of the waveguide was chosen to be 1μm to maximize TE lateral leakage radiation at a wavelength of 1.55 µm [6,9].

Figures 2(a) and (b) show the real and imaginary parts of the vectorial components of the electric field distributions of the guided TM-like mode of the thin-ridge waveguide. The TM-like mode of the waveguide is clearly strongly coupled to the radiating TE-slab mode. The TE radiation amplitude is symmetric on both sides of the waveguide.

 

Fig. 2 Electric field components of the guided mode of the waveguide of Fig. 1 with width 1µm to maximize TE radiation at λ = 1.55μm. (a) real component; (b) imaginary component.

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As a first approximation of the field that would be expected in two parallel thin, shallow ridge SOI waveguides, we simply super-imposed two identical instances of the field solutions presented in Fig. 2, but with a lateral translation and an additional phase of Φ = π/2 applied to the complex field profile on one instance. This simplistic approach of superposition ignores the effect of coupling between the waveguides.

Figures 3(a) and (b) present the magnitude of E_y for the left waveguide and the right waveguide respectively. The radiation into the TE mode is evident in both figures. The field (E_y) of right waveguide (Fig. 3(b)) is identical to the left waveguide, but has been translated to the right by 4.58μm and has been advanced in phase by π/2. Figure 3(c) presents the coherent superposition of the fields of Fig. 3(a) and Fig. 3(b). It is evident that the coupling to the TE radiation is far stronger on the right side than it is on the left. This supports our reasoning that this structure should achieve single side radiation.

 

Fig. 3 The magnitude of the y-component of the electric fields of the TM-like modes of (a) the original SOI waveguide; (b) the additional identical SOI waveguides with a 4.58µm lateral translation and a π/2 phase shift and (c) the coherent superposition of these two modes; (d) the magnitude (in Log scale) of the super-imposed radiation fields of identical modes at two symmetric fixed points (y1 = + 8µm and y2 = −8µm, x1 = x2 = midpoint of the Si film) on right & left hand sides of both waveguides versus the gap (S) between the waveguides and the magnitude of the summation of the TE radiation waves as obtained from Eq. (1) and (2) which account for the phase difference between the two TE waves.

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To examine the relationship between the field radiated to each side of the waveguide structure and the waveguide separation, field distributions were calculated as a function of waveguide separation and the TE field was monitored on both sides of the structure. The locations chosen to monitor the field were at two symmetric locations to the left and right of the parallel waveguide structure. These two locations are vertically in the middle of the silicon core layer and have the same lateral distance to the center of the left or right waveguide respectively.

Figure 3(d) presents the radiated TE field amplitude on the left and right sides of the structure as a function of separation. It is clear that the TE radiation field amplitude measured at each side of the structure has a cyclic dependence on the waveguide separation. For some waveguide separations, the TE radiation on one side is suppressed, while it is at a maximum on the other side. Figure 3(d) also shows the field amplitude predicted by Eq. (1) and (2) using the relative phase between TE radiation from each waveguide. Excellent agreement is evident.

Figure 3 illustrates through the use of superposition that it should indeed be possible to suppress the radiation from one side of the waveguide structure using appropriate waveguide separation and phase shift. Examination of Fig. 3 would suggest that the radiation suppression will be quite sensitive to waveguide separation and it might be expected that the initial phase of the two modes would be equally important. To quantify this sensitivity, Eqs. (1) and (2) were used to calculate the tolerance in initial phase and waveguide separation that would be required to maintain >20dB radiation suppression on one side. It was found that initial phase needs to be maintained within ± 10 degrees which would be quite achievable practically. The waveguide separation should be maintained within ± 20nm of the designed values which, while challenging, is within the range that can be achieved with modern CMOS processing.

The results of Fig. 3 suggests that it should in principle be possible to configure two waveguides such that they radiate to one side only, however, this analysis has neglected the effect of coupling between the waveguides and this may be an important factor limiting the degree of suppression that can be achieved.

4. Simulation using supermodes

In Section 3, we have shown that by selecting the appropriate separation between two parallel waveguides and exciting the TM guided modes in the two waveguides with a π/2 phase difference, the TE radiation is enhanced in one side while it is suppressed in the other side of the waveguide structure. However, in the previous analysis, the coupling between two waveguides was ignored. In this Section we aim to more rigorously model the radiation behavior of two parallel waveguides by including the effect of coupling.

The behavior of a coupled parallel waveguide structure can be rigorously represented as a superposition of the even and odd supermodes of the coupled structure [15]. We have shown previously that mode matching can be used to effectively model coupled waveguide structures that exhibit lateral leakage [9]. In the present case, we wish to excite the two parallel waveguides with modes of equal amplitudes and a π/2 phase difference between them. This π/2 phase difference between the light in each waveguide can be achieved by exciting the odd and even supermodes with a π/2 phase difference.

Using the mode-matching model, the eigenvalues and eigenvectors of the even and odd supermodes were calculated for different waveguide separation. Figures 4(a) and (b) show the real and imaginary parts of the calculated effective indices of the even and odd supermodes as functions of the waveguide separation. For separations of less than 3μm, the effective indices of the odd and even modes become significantly different indicating strong coupling. For separations above 3μm, the odd and even mode effective indices asymptote to approximately the same value indicting that coupling between the waveguides has significantly diminished. Careful inspection of the real part of the effective index of both the supermodes for separations above 3µm reveals that this value actually oscillates with separation. This oscillatory behavior is due to the coupling between the two waveguides through the minor TE radiation field component. This coupling is via the propagating TE wave rather than being evanescent and hence does not reduce with increasing separation.

 

Fig. 4 .Real (a) and imaginary (b) parts of the effective refractive indices of the even and odd supermodes of the structure versus the waveguide separation, respectively. The magnitude of the y-component of the electric fields of the TM-like supermodes of the structure with 4.54µm waveguide separation respectively: (c) even mode (d) odd mode.

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The imaginary component of the effective index for the odd and even supermodes is presented in Fig. 4(b). This corresponds to radiation loss. The loss clearly oscillates as a regular function of separation. This suggests that if the waveguides are independently excited with equal power and are either in phase (even mode) or in anti-phase (odd mode) then there will be distinct waveguide separations at which the radiation of both modes is equal and hence the lateral leakage will be suppressed. This oscillatory lateral leakage appears to be largely unaffected by the evanescent coupling at separations below 3μm. Figures 4(c) and (d) present the magnitude of the electric field eigenvector calculated for the odd an even eigenvalues at a separation of 4.54μm. Radiation is clearly evident in both supermodes.

Figure 5(a) presents the coherent superposition of the odd and even supermodes of Fig. 4(c) and (d) with a relative phase shift of π/2. As expected, the radiating TE field to the left is suppressed while the radiation to the right is enhanced. The impact of the waveguide separation on the amount of radiation that would be expected on the left and right sides was quantified by recording the magnitude of the field at symmetric locations in the centre of the silicon core layer to the left and right of the waveguide structure. Figure 5(b) presents the radiation recorded at these locations as a function of the waveguide separation. It is evident that the radiation oscillates from left to right. Complete cancellation on the left or right is only observed at separations where the amplitude of radiation of the odd and even modes is equal. These specific separations can be determined from Fig. 4(b).

 

Fig. 5 (a) Magnitude of the y-component of the electric field of the superimposed TM-like even and odd supermodes of the structure with a phase difference of π/2 and 4.54µm waveguide separation; (b) magnitude (Log scale) of the radiation field from the superposition of the supermodes for two symmetric fixed points (y1 = + 8µm and y2 = −8µm, x1 = x2 = middle point of the Si film) to the right and left of the structure as a function of waveguide separation.

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It can be seen from Fig. 4(a) that a separation of greater than 3μm is required to minimize the evanescent coupling between the two waveguides. Further, even for larger separations, due to the oscillatory nature of the real part of the effective index, only specific separations will result in equal effective indices for both odd and even modes. Comparing Fig. 4(a) and Fig. 4(b) for large separations, it is evident that when the radiation is equal for the odd and even modes, the difference in the real parts of their effective indices is greatest. Hence, it is not possible to have both the real and imaginary parts of the effective indices of the odd and even supermodes equal simultaneously and thus some compromise must be made.

5. Impact of phase and radiation mismatch between odd and even supermodes

In the previous Section, it was shown that even for large separations, both the radiation and the effective indices of the odd and even supermodes of a laterally leaking waveguide pair would oscillate with waveguide separation. Further, it was shown that it was not possible to achieve equal radiation and phase velocity for the odd and even supermodes simultaneously and hence ideal radiation cancellation on one side of the structure is not possible along the entire propagation length. We anticipate that with radiation balance, but phase mismatch, ideal cancellation would be achieved initially, but as the modes propagated this cancellation would oscillate from side to side as the supermodes accumulate phase difference. Conversely, if phase matching was achieved, but radiation was imbalanced, stable radiation behavior would be observed as the modes propagated, but the cancellation of the radiation would be poor. To investigate this predicted behavior and to explore possible compromises we simulated the evolution of the super-imposed odd and even supermodes and analyzed the radiation.

To perform this analysis we used the supermodes of Section 4 with separations of 4.36μm (perfectly phase matched, but radiation imbalanced), 4.54μm (phase mis-matched, but perfectly radiation balanced) and 4.45μm (a compromise mid point with neither perfect phase match nor radiation balance). To advance the modes along the propagation direction we simply adjusted the phase and amplitude of the two modes according to their complex propagation constants. To quantify the radiation predicted on each side of the structure as a function of propagation distance we chose two positions on the left and right of the parallel waveguide structure and calculate the Poynting vector at these points along the propagation direction. To visualize the propagation and radiation as the mode superposition evolved, we calculated the Poynting vector at each lateral and longitudinal coordinate.

Figures 6(a)-(c) show the power (Poynting vector magnitude) that would be expected throughout the entire structure as a function of propagation distance for separations of 4.36μm, 4.54μm and 4.45μm. In Fig. 6(a) the power does not couple between the waveguides, but there is no evidence of radiation cancellation on either side of the structure. This is

 

Fig. 6 Poynting vector magnitude of Y-Z cross section of the structure for waveguide separations of, (a) 4.36μm, (b) 4.54μm and (c) 4.45μm; Poynting vector magnitude at two symmetric fixed points to the right and left of the structure (y1 = + 8µm and y2 = −8µm, x1 = x2 = midpoint of the Si-film) as a function of propagation distance in the z-direction for (d) 4.36μm, (e) 4.54μm and (f) 4.45μm.

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expected due to the perfect phase match but imperfect radiation balance as can be seen in Fig. 4(a) and (b). Figure 6(d) shows that the radiation is indeed equal on both sides of the structure. The magnitude of both the guided and radiated field reduces with propagation distance as expected since power will be lost due to radiation.

Figure 6(b) shows strong coupling between the light confined in the two waveguides and initially strong cancellation of the radiating field on one side of the structure. However, as the light propagates, this cancellation rapidly degrades and towards the end of the propagation, the cancellation is observed on the other side of the structure. The coupling can be explained since the phase velocities of the odd and even modes are not equal. Figure 6(e) shows the power radiated on each side of the structure and it is clear that initially the cancellation on the right side is near ideal, but rapidly degrades as the odd and even modes de-phase. Towards the end of the propagation the cancellation shifts to the other side. Attenuation of both the guided and radiating fields is again evident with propagation.

Figure 6(c) presents the field evolution for the compromise situation. Here the coupling between the guided light is only subtle. Modest cancellation of the radiation at initial stages is also evident. Figure 6(f) shows the power radiated to the right side is suppressed by about 20dB when compared to the left side. Importantly, this suppression is maintained for approximately 250μm of propagation. After this length, coupling begins to dominate. The results of Fig. 6 show that while it is not possible to achieve ideal cancellation of radiation that is sustained for a long propagation distance, it is possible to strike a compromise where a reasonable degree of radiation cancellation can be maintained over some propagation distance. It should be possible to adjust the separation to trade propagation distance for improved suppression or vice versa. It may also be possible to design longitudinally varying structures that suppress coupling without compromising radiation suppression.

To practically implement this structure, a means of providing the initial π/2 phase difference between the excited waveguide modes must be found. This could be achieved by using the two outputs of a 3dB directional coupler. In addition phase modulators could be inserted between this directional coupler and the waveguide pair to fine tune the initial phase on the two waveguides. Active control of the phase of excitation may also enable dynamic manipulation of the direction of the radiation.

6. Conclusion

In this paper, we have presented the first analysis of directional control of radiation for Silicon-On-Insulator thin-ridge waveguides with lateral leakage. We have employed mode matching to calculate the TM mode solution of the whole structure and the effective indices for different gaps between the two waveguides. We have used both simple superposition and more sophisticated supermode analysis to predict the radiation behavior. Through this analysis we have found that it is not possible to meet both the phase matching and radiation balance conditions required for ideal suppression of radiation on one side of the structure to be sustained over long propagation distances simultaneously. We have examined the evolution of superposed supermodes as they propagate and have shown that it is possible to strike a compromise between phase-match and amplitude balance that allows modest radiation suppression to be sustained over reasonable propagation distances. More sophisticated techniques to control radiation behavior using longitudinally varying structures are currently under investigation. The directional control of radiation could be used to transmit power to another well separated waveguide on one side of the structure with almost no radiation lost to the other side. Devices utilizing this concept are also currently under investigation.