We present a rigorous analysis of the optical gradient force between coupled single-mode waveguides in three dimensions. Using eigenmode expansion we determine the optical mode patterns in the coupled system. In contrast to previous work, the sign and amplitude of the optical force is found to vary along the waveguide with a characteristic beating length. Our results establish design principles for optomechanically tunable directional couplers.
© 2009 OSA
Optical forces in dielectric structures have recently attracted great interest for the realization of optomechanical coupling at a chip-scale level. It was theoretically predicted, that in coupled waveguide structures optical forces could be either attractive or repulsive, depending on the mode symmetry [1–3]. Similarly, it was also theoretically suggested, that the resulting forces could be enhanced by optical resonant structures [4–6]. These results were based on the convenient formalism first suggested by Povinelli et al. , which allows for the derivation of the optical force by determining the eigenenergy of cross-sectional optical modes in dependence of waveguide separation. We previously extended this method to the calculation of substrate coupled nano-mechanical devices that support only one mode . These theoretical results were also confirmed experimentally [8,9].
The approaches mentioned above consider optical modes in two-dimensional waveguide cross-sections. In realistic devices, however, the evolution of the optical mode in the direction of propagation has to be taken into account. Furthermore, it is usually not possible to selectively excite a purely even or odd mode of a coupled waveguide structure. Rather a combination of both modes is launched. In this case the resulting optical force is not simply a superposition of the weighted attractive and repulsive force generated by each eigenmode, but rather a cross-linked force profile that depends on material, geometry and the spatial coordinates.
In this article we present a rigorous method for the calculation of the optical force in realistic three-dimensional coupled waveguides. The mode profile along the coupled beams is determined using eigenmode expansion [10,11], a technique described in detail by Yariv . From the mode profile we obtain the optical gradient force through integration of the Maxwell stress tensor over the surface of the beam . With our method the optical force can be determined for arbitrary phase difference between the input light-waves. We find that continuous tuning of the sign of the force from attractive to repulsive is feasible by adjusting the phase difference between the incoming lightwaves. Our results show a route for realizing improved optomechanically tunable directional couplers with reduced switching power.
2. The force distribution in coupled waveguides using eigenmode expansion
We are interested in obtaining the optical force distribution on two coupled single-mode waveguides, also referred to as a directional coupler. The coupler is assumed to be part of a larger photonic circuit, in which the phase of each waveguide can be controlled individually. This could for example be achieved by using Mach-Zehnder interferometers or phase-shifting elements [14,15]. When looking at the coupling region the problem can be described with the black-box shown in Fig. 1(a) .
Excited are the fundamental mode amplitudes of the left and right input waveguides (L and R). After passing through the coupling region of length l, their new amplitudes will be denoted by L’ and R’. L and R are the eigenmodes of the uncoupled system, i.e. the eigenmodes of the incoming waveguides. We assume that no optical or material loss will occur during the coupling.
In the coupled system, the left and right waveguide modes need to be expanded in terms of the eigenmodes of the directional coupler. These are denoted the even and odd modes E and O, labeled by considering their symmetry properties with respect to the direction of propagation. We consider two parallel silicon waveguides of length l and thickness h as shown in Fig. 1(b). The waveguides are of not necessarily equal width w1, w2. Each individual waveguide is assumed to support only a single fundamental mode.
The coupled system will support a set of two orthogonal modes. We label the modes as E(x,y,d) and O(x,y,d) for the cross-sectional profiles of the even and odd modes, respectively. The modes are assumed to be normalized so that, where * denotes the complex conjugate. The modes propagate along the waveguide in the z direction with propagation constants βe and βo.
When a phase shifter is added before the input waveguides, the modes of the left and right input waveguides may have an adjustable phase difference Φ 0. When both modes are excited with equal amplitude, for identical waveguides two dominant cases can be identified: both waveguide modes are in phase (symmetric) or both modes are 180 degrees out of phase (anti-symmetric). In order to avoid confusion, these situations are clarified in Fig. 2 .
When the fundamental mode of either the left (L) or the right waveguide (R) is excited, the mode profile at position z=0 (the beginning of the coupling region) can be expressed in terms of the even and odd modes of the coupled system using a projection matrix M as
The projection matrix M contains the decomposition coefficients into the even and odd mode. Because of the orthogonality of the eigenmodes, these coefficients can be obtained by evaluating Eq. (2) as
When the coupler is excited with either the even or odd eigenmode, the mode will propagate along the coupler without being perturbed by superimposed fields. In the L,R system, a combination of even and odd modes are excited, which propagate along the coupler with their respective propagation constants.
In practical devices the two beams are normally not identical due to process variations. Therefore we consider a coupler system composed of two silicon beams of slightly different widths w1=300nm and w2=310nm. The thickness of the beams is assumed to be h = 220nm and the beam length is 10μm. These are typical devices parameters for practical silicon devices. Because of the difference of the waveguide cross sections, the even and odd modes are not uniform. Thus the symmetric/anti-symmetric waveguide modes will excite both the even and odd modes. Due to the different propagation constants of the even and odd modes a typical beating pattern will occur along the length of the coupler. This is illustrated in Fig. 3 for two representative cases, the excitation of the symmetric and anti-symmetric modes from the left.
For this particular choice of waveguide cross sections, the beating length is found to be 6.4μm. After one beating period the power from both waveguides is transferred completely from one waveguide to the other. Changing the input phase difference moves the beating pattern along the coupler in the z direction.
3. Optical force distribution along the coupled beams due to optical mode beating
Having obtained the mode distribution in the direction of propagation, we are able to determine the optical force between the beams using the Maxwell stress tensor (MST). The stress tensor T is given as , whereanddenote the electric and magnetic field vectors and the indices i,j are counted over the three field components. Once the field vectors have been obtained, the optical force can be calculated by computing the time average of the surface integral of the MST as, where the brackets denote the time average over one optical period and denotes the outward normal vector. In the following we assume that the time average is always carried out, therefore we drop the bracket notation for readability.
For any cross-section of the coupled waveguide system the force between the beams is given by the contour integral in the following relationEq. (1) into Eq. (3) we find the optical force for an input phase difference Φ0 as
Here denote the electrical field components of the even/odd mode, and denote the magnetic field components, respectively. The three field components can be easily obtained with numerical methods, such as finite-element codes.
From Eq. (4) we see that the optical force is the sum of three contributions: the force resulting from the proportion of fields in the even mode (), the force resulting from the excitation of the odd mode (), and finally a term resulting from the beating between the even and odd modes (). The contribution to the force from the even and odd modes (w e, w o) remains constant over the length of the directional coupler. The z dependence of the force distribution is thus a result of the beating term (with a modulated weight of wb).
We examine the individual contributions to the total force in an illustrative example. We consider two coupled free-standing silicon waveguides with a refractive index of 3.477 (Silicon) surrounded by air. The resulting forces are calculated as a function of the distance g between the waveguides. Results are presented in Fig. 4 , where the optical force has been normalized to the waveguide length and the optical input power.
It is obvious, that the force Fe resulting from the even mode is the dominant term for all separations. This force contribution is always attractive, as also shown by Povinelli et al . The second dominant term Fo results from the odd mode. For gaps smaller than the crossover gap gc (130nm), this force Fo is also attractive, but then changes sign to be repulsive for larger gaps . For identical waveguides, the beating force Fb is only noticeable at small distances, but decays quickly with increasing gap. However, for asymmetric waveguides, this beating term can be significant due to the large difference between the propagation constants.
4. Tunability of the net optical force and directional coupling
From Eqs. (4-7) it is apparent that the net optical force (i.e. the sum of the force contributions in Eq. (4)) depends on the relative phase between the input optical waves. From Fig. 4 we note, that both attractive and repulsive net optical force can be obtained when the beams are separated by a gap greater than gc. Thus from Eq. (4) it is obvious, that the force can be changed to be net repulsive or net attractive depending on the phase difference Φ0. This is illustrated in Fig. 5(a) for calculated force values of the geometry outlined above.
From the calculation, we obtain a maximum repulsive force of 1.9pN/μm/mW for waveguides separated by a distance of 240nm, whereas the maximum attractive force is 3.2pN/μm/mW. When the phase difference between the two input waveguides is varied from 0 to π at this separation, the sign of the force switches from negative to positive. As a result the net optical force can be continuously tuned by adjusting the input phase. The maximum attractive force is found for integer multiples of 2π, whereas the maximum repulsive force is found at the antinodes (π(2n+1)).
By exploiting the force tenability an optically tunable directional coupler can be implemented. The nodal points are the most efficient configurations to operate such a directional coupler, as illustrated in Fig. 5(b). We evaluate the performance of a free-standing silicon coupler of 70μm length, by calculating the required power to achieve 100% switching of the transmission. We compare the tuning efficiency of the coupler, when the coupler is excited from only one waveguide (L mode, magenta curve), or excited from both waveguides. For the single-waveguide excitation a static tuning power of 0.95mW is required to achieve 100% switching. When the coupler is excited simultaneously in both waveguides, the switching power can be reduced to 0.86mW when operating at a phase difference of π (blue line), and further to 0.44mW when operating at a phase difference of 0. If at the same time the force is phase tuned from attractive to repulsive to pull and push the waveguides, the full dynamic range of the opto-mechanical device can be exploited. This situation is shown by the green line in Fig. 5(b). In this case the switching power can be as low as 0.28mW, providing considerable improvement compared to the single waveguide excitation.
We have presented a rigorous approach to determine the three-dimensional force distribution in coupled waveguides. The phase sensitivity of the net optical force presents a convenient tuning parameter to change the sign of the force in situ. Enhanced tunability for opto-mechanical structures can be achieved by exploiting the increased dynamic range provided through phase engineering.
H. X. Tang acknowledges a seedling fund from DARPA/MTO and a NSF CAREER grant. W. H. P. Pernice would like to thank the Alexander-von-Humboldt Foundation for providing a postdoctoral fellowship.
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