A design for a photonic syringe with multimode coupled slot waveguides

A design is proposed that allows non-stationary fi eld distribution with Bragg gratings in multiple slot waveguides. Se lective coupling between modes is achieved or suppressed, according to controllable selection rules, based on mode symmetry. By applyin g such rules, backward pulling radiation pressure i.e. toward the light source can be obtained inside the slots. A mode-switching filter is also p ro osed, which allows the switching between forward and backward direction of radiation pressure. This "light-actuated" syringe could have potential pplications for bidirectional particle trapping and manipulation, o ptofluidics, optomechanics and biotechnology. 2009 Optical Society of America OCIS codes: (130.3120) Integrated optical devices ;(230.0230) Optical devices; (230.1480) Bragg reflectors; (230.7370) Waveguides. References and links 1. A. Ashkin, "Acceleration and Trapping of Particles by Radiation Pressure," Phys. Rev. Lett. 24, 156-159 (1970). 2. A. 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Because they are exponentially decaying, evanescent fields always exhibit intensity gradients, which result in gradient forces.The effect of these forces is to pull the particles toward the interface that sustains the evanescent wave.However, the wave also exerts a force on the particles due to radiation pressure.This force originates from the scattering of the field by the particle.The net scattering force pushes particles along the direction of propagation of the beam [3].
In this paper, I suggest a multimode waveguide design which supports backward (i.e., toward the source) propagating modes with substantial evanescent fields.With this design, the direction of propagation can be flipped by changing the mode of injection.Propulsion of particles can be therefore also inverted in direction.

Signature of supermodes in coupled slot waveguides
If a waveguide supports at least two modes, non-stationary field distributions can be achieved by reflection.Let us consider a waveguide with two modes, whose propagating constants are β 0 and β 1 =β 0 +∆β.The signs of β 0 and β 1 are opposite if they are counter propagating.Let us now suppose that electromagnetic energy, initially travelling in mode β 0 only, is reflected from mode β 0 to backward mode β 1 .This reflection leads to non-stationary field distribution, because the field profiles of mode β 0 and β 1 are necessarily different, since they belong to different modes.Such reflection could be due, for example, to a photonic structure with period The coupling of different counter propagating modes in multimode waveguides has been already suggested for mode conversion, but, to the author's knowledge, without an explicit waveguide design, nor discussing applications for optomechanics [26].
The condition in Equation 1 is a constrain on the period, but gives no hint about how to design the unit cell of the photonic structure.Here, a possible procedure is suggested, which utilizes coupled slot multimode waveguides.Coupled slot waveguides have been already demonstrated, both in horizontal [27] and vertical [28,29] configurations for single mode guiding but, to the author's knowledge, not for multiple mode guiding.
In this work, a multiple n-slot waveguide is considered, which is composed by n+1 coupled homogeneous and equal waveguides.The idea for the design is inspired by the tightbinding method, a very useful approximation in atomic and chemical physics not only to simplify the calculations, but also to gain deeper insight into electron states in complex molecules and crystals [30].In the tight-binding procedure, molecular or crystalline orbitals are described as linear combinations of atomic orbitals (LCAO).The analogy between coupled waveguides and molecular orbitals has been already proposed for calculating transverse optomechanical forces between coupled waveguides [31][32][33][34][35].In the case of waveguides, the uncoupled picture, corresponding to isolated atoms, is the set of all the separate n+1 waveguides, considered at infinite distance from each other.The modes of the separate waveguides are the analogous to atomic orbitals.When these elementary components are brought closer together, the coupling of elementary eigenmodes lifts the degeneracy of the originally degenerate supermodes (i.e. the modes of the whole waveguide, considered as a single entity).
Let us first recall the main ideas of the tight binding approximation for the electronic case.For crystalline orbitals, the electronic wave functions ψ(r) are solutions of an eigenvalue equation (Schrödinger equation) in a periodic potential.Because of the latter condition, the problem possesses translational symmetry, thus the wave functions must satisfy the Bloch theorem.On the other hand, the wave functions ψ(r) should retain a trace of the wave functions ψ n (r) of isolated atoms, at least when the atomic distance is not too close, in some sense.This assumption is the core of the tight binding method.Without loss of generality, an eigenfunction ψ(r) can both satisfy the Bloch theorem and be expressed in terms of functions φ(r) (Wannier functions) centered at atomic sites in the lattice, as [30] , ) ( ) ( where R is a lattice vector, and the Bloch wave vector k has also been used to label the eigenfunction.The isolated atomic wave functions can be introduced by expanding the Wannier functions as , ) ( ) ( where the b n 's are the required expansion coefficients.Generality of Equation 3 is guaranteed by the completeness of the eigenset {ψ n }.At this point, in the tight binding approximation, one looks for the conditions for the Equation 3 to be approximately true with few, possibly only one, ψ n (r).These conditions can be expressed as the vanishing of overlap integrals, given the target wave function ψ n (r), with all the wave functions ψ m (r) [30] , where ∆U is the difference between the crystalline and the single atom potential.
This approximation can be extended to waveguide modes.The expansion in Equation 3 requires the completeness of the eigenmode set, thus in principle all unguided modes must also be included.But, as before, the question is under what conditions it is possible to build a supermode as a superposition of few (possibly only one) displaced elementary guided modes.The conditions can be formulated similarly to Equations 4. The Schrödinger equation must be substituted with the Helmholtz equation for the modes of waveguides, which reads (assuming scalar form) , 0 ) ( 2 where E is the amplitude of the electric field, k 0 is the magnitude of the wave vector, and n c (r) is the refractive index as a function of space.Equation 5 implies that the term ∆U must be substituted by a term k 0 2 [n c (r) 2 -n s 2 ], where n s is the refractive index of the background, and the integral can be limited to waveguide cores, because outside the cores n c =n s .For the overlap integrals to be small, ultimately, Equations 4 require the decay length of evanescent field of the target mode to be small compared to the distance of waveguides.
For this paper, the tight-binding picture is primarily useful to introduce the signature of supermodes' profile of the electric field.The concept of the signature can be outlined as follows.Let us assume for simplicity that the problem is two dimensional (i.e., that the elementary waveguides are slabs), and that the distance d between nearest neighboring elementary waveguides is the same for all the waveguides, and is large enough for a mode of elementary waveguides to satisfy the conditions in Equation 4. Let us indicate with f e (y-md) the (real valued) profile of the electric field of this mode, in the m th elementary waveguide, where y denotes the transverse coordinate.The origin of y is taken to be the middle point of transverse cross section of first waveguide (slab 0).Let us also indicate with f h (y) the field profile of supermode h of the whole waveguide.In the limit of n → ∞, the structure is periodic.Hence, according to Equation 2, the real-valued f h (y) can be approximately expressed as a quasi-Bloch function as where C is a normalization constant, and K h =hπ/nd (for h=0, 1, ..., n).In Equation 6, by referring to a discrete set of K h , I contextually introduced also the quantization of K h , which is a consequence of the finite extension of the system in the y coordinate.The parameter h individuates all the n+1 wave vectors, in the first Brillouin zone, relative to the n+1 supermodes.
By applying the assumptions of the tight binding approximation, I assume that f e (y-md) essentially vanishes in proximity of the neighboring waveguides, or f e (y-md) ≠ 0 only for y ≈ md (on the scale of d), which leads to .
From Equation 7one can define the "signature" of supermode h as the set of the signs of the n+1 coefficients cos(hπm/n).For a given set on n+1 elementary waveguides, there are n+1 signatures, each being characterized by the number of sign changes.The sign can change up to n times.For each supermode, this number is described by parameter h.Hence, it is the magnitude of the Bloch wave vector which determines the number of sign changes in the signature.An example is given in Figure 1, where the number of slots is n = 2, the number of elementary waveguides is n+1=3, and hence the signatures are (+++) for h=0, (+0-) for h=1, and (+-+) for h=2.Thus, the signature of supermodes can be associated to the "dominant sign" of the field profile across the elementary waveguides.The definition is suggested by Equation 7 under quite strict assumptions, but the example in Figure 1 shows that its physical meaning can be robust even in cases for which the coupling is not weak, or n is very small (n=2, in Figure 1).In general, however, one must be careful in evaluating the signature according to Equation 7.This especially applies to slot waveguides: to the take full advantage of the field enhancement in the slots, deeply sub wavelength slot size is required, i.e. strong waveguide coupling.In such case, supermodes cannot be accurately described with tight-binding approximation.Moreover, when n is small, boundary effects can acquire larger importance, and thus the profile of supermodes can deviate significantly from a Bloch function.For these reasons, it is helpful to provide an operative extension of the definition of supermode signatures.A practical way can be the following.First, one analytically [24] or numerically [36] calculates the actual field profile f h (y) of each supermode.The sign of the m th coefficient in the signature of supermode h can be again associated to the "dominant sign" of the m th waveguide, which can be now defined as the sign of an integral of the type where a(m) and b(m) are the boundaries of the m th elementary waveguide.A definition with the integral in Equation 8resolves the issues related to the assumptions for Equation 7.
Once the signature is known, the design of mirrors which couple different supermodes can be performed by considering the required operation on single signs in the signature.Let us apply this idea, for example, to the mentioned case of the 2-slot waveguide of Figure 1, with the goal of coupling, by reflection, supermode +++ to supermode +-+.There are two ways to perform this sign change.The first is by maintaining the sign of the first and last coefficients, and flipping the sign of the central coefficient.The second is by flipping the first and last sign, and maintaining the central sign unaltered (signature -+-is the same signature as +-+).The change or maintenance of sign in each elementary reflection can be achieved according to Fresnel reflection coefficient r, which, for normal incidence, reads , where n 0 (n 1 ) is the refractive index of the medium of incidence (refraction).Thus, there is a 0 (π) phase shift in the reflected beam, if the medium of incidence has higher (lower) index than the medium of refraction.Equation 9 also holds for effective indices n eff =|βc/ω| in a waveguide, where c is the speed of light in vacuum, and ω is the angular frequency [36].In a waveguide, the effective index increases with increasing the core thickness.Hence, a reflection without (with) a change of sign, in a waveguide, can be obtained by an abrupt thinning (thickening) of the waveguide core, as this implies an abrupt decrease (increase) of the effective index n eff (Figure 2).

Two dimensional design and finite-difference time-domain simulation
Because of the phase relationships in the individual slabs, the structure in Figure 2  where λ 0 is the vacuum wavelength.Equations 10 can be interpreted as the conditions to get a π phase shift in a roundtrip in any of the layers.I generalize this condition of a π phase shift per roundtrip by assuming that forward and backward propagation are carried out in different supermodes.Thus, even in the same layer, there are two different refractive indices, one for the forward and one for the backward direction.Moreover, a Bragg mirror has two kinds of "layers".For example, in Figure 2, the layers are identified by the thickness of the three slabs (layer h: thick-thin-thick, and layer l: thin-thick-thin).Therefore, there are 4 relevant refractive indices, which I label here as n (i,j) , where i indicates the layer type (i=h or l), and j the direction, forward or backward (i=f or b).An additional simplification has been introduced in Figure 2, by imposing the corrugation to be small with respect to the thickness.This condition limits the mode mismatch -hence, unwanted mode mixing -between layers of type h and l layers.Small corrugation implies small index contrast, and thus, from Equations 11, d l ≈ d h .The calculated value is d l ≈ d h ≈ 345 nm, thus the period Λ is 690 nm (Figure 2).
Once the period has been defined, just because of the periodicity, additional stop bands are expected to appear in the transmission spectrum.Their central wavelength can be estimated by solving Equations 11 for λ 0 , by inserting the now known values of d l and d h , and all the meaningful combinations of refractive indices.For example, a stop band is expected which reflects supermode +++ to itself.Its central wavelength can be estimated as follows.The numerically calculated effective index of supermode +++ is approximately 1.209 at λ 0 ≈1.7 µm (note that the value 1.243 in Figure 1 is calculated at 1.55 µm).The estimated central wavelength of the +++ to +++ stop band is λ 0 = 4n l d l = 1.668±0.024µm (the error is added assuming a 10 nm tolerance for d l ).
Two 2D FDTD simulations of the structure in Figure 2 with 100 periods have been performed using commercial software (FDTD Solutions by Lumerical Solutions, Inc.).The purpose of the first simulation is the spectral characterization.Hence, a broadband source has been used as stimulus.The injected field had the beam profile of supermode +++ in Figure 1, a central wavelength λ 0 = 1.6 µm, and a full width at half maximum of ∆λ ≈ 400 nm.The geometrical mesh had 10 nm and 5 nm step size resolution, respectively, in x and y directions.The total span of the simulation region in the x direction was determined by the structure itself, whereas in the y direction was set to 20 µm, to simulate the evanescent tails with satisfactory approximation.The material (Si) was simulated by taking into account both real and imaginary part of permittivity according to Palik [37].The time step was 99% of Courant stability limit.The results are shown in Figure 3.
The effectiveness of the design is demonstrated by presence of the stop band at 1.545 µm (supermode +++ reflected to supermode +-+, inset a in Figure 3), and simultaneously by the almost complete suppression of the stop band at 1.645 µm (supermode +++ to itself, inset c in Figure 3).The location of the latter is consistent with the previously estimated 1.668±0.024µm.The suppression cannot be ascribed to a low index contrast, which would also similarly affect the band at 1.545 µm.Instead, suppression is due to the symmetry mismatch between the signatures of the modes.with 100 periods and with supermode +++ excitation (see Figure 1).The stop band is relatively narrow due to the low contrast of effective refractive index.Insets show the profile of electric field Ey.In (a), the reflected beam has the profile of supermode +-+, whereas in (c) the profile of reflected beams matches the profile of incident beam (not shown), as expected, because the Bragg condition is satisfied for the reflection from supermode +++ to itself.However, in (c), reflection is almost completely suppressed by the symmetry mismatch of the mirror (residual reflection is Inset (b) shows that the transmitted beam maintains the profile of supermode +++.Decrease of R+T at lower wavelength is due to coupling by the grating of supermode +-+ to free space propagation.
One can picture this suppression as a selection rule, dictated by symmetry, in close analogy with rules that prevent transitions between electronic states due to symmetry constrains.In such analogy, backward (forward) modes can formally be assimilated to "bra" <+++| ("ket" |+++>) vectors, and the mirror to an "operator" |+-+|.The "matrix element" <+++|+-+|+++> approximately vanishes, whereas <+-+|+-+|+++> encodes the information of the "oscillator strength" of a single interface.The value can be calculated, analytically, from overlap integrals of modes, or numerically from simulations, as the ratio between the integral of reflected field versus the integral of incident field.However, this is not further investigated in this work.Instead, I suggest that this notation can be directly exploited: the vanishing or non-vanishing character of matrix elements can be obtained as the straightforward result of sign multiplications.In the two sample cases above, the sign multiplications yield (+-+) and (+++), respectively.The presence of the minus sign in the first case indicates a symmetry forbidden transition, hence a vanishing matrix element.
The second simulation was performed with similar FDTD settings, except that a narrowband instead of broadband source was used.The wavelength 1.545 µm was chosen for this simulation: at this wavelength, the stop band that generates the backward propagation effect has a maximum (Figure 3).For this simulation, the excitation source was set to supermode +-+, to obtain backward reflection to supermode +++.This choice leads to backward propagation of the evanescent slot-confined field, which is more interesting for the applications.The companion spectrum (forward propagating supermode +-+ and back reflection to supermode +++, not shown) does not deviate significantly from the spectrum in Figure 3.
A snapshot of the resulting 2D map of Poynting vector over a few periods, close to the input end of the waveguide, is shown in Figure 4.The steering effect of the structure can be clearly seen looking at the black arrows, which are the calculated Poynting vectors from the FDTD outputs (red arrows show the result of simulation with no corrugation on the waveguides).The black vectors initially point to the right (top left and bottom left of the figure, outside the slots and of the waveguide structure).This direction is outward with respect to the source (which is located at the left, out of the shown region).Moving to the right, black arrows start turning toward the center of the structure.In the core of outer waveguides, the vectors keep being directed toward the center of the structure (right side of figure), but they also acquire negative component in the x direction.Finally, in the two slots, the x component is negative almost in every sampled point considered.The backward effect is the net result of the fields of backward supermode +++ and of forward supermode +-+.The key idea is that, in the slots, the amplitude of the latter is lower.A snapshot of the movie of the simulation of electric field E y (Media 1) is shown in Figure 5.

Switching of the direction of radiation pressure
The direction of radiation pressure can be switched by switching the supermode of injection.In fact, the direction of the radiation pressure in the slots depends on the direction of propagation of supermode +++.If the injection is directly in supermode +++, the radiation pressure in the slots will be trivially outward (away from the source).However, if the injection is in supermode +-+, this supermode will be back reflected by the structure to supermode +++, and the pressure will be directed toward the source in the slots.In this section, I propose a sample device (Figure 6) to switch the mode of injection.This structure is meant to be placed before the main device of Figure 2.  The backbone of the device in Figure 6 is the set of three coupled waveguides shown in Figure 2, however in this case the waveguides have no corrugation.Instead, two photonic structures are added.The idea is to take advantage of two independently designed band structures.The first structure is constituted by the rows of thinner rectangles inside the slots, and is designed to generate a stop band to reflect supermode +++ to itself.Its period (580 nm) locates the lower-wavelength band-edge of the stop band slightly below 1.545 µm (i.e. the wavelength for which backward effect in the main device is largest, see Figure 3).A FDTD simulation of the device in Figure 6, under excitation of supermode +++, is shown as a black curve in panel a of Figure 7.This figure demonstrates that this first photonic structure acts as a "closed gate" for supermode +++ at 1.545 µm (black dot, transmission is T c ≈0.1%).However, a relatively small red shift of the spectrum can open this gate, and let the supermode pass.This situation is simulated in panel b of Figure 7, where the refractive index of the background has been increased by 2%.The transmission of supermode +++ (black line) increases to about T o ≈49% (black dot in panel b).In this "open gate" configuration, the insertion loss is about 3 dB, whereas the modulation depth (T o -T c )/(T o +T c ) is larger than 99%.There are several possible options to achieve experimentally a shift that is compatible with this simulation.For example, if the background is a controlled fluid, the shift can be obtained by changing the composition of the fluid.If the structure is porous, a more subtle way could be by resorting to capillary condensation, as we have already demonstrated experimentally in a different context [38].Other solutions could include nonlinear materials.1.545 Fig. 7. 2D FDTD simulation of transmission spectra of the structure in Figure 6.Each spectrum has been calculated by exciting only one supermode.Black (red) curve is the relative power transmission of supermode +++ (supermode +-+).The refractive index of Si was modeled as a complex quantity according to Palik [37] (the real part at 1.545 was about 3.48).Panel a shows the simulation with the background refractive index set to n0 = 1.Panel b shows the simulation with an increase of the background index by 2%.The vertical dashed line is the design wavelength for this filter (1.545 µm).Colored dots indicate the transmission of the supermodes at this wavelength.In panel a (b), the power transmission is 0.1% (0.8%) for supermode +++ (supermode +-+), and about 57% (49%) for supermode +-+ (supermode +++).The bands generated by the photonic structure with period 580 nm (667 nm) are marked by blue (green) arrows.Band 1 reflects supermode +++ to +++, band 2 supermode +-+ to +-+, band 3 supermode +-+ to +-+, band 4 is the same as band 3, but occurs under excitation with supermode +++ due to stray coupling with supermode +-+.Band 5 (band 6) appears in both the red and black curve because it is couples supermode +++ to supermode +-+ and it is provided by the inner (outer) photonic structure.
In similar fashion, the second photonic structure is placed in the space outside (two rows of thick rectangles in Figure 6).In this case, the period (667 nm) is designed to introduce a stop band for supermode +-+, with the upper-wavelength band-edge slightly below 1.545 nm.In Figure 7, the FDTD simulation, under excitation of supermode +-+, is shown as a red curve.As opposed to the previous case, in absence of spectral shifts this photonic structure acts as an "open gate" (transmission is T o ≈57%) for supermode +-+.A relatively small red shift of spectrum closes the gate and blocks the supermode +-+.This blocking effect is demonstrated in panel b of Figure 7 (red curve).Transmission drops to about T c ≈0.8%, and modulation depth (T o -T c )/(T o +T c ) in this case is about 97%.Incidentally, the first photonic structure, although designed for supermode +++, unavoidably introduces a stop band also for supermode +-+ (visible in Figure 7, band 3).However, this band has no relevant effect for this design because it is located away at lower wavelength (this is due to the lower effective index of supermode +-+).On the other hand, the second photonic structure has no observable effect at all, on supermode +++.In fact, in this case, additional bands are suppressed, due to the negligible overlap in space between the second structure and supermode +++.
Finally, to excite both +++ and +-+ supermodes simultaneously, the structure of Figure 6 was excited only in the central waveguide (left panel of Figure 8).The output of the device is shown in the central panel of Figure 8.  6 has been prolonged to the left, as shown in the inset, to allow this mode profile).Simultaneous excitation is due to mismatch between the source and the supermodes profiles.Central panel: field profile of the output, with no index shift (black curve) and with 2% shift (blue curve).Right panel: analytical profile of the +++ (blue curve) and +-+ (black curve) supermodes.Comparison between the central and the right panel shows that the output of the device has a field profile almost overlapping a single supermode (either +++ or +-+), depending on the index of the background.Hence, the structure in Figure 6 works as a supermode selector.Residual components of the other modes are observable as deviations from the theoretical profile (particularly visible for |y-y0|>1 µm).
Without any index shift, only the supermode +-+ passes through.This supermode, injected in the device of Figure 2, leads to the backward propagation effect in the slots.If on the other hand the refractive index is shifted by 2%, then only the supermode +++ passes through, and the radiation pressure will be directed forward.

Conclusions
Multimode coupled slot waveguides can be structured for selective mode coupling.Selection of modes can be achieved by exploiting the symmetry of modes.For example, a distributed Bragg mirror has been proposed, which reflects a mode to a different one.The design presented here takes advantage of a generalization of the well known "λ/4 optical thickness" rule for the layers of distributed Bragg mirrors.Finite-difference time-domain simulations show that stop bands obtained with this generalized procedure can appear or be suppressed, according to consistent selection rules, which are reminiscent of selection rules for eigenmodes transitions in quantum mechanics.As an intriguing application, I have proposed a design of slot waveguides, in which backward pulling radiation pressure could be obtained in evanescent fields inside the slots.Furthermore, I have also suggested a front-end filter that can allow the switching between forward and backward direction of this radiation pressure.The integration of the front-end filer with the backward-wave device could behave as a syringe, in which the "needle" is constituted by the corrugated empty slots, thus naturally suggesting potential applications as a tool for biotechnology.

Fig. 1 .
Fig. 1.Signatures of the 3 guided TM supermodes at vacuum wavelength λ0=1.55 µm in a 2slot waveguide resulting from 3 coupled 150 nm thick Si slabs.Air slots are 350 nm thick.For each supermode, the effective index neff is also shown.The thickness of Si slabs is chosen to guarantee single mode operation (for the isolated slab).
is expected to efficiently couple supermode +++ to supermode +-+ with opposite propagation, and vice versa.The conditions for the largest reflection in a 1D Bragg mirror, fabricated by alternating layers with low (high) refractive index n l (n h ) and thickness d l (d h )

Fig. 2 .
Fig.2.Photonic structure to reflect supermode +++ to supermode +-+ (see Figure1).The dashed line indicate one of the reflecting interfaces.The phase shift associated to elementary reflections is shown by bending arrows.The period Λ of the structure (690 nm) and the other values are resulting from the procedure described in the text.Layer h (higher effective index) and layer l (lower effective index) refer to the 2 types of quasi-1D layers which define the Bragg mirror.The graph is to scale.

Fig. 3 .
Fig.3.2D FDTD simulation of transmission and reflection spectra of the structure in Figure2with 100 periods and with supermode +++ excitation (see Figure1).The stop band is relatively narrow due to the low contrast of effective refractive index.Insets show the profile of electric field Ey.In (a), the reflected beam has the profile of supermode +-+, whereas in (c) the profile of reflected beams matches the profile of incident beam (not shown), as expected, because the Bragg condition is satisfied for the reflection from supermode +++ to itself.However, in (c), reflection is almost completely suppressed by the symmetry mismatch of the mirror (residual reflection is Inset (b) shows that the transmitted beam maintains the profile of supermode +++.Decrease of R+T at lower wavelength is due to coupling by the grating of supermode +-+ to free space propagation.

Fig. 4 .
Fig. 4. 2D FDTD simulation of Poynting vector of the structure in Figure 2, with 100 periods, and with supermode -+-excitation (see Figure 1).Light injection is from the left side.Excitation is narrowband (vacuum wavelength λ0≈1.54 µm).Map is to scale.Only a few periods are shown, in proximity of the input side.Red (black) arrows show the result without (with) corrugation.The blue lines are the outline of the structure with corrugation.While the energy in the outer side of the waveguide is always traveling forward (top and bottom of figure), in the slots (i.e. in the 2 regions between the slabs), the component along the waveguide axis is dominantly forward (backward) in absence (presence) of the corrugation.

Fig. 5 .
Fig. 5. Snapshot of the electric field Ey distribution, according to 2D FDTD simulation of the structure in Figure 2, with 100 periods, and with supermode -+-excitation (see Figure 1).Light is injected from the left side.Excitation is narrowband (vacuum wavelength λ0≈1.54 µm).Only a few periods are shown, in proximity of the input side.Map is to scale.Large arrows indicate the direction of energy flux (Media 1).

Fig. 6 .
Fig. 6.Sample 2D photonic structure to selectively filter supermode +++ or supermode +-+ (supermode profiles are shown in Figure 1).Grey (white) areas represent Si (air).The vertical line at x = 0 µm indicates the location of the energy source in FDTD simulations.The arrow indicates the direction of injection.The vertical line at x = 69 indicates the location of monitor.The graph is to scale.

Fig. 8 .
Fig.8.2D FDTD simulation of simultaneous excitation of supermodes +++ and +-+ in the device in Figure6.Left panel: field profile of source (the central waveguide in Figure6has been prolonged to the left, as shown in the inset, to allow this mode profile).Simultaneous excitation is due to mismatch between the source and the supermodes profiles.Central panel: field profile of the output, with no index shift (black curve) and with 2% shift (blue curve).Right panel: analytical profile of the +++ (blue curve) and +-+ (black curve) supermodes.Comparison between the central and the right panel shows that the output of the device has a field profile almost overlapping a single supermode (either +++ or +-+), depending on the index of the background.Hence, the structure in Figure6works as a supermode selector.Residual components of the other modes are observable as deviations from the theoretical profile (particularly visible for |y-y0|>1 µm).