High‐Contrast Switching of Light Enabled by Zero Diffraction

Diffraction allows to change the direction of light. Therefore, controlling the diffraction efficiency with high contrast enables controlling the pathway of light within optical systems. However, a high contrast requires that the diffraction efficiency is tunable close to zero. Probably the most prominent example for zero diffraction in a waveguide grating is a bound state in the continuum (BIC). Herein, zero diffraction of two plane waves under symmetric incidence to a leaky symmetric waveguide grating is found. The phenomenon not only occurs at singular spectral positions but on continuous curves in the energy–momentum space. The relative phase of the two waves enables large contrast control over diffraction in a wide spectral range. The practical meaning of this finding for local switching is demonstrated. Light is trapped into a nonlinear optical waveguide and detrapped at a desired position with electric control. A switching contrast exceeding 1000 is experimentally shown.

energy-momentum space.This enables high-contrast control for Gaussian beams covering a momentum range.
To prove this theory, we conduct an experiment by placing the leaky symmetric waveguide gratings between two lithium tantalate (LiTaO 3 ) wafers of opposite crystal direction as phase shifting elements based on the Pockels effect.In the resulting device, light can be actively and locally trapped and detrapped from a slab waveguide with electric control.We anticipate our work to inspire the design of future optical systems.

Study of a Waveguide Grating under Single Plane Wave and a Standalone Grating under Dual-Plane Wave Incidence
The minimized interaction of a slab waveguide mode with a film or grating at its node position has been used to reduce undesired waveguide losses. [19,29,30]Being very sensitive to permittivity changes, it was also suggested for applications in light concentrators, displays, and sensors. [31,32]By rigorous coupledwave analysis (RCWA) [33,34] we examine if this phenomenon of destructive interference can be mimicked in a simpler geometry without a waveguide.The sinusoidal field profiles of the waveguide modes within the high-index core can be understood as a standing wave formed by interference of plane waves of symmetric incidence.In case of the even and odd modes in a symmetric waveguide, these waves are constructively or destructively interfering at the center plane, respectively.If the aforementioned figure of a standing wave were correct, symmetric dual-plane wave incidence to a standalone grating would enable a diffraction efficiency that could be tuned with the relative phase.
To test this figure, we first determine the propagation lengths l prop, TE 0 and l prop,TE 1 of a TE 0 and a TE 1 mode in a symmetric waveguide grating.The propagation length describes the length in which the intensity of a mode decreases to 1/e.We calculate it from the imaginary part of the modes effective index l prop ¼ λ=ð4 Â Imðn eff ÞÞ.
We then remove the cladding layer from the waveguide grating so a waveguide is not formed anymore.Instead, a standalone grating is considered surrounded by the former core material only.We now mimic destructive (ΔΦ ¼ π) and constructive (ΔΦ ¼ 0) interference at the center plane by symmetric dualplane wave incidence (θ core,1 ¼ θ core,2 , λ 1 ¼ λ 2 ).The occurrence of maximized diffraction efficiency (ΔΦ ¼ 0) has some similarity to achieving perfect coherent diffraction for nanophotonic structures with two incident beams. [27,28]y tuning the relative phase ΔΦ of the incident beams, we calculated a contrast C ¼ η max =η min of the diffraction efficiency.Our study confirms that the contrast is very similar to the ratio l prop, TE 1 =l prop, TE 0 of the waveguide grating modes.Small deviations might result from the fact that the momentum of the TE 0 and a TE 1 mode deviates from each other (Figure 1a).
The contrast under symmetric incidence at a standalone grating of thickness t g scales with t g À4 .Large contrast therefore requires a very thin grating.However, with smaller t g also, the maximum diffraction efficiency η max decreases.As we want to switch light via tuning the diffraction efficiency, it is of course desirable to also reach a certain maximum diffraction.In the following sections, we will therefore use structures that enable a maximum η max > 20% with t g ¼ 200 nm.
Figure 1b shows the geometry of this standalone grating together with the contrast map C ¼ η max =η min (Figure 1c).From the comparison between the waveguide grating under single incidence with the standalone grating under dual-plane wave incidence (Figure 1a-c) one can draw two conclusions.The first conclusion describes the drawback of a resonator.The waveguide grating forms a resonator showing discrete modes.This bandwidth limitation of the resonator is of course undesired.
On the contrary without a cladding (Figure 1b,c), no internal reflection and thus no resonator is present.As no resonant conditions have to be fulfilled, η can be tuned at any energy (wavelength) and momentum (incident angle).
However, this optimized switching performance in an even simpler geometry comes at a price.In practice, the illumination with two perfectly symmetric waves is highly challenging.From this practical point of view, the symmetric waveguide grating is beneficial.It can simply be excited from one side.The internal reflections cause an automatic leveling of the waves, creating perfectly symmetric internal waves incident to the center grating.This second conclusion describes the benefit of a waveguide.Later it will be discussed how the positive leveling in a waveguide can be used without the negative influence of bandwidth limitation.

Waveguide Grating under Symmetric Dual-Plane Wave Incidence
In the next step, we combine the two approaches described above and study a waveguide grating under symmetric dual-plane wave incidence (like in Figure 1b).The geometry is shown in Figure 1d.Maximized and minimized diffraction are again found at ΔΦ ¼ 0 and ΔΦ ¼ π, respectively.The contrast C ¼ η max =η min is increased but still limited (Figure 1e).This increase is only observed along resonant modes, which corresponds to the described bandwidth limitation of a resonator.
In order to allow the waveguide modes to level by passing through the core-cladding interface, we now turn the waveguide grating into a leaky structure.For that purpose, the former cladding is fabricated as a thin buffer and a high index ambience is introduced at both sides (Figure 1f ).The average contrast is significantly increased (Figure 1g).
Remarkably, along curves with a near-elliptic shape, the contrast converges to infinity.Like Figure 1 indicates, this property is not achieved by simple destructive interference.For infinite contrast, a leaky waveguide grating is necessary.Figure 2a emphasizes the contrast in the range marked by the green dashed rectangle in Figure 1g.A line cut at λ ¼ 600 nm shows the corresponding contrast (Figure 2b) with minimized and maximized diffraction efficiencies η min and η max (Figure 2c).The minimized diffraction efficiency converges to 0 and thus the contrast to infinity.That means any line cut through the near-elliptic curve will contain two singularities.At these spectral positions, the two plane waves are tunable to zero diffraction for ΔΦ ¼ π.Remarkably, at these positions of infinite contrast, the diffraction efficiency can be tuned to large values of η max ¼ 0.23 or 0.19 for ΔΦ ¼ 0, respectively, just by tuning the relative phase (Figure 2c).RCWA simulation data confirms that minimized and maximized diffractions correspond to ΔΦ = π and ΔΦ = 0, respectively.A detailed mathematical derivation for zero diffraction is given in the Supporting Information.In short, the leveling of waves in the waveguide grating causes a symmetrizing of its scattering matrix.In consequence, the waveguide grating behaves like a standalone grating with equal diffraction coefficients (d i,0,t ¼ d i,0,r ).

Toward Practical Realization of the Phenomenon
In order to apply the described phenomenon for spatial light control with high contrast, at first, a symmetric waveguide grating is needed.In later shown experiments, we achieve best symmetry, if we use a transfer-printed metal grating.In the following simulations, a thin metal grating t g ¼ 30 nm is assumed instead of the loss-free dielectric grating.Other parameters are identical to the leaky waveguide structure discussed in Figure 1f,g and 2. [35][36][37] The next challenge is to establish perfectly symmetric dual-plane wave incidence.As discussed above, a symmetric waveguide leads to leveling and this way creates internal symmetric plane waves.Hence, we position the leaky symmetric waveguide grating at the center of an outer nonlinear waveguide (Figure 3a).The latter is assumed to consist of two LiTaO 3 wafers of equal thickness t LiTaO 3 orientated in opposite directions of the c-axis (Z þ and Z À parallel to the growth direction), enabling an opposite change of the refractive index of þΔn and ÀΔn under an externally applied electric field.This way the relative phase ΔΦ can be tuned with an electric field.Unfortunately, under plane wave incidence, the outer nonlinear waveguide would function as a second resonator limiting the bandwidth and severely reducing the switching contrast of the device.
Therefore, the outer nonlinear waveguide must be excited with one Gaussian beam with a diameter smaller than the total waveguide thickness.Each Gaussian beam can be considered a superposition of plane waves.As the Gaussian beam covers a certain momentum range the overall switching contrast can be calculated by averaging the plane wave solution found by RCWA. [35]More details are given in the Supporting Information.
In the first step, the beam diffracted into the device levels to form two perfectly symmetric Gaussian beams crossing the waveguide grating at the position that we will term dots (see red beam evolution in Figure 3b).By partial transmission and reflection at the center leaky waveguide grating, the amplitudes equalize after a few interactions.A mathematical derivation is found in the Supporting Information.The beams do not interfere in the outer nonlinear waveguide, as the dots are separated.This way the outer nonlinear waveguide enables leveling without functioning as a resonator.In the second step, the relative phase of the dual beam is switched by ΔΦ ¼ π (see black arrow in Figure 3b).The result is minimized diffraction and thus the dual beam does not leave the waveguide by diffraction.Remarkably this behavior is not only found at the next dot, but for all further dots (see Supporting Information).In other words, after switching the relative phase, the dual beam is trapped which is indicated by changing its color from red to green in Figure 3b.In the third step, we investigate the contrast of when the trapped dual beam is detrapped by another switching of the relative phase by π. Figure 3c shows the contrast map of the dual beam detrapped at dot 1 immediately after trapping.Counterintuitively, this contrast map changes when the detrapping is done after 100 dots (Figure 3d) or after an infinite amount of dots (Figure 3e).This phenomenon can be explained by the relation of space and momentum uncertainty.In the radiative state, the double beam is localized to very few dots, resulting in a relatively large variation of the momentum of angle.On the contrary, the trapped state is localized to the amount of dots between the positions of trapping and detrapping.Hence, with longer propagation, it will eventually become certain in momentum like a plane wave which is indicated on the right side of Figure 3b.Thus, the corresponding Figure 3e has been simulated, assuming only two plane waves.It should be noted that similar near Figure 3. Simulations of Gaussian beams in a geometry for realizing dual-plane wave incidence with tunable phase.a) Mirror symmetric geometry with nonlinear lithium tantalate (LiTaO 3 ) waveguide with crystal orientation Z þ and Z À and silver (Ag) grating in the center of Ormocore/Ormoclad waveguide grating.b) Several interactions with the waveguide grating under single-Gaussian beam incidence with external angle θ ext : leveling of amplitudes (red beam), trapping by switching the relative phase to ΔΦ ¼ π, and several interactions of the trapped beam (green beam).c) Contrast map after first interaction, d) after 100th interaction, and e) after infinite interactions mimicked by plane waves.The green area in (c)-(e) is not considered here, the dual beam does not cross the waveguide grating due to total internal reflection at the LiTaO 3 /Ormoclad interface.
ellipsoids of infinite contrast are found like in Figure 1g and 2 although now a silver instead of a loss-free air grating is assumed.
In short summary, the device concept shown in Figure 3a theoretically enables to prepare symmetric dual-plane wave incidence with controlled phase to a leaky symmetric waveguide grating.While in theory plane wave excitation will always require infinite lateral size, Figure 3 underlines the performance of a real device of limited lateral dimensions.The transition from Figure 3c,d,e shows how spatial localization leads to averaging of the ideal contrast map (Figure 3e) over a larger momentum range and thus to smaller contrast for the Gaussian dual beam.However, remarkably even when detrapping is introduced immediately after trapping (Figure 3c), a large contrast in the range of 10 3 -10 4 is found in wide spectral ranges with regard to both the energy (the wavelength of the Gaussian beam) and momentum (the external excitation angle).Numerical calculations support these findings in the Supporting Information.

Experimental Realization
The fabricated device (according to Figure 3a) is shown in Figure 4.After diffraction into the waveguide, the beam is split by the waveguide grating and leveled to create a symmetric dualbeam incidence.Figure 4 now emphasizes two phase switching events as described earlier.For that purpose, tip electrodes are placed between selected dots at the positions, marked by an 'x'.With both electrodes, the diffraction efficiency of the Gaussian dual beam can be controlled.Without diffraction, the dual beam is guided and thus trapped within the outer nonlinear waveguide.On the other hand, diffracted beams fall to an angle below the critical angle of total reflection so they can leave the lithium tantalate.This means by diffraction the dual beam is detrapped as it leaves the device.Figure 4b,d shows such switching between the detrapped and trapped state.Two electrodes are on the same potential and shift the local phase by ΔΦ, each.Figure 4b shows the dual beam diffracted at a global value of ΔΦ ¼ 0, so all dots are visible in the view on the surface of the device and projected onto a diffusive screen.Figure 4d shows the case of a local phase shift of ΔΦ ¼ π at both electrode positions, marked by 'x'.After the first phase shift, the dual beam is trapped so that they propagate largely undisturbed despite the presence of the grating.The phase shift at the second electrode causes detrapping, and the dual beam is again diffracted.Thus, the dots appear dark between the electrodes (trapped) but bright again behind the second electrode (detrapped).Therefore, Figure 4d shows that the trapped Gaussian dual beam is still transporting power.

Measurement and Simulation of the Contrast
To verify the contrast of the described device, we measure the detrapped power as a function of a globally applied field strength applied by water electrodes (see Figure 5a).These electrodes have two advantages.First, they avoid measurement-distorted scattering induced by electrode edges.Second, they allow a homogeneous external electric field strengthE g in the direction perpendicular to the device surface.
The simulation is performed using a finite beam RCWA (FB-RCWA) [33,34,38] and assuming Gaussian beams with a diameter of 500 μm, which correspond to the laser beam used in the experiments.It assumes L dots outside and further M dots inside the homogeneous electric field.The contrast at that m th dot is than calculated and compared with the experimental data.More details on this simulation can be found in the Supporting Information.
The schematic setup to control detrapping is shown in Figure 5a.For the contrast measurements of the fabricated device, we selected two exemplary wavelengths (λ ¼ 532 nm and λ ¼ 632.8 nm).The diffracted power could be controlled within a maximal contrast ratio of C ¼ 511 for λ ¼ 532 nm at L ¼ 4, m ¼ 13 (Figure 5b) and C ¼ 1236 for λ ¼ 632.8 nm at L ¼ 1, m ¼ 15 (Figure 5c).The observation of high-contrast values despite these small values of L indicates that the leveling of amplitudes occurs over a few interactions already.This short-spaced leveling is a result of the excitation of resonances in the waveguide grating (see Supporting Information).In total, we found excellent agreement between the theoretical model modified for the homogeneous electrodes (dashed lines) and the obtained experimental data (solid lines).In a first experiment, we have also examined the applicability of the concept at different wavelengths in the visible spectrum in order to verify a possible application in the field of laser displays (see Video, Supporting Information).The experimental results confirm that the phenomenon of zero diffraction theoretically described above can indeed be applied in practical applications.

Conclusion
Inspired by node-aligned waveguide gratings, we have investigated symmetric dual plane wave incidence to standalone gratings, classical, and leaky symmetric waveguide gratings.Only the latter enable zero diffraction and thus infinite contrast not only at singular spectral positions but along curves in the energy momentum space.Importantly, finite but very large contrast is found in wide spectral ranges with respect to both energy and momentum.As an example, it was demonstrated how this phenomenon can be applied in a nonlinear waveguide in order to control trapping and detrapping of light with high contrast.This way the position where a directed laser beam is detrapped from the surface of that device can be selected without any mechanics.Thus, the long-desired goal of electrical high-contrast control of radiation from a slab waveguide has been achieved.Of course, for a large-area implementation such as laser displays, new approaches to control the phase shifting have to be developed that enable scalability beyond wafer size and low operation voltages.Such laser displays would be more efficient than liquid crystal displays as they extract the light only where it is needed, without absorbing it.
From a wider perspective, our finding suggests to avoid resonances in the outer nonlinear waveguides, limiting the bandwidth in order to reach phenomena like zero diffraction not only at singular spectral positions.While the mentioned example uses spatial separation of beams to avoid resonances, we are confident that better methods will emerge that can be implemented in ultrathin integrated optics.

Experimental Section
Fabrication: The device stack (Figure 4) was fabricated by laminating two pieces of a coated c-oriented LiTaO 3 substrate and enforcing mirror symmetry.The whole substrate t LiTaO 3 ¼ 500 μm was coated with a layer of Ormoclad t Ormoclad ¼ 70 nm and fully crosslinked using a UV curing step.On the upper half of the substrate, a line-shaped silver grating coupler with a thickness of t coupler ¼ 80 nm was added by transfer printing.After coating this stack with a layer of Ormocore t Ormocore ¼ 1900 nm and leaving it uncured, one half of the substrate was mechanically removed, UV cured, and coated with a large area of silver grating of a nominal thickness of t grating ¼ 30 nm by transfer printing.On the second half, the Ormocore film was cured with %15% of the full crosslink exposure dose and, thus, partially crosslinked.Subsequently, both substrate halves were laminated together in a lab press (p = 100 bar) and simultaneously UV cured, forming a symmetric LiTaO 3 /Ormoclad/Ormocore/grating/Ormocore/ Ormoclad/LiTaO 3 stack.This procedure was developed to optimize the thickness homogeneity of all included films.A flowchart of the process is found in the Supporting Information.
Simulation: All simulations in the manuscript were performed by RCWA.From Figure 1 and 2, we used ideal mirror symmetric stacks.First, we studied a waveguide under single-beam incidence consisting of Ormocore t Ormocore ¼ 1900 nm with a grating of variable thickness t g , with a unit cell period of 555 nm and a filling factor of FF ¼ 0.35 in its center and Ormoclad as its cladding.For the dual-beam incidence, we removed the cladding forming a standalone grating surrounded by Ormocore.For the following contrast simulations in Figure 1c, the grating thickness was set to t g ¼ 200 nm.To the standalone grating, we again added a cladding and studied the waveguide under dual-beam incidence.For the leaky waveguide, we set the thickness of Ormoclad to t Ormoclad ¼ 70 nm and added lithium tantalate on the outside.
The symmetric device stack in Figure 3 was modeled using an ideal 30 nm-thick rectangular grating, consisting of alternating blocks of silver and Ormocore with a unit-cell period of 555 nm and a FF ¼ 0.35, positioned at the center of the Ormocore film.The Ormocore films had a thickness of 1900 nm; the Ormoclad layers had a thickness of 70 nm.The optical constants of Ormocore and Ormoclad were modeled using the Cauchy formula on the basis of data given by the manufacturer (micro resist technology GmbH) where A Ormocore ¼ 1.53, B Ormocore=Ormoclad ¼ 8000 nm 2 , C Ormocore=Ormoclad ¼ 0.70 nm 4 , and A Ormoclad ¼ 1.5.The Gaussian beams were modeled assuming an angle of divergence of 0.5 mRad and an aperture (dot size) of 0.5 mm.Assuming an abrupt change of the refractive index due to switching, one would expect parasitic reflectivity at the interfaces between regions in the range of around 10 À5 .In reality, due to parasitic electric field at the electrode edges, the index will change gradually.In experiment, we did not observe such parasitic reflections and therefore neglected this effect in the simulation.The wavelength-dependent optical constants of silver (Ag, n = 0.12 þ 3.65i) and lithium tantalate (LiTaO 3 , n 0 = 2.1817, r 13 = 8 pm V À1 ) as well as the electro-optic coefficients were used for all simulations. [39,40]easurement of Diffracted Power: The electrically controlled diffracted power was measured using water electrodes on both sides of the device.Therefore, the impact of electrodes was minimized.We measured the diffracted power of individual dots using a referenced power detector (Thorlabs, PM100USB, S150C) and a slit aperture.To exclude any influence of stray light, a noise reduction tube was used.The setup, consisting of electrodes and the sample, was mounted on an angular stage.A schematic sketch of the measurements is shown in the Supporting Information.

Figure 1 .
Figure1.Diffraction at different loss-free structures by RCWA.a) Ratio of the propagation length l prop of TE 0 and TE 1 at small grating thickness t g in a waveguide grating compared with diffraction efficiency contrast C ¼ η max =η min under symmetric dual-plane wave incidence (internal angle θ core,1 ¼ θ core,2 , wavelength λ 1 ¼ λ 2 ) on a standalone grating plotted over grating thickness t g =λ, dashed black line shows ðt g =λÞ À4 .b) This standalone grating at t g ¼ 200 nm with c) average contrast C % 24, demonstrating that destructive interference at the grating alone does not enable high contrast; d) waveguide grating with e) C % 41; and f ) leaky waveguide grating with g) C % 420.000 and near ellipsoids of infinite contrast, not present for a standalone grating.

Figure 2 .
Figure 2. A near ellipsoid of infinite contrast.a) Contrast C ¼ η max =η min of leaky waveguide with b) linecut at λ ¼ 600 nm (dashed green line in (a)) and c) minimized η min and maximized η max diffraction efficiency.

Figure 4 .
Figure 4. Experimental realization with nonlinear waveguide shown in Figure 3a with phase control by two tip electrodes (marked by an x) and a counter water electrode under single-beam incidence (θ ext = 35.5°andλ = 532 nm), the symmetric dual beam is formed, as shown in Figure 3b.a,b) A global phase value of ΔΦ ¼ 0 and diffraction at every interaction projected on a diffusive screen (inset) and c,d) ΔΦ ¼ π between both electrode positions, marked by an 'x' and no diffraction between the electrodes (dots 1…4).

Figure 5 .
Figure 5. High-contrast electrical control of the diffraction with a) schematic sketch of the experiment with water electrodes on both sides and global phase control and b,c) normalized measured (solid line) and simulated (dashed line) power for green (532 nm) and red (632.8nm) light.