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We present a new principle for tuning the diffraction efficiency of an optical grating and its implementation in a micro-optical device. The overlap of two phase gratings is used to vary the effective phase shift and hence the diffraction efficiency. We study the working principle using Fourier Optics to simulate the diffraction pattern in the far field and design and realize a device based on integrated piezo actuation. We find good agreement between simulation and experiment and observe a suppression of the first diffraction order intensity by more than 97% and response times of less than 3 ms.
© 2016 Optical Society of America
1. Introduction and working principle
The properties of optical gratings can be tuned in two different ways: varying the grating constant or the diffraction efficiency. Adjusting the grating constant leads to a variable diffraction angle which can be used e.g. for sensing applications or in spectrometers. Many different devices with integrated actuation for tuning the grating constant are discussed in literature, mainly with elastomeric gratings, for example [1–3].
In this work we focus on tuning the grating efficiency, i.e., the distribution of energy in the different diffraction orders. Applications for this behaviour comprise optical switches [4] and projection systems [5] but could also include an imaging spectrometer that is switchable between a spectral and a lateral imaging mode as described in [6].
These applications require miniaturized and compact systems such that the actuation mechanism to vary the effective phase difference of the grating should be integrated. While there exists a tuning concept for reflection and transmission by varying the plasmonic and photonic mode interaction between two grating structures [7] we address a straightforward modification of the phase shift introduced by the grating. This can, on the one hand, be realized by a variation of the refractive index of the medium that surrounds the grating, as it is achieved for example by pumping different fluids over the grating [4] or by using the temperature dependency of the refractive index and employing an integrated heater [8]. The latter is, however, rather slow. Another mechanism to tune the effective phase is a mechanical change of the grating geometry. This was studied in principle in [9,10] where elastomeric gratings are compressed between two glass plates, however, without an integrated actuator. Further approaches use a direct deformation of the grating geometry either by electrostatic [5,11] or electrothermal actuation [12]. A completely different approach is the phase variation by liquid crystal elements that might impose certain restrictions due to their polarization dependency [13].
As an alternative, we present here a tuning principle for gratings that works by superposition of two gratings as shown in Fig. 1. Compared to the before mentioned devices this works with transmissive rather than only with reflective gratings.
Fig. 1 Schematic illustration of the working principle. The overlap p determines the intensity in the diffraction orders.
For manufacturing reasons we chose binary gratings, however, the method itself is not restricted to binary gratings and could also be transferred to blazed gratings. The overlap p (relative lateral position) between the gratings defines the effective phase difference: Without overlap (p = 0%, Fig. 1, left) the phase all over the grating is the same and no diffraction occurs in a simplified model, while maximum diffraction efficiency is achieved at p = 100% (Fig. 1, right). A change from p = 0% to 100% is therefore equal to a lateral shift of half the grating constant.
In contrast to the above mentioned approaches this method just needs an actuator stroke of half the grating constant, i.e., arbitrarily sized grating areas can be tuned homogeneously without position-dependent delays. Therefore, we propose a piezo actuator for our prototype that allows for faster tuning than many of the other published working principles.
We have demonstrated a first prototype using amplitude gratings in [14]. In this paper we study the concept in more detail, both in simulation and in experiment and improve the concept with the use of phase gratings that allow for much larger diffraction efficiency.
In section 2 we explain and investigate the working principle using a discrete Fourier approach and obtain an estimate of the achievable tuning capabilities. Section 3 introduces the design and the fabrication process for the prototypes that are characterized and compared to the simulation results in section 4 before the work is summarized in section 5.
2. Simulation
To obtain an estimate for the achievable tuning range we simulated the expected diffraction efficiency with a discrete Fourier approach in two dimensions [15]. This approach neglects any near-field effects which we assume is a valid assumption because for production reasons the gratings are separated by a distance a ≫ λ. The effective optical path difference Δl is given depending on the overlap p of the two identical gratings
where h is the height of the grating lines and n their refractive index, while x defines the lateral position within one unit cell of the grating with period g as shown in Fig. 2.The optical path difference is used to calculate the phase shift of the optical wave function Ψ at the grating along the lateral direction x
with the wavelength λ and I0 being the intensity distribution of the incident plane wave beam, in our case a Gaussian distribution, cropped to a beam width of 2 mm. We then propagate the wave function and find the intensity distribution I at a distance z from the grating using the wave vector and discrete Fourier transformations :We consider the finite width of the detector that we use in our experiment by integrating the intensity values over the detector width and normalize the results to the intensity of the incident beam on the grating. The results of this simulation, namely the relative intensities of the 0th to 2nd diffraction orders, are shown in Fig. 3 as a function of the relative overlap p for conditions also found in the experiment: λ = 635 nm, g = 10 µm, n = 1.46 (fused silica) and a width of the detector of 5 mm in a distance of 25 cm. We show the results for two different grating heights h: the optimal height of 345 nm corresponding to a λ/2 phase shift caused by the two gratings and the real height of 386 nm, which is the mean value of the grating depth of the prototype for the experimental results.
Fig. 3 Simulated intensity in the 0th to 2nd diffraction orders (d.o.) for optimal and measured grating depths as a function of the overlap factor p that defines the effective phase.
With 0% overlap there is no phase change along the lateral direction, i.e., no diffraction occurs and we see only the 0th order. For rising p, the 0th order drops and 1st and 2nd orders appear. In this ideal simulation, the intensities in negative and positive diffraction orders are fully equal. Comparing the two grating heights, we see that the 0th order of the real grating does not disappear at 100% overlap but at lower values instead, while the maximum of the 1st order intensity is still achieved at p = 100% but yields less intensity. Furthermore, we also note that the 2nd diffraction order disappears at p = 100%. This effect is expected for gratings with slits with exactly half of the grating period because the intensity pattern is essentially a multiplication of the diffraction pattern of a grating with infinitely narrow slits with an envelope function – the diffraction of a single slit – that has in this case a zero exactly at the 2nd order.
3. Prototype fabrication
To demonstrate the working principle experimentally and to show that a setup with integrated actuation is feasible, we fabricated a prototype with a cross section as shown in Fig. 4. One of the gratings is used as a substrate while the other grating is mounted on a piezo actuator that moves the grating laterally. The actuator itself is mounted to the substrate grating via a spacer.
We structured the binary gratings with an area of 7.5×7.5 mm2 and a grating constant of 10 µm into fused silica substrates using reactive ion etching. As a masking layer, we sputtered a 100 nm chromium layer and patterned it photolithographically with AZ 5214 resist. After a softbake for 2 min at 110 °C the photoresist was exposed with a dose of 37 mW/cm2, developed for 15 s in AZ 726 developer and it was hard baked for 2 min at 120 °C. To ensure that the two fabricated gratings match each other perfectly, we actually used the chromium structure of the first sample as a lithography mask for the second one. The chromium layers were etched in a 45 °C hot solution that contains perchloric acid and ceric ammonium nitrate and the photoresist was then removed with AZ 100.
The reactive ion etching took place in a parallel plate reactor with capacitively coupled plasma (Vacutec Plasmatch 650) with 20 sccm SF6, 150 W power and a chamber pressure of 8 Pa. These parameters give an expected etch rate of 27.1 nm/min. After removing the chromium layer, we measured a mean etching depth of 386 ± 2 nm with a white light interferometer. This is larger than the optimal depth of 345 nm and therefore needs to be considered in the simulation and characterization of the device.
We first measured the angle dependent intensity distribution of a single grating element (Fig. 5). For comparison, the graph also contains the results obtained by a simulation of this element. The results are normalized to an incident beam that passes through a blank glass slide to account for the intensity loss by reflections. We can see that the measured intensities and angles of the 0th and 1st diffraction orders match well with the simulation. From second order upwards there are mismatches, however, higher diffraction orders are more sensitive to small errors than lower orders. For example, the exact cancellation of the second order occurs only for a perfect width of the grating line, as discussed before. Furthermore, there are effects that are not included by our simplified model such as a finite edge slope, the roughness of etched surfaces or near field effects in the structure of the grating.
To displace the gratings horizontally to each other, we chose piezo actuation, which allows for precise and fast movement. We use a commercial PZT sheet (Physik Instrumente, PIC151) with 200 µm thickness and out-of-plane polarization, such that the transverse piezo effect displaces the grating in the in-plane direction. To obtain the necessary displacement of at least half of the grating constant, i.e. 5 µm, with a reasonably compact footprint of 24 × 20 mm2, we structured the piezo to a meander shape as shown in Fig. 6(a) using a UV ns-laser (Trumpf Trumark 6330, 355 nm). The electrode on the upper surface of the meander was segmented into passive and active areas which contribute to the deflection, such that the total deflection adds up in one direction. The FEM simulation with the datasheet value of d31 = −180 pm/V at a moderate field ranging from −375 to 750 V/mm (which corresponds to −75 to 150 V at the actuator electrodes) yields a maximum deflection of 6.5 µm.
Fig. 6 (a) Result of a FEM simulation of the actuator deflection with 1125 V/mm (results only for one half of actuator, symmetric). (b) Measured deflection across that field strength range, hysteresis caused by piezo actuator.
While we could achieve higher deflections by applying an opposite voltage to the passive parts of the actuator, we refrained from doing so in this work, on the one hand to avoid additional wiring effort. In addition, we already achieved a horizontal displacement of the actuator of 12 µm (Fig. 6b), that is much larger than expected and is more than sufficient for a full tuning. This is presumably due to a conservative value for the d31 coefficient provided by the manufacturer. The two meanders are connected individually, allowing us to apply different voltages if needed and thus also rotate the upper grating with respect to the lower one to compensate for alignment errors of the gratings.
For the assembly of the device, we first glued the upper grating with a two-component epoxy glue (Araldite 2020) to the piezo actuator with the structured side facing downwards, away from the actuator (cf. Fig. 4). Then, we mounted the actuator itself to the lower grating substrate via a spacer of the same thickness as the gratings, such that the distance between the gratings (ignoring gravity) is of the order of the glue layer. The parallel alignment of the grating lines during this step is crucial for the functioning of the device since an angular mismatch between the lines of upper and lower grating leads to an overlap that varies in direction of the grating lines and hence to an inhomogeneous diffraction efficiency, up to a complete averaging of the switching effect if the grating is illuminated with sufficiently large spots.
For accurate alignment, we developed a procedure in which we use the moiré lines that appear when two patterns with parallels lines are rotated relative to each other: We illuminated the device with a 5 mm laser spot and observed the diffraction pattern on a screen as shown in Fig. 7(a). Using precision stages, we rotated the upper grating until no lines were visible anymore and kept the position while the glue cures. This procedure provides an accuracy of a fraction of the grating period, and any remaining misalignment can be accounted for in the actuation as described above.
Fig. 7 (a) Diffraction pattern in the −1st to 1st diffraction order for different angles between the gratings. (b) Photograph of the final device with connections to a carrier PCB.
Finally, the device was mounted on a PCB for handling and mounting in the experimental setup as shown in Fig. 7(b) and the electrodes of the piezo were contacted by manual soldering to the PCB. Thin wires of 25 µm diameter ensure sufficient flexibility such that the deflection of the actuator is not restricted by the wires.
4. Characterization
To characterize the diffraction efficiency, we used an integrating sphere photodiode with an opening of 5 mm and a further lens with f = 20 mm that was mounted in front of the opening, focusing the light into the sphere. This configuration reliably covers the intensity also from larger beams and is relatively insensitive to alignment errors. We swept this photodiode on a circle with approx. 25 cm radius around the grating to measure the different diffraction orders. The results are given relative to the intensity of the incident beam, correcting for the reflections at the interfaces as described in section 3. We illuminated the grating with a fiber-coupled diode laser with λ = 635 nm that was collimated with a beam diameter of 10 mm. A mechanical iris aperture underneath the grating was then used to control the width of the incident beam.
During the fabrication, we have identified the parallel alignment of the grating lines as a critical step which influences the usable area of the device. The moiré pattern during the assembly was used to reduce the angular misalignment, however, after releasing the system from the assembly tool there was a remnant error of a fraction of the grating constant. In order to correct this error and fine-tune the alignment we varied the voltages at the lower and upper meander Vlow and Vup and observed the diffraction intensity of a 5 mm wide spot in the different diffraction orders. The influence of hysteresis was filtered out in these results by only considering the rising slopes of the actuation voltages. The data (averaged over ten repetitions) is given at the example of the −1st order in Fig. 8(a) and shows that the maximum and minimum intensity do both not occur at a symmetric actuation voltage (indicated by the dashed line) as it would be expected for equally strong actuator meanders and perfectly aligned gratings. The voltages (Vlow, Vup) that lead to extrema of the intensity were obtained from a polynomial regression of the intensity data near the extrema for −1st to 1st orders and are indicated by −, o and + for the −1st, 0th and 1st order, respectively. A straight line was fitted to these voltage ratios that lead to an optimized voltage trajectory that compensates for any asymmetry of the actuator and misalignment of the gratings as indicated by the dotted line in the graph from which we expect a larger tuning capability. An angle between symmetric voltages and optimized trajectory (dashed and dotted lines) corresponds to a general difference between the displacements of the two actuators parts, i.e., that one of them is stronger than the other, presumably due to production and material tolerances. Since the area on the actuator where the grating is mounted is electrically passive, we do not expect any deformation of the grating itself. Hence, we assume that all non-uniformity of the piezo material is captured by this rotation. A parallel shift compensates for the non-parallel grating lines, i.e., an angular error in the alignment. Both effects are visible for the dotted line in Fig. 8(b), but they can be compensated if we actuate the lower actuator arm accordingly with the relation Vlow = 1.14Vup + 59 V.
Fig. 8 (a) −1st diffraction order intensity as a function of voltage at both actuator meanders. Dashed line indicates Vlow/Vup = 1, dotted line is the linear regression to the extrema intensities in −1st to 1st order (indicated by −, o and +). (b) Tuning contrast as a function of the spot size, for symmetric and optimized actuation voltage. Positive and negative diffraction orders are averaged, the data shows the mean value of the measurement at three different illuminated locations on the grating, errorbars indicate the standard deviation.
To analyze this improvement, we varied the spot size of the incident beam and obtained the tuning contrast that is given by the difference between maximum and minimum relative intensity in the individual diffraction orders for a symmetric actuation voltage and for the optimized voltage trajectory. The measurement was repeated at three different locations on the grating. An averaging over these three results is possible since the optimized voltage trajectory improves parallelity of the grating lines independent of the location. Hence a sufficiently large actuation range ensures that the minimum and maximum intensity can be reached at all points. Figure 8(b) shows that a symmetric actuation leads to a significant decrease of the tuning contrast with increasing spot sizes. This effect is more apparent in the 2nd order than in the 1st order, as the intensity of the 2nd order varies twice over a tuning cycle, so it is twice as sensitive as the 1st order. For the optimized voltage trajectory, the curves are flat within errors, so it completely suppresses this drop in the contrast. The difference between maximum and minimum intensity in 1st and −1st diffraction at 5 mm spot size order rises by 0.9% compared to a symmetric actuation, and by 0.5% in the 0th order.
Next, let us look in Fig. 9 at how the intensity depends on the actuation voltage with a quasi-static trajectory (0.5 Hz sinusoidal). Here, we used a 2 mm spot for which we have seen in the previous measurement no significant influence of the angular error, such that no corrected voltage trajectory was necessary and we applied a symmetric voltage to both meander arms. From the mechanical measurements in Fig. 6, we expect a displacement of around 12 µm, which is larger than the grating constant. For the measurement in Fig. 9(a) we have therefore decreased the actuation voltage range to a trajectory between −29 V and 91 V which yields half of the tuning period in the 0th order. This approximately corresponds to a loop between p = 100% and p = 0% and can therefore be compared to the simulation results in Fig. 10. A higher number of periods is observed for the 2nd diffraction order as was also expected from the simulation in Fig. 3 where its intensity always reaches a minimum either when the grating is not active at all (p = 0%) or when the grating is fully active (p = 100%) while it has a maximum at p = 50%. This is caused by a periodic appearing of grating parts that cause a λ/2 phase-shift as well as parts that lead to a λ/4 shift. For clarity the measurements are only shown up to the 2nd diffraction orders, although higher orders are also visible. To illustrate the switching of individual diffraction orders with this device, we further manually varied the voltage range for each diffraction order, such that maximum and minimum intensity can just be reached, which leads to simple hysteresis loops as shown in Fig. 9(b).
Fig. 9 (a) −2nd to 2nd diffraction order intensities as a function of the actuation voltage between −29 V and 91 V. (b) Intensities with adapted voltage range for complete range switching. All results are normalized to the intensity of the incident beam with 2 mm width.
Fig. 10 (a) Simulated intensity in the 0th to 2nd diffraction orders (d.o.) for an air gap of a = 10 µm and the measured grating depth of 386 nm as a function of the overlap factor p. (b) Shift Δp between positive and negative diffraction orders as a function of the air gap a between the two gratings. The values of the 2nd diffraction order were omitted near 20, 60 and 90 µm, as the distinct maximum disappears at these points.
In all results we observed a shift between the curves of positive and negative diffraction orders that was not expected from the previous simulation. This asymmetric behaviour can be explained by a gap a between the two gratings which leads to a phase shift between the diffracted wavefront of the first and second grating. To study this shift quantitatively, let us consider a grating illuminated by a plane wave. A path from a point on the grating (at x, z = 0) to some point in the far field can be expressed as
If we displace the grating by a distance a in the direction of the incoming wave, and by a transverse shift Δx, the path of the wavefront that travels through the same point on the grating will be
and the condition for equal path length, i.e. constructive interference, is l1 = l2. Since we consider the very far field with x, z ≫ a, Δx, we only keep the leading order in a and Δx and obtainFor small diffraction angles θ = arcsin (x/z) (i.e. z ≫ x), a Taylor approximation can be used to express
This corresponds to a shift between positive and negative diffraction orders which can also be found in Fig. 10(b). Such an air gap is introduced during the assembly process, e.g., as a result of an inhomogeneous thickness of the glue layer or particle contamination. We estimated the air gap with an optical microscope of the order of 10 – 15 µm for our prototype. The effect of this gap was studied with another simulation using the same method as in section 2 but simulating two discrete binary gratings at a distance to each other, each of which generating half of the total phase shift. First, we obtained the diffraction after one grating and then the resulting wave function at a position a was used as an input for the diffraction calculation of the second grating that is laterally displaced according the overlap factor p. The resulting intensity in the far field of the second grating with a gap a = 10 µm is shown in Fig. 10(a). The asymmetry between positive and negative diffraction orders is now also visible in the simulation and agrees qualitatively with the experimental results. In the 0th order, the gap leads to a reduced maximum and increased minimum intensity and to a change in the overall shape of the curve while the 1st orders maintain their sinusoidal shape and can individually still be reduced to 0 intensity but their maximum intensity decreases. The simulated dependence of the relative shift Δp on the gap a is plotted in Fig. 10(b) and shows a good match with the analytic prediction. The 2nd order shift rises twice as quickly as the shift between the 1st orders as the diffraction angle is also twice as large. Furthermore, also the tuning range, i.e., the difference between maximum and minimum intensities of 0th, 2nd and higher orders decreases as a result of the air gap, while the intensities remain constant for the 1st diffraction orders. The maximum overall tuning contrast can therefore be achieved at a distance of a = 0. However, to avoid sticking and wear and to ensure negligible near-field effects, the gap should be at least in the range of a few micrometers.
As mentioned in the beginning of the section, the intensities in Fig. 9 are normalized to the incident beam that passed through two glass slides from the same fused silica material as the grating such that the reflections are not accounted for. While the overlap factor in Fig. 10 cannot be directly mapped to the voltage in Fig. 9 due to the hysteresis and non-linearity of the piezo effect, we see that there is a good qualitative agreement of the intensity. The maximum intensity of the 0th order is reached near 80 V, which corresponds to p = 0, while the minimum that is expected at p = 100 and can be seen at −20 V. The 0th order reaches a relative tuning capability in the experiment from 4 to 74% while the simulation with 10 µm results in a range from 1 to 77 %. For the first orders we find accordingly 1 to 37% in experiment and 0 to 39% in simulation. Considering the deviation that we already observed for the single grating element (Fig. 5) those values show good consistency of the simulation model and experiment even though the curve shape does not show a fully sinusoidal behaviour, in particular in the 1st order. An explanation can be obtained from the fact that there is not only a horizontal displacement upon actuation but also a small vertical deflection. This may be caused by bending effects due to a non-ideal electric field distribution in the actuator as well as by an external force on the actuator for example due to the wiring. We measured the deflection in the used voltage range and obtained maximum deflections depending on the position of the actuator ranging from 1 to 2 µm which is less than the gap between the gratings. A more likely effect is hence simply the hysteresis and non-linearity of the piezo material.
For switching applications, the response time of the system is an important performance measure. Due to the piezo actuator we expect a quick response which we evaluated by measuring the intensity in −1st to 1st order upon an actuation with a voltage step that switches from minimum to maximum observable intensity. The measured intensities are given in Fig. 11 as a function of the time after switching the voltage step in both switching directions. For this measurement the photodiode amplifier was driven in a high bandwidth mode that causes a higher noise level, therefore the data was obtained by averaging over 100 actuation periods.
Fig. 11 Step response of the intensity in the −1st to 1st diffraction orders upon application of a positive (a) and negative (b) voltage step at t = 0 ms.
We observe an underdamped behaviour with an oscillation at a frequency of approximately 2.1 kHz, which settles for both investigated voltage step directions and for all diffraction orders after 3 ms. The system transfers the mechanical oscillation to the optical signal, the visible period of the intensity therefore occurs at twice the frequency of the mechanical oscillation. This leads to the conclusion that with a suitable voltage ramp the maximum driving frequency is in the kHz range which makes the system much faster than fluidic actuation (55 ms reported in [4]) and comparable to thermal switching (0.5 ms in [8]) and liquid crystals (0.5 ms in [13]).
5. Conclusions
We presented a principle for tuning the diffraction intensity by the relative shift of two parallel gratings that are superposed to each other, requiring a displacement of just half of the grating constant, independent of the grating area. We successfully demonstrated this with a compact (< 1.3 mm thickness) prototype with integrated piezo actuation that showed a maximum intensity tuning of more than 97% in the 1st diffraction order. While our first simplified simulation model already showed a good qualitative agreement with experiments, it could not explain asymmetries between positive and negative diffraction orders that were observed in the experiment. We did reproduce these, however, with an extended model that also took an air gap between the two gratings into account and yielded a good qualitative and quantitative agreement.
Good alignment of both gratings is necessary for this working principle. We achieved this with an accuracy of a fraction of the grating constant in the fabrication using the moiré pattern during assembly. The fine tuning was then performed with the same actuator that was used for the efficiency tuning by applying slightly different voltages to the two meanders causing a corrective rotation. An improvement of the contrast between maximum and minimum diffraction intensities by more than 0.5% could be achieved with an optimized voltage trajectory.
We have demonstrated the working principle at the example of a single prototype. While the manufacturing of the gratings is already very reproducible due to a well-defined lithography process it would be necessary for the production of reliable devices to obtain a good control especially of the assembly which determines the gap size and the parallelity. A precisely set gap size could be achieved with an appropriate guiding structure and one can include a calibration to compensate the angular misalignment as we describe in section 4 or one can improve the parallel alignment during assembly by an additional grating with smaller grating constant next to the actual used grating and align using the moiré pattern of the finer grating as this will be more sensitive to angular errors.
Due to the piezo actuation we saw a strong hysteresis of the intensities as a function of the actuation voltage. Hence, the piezo should be operated with a strain control for applications where a precisely defined intensity is needed. Aside from optimization of the actuator the device could also be improved by an overall increased diffraction efficiency using a blazed grating to which the tuning principle could also be applied. We already found a response time of just 3 ms, which enables high speed switching applications.
Funding
This research was supported by Deutsche Forschungsgemeinschaft (DFG) grant WA 16547/1-2 within the Priority Program “Active Micro-optics” and the Cluster of Excellence BrainLinks-BrainTools, DFG grant EXC 1086.
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