1. Introduction

Scanning light using two orthogonal sinusoidal motions results in a complex trajectory pattern called the Lissajous curve. Lissajous scanning is applied in many different imaging techniques, including magnetic resonance imaging [1] and magnetic particle imaging [2], atomic force microscopy [3,4] and scanning imaging microscopy [5]. Scanning-based optical projection can be realized with as little as a collimated light source, a (micro)mechanical mirror, and only a few or even no additional optical elements. A low number of required components and the corresponding low complexity enables these systems to be highly miniaturized, cost-effective, and mass-producible. These advantages apply to a variety of different laser beam scanning implementations. Lissajous scanning, where two perpendicular resonant motions are used to scan the field of view, incorporates additional advantageous properties. The resonant nature of the motions allows for reaching high scan angles at high frequencies by applying relatively low forces (low actuation voltages). Furthermore, designing resonators with high quality factors allows for the operation of scanning devices with lower power consumption compared to other scanning approaches [6–8]. The advantages of laser beam scanning in general and Lissajous scanning specifically facilitated the emergence of many fascinating applications like light detection and ranging [9], smart headlights for road lighting [10,11], image projection and acquisition systems [6,12,13], biomedical diagnostics such as OCT [5,14–17], as well as small pico-sized projections for virtual and augmented reality [18]. Driven by these applications, the design and fabrication of micro-electromechanical scanners have seen tremendous growth in recent years and are now transitioning from research labs to the industry. The development of the actual imaging systems based on Lissajous scanning is still lagging behind. So far only a few projected examples with rather low quality were reported in the literature and high-resolution projection are yet to emerge. As a consequence, the intrinsic limits of Lissajous-based image formation are not fully studied. In practice, many factors contribute to the total image contrast budget, like amplitude, frequency, and phase stability of the mechanical motions and synchronization issues between mirror motion and light detection (imaging) or emission [19], as well as laser pulse-width jitter and dynamic deformation [20,21] to name a few. While these factors might all affect the final image quality, they do not present inherent theoretical limitations. Several authors studied the imaging and projection performance of Lissajous scanners from a geometrical point of view and established an important foundation for later works. Urey et al. have shown that the achievable imaging resolution is a function of mirror diameter $D$ and the scan angle $\theta$ [22] and introduces the spot overlap and fill factor. Inspired by this pioneering work, several authors have later provided the guidelines for selecting frequency pairs for dense Lissajous patterns achieving high fill factors and high refresh rate [12,13,23,24]. These studies treated image formation as a fill-factor maximization problem and do not provide a definite answer to the inherent limits of the achievable image quality. So far, systems with an equal fill factor, spot overlap, and optical scan angles are considered to deliver comparable if not the same optical performance, irrespective of the choice of resonant frequencies. It is generally believed that at comparable parameters raster scanning can deliver the same image quality as Lissajous-based scanning. As a consequence, mirrors’ performances have so far been compared only through a simple figure of merit derived for a single scanning axis as a product of diameter, frequency, and scanning angle [6,9,25,26] Several authors later introduced additional important terms like current consumption, input power [27] or dynamic deformation [20,21] to allow a fair comparison in certain highlighted aspects. Still, this approach fails to link the established figure of merits to the achievable image quality.

This paper demonstrates that the state-of-the-art methods to estimate the image quality achievable by a system based on a given scanning device neglect important aspects that are inherent to scanning-based projection systems. As the nature of image formation of Lissajous scanners fundamentally differs from classical lens-based systems, the established evaluation metrics for assessing optical performance, i.e. presence of optical aberrations, pixel resolution and contrast evaluation are not transferable to Lissajous scanning systems in a straightforward fashion. In this paper, we present a new method for consistent and comparable assessment of the imaging quality performance of the scanning system. Based on this, we discuss the image quality and the associated contrast budget as a function of different scanning frequency ratios. We report aliasing effects associated with the appearance of Lissajous induced Moiré patterns, which give rise to large spatial and directional dependence of the modulation transfer function. Further, different Lissajous scanning approaches were investigated based on the choice of frequency pairs which allowed side-by-side comparison of different double resonant scanning configurations. The results of this study highlight the importance of the choice of oscillation frequencies in a completely new way, laying an important foundation for building practical Lissajous-based image projection systems.

2. Methods

2.1 Contrast calculation

The simulations for this study were performed using a self-written ray tracing toolbox implemented in Python. Commercial optical simulation products like LightTrans or Zemax OpticsStudio are optimized for propagating a high number of rays through a fixed optical system to quantify optical aberrations and optimize optical systems accordingly. For classical lens-based imaging systems, this method is well suited to evaluate the achievable image quality. However, for Lissajous-based projection systems, the Lissajous pattern itself limits the achievable image resolution. While there are possibilities to implement moving elements like MEMS mirrors in commercially available simulation software, it is computationally very expensive to simulate hundreds of thousands or even millions of MEMS positions that are required to accurately resemble a Lissajous pattern like intensity distribution. The self-written Python code is optimized to propagate one emitted beam of a defined origin and direction through an optical system with up to several millions of different MEMS deflection angles. A detailed description of the applied simulation method is available in the Supplement 1 and in the literature [28]. Figure 1 shows a typical image projection scenario used in all simulations in this study. The projection plane and the equilibrium mirror surface are conjugate so the incident beam hits the equilibrium mirror surface at 90°. For the simulations in this study, the location of the projection screen $z$ and the maximum mechanical deflection angle were kept constant. The maximum mechanical deflection angles are set to $\pm {12.5}^{\circ }$, which is a typical value of real-world systems. [6,29]. In the simulated configuration, the one-dimensional scanned distance on the screen is equal to $x = 2 \tan { \frac {\theta _{opt}}{2}}z = 0.828 z$. In the results below, the spot size and spatial frequencies are described relative to the scanned distance on the screen $x$.

Figure 2 illustrates the overview of the algorithm, proposed to calculate the spatially resolved contrast. First, the center beam intensity distribution of the full Lissajous pattern is calculated (Figure 2(a)). The angle $\alpha$ marked in the image describes the angle at which the Lissajous pattern crosses the center of the projected area. For the projection geometry, illustrated in Figure 1, $\alpha$ can be described according to Eq. (1).

 

Fig. 1. Visualization of the simulated projection geometry. The incidence angle on the equilibrium mirror surface is 90°. The projection surface and the equilibrium mirror surface are anti-parallel. Arrows indicate the distance of the mirror to the projection surface $z$ and the scanned distance on the screen $x$.

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Fig. 2. Visualization of the contrast retrieval algorithm. a): Full Lissajous pattern projected onto the virtual detector. b): Target binary line pattern to be displayed using the Lissajous pattern. c): Line pattern projected using the Lissajous pattern. d): Laser spot intensity profile (top hat). e): Projected line pattern convoluted with the laser intensity profile. f): 1-dimensional intensity profile along on row of detector pixels (red) and target line pattern (gray/white). g): Assignment of projected intensities to $I_{\mathrm {min}}$ (blue) and $I_{\mathrm {max}}$ (green) according to the target pattern. h): Contrast profile along the detector pixel row computed from $I_{\mathrm {min}}$ and $I_{\mathrm {max}}$. i): 2-dimensional spatially resolved contrast average over all test pattern phase angles. j): Spatially resolved contrast phase sensitivity (standard deviation) to the test pattern phase angle.

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The intensity distribution of the binary test pattern $I_{test}$ is mathematically described in Eq. (2).

For classical imaging systems $I_{\mathrm {min}}$ and $I_{\mathrm {max}}$ represent the intensity minimum and maximum respectively. For Lissajous-based imaging systems, the non-unitary fill factor causes intensity minima to locally drop to very low values, thereby maximizing the contrast, even if the original feature does not resemble well. For this reason, different interpretations of $I_{\mathrm {min}}$ and $I_{\mathrm {max}}$ have to be applied here. The intensities in the areas corresponding to dark and bright areas in the test pattern in a range of $\pm$ 0.5 line pattern periods are integrated to retrieve $I_{\mathrm {min}}$ and $I_{\mathrm {max}}$. These integrations are described in Eq. (4) and Eq. (5) and illustrated by the blue ($I_{\mathrm {max}}$) and green ($I_{\mathrm {max}}$) areas in part g) of Figure 2.

We note that ($\frac {I_{\mathrm {max}}+I_{\mathrm {min}}}{2}$) still describes the average luminance and ($\frac {I_{\mathrm {max}}-I_{\mathrm {min}}}{2}$) reflects half the difference in intensity between dark and bright areas, hence the amplitude. Therefore the formulation of contrast employed in this study remains in line with the original definition of the Michelson contrast. Part h) of Figure 2 shows the spatially resolved contrast calculated by Eq. (3) for the pixel row marked in part e). The black line shows the contrast for the pattern phase illustrated in the subfigures before. The position-resolved contrast behavior for other test pattern phases is indicated by light gray lines. The red line represents the mean contrast value averages over all 20 test pattern phase angles. The width of the purple band shows the contrast standard deviation over all phase angles at the given pixel. This standard deviation is referred to as phase sensitivity. In the way described above, contrast values for all $x,y$-positions, and phases $\varphi$ are calculated to generate a three-dimensional contrast map $C\left (x,y,\varphi \right )$. Calculating the mean contrast along the $\varphi$ dimension yields the two-dimensional contrast distribution shown in part i) of Figure 2. The phase sensitivity distribution shown in part j) is retrieved from calculating the corresponding standard deviation along the $\varphi$ dimension (Eq. (19)). To further evaluate the two-dimensional maps of contrast and phase sensitivity, the mean contrast $\mu$ (Eq. (6)), the overall phase variability $\sigma _{\varphi }$ (Eq. (7)), spatial variability $\sigma _{x,y}$ (Eq. (8)) and the total variability $\sigma _{tot}$ (Eq. (9)) are calculated as described in the equations below.

2.2 Frequency pair selection

The objective of this study is to compare Lissajous curves of different crossing angles $\alpha$ for their ability to resolve structures of given spatial frequencies and orientations. For this comparison, it is necessary to define a metric quantifying the general resolution capability associated with a Lissajous pattern. We propose the pattern’s arclength in the two-dimensional deflection angle space for this purpose. The arclength describes the distance covered by the pattern until it repeats itself. This metric is in a linear relationship with the distance between two quasi-parallel line segments at the center of the pattern, hence the maximum void distance within the Lissajous pattern. Hence, the arclength is also an indicator of the minimum spot size required to fill the whole projection area without any dark spots. The calculation of the arclength and its correlation with the maximum void distance in the pattern is elaborated in the Supplement 1. Figure S1 in the Supplement 1 illustrates the maximum void distance for two different Lissajous curves. Table 1 lists the frequency pairs analyzed in this study, their corresponding number of lobes $n_{lobes}$, Lissajous crossing angle $\alpha$, and the path lengths of the pattern in deflection angle space $s_{path}$. The range of Lissajous angles is restricted to $\alpha = {0}^{\circ} \; to \; {45}^{\circ }$ for symmetry reasons. For practical reasons, a target path length of 2650.0° was chosen for the comparison of the different Lissajous curves. This length allows for studying imaging effects at reasonably low computational costs without any loss of generality. Choosing a higher target path length would result in denser Lissajous patterns. The underlying effects responsible for the decay of resolution are the same denser Lissajous curves, but these effects will appear at higher spatial frequencies of test patterns and image features.

Table 1. Table of frequency pairs used for simulation. The simulation results of the frequency pairs marked with asterisks are visualized in the results section. The full data set comprising the results of all frequency pairs is available in Data File 1.

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The discrete nature of integer fractions defining Lissajous curves also limits the available path lengths for a certain set of maximum deflection angles to discrete numbers and the target path length can usually not be exactly met. For the frequency pairs investigated in this study, the path lengths vary between 2612.7° and 2675.2°. This means maximum deviation from the target path length of 2650.0° by −1.4 % and 1.0 %. It is shown in the Supplement 1 that there is a linear relationship between the path length and the reciprocal minimum spot size required to fill the whole field of view. Therefore, the differences in path lengths also translate linearly to the achievable resolution. These relatively small deviations from the target path do not significantly compromise the quantitative contrast results presented and discussed below because absolute contrast values achieved for different frequency pairs are not compared and the contrast variations caused by spatial and phase variation dominate the variations introduced by the different path lengths.

The initial phase difference of the mechanical motions has been chosen to maximize the density of the Lissajous pattern [24]. The initial beam was propagated through the system for 1×10⁶ equidistant time steps and corresponding 2D mirror surface orientations to simulate the intensity distributions on the detector. At a target arclength of 2650° one time-step corresponds to a mean change of deflection angle of 0.002 65°(46 µrad). For all frequency pairs investigated, the timing resolution is more than a factor of 140 above the limit of $4f_1 f_2$ where aliasing effects can appear [15]. This excludes limited timing resolution as a source of aliasing effects in the simulation performed in this study. The projected line patterns used to compute the space-resolved contrast distribution are varied in spatial frequency and orientation angle $\beta$. The simulations described above are performed to investigate the fundamental potential of Lissajous patterns to resolve features of given spatial frequencies and orientations. Therefore, retrieved results represent the best-case scenarios. In real-world projection systems, additional effects like the ones listed below can lead to further deterioration of the image quality:

3. Results and discussion

In this section, the results of the simulations described above are visualized and discussed. For the sake of clarity, the results shown in this section are restricted to the four frequency pairs marked in Table 1. The observed trends and dependencies are well reflected in the results associated with the four selected frequency pairs. The full data set comprising the results of all frequency pairs listed in Table 1 is available in Data File 1.

3.1 Spatial variability

Column a) of plots in Figure 3 shows the mean contrast $\mu$ (solid line) and the spatial variability $\sigma _{x,y}$ (width of filled band) for different pattern orientations $\beta$ at a spatial frequency of 29.6 line pairs per image width. The lines and bands of different colors represent different spatial frequencies relative to the scanned distance $x$. Columns b) to e) of plots in Figure 3 show the spatial contrast and intensity (one selected pattern phase angle) distributions of the combinations of $\alpha$ and $\beta$ leading to minima (green) and maxima (red) in spatial contrast variation. The lines indicating the mean contrast as a function of $\beta$ exhibit no prominent features only deviating slightly from straight horizontal lines. This indicates that the contrast averaged over the region of interest (ROI) and all target pattern phase angles are not significantly impacted by the angles $\alpha$ and $\beta$ or combinations thereof. However, the influence of $\alpha$ and $\beta$ is significant for the spatial contrast variability inside the region of interest which is indicated by the widths of the semitransparent bands. The obtained results show that maxima in the spatial contrast variation appear where $\beta = {90}^{\circ }-\alpha$. This means that the spatial contrast variation inside the ROI peaks if the test pattern lines are perpendicular to the lines of the Lissajous curve at the center of the image. When comparing the spatial contrast distributions in column 4 (minima) and column 5 (maxima) of images in Figure 3, the differences in contrast variation indicated by the plots in column 1 are clearly visible. There are pronounced peaks and valleys of contrast in the images in column c) while the contrast relief in the images in column b) is significantly smoother. However, when looking at the corresponding intensity distributions in columns d) and e), there appears to be no significant influence of the spatial contrast variation on the quality features can be resolved with. There are combinations of $\alpha$ and $\beta$ yielding high spatial contrast variation and compromised structure resolution (e.g. $\alpha = {44.3}^{\circ }$, $\beta = {45}^{\circ}$) but there are also combinations of $\alpha$ and $\beta$ that lead to a relatively smooth spatial contrast distribution but similarly bad structure resolution (e.g. $\alpha = {28.9}^{\circ}$, $\beta = {30}^{\circ}$). Despite the strong variability of the Lissajous curves’ densities and trajectory directions, the results show that spatial contrast variability only has a minor influence on the resolution quality of features. I.e., although the field of view is scanned in a non-homogeneous fashion, it only has a minor influence on the resolution quality where in the field of view a feature is placed.

 

Fig. 3. Visualization of simulation results evaluated for the mean contrast (averaged over all line pattern phase angles) in the region of interest. Rows of plots correspond to fixed Lissajous pattern angles $\alpha$. Column a: Mean contrast and standard deviation as a function of the pattern angle $\beta$ and a spatial frequency of 29.6 lp. Column b, c: Spatial contrast distribution for the minimum (column c, green) and maxima (column c, red) in spatial contrast variability for the given Lissajous angle $\alpha$. Column d, e: Spatial intensity distribution (one selected test pattern phase angle) for the minimum (column d, green) and maxima (column e, red) in spatial contrast variability.

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3.2 Lissajous aliasing / phase variability

Spatial sampling along the Lissajous curve gives rise to artifact formation and can significantly compromise the resolution of features. Poorly sampled periodic structures give rise to artifacts, known as Moiré patterns. Artifact formation due to under-sampling is generally referred to as aliasing and commonly appears in Cartesian grid imaging. Analogously to Cartesian grid imaging also in Lissajous projection, the prominence of aliasing effects depends on the relative orientation and the spatial frequencies of the features and the sampling structure (Lissajous curve). The phase variability $\sigma _{\varphi }$ introduced above is a measure of the intensity of aliasing in the region of interest. The plot column a) in Figure 4 shows the phase variability dependence on the test pattern orientation $\beta$ and different spatial frequencies for the four selected frequency pairs and a spatial frequency of 29.6 lp. The lines in the plot indicate the phase variability $\sigma _{\varphi }$. The columns b) to e) in Figure 4 show the spatial phase sensitivity and intensity distributions for the combinations of $\alpha$ and $\beta$ leading to minima (green) and maxima (red) in contrast phase sensitivity. Among the frequency pairs investigated, the Lissajous curve at $\alpha = {1}^{\circ}$ shows the largest impact of the feature orientation $\beta$ on the contrast phase sensitivity. While features with $\beta$ close to 0° can significantly vary in contrast depending on their phase, features with a $\beta$ close to 90° will be displayed with the same quality irrespective of small spatial shifts. For the combination of $\alpha = {1}^{\circ}$ and $\beta = {0}^{\circ}$ this effect is already significant at relatively moderate spatial frequencies. The Lissajous pattern at $\alpha = {1}^{\circ}$ is based on oscillation frequencies of 1 and 59. This means that a vertical line is intersected $2*59=118$ times by the Lissajous curve. The representation of 21 horizontal lines, which is only about a sixth of this theoretical upper limit of vertical supporting points already suffers significantly from elevated contrast phase sensitivity. With a mean contrast of 0.54 and a phase sensitivity of 0.17, a feature’s contrast of a random phase is expected to deviate by approximately 30 % from the mean value. When comparing the angle dependencies of the phase sensitivity for the four different frequency pairs some clear trends can be deduced. The closer the Lissajous angle $\alpha$ is to 45°, the less the phase sensitivity depends on the feature angle $\beta$. The Lissajous curve with $\alpha = {44.3}^{\circ}$ still shows a distinct maximum in phase sensitivity around $\alpha = \beta$ but the peak-to-valley difference is reduced by a factor of more than 3 compared to the $\alpha = {1}^{\circ}$ Lissajous curve for all spatial frequencies shown. This trend of increasing phase angle robustness with higher Lissajous angles $\alpha$ is also visible in the 2D phase sensitivity maps for the minima and maxima in the columns b) and c) of Figure 4. For the Lissajous curve with $\alpha = {1}^{\circ}$ the 2D phase sensitivity maps corresponding to the minimum ($\beta = {90}^{\circ}$) and the maximum phase sensitivity ($\beta = {0}^{\circ}$) show radically different profiles. While the phase sensitivity map corresponding to the minimum mean value (1b) is flat and close to zero over the whole projected area, the map corresponding to the maximum mean value (1c) shows distinct structures and maxima up to 0.6. With increasing Lissajous angles $\alpha$ some structures in the minima maps appear and the average values rise. For the maxima maps, the opposite trend can be observed. The peaks and valleys get less pronounced and the average values drop with increasing $\alpha$. Comparing the phase sensitivity maps with the corresponding intensity maps in columns d) and e) of Figure 4 clearly shows that there is a strong correlation between the phase sensitivity and the feature accuracy. The intensity profile with green frames corresponding to the minima in phase sensitivity show line patterns that are limited in contrast but still recognizable for all four Lissajous patterns. Analogous to the trend in phase sensitivity, the pattern 1d) for $\alpha = {1}^{\circ}$ and $\beta = {90}^{\circ}$ is resolved the best and the quality decreases with increasing $\alpha$. In the intensity maps with red frames (column e), maximum phase sensitivity) the test patterns are barely recognizable and artifact structures are visible. The prominence of the artifact structures is in correlation with the respective phase sensitivity. These structures can be classified as Moiré-patterns.

 

Fig. 4. Visualization of simulation results evaluated for the contrast phase variability $\sigma _\varphi$ in the region of interest. Rows of plots correspond to fixed Lissajous pattern angles $\alpha$. Column a: Mean phase variability as a function of the pattern angle $\beta$ for a spatial frequency of 29.6 lp. Column b, c: Spatial phase variability distribution and the minimum (column b, green) and maxima (column c, red) in phase variability for the given Lissajous angle $\alpha$. Column d, e: Spatial intensity distribution (one selected test pattern phase angle) for the minimum (column d, green) and maxima (column e, red) in phase variability.

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As both the angle $\alpha$ and the spacing of the lines in the Lissajous curve vary over the projected area, the formed Moiré patterns are not periodic. Furthermore, the symmetric nature of Lissajous curves mandates not only to consider the set of lines at the angle $\alpha$ but also the mirrored set of lines at the angle ${180}^{\circ}-\alpha$. Therefore, two overlapping Moiré patterns of different periodicities are formed when a periodic test pattern is represented using a Lissajous pattern. In general, the spatial frequency of the Moiré patterns is inverse proportional to the difference between $\alpha$ and $\beta$. Small differences between $\alpha$ and $\beta$, therefore, lead to maxima in phase sensitivity. The second row of plots in Figure 4 corresponding to the $\alpha = {14.3}^{\circ}$ Lissajous patterns constitutes an exception to that rule. Here the maximum phase sensitivity is reached at $\alpha = {0}^{\circ}$ and not at $\alpha = \beta$ as one might expect. The reason for this shifted maximum is the fact discussed above, that always two overlapping Moiré patterns are formed. Due to the Lissajous patterns being symmetric around the $x$ and $y$ axis, two Moiré patterns that are mutual mirror images are formed if $\beta = {0}^{\circ}$ or $\beta = {90}^{\circ}$. In the case of $\alpha = {14.3}^{\circ}$ and a spatial frequency of 29.6 lp, the addition of these two mutually mirrored Moiré patterns leads to a slightly higher phase sensitivity than the expected maximum at $\alpha = \beta$. Comparing the intensity maps 2e) and 3e) clearly shows that there are two overlapping Moiré patterns. In the case of 2e), where $\beta = {0}^{\circ}$, the intensity artifact structures are based on two equally contributing mirrored patterns ($\alpha = {14.3}^{\circ}$ and $\alpha = {180}^{\circ} - {14.3}^{\circ} = {165.7}^{\circ}$). In the intensity map 3e), the Moiré pattern originating from the interference between $\alpha = {28.9}^{\circ}$ and $\beta = {30}^{\circ}$ dominates the intensity distribution and the pattern from $\alpha = {180}^{\circ}- {28.9}^{\circ} = {151.1}^{\circ}$ is negligible.

The contrast distributions in Figure 3 support the identified trend that Lissajous patterns with shallow angles $\alpha$ lead to higher contrast variation in the projected image and exhibit a higher impact of the feature orientation on the contrast compared to patterns with $\alpha$ close to 45°. The prominence of the Moiré patterns in Figure 3 is in agreement with the evaluated phase sensitivity of the contrast in Figure 4. This leads to the final conclusion, that in order to estimate the quality a feature can be resolved with, not only the spatially resolved mean contrast but also the contrast’s phase sensitivity has to be considered. If the phase sensitivity is low relative to the mean contrast the features representation is robust against small spatial shifts. If the phase sensitivity is in the range of the mean contrast, small spatial shifts of the feature must be expected to change the contrast between 0 and twice the mean value.

3.3 Total variability

Figure 5 shows the mean contrast $\mu$, the total variability $\sigma _{tot}$, the spatial standard deviation $\sigma _{x,y}$ and the phase variability $\sigma _{\varphi }$ as a function of $\beta$ for the four Lissajous curves investigated and five different spatial frequencies. To show the evolution of contrast and aliasing effects intensity maps in the background of the plots show the representations of the line patterns at the given spatial frequencies by the corresponding Lissajous patterns and a rotation angle of $\beta = {30}^{\circ}$. The mean contrast achieved for a set of parameters shows a clear monotone trend of decreasing contrast for increasing spatial frequency. The phase variability dominates the total contrast variability, hence the image quality impairment, for the vast majority of $\alpha$ and $\beta$ combinations shown. Only in the cases where $\alpha = {1.0}^{\circ}$ and $\beta >{80}^{\circ}$, the spatial variability is significantly larger than the phase variability. The phase variability dominating the overall variability means that a spatial shift that is small with respect to the feature’s spatial period, changes the contrast more than placing the feature in a different area of the image. This observation is well in line with the results shown in Figure 3 and Figure 4 where it is shown that a high phase variability is correlated with limited feature resolution. For $\alpha ={44.3}^{\circ}$ the maxima in spatial variability and phase variability are both at around $\alpha =\beta$. This coincidence leads to a significant increase of the total variability $\sigma _{tot}$ for the two highest spatial frequencies shown in Figure 5. For lower spatial frequencies the total variability is dominated by the phase sensitivity and there is no significant amplification effect. The intensity maps in the backgrounds of the plots show that the line patterns are represented by the different Lissajous curves in comparable qualities up to a spatial frequency of 23.7 lp. For the two highest frequencies shown, significant differences caused by aliasing effects are visible between the line pattern projections using different Lissajous patterns. As expected, for the chosen example of $\beta ={30}^{\circ}$, the Moiré patterns are the most pronounced and at the lowest spatial frequency for the Lissajous pattern with $\alpha ={28.0}^{\circ}$. The projections using $\alpha = {14.3}^{\circ}$ and $\alpha = {44.3}^{\circ}$ show very similar aliasing artifacts rotated by ${90}^{\circ}$ due to the symmetry around $\beta = {30}^{\circ}$. The projections using $\alpha = {1.0}^{\circ}$ suffer the least from Moiré pattern creation because of the relatively high difference between $\alpha$ and $\beta$ for the chosen examples.

 

Fig. 5. Mean contrast, phase sensitivity, spatial variability, and total contrast standard deviation as functions of the test pattern angle $\beta$ for the four Lissajous curves investigated and five different test pattern spatial frequencies. The intensity maps in the background of the plots show the representations of the line patterns at the given spatial frequencies by the corresponding Lissajous patterns and a rotation angle of $\beta = {30}^{\circ}$.

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3.4 Image projection example

Figure 6 visualizes the effect of contrast phase sensitivity on the representation of two images, where a manifold of spatial frequencies and feature orientations appear. The original images are shown on the left side of the figure. For the simulated projection of the images using Lissajous scanning the original images are binarized (black and white). The second and third columns of images show the projections of the images using the Lissajous patterns with $\alpha = {1.0}^{\circ}$ and $\alpha = {44.3}^{\circ}$ respectively. To demonstrate the feature orientation dependence of the contrast the unrotated image ($\gamma = {0}^{\circ}$) and the same images rotated by 90° ($\gamma = {90}^{\circ}$) are projected. For both image projection examples, it is clearly visible that the image rotation has a significantly larger impact on the projection with $\alpha = {1.0}^{\circ}$ compared to the projection with $\alpha = {44.3}^{\circ}$. For $\alpha ={1.0}^{\circ}$, the quasi-horizontal lines in the SAL logo suffer from strong aliasing effects for and $\gamma ={0}^{\circ}$ while they are resolved reasonably well if the image is rotated by $\gamma ={90}^{\circ}$. The small gaps between the quasi-horizontal lines however are resolved better for $\gamma ={0}^{\circ}$ than for $\gamma ={90}^{\circ}$. The same effects can be observed in the representations of the zebra’s stripes in the second example. At the center of the image, the stripes are orientated approximately vertically. Therefore, their representation is superior for $\gamma ={0}^{\circ}$ establishing a large difference between the Lissajous angle $\alpha$ and the feature orientation $\beta$.

 

Fig. 6. Visualization of the feature angle dependence on the resolution for Lissajous patterns with $\alpha = {1.0}^{\circ}$ and $\alpha = {44.3}^{\circ}$ on the example of the Silicon Austria Labs logo and an image of a drawn zebra. The angle $\gamma$ denotes the rotation of the projected image.

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The projection examples for $\alpha ={44.3}^{\circ}$ demonstrate the lower dependence of the image quality on the feature orientations for a Lissajous pattern with $\alpha$ close to ${45}^{\circ}$. The projected images for $\gamma ={0}^{\circ}$ and $\gamma ={90}^{\circ}$ differ only in minor details. The achieved feature resolution quality is inferior to the quality that can be achieved by the Lissajous pattern with $\alpha ={1.0}^{\circ}$ if the features are orientated approximately vertically. However, pronounced aliasing effects, similar to the ones observed for $\alpha ={1.0}^{\circ}$ and quasi-horizontal features are not observed. In the case of a Lissajous curve generated by two similar frequencies, the lines at the center of the curve meet at an angle close to 90°. This means that features can never be oriented within a small angle to all lines of the Lissajous curve. Therefore, the effects of contrast phase sensitivity and Moiré formation are much less pronounced compared to Lissajous patterns with high frequency ratios.

4. Conclusion and outlook

This paper focuses on the theoretical limitations of Lissajous-based projection systems and highlights the effect of choice of scanning strategies and the spatial and directional dependence of the modulation transfer function in a way that is not considered in the existing literature. To this end, we report a method to quantify the achievable contrast in scanning-based projection systems and apply it to show that Lissajous scanning-based projection systems can resolve features in significantly varying quality depending on the features’ orientations. Aliasing effects can cause a small spatial shift of features oriented within a small angle to the lines of the Lissajous curve to dramatically change the contrast they are resolved with. The aliasing effects, which are correlated to the feature phase sensitivity cause significant degradation of the perceived image quality. For periodic patterns, this phase sensitivity shows in the form of Moiré patterns. Provided that the system’s timing resolution is high enough [15], features that are perpendicular to the Lissajous lines exhibit a more continuous decay in contrast with increasing spatial frequencies that is reminiscent of classical lens-based imaging systems. A systematic study varying the orientation of a binary test pattern over its full range reveals that Lissajous curves where the ratio of the underlying frequencies is either much smaller or much larger than 1, show a pronounced feature angle dependence of the way the contrast decays with increasing spatial frequencies. Lissajous curves with nearly unitary frequency ratios are significantly more robust in this regard. Furthermore, we found that the location of a feature within the scanned field of view only has a minor influence on the contrast it can be resolved with.

These findings represent the first step toward systematic analysis of the image quality of Lissajous-based projection systems. There are numerous ways in which the methods presented here can be expanded to study other real-world effects on image quality, such as dynamic deformation, beam positional accuracy, phase stability, laser synchronization issues, and laser intensity distribution functions, to name a few. We believe the developed quantitative characterization methods represent an important step towards building practical high-quality Lissajous projection systems and will in the future shape the design considerations of next-generation scanning mirrors.