One design of the state-of-the-art laser scanner systems in automotive applications is based on oscillating mirror modules. The requirement of a large mirror surface for eye-safe transmission beams and long measurement distances is a major drawback for fast and reproducible scanning. Tolerances of angular positioning, position sensing, and vibrational perturbations limit the position accuracy of such a mirror and, thus, the accuracy of the transmission spot position in the field of view (FoV). Our approach for a scanner module with maximum transmission beam diameter combines a microlens array with an objective lens for generating one optical telescope assembly for each angular scan position exclusively. Aperture stops define the beam positions in the FoV and avoid positioning errors caused by angle deviations of the scanner mirror. This increases the reliability of the angular position accuracy of the scanner module significantly. To minimize the shadings between adjacent scan spots in the target distance, created by beam cutoffs at the aperture stop of the objective lens, an array of optimized microwedge prisms is provided in combination with the microlens array. Therefore, we can increase the throughput of transmission power into the FoV and improve the measurement distance, especially at large scan angles.

## 1.

## Introduction

Typically, there are two system principles of time-of-flight distance measurement, namely, scanner systems and three-dimensional camera/flash light detecting and ranging (LiDAR) systems. While scanner systems illuminate and collect each object point in the field of view (FoV) sequentially,^{1}^{,}^{2} flash LiDAR systems measure the entire FoV simultaneously in one snapshot.^{3}^{,}^{4} Hence, high frame rates are realized without smearing effects of moving objects. Owing to a solid-state setup, each measurement point relates to a fixed field angle and can be detected by one certain receiver element. This allows a precise spatial assignment of object points in the FoV, which leads to a high absolute angular accuracy of the system. The maximum measurable range of flash LiDAR systems is smaller with respect to scanner modules if the FoV size and transmission power per pulse are identical. The advantage of scanner modules regarding measurement ranges results from a higher level of receiving signal due to the concentration of the laser power to one measurement point, while flash LiDAR requires simultaneous illumination of the entire FoV. In terms of realizing the specifications of obstacle recognition and FoV surveillance in automotive applications, typically, scanner systems are the preferred technology in the market.^{5}^{,}^{6} Concerning transmission power consumption, the trade-off between meaningful measurement distances ($<100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}$)^{6}^{,}^{7} and horizontal FoV dimensions ($>\pm 20\text{\hspace{0.17em}}\mathrm{deg}$)^{8} is obligatory for flash LiDAR systems. The power consumption exceeds by factors compared to that of scanner systems, which is why the common operating range of flash systems is around 0.1 to 25 m.^{3}^{,}^{7}

The state-of-the-art laser scanner systems in automotive applications are based on transmission signal deflection modules with large, heavy and expensive oscillating or rotating mirrors. For transmitting eye-safe laser beams and collecting maximum amounts of scattered and reflected signal of measured targets in the FoV, large sending and receiving apertures, which means large scanning mirror surfaces, are essential. The reachable distance up to 200 m and the corresponding distance uncertainty is typically in the range of 2 mm to 10 cm, depending on the environmental conditions and ambient illumination. The range of angular resolution is around 0.01 deg to 0.5 deg with an angular uncertainty up to 0.015 deg.^{5}^{–}^{9} However, the large construction size, the fine mechanics setup, as well as low potential of automated high volume manufacturing seem to be the major drawbacks of this kind of scanner design.

Replacing the fine-mechanical deflection mirror by a microelectromechanical system (MEMS) scanning mirror overcomes the large construction size. However, a smaller scanning mirror size reduces the sensor aperture and, therefore, the measurement distance. Consequently, a mirror array or a very large single MEMS mirror setup is in the range of typically 1- to 11-mm diameter.^{9}^{–}^{12} On the one hand, the cost factor rises directly with the active mirror area.^{13} On the other hand, the requirement of a large MEMS mirror surface is a major drawback for fast and reproducible scanning. Tolerances of angular positioning, position sensing, as well as vibrational perturbations, e.g., externally induced shock lead to mirror-deflection noise, which is projected directly into the FoV. Thus, the resulting position noise of measurement spots in the FoV is the main reason for the limitation of the accuracy of spot positions.

Add-on closed-loop feedback circuits realize the measurement of the actual mirror position. The position can be readout either optically on the basis of the position of reflected signal on a monitor diode at the backside of the mirror^{14} or piezoelectrically through the measurement of torsion at the mirror springs while oscillating.^{15} However, with this technique several vibration modes can cause ambiguous position feedback. A further possibility is to measure the capacity variation depending on the overlapping depth of finger structures at the mirror.^{16}^{,}^{17} The reachable angular uncertainty in closed-loop circuit modules is located in the range around 0.015 deg to 0.1 deg.

By increasing the mirror surface, the inherent dynamic deformation ${\delta}_{\mu \mathrm{M}}$ increases with the mirror diameter ${d}_{\mu \mathrm{M}}$ (${\delta}_{\mu \mathrm{M}}\propto {d}_{\mu \mathrm{M}}^{5}$) while oscillating, especially in the turning points. In addition, the static deformation induced by tensions between particular mirror layers becomes more relevant.^{18}^{,}^{19} To scale-up the mirror thickness ${t}_{\mu \mathrm{M}}$ (${\delta}_{\mu \mathrm{M}}\propto 1/{t}_{\mu \mathrm{M}}^{2}$) is one possibility of reducing the deformations ${\delta}_{\mu \mathrm{M}}$. This again leads to higher mass of the mirror ${m}_{\mu \mathrm{M}}$ and, respectively, higher inertia ${I}_{\mu \mathrm{M}}$ (${I}_{\mu \mathrm{M}}\propto {d}_{\mu \mathrm{M}}^{2}$, assumed a round mirror surface). Higher inertia and higher rotating angles ${\phi}_{\mu \mathrm{M}}$ correspond to increasing torsional stress ${\kappa}_{\mathrm{S}}$ in the mirror springs (${\kappa}_{\mathrm{S}}\propto {I}_{\mu \mathrm{M}}\propto {\phi}_{\mu \mathrm{M}}$). For fast retraction, a higher stiffness of springs is advantageous, which increases the stress level and the postoscillation. Complementarily, this reduces the shock and vibration resistivity.^{19}^{–}^{21} In conclusion, regarding lifetime and robustness aspects, the target for the operation of a scanning mirror module is minimum dimensions ${d}_{\mu \mathrm{M}}$ and low deflection angles ${\phi}_{\mu \mathrm{M}}$.

As presented in Refs. 22 and 23, scanner systems, including micro-optics, are an alternative approach for maximization of the sending and receiving apertures without increasing the mirror surface. Here, the assembly of one micro-optical element with an objective lens generates an optical telescope for each separate scan position exclusively.^{24}^{–}^{26} The maximum transmission beam diameter and the maximum receiving aperture of these systems accordingly depend on the combination of microlens and objective lens. Transmission beam cutoffs at the aperture stop of the objective lens turn out to be an inherent system drawback while scanning over a microlens array. The consequence is a significant loss of measurement distance and an intermittently sampled FoV at large field angles.

In this paper, we present the approach for a scanner module based on prior work.^{22}^{,}^{23} This scanner module improves the position accuracy of measurement spots in the FoV without any shadings in between. Among others, we show in detailed simulation and measurement the influence of spot position noise and beam cutoffs on the transmission power. Usually, mirror-deflection noise is the dominant limitation for the position accuracy of the measurement spots. We design the dimensions of aperture stops between the microlenses and the transmission beam diameter on the microlens array carefully in order to suppress the direct projection of the angular noise of the rotating mirror into the FoV. Therefore, our new design approach leads to a system with inherent high measurement spot-position accuracy. In addition, the design of a specific microwedge prism for each particular microlens avoids previously described transmission beam cutoffs at the aperture stop of the objective lens. This avoids resulting shadings between adjacent scan spots in the FoV and increases the throughput of transmission power into the FoV. The consequence is the improvement of measurement distance, especially at large scan angles.

## 2.

## Laser Scanner Module

The geometrical correlations of the actual scanning principle are described in Sec. 2. We also show the system limitations regarding reachable transmission beam diameters and the correlated maximum allowed transmission power. This consideration is in conformity to the IEC 60825-1:2014 standard.

## 2.1.

### Principle of the Laser Scanner Module

In this paper, we present a laser scanner system (Fig. 1), which realizes large transmission beam diameters with minimum mirror dimensions and rotation angles. We derive an equation for the two-dimensional (2-D) rotating angles ${\phi}_{\mu \mathrm{M}}$ of the oscillating mirror $\mu \mathrm{M}$ to hit the center position of each microlens on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ with the deflected transmission beam optimally [see Fig. 2(a)]. The necessary angle depends on the incoming transmitter vector ${\overrightarrow{V}}_{{\mathrm{T}}_{\text{in}}}$ on mirror $\mu \mathrm{M}$, the inclination angle ${\beta}_{\mathrm{MT}}$ of mirror MT, and the center positions of the microlenses on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$. For better clarity, Figs. 1(a) and 2 just show the $yz$-plane. The rotation angle of the mirror $\mu \mathrm{M}$ around the $y$-axis with ${\phi}_{\mu \mathrm{M},y}$ in the $xy$-plane is derived analog to the rotation angle around the $x$-axis with ${\phi}_{\mu \mathrm{M},x}$ in the $yz$-plane. We also show the derivation of the divergence angle ${\theta}_{T}$ in dependence on the focal length ${f}_{\mu \mathrm{L}}$ and the diameter ${d}_{\mu \mathrm{L}}$ of the microlens. It also depends on the distance ${a}_{\mu \mathrm{L}}$ of the microlens to the objective lens (Lens LO) and its focal length ${f}_{\mathrm{LO}}$.

The scanner module provides the classification of laser class 1 in conformity with the IEC 60825-1:2014 standard. Our system incorporates an optimized micro-optical array for noise suppression of the transmitted spot position in the FoV. The optics solution additionally avoids shadings in the FoV for high light-collecting efficiency even at large field angles. These characteristics allow high robustness against vibrational perturbations and front lens contaminations, as well as constant angular resolution within maximum measurement range. The separation of optical paths in the setup gives a high stray light insensitivity of transmission signal reflections at components in the transmitter path. In addition, the setup allows separate optical design of the microlens arrays for particular transmission and receiving requirements.

Figure 1(a) shows the optical operating principle of the scanner module and, respectively, the equivalent demonstrator setup [see Fig. 1(b)]. Outgoing from the laser source (transmitter), the transmission beam (${T}_{0}$ and ${T}_{n}$ for one exemplarily angular mirror position each) is deflected via a MEMS mirror $\mu \mathrm{M}$ to the lens LT, which converts the rotational movement of mirror $\mu \mathrm{M}$ to a parallel shift to the optical axis. This again realizes a sampling of each microlens perpendicular to its principal plane on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$. Depending on the focal length ${f}_{\mathrm{LT}}$ of lens LT, the necessary rotation angle of mirror $\mu \mathrm{M}$ correlates directly to the resulting parallel shift of ${T}_{0}$ to ${T}_{n}$. The beam-shaping section, consisting of one microlens and the objective lens LO, generates an optical telescope assembly for each scan position and, respectively, spot position in the FoV. Therefore, the amount of microlenses on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ is equal to the amount of spot positions in the FoV, assuming no subsampling on the detector. Concerning eye-safety requirements, each microlens widens the transmission beam to the size of a less critical diameter ${d}_{{\mathrm{T}}_{\mathrm{LO}}}$ at the system exit plane of the objective lens LO [see Fig. 2(b)]. Despite a beam widening, the advantage compared to a conventional telescope setup is no loss of field angle in the FoV at all. After scattering at the target, the receiving signal (${R}_{0}$ and ${R}_{n}$) reaches a separate receiver path via the objective lens LO and the beam splitter. Each angular mirror position of the mirror $\mu \mathrm{M}$ corresponds to one particular incidence angle of the receiving signal (see ${R}_{0}$ and ${R}_{n}$). Accordingly, the deflection of the receiving signal can be compensated to one static position on the detector with the advantage of minimum necessary detector dimensions. Such a system setup is favorable for the design of a microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ in the transmission path with additional aperture stops between each microlens. This design suppresses spot position noise in the FoV caused by the limited angular accuracy of the mirror $\mu \mathrm{M}$. However, the microlens array ${\mathrm{MLA}}_{\mathrm{R}}$ in the receiving path can be designed with an optimized diameter of each microlens without additional aperture stops for maximum light-collecting efficiency.

The relation of the necessary 2-D rotation angles ${\phi}_{\mu \mathrm{M},x}$ and ${\phi}_{\mu \mathrm{M},y}$ for deflecting the transmission beam depending on the corresponding center position of each sampled microlens is illustrated in Fig. 2(a). For the following calculation, the rotation point of the rotating mirror $\mu \mathrm{M}$ is defined as the point of origin of the coordinate system (marked with the red point).

The vector ${\overrightarrow{V}}_{{\mathrm{T}}_{\mathrm{Res}},n}$ describes the position vector from the rotating point of mirror $\mu \mathrm{M}$ to the center position of each single microlens. In consequence, this vector is defined with the given coordinates of the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ (${x}_{{\mathrm{MLA}}_{\mathrm{T}},0}$, ${y}_{{\mathrm{MLA}}_{\mathrm{T}},0}$, and ${z}_{{\mathrm{MLA}}_{\mathrm{T}}}$) as well as the pitch between each microlens ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}$ and the number ${n}_{\mu \mathrm{L}}$ of each particular microlens in $x$- and $y$-directions

## Eq. (1)

$${\overrightarrow{V}}_{{\mathrm{T}}_{\mathrm{Res}},n}=\left(\begin{array}{c}{n}_{\mu \mathrm{L},x}\xb7{\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}},x}+{x}_{{\mathrm{MLA}}_{\mathrm{T}},0}\\ {n}_{\mu \mathrm{L},y}\xb7{\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}},y}+{y}_{{\mathrm{MLA}}_{\mathrm{T}},0}\\ {z}_{{\mathrm{MLA}}_{\mathrm{T}}}\end{array}\right).$$The vector ${\overrightarrow{V}}_{\mathrm{T},n}$ with ${\overrightarrow{V}}_{\mathrm{T},n}={\overrightarrow{V}}_{{\mathrm{T}}_{\mathrm{Res}},n}-{\overrightarrow{V}}_{{\mathrm{T}}_{\mathrm{LM}},n}-{\overrightarrow{V}}_{{\mathrm{T}}_{\mu \mathrm{L}},n}$ describes the transmission beam from the rotating point of mirror $\mu \mathrm{M}$ to lens LT to hit finally the center position of each microlens. The vector ${\overrightarrow{V}}_{{\mathrm{T}}_{\mathrm{LM}},n}$ describes the transmission beam after refraction on lens LT and ${\overrightarrow{V}}_{{\mathrm{T}}_{\mu \mathrm{L}},n}$ is the vector after deflection at the mirror MT. The inclination angle ${\beta}_{\mathrm{MT}}$ of mirror MT, the angle of lens LT ${\phi}_{T0,x}=2\xb7{\beta}_{\mathrm{MT}}$, and its focal length ${f}_{\mathrm{LT}}$ are given by the system design as well. Hence, ${\overrightarrow{V}}_{\mathrm{T},n}$ can be expressed in dependence on these known system design parameters as

## Eq. (2)

$${\overrightarrow{V}}_{\mathrm{T},n}=\left[\begin{array}{c}{x}_{\mu \mathrm{L},n}\\ \mathrm{sin}({\phi}_{\mathrm{T}0,x})\xb7({f}_{\mathrm{LT}}-\begin{array}{c}{\mathit{C}}_{1}\end{array})\xb7\mathrm{sin}\left(\frac{{\phi}_{\mathrm{T}0,x}}{2}\right)\xb7\frac{1}{\mathrm{tan}({\phi}_{\mathrm{T}0,x})}\\ \mathrm{cos}({\phi}_{\mathrm{T}0,x})\xb7({f}_{\mathrm{LT}}+{\mathit{C}}_{1})\xb7\mathrm{sin}\left(\frac{{\phi}_{\mathrm{T}0,x}}{2}\right)\xb7\mathrm{tan}({\phi}_{\mathrm{T}0,x})\end{array}\right],$$Now, we can determine the necessary rotation angle ${\phi}_{\mu \mathrm{M}}$ of mirror $\mu \mathrm{M}$ to deflect the known incoming transmitter vector ${\overrightarrow{V}}_{{\mathrm{T}}_{\mathrm{in}}}$ to the vector ${\overrightarrow{V}}_{\mathrm{T},n}$. In a first step we have to calculate the normalized normal vector to the mirror surface $\mu \mathrm{M}$ ${\overrightarrow{V}}_{\mathrm{u},\mathrm{N}}=-{\overrightarrow{V}}_{\mathrm{u},{\mathrm{T}}_{\mathrm{in}}}+{\overrightarrow{V}}_{\mathrm{u},\mathrm{T},n}$, where ${\overrightarrow{V}}_{\mathrm{u},{\mathrm{T}}_{\mathrm{in}}}$ is the given normalized vector of the laser source and ${\overrightarrow{V}}_{\mathrm{u},\mathrm{T},n}$ is the known normalized vector of the transmission beam after deflection on mirror $\mu \mathrm{M}$. Then, we calculate ${\overrightarrow{V}}_{\mathrm{Rot}}$ with the inverse direction cosine matrix ${\overrightarrow{V}}_{\mathrm{Rot}}=DC{M}_{y}^{-1}\xb7{\overrightarrow{V}}_{\mathrm{u},\mathrm{N}}$, which describes the rotated vector ${\overrightarrow{V}}_{\mathrm{u},\mathrm{N}}$ to the optical axis in $z$-direction. The 2-D rotation ${\overrightarrow{\phi}}_{\mu \mathrm{M}}$ to deflect the transmission beam to the center position of each microlens finally can be described as

## Eq. (3)

$${\overrightarrow{\phi}}_{\mu \mathrm{M}}=\text{\hspace{0.17em}\hspace{0.17em}}\left(\begin{array}{c}{\phi}_{\mu \mathrm{M},x}\\ {\phi}_{\mu \mathrm{M},y}\end{array}\right)=\left[\begin{array}{c}{\mathrm{tan}}^{-1}\left(\frac{{V}_{\mathrm{Rot},y}}{{V}_{Rot,z}}\right)\\ {\mathrm{tan}}^{-1}\left(\frac{{V}_{\mathrm{u},\mathrm{N},x}}{{V}_{\mathrm{u},\mathrm{N},z}}\right)\end{array}\right].$$In Fig. 2(b), the beam-shaping section of the scanner module is depicted in simplified form with just one microlens. It shows the correlation of the distance ${a}_{\mu \mathrm{L}}$ in $z$-direction between one microlens and the objective lens LO. The resulting divergence angle ${\theta}_{\mathrm{T}}$, with help of the paraxial approximation, is expressed as

## Eq. (4)

$${\theta}_{\mathrm{T}}={\mathrm{tan}}^{-1}\left[\frac{\frac{{d}_{\mu \mathrm{L}}}{2\xb7{f}_{\mu \mathrm{L}}}-\frac{{a}_{\mu \mathrm{L}}\xb7{d}_{\mu \mathrm{L}}}{2\xb7{f}_{\mathrm{LO}}\xb7{f}_{\mu \mathrm{L}}}}{1+{\left(\frac{{d}_{\mu \mathrm{L}}}{2\xb7{f}_{\mu \mathrm{L}}}\right)}^{2}\xb7\frac{{a}_{\mu \mathrm{L}}}{{f}_{\mathrm{LO}}}}\right],$$## 2.2.

### Limitations of the Transmission Beam Diameter and Transmission Power

The geometrical correlation between reachable FoV dimensions and angular resolution ${\alpha}_{\mathrm{Tr}}$, coupled with large transmission beam diameter ${d}_{{\mathrm{T}}_{\mathrm{LO}}}$, is one fundamental limitation of scanner systems including micro-optics for transmission beam widening. The theoretically feasible transmission beam diameter on the principal plane of the objective lens LO is limited by its diameter ${d}_{\mathrm{LO}}$. We assume planoconvex spherical lenses with a refraction index ${n}_{{\mathrm{MLA}}_{\mathrm{T}}}$ of 1.5 and a maximum radius of curvature ${R}_{\mu \mathrm{L}}$ of half the diameter ${d}_{\mu \mathrm{L}}$ of one microlens on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$. In consequence the lower limitation of focal length of one microlens is ${f}_{\mu \mathrm{L},\mathrm{min}}=2\xb7{R}_{\mu \mathrm{L}}$, which results in a $f$-number $F{\#}_{\mu \mathrm{L}}={f}_{\mu \mathrm{L}}/{d}_{\mu \mathrm{L}}\ge 1$.^{27} Here, a microlens with $F{\#}_{\mu \mathrm{L}}=1$ defines the minimum possible focal length ${f}_{\mu \mathrm{L},\mathrm{min}}$ and, thus, a maximum beam widening of the transmission beam. In this scanner system, the number of scan spots in the FoV is equal to the number of microlenses on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$, with no subsampling on the detector assumed. Consequently, the pitch of the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ corresponds to the projected ${\text{Pitch}}_{\mathrm{Tr}}$ between the measurement spots in the FoV as well. This again is related to the system angular resolution ${\alpha}_{\mathrm{Tr}}={\mathrm{tan}}^{-1}({d}_{\mathrm{Tr}}/{z}_{\mathrm{Tr}})$, with the transmission spot diameter on the target ${d}_{\mathrm{Tr}}$ at the target distance ${z}_{\mathrm{Tr}}$ [see Fig. 2(b)]. For the following calculations, the ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}$ of the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ equates to the diameter of one microlens ${d}_{\mu \mathrm{L}}$. Outgoing from a chosen focal length ${f}_{\mathrm{LO}}$ of the objective lens LO and the resulting horizontal FoV ${\mathrm{FoV}}_{h}$, the system-required pitch of the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ is given as

## Eq. (5)

$${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}=\frac{{d}_{\mathrm{LO}}\xb7\mathrm{tan}({\alpha}_{\mathrm{Tr}})}{2\xb7\mathrm{tan}\left(\frac{{\mathrm{FoV}}_{\mathit{h}}}{2}\right)}.$$This is illustrated in Fig. 3(a) with the example of one standard objective lens ${\mathrm{LO}}_{1}$ with a ${\mathrm{FoV}}_{h}$ of 47 deg and two wide-angle objective lenses ${\mathrm{LO}}_{2}$ and ${\mathrm{LO}}_{3}$ with 57 deg and 84 deg, respectively. The necessary focal length ${f}_{\mu \mathrm{L}}$ of the microlens with

## Eq. (6)

$${f}_{\mu \mathrm{L}}=\frac{{d}_{\mathrm{LO}}^{2}\xb7\mathrm{tan}({\alpha}_{\mathrm{Tr}})}{4\xb7\mathrm{tan}{\left(\frac{\mathrm{F}\mathrm{o}{\mathrm{V}}_{\mathit{h}}}{2}\right)}^{2}\xb7{d}_{{\mathrm{T}}_{\mathrm{LO}}}},$$Theoretically, our scanner concept realizes transmission beam diameters up to 20 mm over the entire FoV, limited by the clear aperture at the system exit plane on the objective lens LO. Because of the micro-optical-based beam widening, the transmission beam can be deflected with a dimensionally small oscillating mirror. In conformity with the IEC 60825-1:2014 standard, our scanner system allows, with an assumed spot diameter of 20 mm, maximum permissible exposures (MPEs) of $<4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kW}$ peak power within 2.5-ns full width at half maximum (FWHM) at a wavelength of 900 nm. A high level of transmission power is indispensable for long measurement distances, considering a minimum measurable value of the receiving signal ${P}_{\mathrm{R}}$ in a range of ${10}^{-9}$ to ${10}^{-12}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{W}$ at the detector. This is based on a huge loss of backscattered signal power even at ideal 100% scattering targets in the FoV with respect to the transmission signal ${P}_{\mathrm{T}}$ (e.g., ${P}_{\mathrm{R}}\approx {P}_{\mathrm{T}}\xb7{10}^{-8}$ at ${z}_{\mathrm{Tr}}=100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}$). We can express the behavior of the receiving signal depending on the ratio of etendues^{28} ${G}_{\mathrm{LO}}$ to ${G}_{\mathrm{Tr}}$ of the receiving aperture and that of the backscattered signal at the target $({P}_{\mathrm{R}}\propto \frac{{G}_{\mathrm{LO}}}{{G}_{\mathrm{Tr}}})$. The MPE power^{29}^{,}^{30} ${P}_{\mathrm{T},\mathrm{max}}$ scales with the beam diameter ${d}_{{\mathrm{T}}_{\mathrm{LO}}}$ with ${P}_{\mathrm{T},\mathrm{max}}\propto {d}_{{\mathrm{T}}_{\mathrm{LO}}}^{2}$.

For our experimental setup we choose a light-sensitive objective lens LO with a $f$-number of 1.2 and a ${\mathrm{FoV}}_{h}$ of $\pm 23.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ (necessary ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}=124.8\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$) (see Table 1). The microlenses used for the transmission beam widening allow a transmission beam diameter ${d}_{{\mathrm{T}}_{\mathrm{LO}}}$ of 10.41 mm and consequently a possible transmission pulse-power maximum ${P}_{\mathrm{T},\mathrm{max}}$ of up to 1 kW, consistent with the IEC 60825-1:2014 standard.

## Table 1

System specifications of the experimental setup of the laser scanner module.

Symbol | Value | Description |
---|---|---|

${\mathrm{FoV}}_{h}$ | $\pm 23.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ | Horizontal FoV |

${z}_{\mathrm{Tr}}$ | 0.9 m | Target/object distance |

${d}_{{\mathrm{T}}_{\mathrm{LO}}}$ | 10.41 mm | Transmission beam diameter at objective lens LO |

${P}_{\mathrm{T}}$ | $<1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kW}$ | Transmission pulse power at 900-nm wavelength |

${t}_{\text{Pulse}}$ | 2.5 ns | Transmission pulse length |

## 3.

## Suppression of Spot Position Noise

In Sec. 3, we show the geometrical correlation between the mirror-deflection noise of the oscillating mirror and the position accuracy of measurement spots in the FoV. We describe how to suppress this position noise in the FoV with the approach of a careful design of the aperture stop dimensions between the microlenses and the transmission beam diameter on the microlens array. Outgoing from the mirror-deflection noise, we show the effect on varying transmission power and spot homogeneity. Further, we give an estimation of the transmission power consumption for a position-noise-suppressed system.

## 3.1.

### Correlation between Mirror-Deflection Angle and Spot Position in the Field of View

One deflection angle of the oscillating mirror $\mu \mathrm{M}$ correlates with one particular position of the transmission beam on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ in the $xy$-plane (see Fig. 4). Outgoing from this position, the transmission beam is projected through the objective lens LO into the FoV and directly defines the field angle ${w}_{\mathrm{FoV}}$ and spot position on the target. Consequently, angular mirror-deflection noise of the oscillating mirror leads directly to spot position noise on the microlens array and finally to spot position noise in the FoV. Next to externally induced perturbations, mirror-deflection noise is the dominant limitation for the position accuracy of measurement spots.

On dependence of an arbitrary rotation angle ${\phi}_{\mu \mathrm{M}}$, we can express the functional correlation to the corresponding spot positions in the $xy$-plane on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ and in the FoV. Through the DCM rotation convention with ${\overrightarrow{V}}_{\mathrm{u},\mathrm{N}}={\mathrm{DCM}}_{y}[{\mathrm{DCM}}_{x}\xb7{\overrightarrow{V}}_{\mathrm{u},z}]$, in a first step, we calculate the normalized normal vector of the mirror surface ${\overrightarrow{V}}_{\mathrm{u},\mathrm{N}}$, where ${\overrightarrow{V}}_{\mathrm{u},z}$ is the unit vector in $z$-direction. With the normalized transmission beam vector

## Eq. (7)

$${\overrightarrow{V}}_{\mathrm{u},\mathrm{T}}=-{\overrightarrow{V}}_{\mathrm{u},{\mathrm{T}}_{\mathrm{in}}}+2\xb7({\overrightarrow{V}}_{\mathrm{u},\mathrm{N}}\xb7{\overrightarrow{V}}_{\mathrm{u},{\mathrm{T}}_{\mathrm{in}}})\xb7{\overrightarrow{V}}_{\mathrm{u},\mathrm{N}},$$## Eq. (8)

$${\overrightarrow{V}}_{\mathrm{T}}={\overrightarrow{V}}_{\mathrm{u},\mathrm{T}}\xb7\frac{{f}_{\mathrm{LT}}}{\mathrm{cos}({\phi}_{\mathrm{T},\mathrm{T}0})}.$$## Eq. (9)

$${\overrightarrow{h}}_{\mathrm{T}}=\text{\hspace{0.17em}\hspace{0.17em}}\left[\begin{array}{c}{V}_{\mathrm{T},x}\\ -\left(\frac{{V}_{\mathrm{T}0,y}-{V}_{\mathrm{T},y}}{\mathrm{cos}({\phi}_{\mathrm{T}0,x})}\right)\end{array}\right]$$## Eq. (10)

$${\overrightarrow{h}}_{\mathrm{FoV}}=-({z}_{\mathrm{Tr}}-{f}_{\mathrm{LO}})\xb7\left(\begin{array}{c}\frac{{h}_{\mathrm{T},x}}{{f}_{\mathrm{LO}}}\\ \frac{{h}_{\mathrm{T},y}}{{f}_{\mathrm{LO}}}\end{array}\right).$$The angle between the lens LT and the optical axis of the system is given as ${\phi}_{\mathrm{T}0,x}$. The variable ${z}_{\mathrm{Tr}}$ represents the target distance and ${f}_{\mathrm{LO}}$ is the focal length of the objective lens LO. The side inversion because of the reflection on mirror MT and the projection through the lens LO are considered with a negative sign each.

With Eq. (9) we can derive the propagation of the angular error $D{\phi}_{\mu \mathrm{M}}$ of the oscillating mirror $\mu \mathrm{M}$, which is based on the mirror-deflection noise, to the resulting position error $D{h}_{\mathrm{T}}$ on the microlens array in the $xy$-plane, as

## Eq. (11)

$$D{h}_{\mathrm{T},y}=\left|\frac{\partial {h}_{\mathrm{T},\mathrm{y}}({\phi}_{\mu \mathrm{M},x},{\phi}_{\mu \mathrm{M},y})}{\partial {\phi}_{\mu \mathrm{M},x}}\right|\xb7D{\phi}_{\mu \mathrm{M},x}+\left|\frac{\partial {h}_{\mathrm{T},\mathrm{y}}({\phi}_{\mu \mathrm{M},x},{\phi}_{\mu \mathrm{M},y})}{\partial {\phi}_{\mu \mathrm{M},y}}\right|\xb7D{\phi}_{\mu \mathrm{M},y}.$$Here, exemplarily the $y$-component is shown (see Fig. 4). We assume an equal probability density distribution of this mirror-deflection noise for calculating the angular uncertainty of the mirror $\mu \mathrm{M}$ with ${u}_{{\phi}_{\mu \mathrm{M}}}=D{\phi}_{\mu \mathrm{M},x}/\sqrt{3}$. Accordingly, the combined uncertainty of the transmission beam position on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ for two uncorrelated input parameters is expressed as

## Eq. (12)

$${u}_{{\mathrm{h}}_{\mathrm{T},y}}=\sqrt{{({\mathit{C}}_{2})}^{2}\xb7{u}_{{\phi}_{\mu \mathrm{M},x}}^{2}+{({\mathit{C}}_{3})}^{2}\xb7{u}_{{\phi}_{\mu \mathrm{M},y}}^{2}},$$Based on Eq. (10), the position error in the FoV $D{h}_{\text{FoV},y}$ and the spot position uncertainty in the FoV ${u}_{{\mathrm{h}}_{\mathrm{FoV},y}}$ are derived analog to this.

## 3.2.

### Approach for Suppression of Spot Position Noise with Aperture Stops

In a scanner system with a microlens array in the beam-shaping section with high optical fill factor as shown in Figs. 5(a) and 6(a), the spot position noise on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ is directly projected into the FoV. Here, a position error of the transmission beam ${T}_{0,D{h}_{\mathrm{T}}}$ from the reference position ${T}_{0}$ on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$, leads to a position error $D{h}_{\mathrm{FoV}}$ in the FoV. In our approach, we suppress this direct projection of spot position noise on the microlens array into the FoV with aperture stops between the microlenses and a reduction of the optical fill factor [see Fig. 5(b)].

The diameter of these aperture stops equates to the diameter of the microlenses ${d}_{\mu \mathrm{L}}$. We design the center-to-center distance of adjacent microlenses ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}}$ to twice the value of the maximum position error $D{h}_{\mathrm{T},y}$ of the transmission beam on the microlens array and coat it with light impermeable material

## Eq. (13)

$${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}}={d}_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}=2\xb7(\frac{{d}_{\mu \mathrm{L}}}{2}+D{h}_{\mathrm{T},y}).$$Considering this range of the spot position noise, we have to increase the transmission spot diameter ${d}_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}$ on the microlens array to the same extent, to ensure fully illuminated microlenses at any time.

Hence, the range of suppression of spot position noise depends on the aperture stop dimension, the pitch between adjacent microlenses and the transmission spot diameter pointing on the microlens array ${\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}$. Our microlens design offers a pitch of twice the diameter of the microlens ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}}=2\xb7{d}_{\mu \mathrm{L}}$ [see Fig. 6(b)].

With such a microlens array and a focal length ${f}_{\mathrm{LT}}=75\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ of lens LT [see Figs. 1(a) and 2(a)], we achieve to suppress a spot position noise in the FoV with a position error of $D{h}_{\mathrm{FoV}}\approx 0.25\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}$ ($D{h}_{\mathrm{t}}\approx 62.5\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ on ${\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}$). This means a suppression of mirror-deflection noise with an angular uncertainty of ${u}_{{\phi}_{\mu \mathrm{M},x}}=0.013\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$. For small deflection angles, the spot position uncertainty can be assumed as constant. In a system without a position-noise-suppressing microlens array and with an angular resolution ${\alpha}_{\mathrm{Tr}}=0.286\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ and a ${\mathrm{FoV}}_{h}=\pm 23.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ (see Sec. 2.2), this spot position error $D{h}_{\mathrm{FoV}}$ equates to half the target diameter ${d}_{\mathrm{Tr}}$, which is to be resolved. This means a spatial degradation of angular resolution up to ${\alpha}_{{\mathrm{Tr}}_{\mathrm{act}}}=\xbd\xb7{\alpha}_{\mathrm{Tr}}$. In worst case, targets within the diameter ${d}_{\mathrm{Tr}}$ can be missed, because of emerging gaps between sequential scan spots.

## 3.3.

### Effects of Spot Position Noise on Transmission Power and Spot Homogeneity

In addition to spot position noise in the FoV, a reduction of the actual transmission power and spot homogeneity is the consequence, if illuminating a microlens not centrically. This reduction depends on the dimension of spot position error of the transmission beam position on the microlens array to the center position of the microlens in the $xy$-plane. Variations of transmission power and spot homogeneity directly affect the maximum reachable measurement distance and eye safety.

To evaluate the effect of spot position noise on the transmission power, with a static setup we consciously displace the transmission spot on the microlens array to the center position of the microlens in the $y$-direction. In Figs. 7(a) and 7(b), we experimentally can show this reduction of transmission power. Outgoing from the reference spot ${T}_{0}$ with ${h}_{\mathrm{T},y}=0$, we displace the spot up to ${\mathrm{T}}_{0,{h}_{\mathrm{T}}}$ with ${h}_{\mathrm{T},y}=\xbd\xb7{d}_{\mu \mathrm{L}}$ of the microlens diameter. For a microlens array without a position-noise-suppressing design [see Figs. 6(a) and 6(c)], eccentric illumination of the microlens means partial illumination of translucent transition zones between the microlenses on the array ${\mathrm{MLA}}_{\mathrm{T}}$. We can measure a reduction of transmission power ${P}_{\mathrm{T}}$ of about 25% from centric to eccentric illumination of the microlens.

To express the homogeneity ${H}_{\mathrm{I}}$ with respect to the spot intensity ${I}_{\text{Spot}}$, we use the mean absolute deviation of all intensity values along the spot cross section in the $y$-direction to the arithmetic mean value ${\overline{I}}_{\text{Spot}}$ of the intensity. We just consider values inside the microlens aperture. Related to the measurement, the parameter $i$ defines camera pixels in the range from ${i}_{\mathrm{min}}$ to ${i}_{\mathrm{max}}$ with $\pm \xbd\xb7{d}_{\mu \mathrm{L}}$ [see the red dashed lines in Figs. 7(c) and 7(d)]. With normalization of the mean deviation to ${\overline{I}}_{\text{Spot}}$, we can write the homogeneity as

## Eq. (14)

$${H}_{\mathrm{I}}=1-(\frac{1}{N}\sum _{i={i}_{\mathrm{min}}}^{{i}_{\mathrm{max}}}|{I}_{\text{Spot},i}-{\overline{I}}_{\text{Spot}}|)\xb7\frac{1}{{\overline{I}}_{\text{Spot}}},$$For a microlens array with a position-noise-suppressing design [see Fig. 6(b)], eccentric illumination of the microlens means a direct increase of the illuminated area on the light impermeable aperture stops. This results in a transmission beam that only partially transmits through the microlens array, which is shown simulatively in Fig. 8.

The consequence is a stronger reduction of transmission power ${P}_{\mathrm{T}}$ in the FoV compared to the microlens design shown in Fig. 6(a). The transmission power ${P}_{\mathrm{T}}$ in dependence on the displacement ${h}_{\mathrm{T},y}$ of the illuminated position to the center position of the microlens can be expressed as

## Eq. (15)

$${P}_{\mathrm{T}}=\iint {I}_{\text{Spot}}(x,y)\xb7{A}_{\mu \mathrm{L}}[x,(y-{h}_{\mathrm{T},y})]\mathrm{d}y\text{\hspace{0.17em}}\mathrm{d}x.$$## Eq. (16)

$${I}_{\text{Spot}}(x,y)={I}_{\text{Spot},0}\xb7\frac{1}{2\pi \xb7{\sigma}^{2}}\xb7{\mathrm{exp}}^{-\frac{({x}^{2}+{y}^{2})}{2\xb7{\sigma}^{2}}},$$^{31}Figures 9(a) and 9(b) show the spot intensity ${I}_{\text{Spot}}$ and the transmission power ${P}_{\mathrm{T}}$. This is illustrated for two ratios of beam diameter to microlens diameter, A: ${d}_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}={d}_{\mu \mathrm{L}}$ and B: ${d}_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}=2\xb7{d}_{\mu \mathrm{L}}$ with identical transmission power ${P}_{{\mathrm{T}}_{\mathrm{A}}}={P}_{{\mathrm{T}}_{\mathrm{B}}}$. On this example, we evaluate the behavior of the spot homogeneity and the transmission power depending on the displacement ${h}_{\mathrm{T},y}$. The spot diameter ${d}_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}$ is defined with $\pm 2\sigma $.

In Table 2, we show the summary of the simulative evaluation based on the graphs given in Fig. 9. With respect to the spot homogeneity ${H}_{\mathrm{I}}$ and the relative loss of transmission power ${P}_{{\mathrm{T}}_{\text{Loss}}}$, we can see the advantageous effect of a diameter ratio B. One negative effect of position-noise-suppressing design of the microlens array is a significant increase of the necessary transmission power compared to a microlens design shown in Fig. 6(a). Thus, the light loss at the aperture leads to a necessary increase of transmission power about the factor 2.19 for a design illustrated in Fig. 8(b).

## Table 2

Evaluation of the spot homogeneity HI and the loss of transmission power PTLoss. Here, these values are given in dependence on the transmission spot diameter dT,MLAT (A and B) on the microlens array MLAT,NS and its displacement hT,y regarding the microlens center position.

Transmission spot diameter on MLAT | HIT0 in % | HIT0,hT in % | PT0 in % | PT0,hT in % | rel. PTLoss(T0−T0,hT) in % |
---|---|---|---|---|---|

A: ${d}_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}={d}_{\mu \mathrm{L}}$ | 56.5 | 2.1 | 86.5 | 40.2 | 53.5 |

B: ${d}_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}=2\xb7{d}_{\mu \mathrm{L}}$ | 87.2 | 57.0 | 39.3 | 26.9 | 31.6 |

## 4.

## Compensation of Spot Shadings

In Sec. 4, we show the drawback of partial shading of measurement spots, while scanning over the FoV. We describe how to avoid this drawback by an optimized optical design of a wedge prism array in the transmission path of the scanner system. Further, we evaluate the effect of spot shadings on the transmission power and consequently on the reachable measurement distance.

## 4.1.

### Principle of Transmission Beam Cutoffs while Scanning over the Field of View

An inherent characteristic of scanner systems, including micro-optics for beam widening, is the emerging of transmission beam cutoffs at the aperture stop of the objective lens while scanning over the microlens array. The consequences are not only gaps between adjacent spots in the FoV. This also leads to a significant reduction of transmission power with coupled degradation of reachable measurement distance. Beam cutoffs at the aperture stop of the objective lens start at a system-specific incidence height of the transmission beam on the objective lens. This incidence height is transformed into a field angle, which we call the critical field angle ${w}_{{\mathrm{FoV}}_{\mathrm{crit}.}}$. Based on these cutoffs, shadings or gaps ${h}_{\text{Gap},\mathrm{Tr}}$ arise between adjacent measurement spots in the FoV and increase proportional to the tangent of the field angle ${w}_{\mathrm{FoV}}$ [${h}_{\mathrm{Gap},\mathrm{Tr}}\propto \mathrm{tan}({w}_{\mathrm{FoV}})$]. The consequence is possible nondetection of targets with higher spatial frequency and the reduction of measurement distance within higher field angles of the scanner system.

By realizing a maximum eye-safe transmission beam with a diameter equal to that of the objective lens LO, the cutoff already starts by the illumination of the first microlens aside the one to the optical axis. At the maximum field angle, the cutoff equates inherently always to half the spot diameter ${d}_{{\mathrm{T}}_{\mathrm{LO}}}$, which inevitably leads to a spot shading in the FoV of half the target diameter ${d}_{\mathrm{Tr}}$. In Fig. 10(a), this is illustrated with the help of two exemplary transmission beams ${T}_{0}$ and ${T}_{n}$.

Figure 10(b) shows the correlation between the increase of field angle ${w}_{\mathrm{FoV}}$ and the reduction of the residual spot height ${h}_{\text{Spot},\mathrm{Tr}}$ up to 0.25- at 100-m target distance. On the basis of an objective lens with a ${\mathrm{FoV}}_{h}=\pm 23.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ and a diameter ${d}_{\mathrm{LO}}=20.82\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, we explain this characteristics for two spot diameters ${d}_{{\mathrm{T}}_{\mathrm{LO}}}=10.41\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ and ${d}_{{\mathrm{T}}_{\mathrm{LO}}}={d}_{\mathrm{LO}}=20.82\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ in the plane of the objective lens. As we can see, in contrast to a fully illuminated objective lens, a spot diameter of 10.41 mm results in a critical field angle of 12.14 deg, which allows a gapless illuminated FoV of 56.26%. Hence, the percentage of the FoV, which is possible to be scanned without provoking shadings caused by these cutoffs, depends on the optical design of the beam-shaping section and the specified transmission spot diameter.

## 4.2.

### Approach for Compensation of Spot-Shadings with Wedge Prisms

To avoid this substantial drawback, our optics solution includes a microwedge prism array MWA in addition to the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$. The microwedge prism array consists of one with the microlens array combined module or of two separate optical components optionally. In Figs. 10(a) and 11(a), the combined and the separate versions are illustrated. We design each wedge prism for one corresponding microlens exclusively. The wedge angle ${\gamma}_{\mathrm{w},n}$ and the offset ${d}_{\mathrm{w},n}$ to the optical axis of the corresponding microlens depend on the radial distance ${h}_{\mu \mathrm{L}}$ to the optical axis of the objective lens LO [see Fig. 11(c)]. The microwedge prism array refracts the divergent light cone of the transmission signal after the microlens for each scan position by exactly the amount that it can pass the objective lens LO without any cutoff at the aperture stop. We show in Fig. 11(b) the geometrical fundamentals for the calculation of the necessary tilt angle ${\tau}_{\text{tilt},n}$ and the resulting wedge angle ${\gamma}_{\mathrm{w},n}$.

From Fig. 11(c), we can calculate the radial distance of the aperture of each microlens ${h}_{{\mathrm{r}}_{\mu \mathrm{L}},n}$ to the optical axis of the objective lens LO with ${y}_{\mu \mathrm{L},n}={n}_{\mu \mathrm{L},y}\xb7{\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}},y}$, ${x}_{\mu \mathrm{L},n}={n}_{\mu \mathrm{L},x}\xb7{\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}},\mathrm{x}}$, and ${h}_{\mu \mathrm{L},n}=\sqrt{{x}_{\mu \mathrm{L},n}^{2}+{y}_{\mu \mathrm{L},n}^{2}}$, which can be expressed as

Here, ${n}_{\mu \mathrm{L},x}$ and ${n}_{\mu \mathrm{L},y}$ is the number of the particular microlens (in $x$- and $y$-directions), starting the numeration with the element on the optical axis of the objective lens LO.The radial distance of the center of each particular microlens to this optical axis is given with ${h}_{\mu \mathrm{L},n}$ and the radius of the microlens with ${r}_{\mu \mathrm{L}}=\xbd\xb7{d}_{\mu \mathrm{L}}$. The necessary tilt angle ${\tau}_{\text{tilt},n}$ of the marginal ray of each microlens is expressed as

## Eq. (18)

$${\tau}_{\text{tilt},n}={\tau}_{\mu \mathrm{L}}-{\tau}_{\mathrm{lim},n}={\mathrm{tan}}^{-1}\left(\frac{\frac{{r}_{\mu \mathrm{L}}}{{f}_{\mu \mathrm{L}}}-\frac{{r}_{\mathrm{LO}}-{h}_{{\mathrm{r}}_{\mu \mathrm{L}},n}}{{a}_{\mu \mathrm{L}}+{f}_{\mu \mathrm{L}}}}{1+\frac{{r}_{\mu \mathrm{L}}}{{f}_{\mu \mathrm{L}}}\xb7\frac{{r}_{\mathrm{LO}}-{h}_{{\mathrm{r}}_{\mu \mathrm{L}},n}}{{a}_{\mu \mathrm{L}}+{f}_{\mu \mathrm{L}}}}\right),$$## Eq. (19)

$${\gamma}_{\mathrm{w},n}={\mathrm{sin}}^{-1}\sqrt{\left(\frac{(1+{n}_{\mathrm{w}}^{2})\xb7\mathrm{sin}({\vartheta}_{\mathrm{w},n})\pm 2\xb7\mathrm{sin}{({\vartheta}_{\mathrm{w},n})}^{2}\text{\hspace{0.17em}}{\mathit{C}}_{4}}{4\xb7\mathrm{sin}{({\vartheta}_{\mathrm{w},n})}^{2}+{(1+{n}_{\mathrm{w}}^{2})}^{2}}\right)},$$The refraction index of the wedge prism material is given with ${n}_{\mathrm{w}}$.

## 4.3.

### Effects of Beam Cutoffs on Transmission Power and Residual Spot Height

We experimentally investigate the optimized micro-optical component, to demonstrate the improvement of a ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ combination in a scanner system and to prove the theoretical calculations. The scanner setup provides a FoV of $\pm 23.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$, a beam diameter of ${d}_{{\mathrm{T}}_{\mathrm{LO}}}=10.41\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, and a measurement distance of 0.90 m. Our tests reveal a high consistency between theoretical simulation and measurement [see Fig. 12(a)]. We can show that a setup with a microwedge prism array in the optical path increases the transmission signal in the FoV by the factor of 2.47 compared to that without such a component. In consequence, this theoretically leads to a rise of measurement distance ($\sqrt{{P}_{\mathrm{T}}}\propto {z}_{\mathrm{Tr}}$)^{26} of about the factor of 1.57, assuming a fully illuminated target with a minimum diameter ${d}_{\mathrm{Tr}}$.

The drop of the curve with ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ combination is based on the incidence angle ${\vartheta}_{\mathrm{w},n}$ into the microlens, which increases the spot diameter with a cosine function in this plane. The result is less effective signal transmitting through the microlens. However, this can be avoided by rotating the microlens with the equal value to the value of ${\vartheta}_{\mathrm{w},n}$ or inverting the sequence of the components itself, which again leads to higher effort in appropriate lens design.

The gradient of the transmission power loss ${P}_{{\mathrm{T}}_{\text{Loss}}}$ in the FoV is given in Fig. 12(b). The curve shows the characteristic for an axial displacement in $y$-direction [similar to Fig. 10(a)] of a rectangular light cone to an objective lens with a round aperture stop. Because of the experimentally used rectangular microlens aperture, the critical field angle ${w}_{{\mathrm{FoV}}_{\mathrm{crit}.}}=8.67\text{\hspace{0.17em}}\mathrm{deg}$ (red dashed line) is lower than the calculated angle of ${\mathrm{LO}}_{1}$ with a round microlens aperture, which is shown in Fig. 10(b). In the experiment, we see that the reduction of transmission power ${P}_{\mathrm{T}}$ of the measured curve is higher compared to the simulated curve. The reason for this is the telecentric characteristics of the objective lens LO utilized in the experiments. In such a lens, the clear aperture decreases significantly with higher scan angles ${w}_{\mathrm{FoV}}$. This also causes the increasing deviation between simulation and measurement curves with increasing ${w}_{\mathrm{FoV}}$. We can avoid this issue with appropriate lens design of the objective lens LO as well.

In Fig. 13, we try to clarify the effects of transmission beam cutoff at the aperture stop of the objective lens LO on the residual spot height ${h}_{\text{Spot},\mathrm{Tr}}$, which is projected on the target at 0.45-m target distance. In the first row, we see the decrease of ${h}_{\text{Spot},\mathrm{Tr}}$ for defined field angles ${w}_{\mathrm{FoV}}=0.00\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$, 9.09 deg, 16.51 deg, and 22.16 deg without ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ combination. Starting at the critical field angle ${w}_{{\mathrm{FoV}}_{\mathrm{crit}}}$, the value of ${h}_{\text{Spot},\mathrm{Tr}}$ shows the behavior of ${h}_{\mathrm{Spot},\mathrm{Tr}}\propto -\mathrm{tan}({w}_{\mathrm{FoV}})$. In the second row with the ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ combination, the upper cutoff of the corners arises from rounding errors between the calculated radial distance to the optical axis and the utilized wedge prism with given wedge angle ${\gamma}_{\mathrm{w},n}$. The lower cutoff starting at 16.51 deg and increasing significantly at 22.16 deg is based on the already-mentioned telecentric characteristics of the utilized objective lens LO.

In Fig. 14, we show the measurement of the decreasing residual spot height, while scanning up to a field angle of 22.16 deg without a microwedge prism component. Here, the measurement result shows good consistency to simulated values. The reduction of the spot height starts at a smaller field angle with 10.73 deg compared to that of the simulation with 12.14 deg. This is based on a minimal smaller diameter of the aperture stop of the objective lens used in the experiments as assumed in the simulation.

## 5.

## Conclusion

For an eye-safe scanner system, we show principal limitations regarding the angular resolution and the maximum possible diameter of the transmission beam. The upper limit for the transmission beam diameter is about 20 mm with a diameter of microlenses of $99.6\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$. This is calculated for a micro-optics-based scanner system with an angular resolution of 0.286 deg. Such an angular resolution equates to the resolution of a target with the diameter 0.5-m at 100-m target distance, assuming an $f$-number of the microlens of $\ge 1$. The system allows MPEs of $\ge 4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kW}$ peak power within 2.5 ns FWHM at 900-nm wavelength. Here, a FoV of $\pm 28.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ is reachable with a diameter of the objective lens of 20.82 mm.

To suppress spot position noise in the FoV, we present an approach of an optimized microlens array with carefully designed aperture stops in the transmission path. We are able to suppress a mirror-deflection noise with an angular uncertainty of 0.013 deg. This equates to a suppression of a spot position error of about 0.25-m at 100-m target distance. With our approach, we additionally improve the transmission power loss by about 21.6% and the transmission spot homogeneity by about 54.9%, depending on the level of spot position error on the microlens array. The power consumption of a setup with position-noise-suppressing microlens design is about a factor 2.19 higher compared to a setup without this design.

The scanner system, presented in this paper, allows a small rotating mirror and provides a high spot position accuracy of measurement spots in the FoV without any shadings in between. With the combination of a microlens array and a microwedge prism array in the optical transmission path, we can show a nearly perfect compensation of transmission beam cutoffs at the aperture stop of the objective lens. In consequence, we avoid the resulting shadings between adjacent spots in the FoV up to half the specified target diameter, which is to be resolved. This realizes not only a gapless illuminated FoV but also a significant increase of transmission signal by a factor of 2.47. Theoretically, this leads to a rise of measurement distance of about a factor of 1.57, derived from the radiometric inverse square law.^{26}

Our experimental setup offers an objective lens with 20.82 mm diameter and a FoV of $\pm 23.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ at 0.90-m target distance. The FoV can be scanned with a deflection angle of the oscillating mirror of about $\pm 3.95\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ at 75-mm focal length of the lens LT. We utilize a microlens with an $f$-number of 2.4, which generates a spot diameter of 10.41 mm at the system exit plane on the objective lens. The MPE of this setup is about 1 kW transmission peak power.

## Acknowledgments

The presented work has been developed in collaboration with Robert Bosch GmbH and the Clausthal University of Technology. The authors sincerely acknowledge the support and want to thank all contributors and colleagues for their helpful inputs and discussions to realize this project.

## References

## Biography

**Jürgen Hasselbach** received his diploma degree in mechatronics and microsystems from Heilbronn University of Applied Sciences, Germany, in 2010. From 2010 to 2016, he worked at the Robert Bosch GmbH as development engineer in optical testing for video-based automotive driver assistance systems. Since 2016, he has been a PhD student at Clausthal University of Technology in cooperation with the Robert Bosch GmbH. The subject of his work is the development and investigation of a LiDAR scanner module with large sending/receiving apertures and high angular position accuracy.

**Siegwart Bogatscher** received his diploma in electrical engineering from the University of Karlsruhe (TH), Germany, in 2009. He was a PhD student at the Karlsruhe Institute of Technology (KIT) from 2009 to 2014 and received his doctorate degree in electrical engineering. In 2015, he joined Robert Bosch GmbH. As a system architect, he works in the field of LiDAR development for autonomous driving applications.

**Christian Rembe** received his diploma in physics from the University of Hanover, Germany, in 1994. From 1994 to 1999, he was a PhD student at the University of Ulm, Germany, where he earned a doctorate degree in electrical engineering. In 1999, he joined the Berkeley Sensor & Actuator Center at the University of California, Berkeley, as a postdoctoral research engineer with the support of a Feodor-Lynen-Scholarship of the Alexander von Humboldt-Foundation and a UC Berkeley MacKay-Lecturer Fellowship. From 2001 to 2014, he was the manager of the Development Optics group at the Polytec GmbH in Waldbronn, Germany. Since 2015, he has been a professor for applied metrology at the Clausthal University of Technology, Germany. His interests are optics, optical measurement techniques, sensors, and microelectromechanical systems.