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 $f_{\mathrm{LO}}$.

Fig. 1

Setup of the laser scanner module. For better clarity, just the $yz$-plane is shown here. (a) Schematic depiction of the optical path inside the scanner module, with the help of two sequentially transmitted laser beams ($T_0$ and $T_n$ for one mirror position each) and the corresponding received signal on the detector after scattering at the target ($R_0$ and $R_n$). (b) Demonstrator setup with (1) Crylas FDSS 532-Q laser, (2) Thorlabs CPS532-C2 alignment laser, (3) Smaract STT-2013 2-D motorized optical mount, (4) Olympus M. Zuiko Digital ED 25 mm F1.2 Pro standard objective lens, and (5) IDS UI-1450-C USB 2.0 camera (for testing issues).

Fig. 2

Geometrical operation principle of the laser scanner module. For better clarity, just the $yz$-plane is shown here. (a) Scanner section with the geometrical correlation of the rotational angle ${\phi }_{\mu \mathrm{M},x}$ of the rotating mirror $\mu \mathrm{M}$ depending on the transmission vector ${\overrightarrow{V}}_{{\mathrm{T}}_{\mathrm{Res}},n}$. This position vector describes the vector from the rotating position of mirror $\mu \mathrm{M}$ to the center position of each particular microlens on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$. (b) Simplified beam-shaping section with correlation between the divergence angle ${\theta }_{\mathrm{T}}$ and the $z$-positioning $a_{\mu \mathrm{L}}$ of the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$ to the objective lens 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

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 ${\phi }_{T0,x}=2·{\beta }_{\mathrm{MT}}$, 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

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 ${\overrightarrow{V}}_{\mathrm{Rot}}=DC{M}_y^{-1}·{\overrightarrow{V}}_{\mathrm{u},\mathrm{N}}$, 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

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

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 $f_{\mu \mathrm{L},\min }=2·R_{\mu \mathrm{L}}$, 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 $f_{\mu \mathrm{L},\min }$ 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 ${\alpha }_{\mathrm{Tr}}={\tan }^{-1}(d_{\mathrm{Tr}}/z_{\mathrm{Tr}})$, 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

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

Fig. 3

Geometrical correlation between microlens parameters, horizontal FoV ${\mathrm{FoV}}_h$, and transmission beam diameter $d_{{\mathrm{T}}_{\mathrm{LO}}}$ with predefined angular resolution ${\alpha }_{\mathrm{Tr}}=0.286\mathrm{deg}$ (for three objective lenses ${\mathrm{LO}}_1$, ${\mathrm{LO}}_2$, and ${\mathrm{LO}}_3$ with a ${\mathrm{FoV}}_h$ of 47 deg, 57 deg, and 84 deg). (a) The resulting ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}$ to reach the specified resolution with given objective lenses and their focal lengths $f_{\mathrm{LO}}$. (b) The resulting focal length $f_{\mu \mathrm{L}}$ and its lower limit $f_{\mu \mathrm{L},\min }$ of one microlens to realize a specific transmission beam diameter $d_{{\mathrm{T}}_{\mathrm{LO}}}$, exemplarily for $d_{{\mathrm{T}}_{\mathrm{LO}}}=20\mathrm{mm}$. This is based on the resulting ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}$ from (a), which is given by the defined objective lenses ${\mathrm{LO}}_1$, ${\mathrm{LO}}_2$, and ${\mathrm{LO}}_3$.

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 $<4\mathrm{kW}$ 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 ${10}^{-12}\mathrm{W}$ 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., $P_{\mathrm{R}}\approx P_{\mathrm{T}}·{10}^{-8}$ at $z_{\mathrm{Tr}}=100\mathrm{m}$). We can express the behavior of the receiving signal depending on the ratio of etendues²⁸ $G_{\mathrm{LO}}$ 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 $P_{\mathrm{T},\max }$ scales with the beam diameter $d_{{\mathrm{T}}_{\mathrm{LO}}}$ with $P_{\mathrm{T},\max }\propto d_{{\mathrm{T}}_{\mathrm{LO}}}^2$.

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 $\pm 23.5\mathrm{deg}$ (necessary ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}=124.8\mu \mathrm{m}$) (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 $P_{\mathrm{T},\max }$ 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.

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.

Fig. 4

Scanner section with the geometrical correlation of the resulting spot position on the microlens array and in the FoV in the $xy$-plane depending on the mirror-deflection angle ${\phi }_{\mu \mathrm{M},x}$ of the rotating mirror $\mu \mathrm{M}$. These positions are given with the radial distance $h_{\mathrm{T},y}$ and $h_{\mathrm{FoV},y}$ from the optical axis of the objective lens LO. For better clarity, just the $yz$-plane is shown here.

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 ${\overrightarrow{V}}_{\mathrm{u},\mathrm{N}}={\mathrm{DCM}}_y[{\mathrm{DCM}}_x·{\overrightarrow{V}}_{\mathrm{u},z}]$, 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

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

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

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)].

Fig. 5

Schematic principle of the effect of the mirror-deflection noise on the measurement position in the FoV. (a) Transmission path with a microlens array without additional aperture stops; here ${\mathrm{MLA}}_{\mathrm{T}}$ with $d_{\mu \mathrm{L}}={\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}=d_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}$. (b) Transmission path with a microlens array with additional aperture stops; here, ${\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}$ with $2·d_{\mu \mathrm{L}}={\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}}=d_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}$ for suppression of spot position noise in the FoV.

Fig. 6

Microlens arrays utilized in the experiments. (a) ${\mathrm{MLA}}_{\mathrm{T}}$ with ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}}}=d_{\mu \mathrm{L}}$ (Zeiss SmartZoom m5, $34\times$ Zoom, Plan Apo D $1.6\times /0.1$ FWD 36 mm, MLA PowerPhotonics PP-FRF-1464). (b) ${\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}$ with ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}}=2·d_{\mu \mathrm{L}}$ (Zeiss SmartZoom m5, $34\times$ Zoom, Plan Apo D $1.6\times /0.1$ FWD 36 mm, MLA PowerPhotonics PP-FRF-1465). (c) Transmission beam directed on the center of a microlens (1), and on the transition zones between the microlenses (2) and (3), with HRS015B stabilized HeNe laser.

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

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 ${\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}}=2·d_{\mu \mathrm{L}}$ [see Fig. 6(b)].

With such a microlens array and a focal length $f_{\mathrm{LT}}=75\mathrm{mm}$ 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 $D{h}_{\mathrm{FoV}}\approx 0.25\mathrm{m}$ ($D{h}_{\mathrm{t}}\approx 62.5\mu \mathrm{m}$ on ${\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}$). This means a suppression of mirror-deflection noise with an angular uncertainty of $u_{{\phi }_{\mu \mathrm{M},x}}=0.013\mathrm{deg}$. 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 ${\alpha }_{\mathrm{Tr}}=0.286\mathrm{deg}$ and a ${\mathrm{FoV}}_h=\pm 23.5\mathrm{deg}$ (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 ${\alpha }_{{\mathrm{Tr}}_{\mathrm{act}}}=½·{\alpha }_{\mathrm{Tr}}$. 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 $h_{\mathrm{T},y}=½·d_{\mu \mathrm{L}}$ 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.

Fig. 7

Transmission power $P_{\mathrm{T}}$ on the target in the FoV, depending on the displacement $h_{\mathrm{T},y}$ (in $y$-direction) of the transmission beam on the microlens array ${\mathrm{MLA}}_{\mathrm{T}}$. The illumination is shown for a centric ($T_0$) position and an eccentric ($T_{0,h_{\mathrm{T}}}$) position relative to the microlens position. The laser source is a Crylas FDSS 532-Q laser and the transition zones of the microlens array are translucent. (a) The correlation of the transmission spot position on the microlens and its projection into the FoV. (b)The resulting curve of transmission power $P_{\mathrm{T}}$ of the measurement spot in the FoV at $z_{\mathrm{Tr}}=0.90\mathrm{m}$, depending on axial displacement of the transmission beam from the center $h_{\mathrm{T},y}=0$ to $½·d_{\mu \mathrm{L}}$ of the microlens diameter. (c) Cross section of the transmission spot intensity $I_{\text{Spot}}$, in gray value GV in the FoV at $z_{\mathrm{Tr}}=0.90\mathrm{m}$, $T_0$: centric illumination of the microlens $h_{\mathrm{T},y}=0$. (d) Cross section of the transmission spot intensity $I_{\text{Spot}}$ in gray value GV in the FoV at $z_{\mathrm{Tr}}=0.90\mathrm{m}$, $T_{0,h_{\mathrm{T}}}$: eccentric illumination of the microlens $h_{\mathrm{T},y}=½·d_{\mu \mathrm{L}}$ (8-bit IDS UI-1450-C USB 2.0 camera).

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 $i_{\min }$ to $i_{\max }$ with $\pm ½·d_{\mu \mathrm{L}}$ [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

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.

Fig. 8

Transmission spot through the microlens, depending on the displacement $h_{\mathrm{T},y}$ of the transmission beam on the microlens array ${\mathrm{MLA}}_{\mathrm{T},\mathrm{NS}}$. The illumination is shown exemplarily with two ratios (A and B) of beam diameter $d_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}$ to microlens diameter $d_{\mu \mathrm{L}}$ on a microlens array with position-noise-suppressing design. This is illustrated for a centric illumination of the microlens with $h_{\mathrm{T},y}=0$ (upper figure) and an eccentric illumination with $h_{\mathrm{T},y}=½·d_{\mu \mathrm{L}}$ (lower figure). The figures on the right of (a) and (b) show the detailed view of one microlens. (a) A: $d_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}=d_{\mu \mathrm{L}}$. (b) B: $d_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}=2·d_{\mu \mathrm{L}}$.

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

Fig. 9

Effects of spot position noise on transmission power and spot homogeneity. (a) Cross section $I_{\text{Spot}}$ in $y$-direction. (b) Transmission power $P_{\mathrm{T}}$ transmitting through one microlens. With two ratios of beam diameter $d_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}$ to microlens diameter $d_{\mu \mathrm{L}}$,. A: $d_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}=d_{\mu \mathrm{L}}$ and B: $d_{\mathrm{T},{\mathrm{MLA}}_{\mathrm{T}}}=2·d_{\mu \mathrm{L}}$. This is shown in dependence on the displacement $h_{\mathrm{T},y}$ of the transmission beam from the center $h_{\mathrm{T},y}=0$ to $h_{\mathrm{T},y}=½·d_{\mu \mathrm{L}}$ relative to one microlens, described with $T_0$ and $T_{0,h_{\mathrm{T}}}$.

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.

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 $w_{\mathrm{FoV}}$ [$h_{\mathrm{Gap},\mathrm{Tr}}\propto \tan (w_{\mathrm{FoV}})$]. 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 $T_n$.

Fig. 10

Influence of transmission beam cutoff on the residual spot height $h_{\text{Spot},\mathrm{Tr}}$ and the emerging gap height $h_{\text{Gap},\mathrm{Tr}}$ between adjacent spots on the target. (a) Schematic functional principle of the beam-shaping section with and without ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ combination. (b) Graph of the residual spot height in the FoV for an objective lens ${\mathrm{LO}}_1$ with a ${\mathrm{FoV}}_h$ of $\pm 23.5\mathrm{deg}$ without a ${\mathrm{MLA}}_{\mathrm{T}}$/MWA combination at 100-m target distance. This is shown for two spot diameters $d_{{\mathrm{T}}_{\mathrm{LO}}}=10.41\mathrm{mm}$ and $d_{{\mathrm{T}}_{\mathrm{LO}}}=d_{\mathrm{LO}}=20.82\mathrm{mm}$ in the plane of the objective lens, where $d_{\mathrm{LO}}$ is the diameter of the objective lens.

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 ${\mathrm{FoV}}_h=\pm 23.5\mathrm{deg}$ and a diameter $d_{\mathrm{LO}}=20.82\mathrm{mm}$, we explain this characteristics for two spot diameters $d_{{\mathrm{T}}_{\mathrm{LO}}}=10.41\mathrm{mm}$ and $d_{{\mathrm{T}}_{\mathrm{LO}}}=d_{\mathrm{LO}}=20.82\mathrm{mm}$ 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 ${\gamma }_{\mathrm{w},n}$.

Fig. 11

Detailed functional principle of the microlens array/microwedge prism array ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ combination. (a) Comparison of the optical transmission path between using a ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ component and a single ${\mathrm{MLA}}_{\mathrm{T}}$ component without MWA. (b) Geometrical correlation of the necessary tilt angle ${\tau }_{\text{tilt},n}$ and the resulting wedge angle ${\gamma }_{\mathrm{w},n}$ of the wedge prism. (c) Schematic design of a ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ component.

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 $y_{\mu \mathrm{L},n}=n_{\mu \mathrm{L},y}·{\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}},y}$, $x_{\mu \mathrm{L},n}=n_{\mu \mathrm{L},x}·{\text{Pitch}}_{{\mathrm{MLA}}_{\mathrm{T}},\mathrm{x}}$, and $h_{\mu \mathrm{L},n}=\sqrt{x_{\mu \mathrm{L},n}^2+y_{\mu \mathrm{L},n}^2}$, which can be expressed as

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 $r_{\mu \mathrm{L}}=½·d_{\mu \mathrm{L}}$. The necessary tilt angle ${\tau }_{\text{tilt},n}$ of the marginal ray of each microlens is expressed as

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 $\pm 23.5\mathrm{deg}$, a beam diameter of $d_{{\mathrm{T}}_{\mathrm{LO}}}=10.41\mathrm{mm}$, 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 $d_{\mathrm{Tr}}$.

Fig. 12

Measurement of transmission power $P_{\mathrm{T}}$ over the field angle $w_{\mathrm{FoV}}$ for an objective lens ${\mathrm{LO}}_1$ with a ${\mathrm{FoV}}_h$ of $\pm 23.5\mathrm{deg}$. (a) $P_{\mathrm{T}}$ over $w_{\mathrm{FoV}}$ in comparison between simulation and measurement values. (b) Gradient of the loss of transmission power $P_{T_{\mathrm{Loss}}}$ over $w_{\mathrm{FoV}}$. The red dashed line gives the critical angle $w_{{\mathrm{FoV}}_{\mathrm{crit}.}}$, where the beam cutoff starts theoretically.

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 $w_{{\mathrm{FoV}}_{\mathrm{crit}.}}=8.67\mathrm{deg}$ (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 $w_{\mathrm{FoV}}=0.00\mathrm{deg}$, 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 $h_{\mathrm{Spot},\mathrm{Tr}}\propto -\tan (w_{\mathrm{FoV}})$. 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.

Fig. 13

Effect of transmission beam cutoff on the spot geometry and the residual spot height $h_{\text{Spot},\mathrm{Tr}}$ on the target. Here, the figures show the comparison between the spots for several field angles $w_{\mathrm{FoV}}$ at 0.45-m target distance with and without a microlens array/microwedge prism array combination (${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$).

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.

Fig. 14

Measurement of spot height $h_{\text{Spot},\mathrm{Tr}}$ over the field angle $w_{\mathrm{FoV}}$ for an objective lens ${\mathrm{LO}}_1$ with a ${\mathrm{FoV}}_h$ of $\pm 23.5\mathrm{deg}$ without a ${\mathrm{MLA}}_{\mathrm{T}}/\mathrm{MWA}$ combination. The red dashed line gives the critical angle $w_{{\mathrm{FoV}}_{\mathrm{crit}.}}$, where the beam cutoff starts theoretically.