Monolitic Hybrid Transmitter-Receiver Lens for Rotary On-Axis Communications

Abstract

High-speed rotary communication links exhibit high complexity and require challenging assembly tolerances. This article investigates the use of optical wireless communications (OWC) for on-axis rotary communication scenarios. First, OWC is compared with other state-of-the-art technologies. Different realization approaches for bidirectional, full-duplex links are discussed. For the most promising approach, a monolithic hybrid transmitter-receiver lens is designed by ray mapping methodology. Ray tracing simulations are used to study the alignment-depended receiver power level and to determine the effect of optical crosstalk. Over a distance of $12.5$ $\mathrm{m}$$\mathrm{m}$, the lens achieves an optical power level at the receiver of $-16.2$ $\mathrm{dBm}$ to $-8.7$ $\mathrm{dBm}$ even for misalignments up to 3 $\mathrm{m}\mathrm{m}$.

1. Introduction

Reliable, real-time connectivity is the backbone of industrial automation. Data transmission over rotating parts is required in a broad range of applications such as wind turbines [1], industrial communications [2,3], surveillance radars [3,4,5,6,7], military [2,8], aerospace [9] and many more.

Table 1. Common principles for rotary data transmission and their typical state-of-the-art performance. Values should be understood as orders of magnitude rather than exact values. * Higher values are possible but have not been published yet; ${}^{‡}$ this work is limited to on-axis scenarios.

Group	Contact	Contactless
Type	Slip Ring	Capacitive	RF	Fiber (FORJ)		OWC
				Single	Multi
Ref.	[5,6,16,17,18]	[6,9,11,13]	[1,6,10,11,13]	[4,7,8]	[6,13]	[19]
data rate/$\mathrm{Gbit}/\mathrm{s}$	$0.1$... 3	$5·{10}^{-4}$...$>1$	$0.054$...5	10...40	$>40$	$>10$ *
max speed / rpm	${10}^1...{10}^4$	${10}^3...{10}^4$	${10}^3...{10}^4$	${10}^3...·{10}^4$	${10}^2...{10}^3$	>1400 *
max revolutions	${10}^7...2·{10}^8$	>${10}^8$	>${10}^8$	>${10}^8$	>${10}^8$	>${10}^8$
cost (initial)	medium	low/medium	low/medium	high	very high	low
cost (maintenance)	high	low	low	low	high	low
RF immunity	weak	medium	weak	strong	strong	strong
on-/off axis	off-axis	off-axis	off-axis	on-axis	on-axis	on-axis ${}^{‡}$

gives an overview of the most important transmission principles used in rotary communication links. Slip rings were widespread in the past. However, due to mechanical contact they suffer wear and tear which limits their durability [1,2,5,6,9,10,11] and thus increases maintenance costs. Precious brush materials and lubricants are used to extend lifetime [12] at the expense of increased system complexity and higher costs. Therefore, contactless data transfer is favored nowadays [6,10]. Life times of several hundreds of revolutions are common and rotation speeds in the order of magnitude of ${10}^3\mathrm{rpm}$ or even ${10}^4\mathrm{rpm}$ are reached with contactless methods.

Capacitive-based near-field transmission links are known as reliable and cost-effective [11]. However, the system proposed by Doleschel et al. [13] shows that system complexity of modern solutions is clearly not negligible. Data rates of up to several $\mathrm{Gbit}/\mathrm{s}$ are possible [11,13]. Practical systems support conventional Gigabit-Ethernet and industrial protocols like ProfiNET and EtherCAT for instance. Thus, devices with data rates ranging from 500 $\mathrm{kbit}/\mathrm{s}$ to 1 $\mathrm{Gbit}/\mathrm{s}$ were developed [13]. The maximum transmission distance is in the range of 1 $\mathrm{c}$$\mathrm{m}$. Inductive coupling is mainly employed for power transfer and only rarely used for high-speed data transmission [14,15].

Several systems using conventional low-power radio-frequency (RF) technologies were proposed [1,10] with data rates in the range of tens of $\mathrm{Mbit}/\mathrm{s}$ or below. Standards like 802.11ac and 802.11ad might be able to provide data rates in the range of $\mathrm{Gbit}/\mathrm{s}$. Their main problem is reliability and robustness in terms of ensuring a bandwidth and low-latency data transfer [2,3,13]. Future millimeter-wave based communication [20] standards like IEEE 802.11ay might even reach data rates in the range of several $\mathrm{Gbit}/\mathrm{s}$ to several tens of $\mathrm{Gbit}/\mathrm{s}$ [11]. However, their practicality and cost-effectiveness has to be proven.

Highest data rates are reached with fiber optical rotary joints (FORJ). Besides their superior data rate in the range of $\mathrm{Gbit}/\mathrm{s}$ up to multiple tens of $\mathrm{Gbit}/\mathrm{s}$ per channel [6,7,13], these links provide immunity against RF. Single-fiber systems only consist of an optical transceiver at both sides and optical fibers in-between. However, due to sophisticated mechanical alignment [8], these systems are expensive. Multi-fiber links offer even higher data rates but exhibit a very high complexity [6].

Optical wireless communications (OWC) aim to combine the advantages of rotary FORJ with relaxed mechanical tolerances, reduced system complexity and thus lower costs. Initially only light emitting diode (LED) based systems with data rates in the lower $\mathrm{Mbit}/\mathrm{s}$ range [2,3] or laser diode based uni-directional links were demonstrated [21]. Faulwaßer et al. [19] introduced a full-duplex link for data rates of up to $10\mathrm{Gbit}/\mathrm{s}$ for on-axis rotary data transfer. Similar to fiber-based communications, data rates are likely to increase in the future. The demonstrated rotation speed from $0 \mathrm{rpm}$ to $1400 \mathrm{rpm}$ [19] was limited by the test equipment. There is no OWC-exclusive factor limiting the speed. The lifetime of OWC links is expected to be similar to other contactless methods, since there are no significant aging effects. The optoelectronic components, i.e., light emitting diodes (LEDs), laser diodes (LDs) and photodiodes (PDs) are known for high reliability and long lifetime [22,23,24]. The key element of the transceiver in [19] is a monolithic hybrid lens that acts as transmitter (TX) and receiver (RX) optics in parallel and thereby relaxes the mechanical alignment to several millimeters. The form factor of the transceiver is only $5 \mathrm{m}\mathrm{m}\times 5 \mathrm{m}\mathrm{m}\times 5 \mathrm{m}\mathrm{m}$ [19].

This article investigates the potential of OWC for bidirectional, full-duplex, on-axis rotary scenarios and describes how to design a monolithic, hybrid TX-RX lens. In Section 2 a channel model is used to derive some adequate figures of merit. The use of a hybrid lens is motivated by discussing several realization approaches of rotary OWC. Next, the design procedure of a hybrid lens is described and the choice of design parameters is discussed. In Section 3, the performance of the lens and a second system is evaluated and compared using optical ray tracing simulations. Thereby, the alignment-depended optical signal power at the receiver and optical crosstalk is studied. The results are discussed in Section 4. Finally, Section 5 provides a short summary.

2. Materials and Methods

2.1. Fundamental Concepts

2.1.1. Channel Model

OWC use optical emitters like LEDs or LDs at the TX to convert an electrical signal into the optical domain. A PD is used for back-conversion at the RX. Similar to FORJ, both transceivers are placed in front of each other [25]. OWC use lenses instead of optical fibers to enable larger mechanical tolerances. The key goal for the designer is to increase the optical power $P_{\mathrm{PD}}$ that falls onto the PD in order to improve the RX signal-to-noise ratio (SNR) [26] and to minimize the bit-error-rate (BER). If the SNR is already sufficient, the excess can be converted into a higher data rate by increasing bandwidth or applying a multilevel modulation scheme.

There will always be a misalignment between both transceivers due to positioning, assembly tolerances or vibrations. The rotation axis might even exhibit a nutation, i.e., nonideal motion around the ideal rotation axis. The combination of these nonidealities and the communication distance leads to a minimum field of view (FOV) that is required for robust operation. The FOV sizes are denoted by the half-opening angles ${\theta }_{\mathrm{TX}\mathrm{FOV}}$ and ${\theta }_{\mathrm{RX}\mathrm{FOV}}$. In order to ensure eye-safe operation, i.e., to classify the system as laser class 1 according to IEC 60825-1:2014 (DIN EN 60825-1:2015-07) [27], the optical transmitter power $P_{\mathrm{TX}}$ is limited. Therefore, efficient transceiver design is required to meet link-budget requirements, ensure eye-safety and to maximize data rate. The designer tries to maximize the dynamic range by keeping the TX and RX performance constant within a plane perpendicular to the optical axis [28]. In other words, TX has to provide constant irradiance $E_{\mathrm{TX}}$ and RX has to detect the same signal level within this plane.

For a moment we assume the distance z between both transceivers is large compared to their apertures. Although this does not fully apply for short distances, the assumption is useful to show the fundamental dependencies. As a result, we can assume that parallel rays are incident onto RX. For a homogeneous FOV, the optical power $P_{\mathrm{PD}}$ can be expressed as product of the irradiance $E_{\mathrm{TX}}\left(z\right)$, the effective receiver input aperture $A_{\mathrm{RX},\mathrm{eff}}$ and the geometrical coupling coefficient ${ϵ}_{\mathrm{c}}$ as shown in Equation (1). For the ideal optical link, TX and RX have the same FOV size and both FOVs are placed on the optical axis. In practice, both FOVs may differ or they might be misaligned. The geometrical coupling coefficient ${ϵ}_{\mathrm{c}}$ quantifies this misalignment. It becomes crucial for short-distance communication as we know from the challenging assembly tolerances of FORJ. ${ϵ}_{\mathrm{c}}$ is defined as the fraction of the illuminated area of TX, which is overlapping with the FOV of the RX divided by the total illuminated area $A_{\mathrm{FOV}\mathrm{TX}}$. For the ideal optical channel, i.e., equally sized TX and RX FOVs with no misalignment, ${ϵ}_{\mathrm{c}}=1$ applies.

$$P_{\mathrm{PD}}=E_{\mathrm{TX}}(z,{\theta }_{\mathrm{TX}\mathrm{FOV}})·A_{\mathrm{RX},\mathrm{eff}}\left({\theta }_{\mathrm{RX}\mathrm{FOV}}\right)·{ϵ}_{\mathrm{c}}$$

(1)

The effective receiver input aperture $A_{\mathrm{RX},\mathrm{eff}}\left({\theta }_{\mathrm{RX}\mathrm{FOV}}\right)$ is expressed as product of active PD area $A_{\mathrm{PD}}$, optical gain $g\left({\theta }_{\mathrm{RX}\mathrm{FOV}}\right)$ and efficiency ${\eta }_{\mathrm{RX}}\left({\theta }_{\mathrm{i}}\right)$ at the angle of incidence ${\theta }_{\mathrm{i}}$ as it is shown in Equation (2). A large PD area $A_{\mathrm{PD}}$ is favorable for the link budget but goes along with a large PD capacitance, which limits the modulation bandwidth (BW) [26,29]. Consequently, a PD with large $A_{\mathrm{PD}}$ but sufficient BW is chosen.

$$P_{\mathrm{PD}}=E_{\mathrm{TX}}(z,{\theta }_{\mathrm{TX}\mathrm{FOV}})·A_{\mathrm{PD}}·g\left({\theta }_{\mathrm{RX}\mathrm{FOV}}\right)·{\eta }_{\mathrm{RX}}\left({\theta }_{\mathrm{i}}\right)·{ϵ}_{\mathrm{c}}$$

(2)

The RX FOV should always be chosen as small as possible, since the optical gain $g\left({\theta }_{\mathrm{RX}\mathrm{FOV}}\right)$ decreases with increasing FOV due to conservation of Ètendue [30]. Moreover, a restricted RX FOV improves the robustness against noise and interchannel interference. Next, $g\left({\theta }_{\mathrm{RX}\mathrm{FOV}}\right)$ is substituted by the maximum theoretical optical gain [30] as shown in Equation (3). Now ${\eta }_{\mathrm{RX}}\left({\theta }_{\mathrm{i}}\right)$ is used as a figure of merit for the optical RX efficiency. The ideal lossless receiver achieves ${\eta }_{\mathrm{RX}}\left({\theta }_{\mathrm{i}}\right)=1$ for all ${\theta }_{\mathrm{i}}\in [0,{\theta }_{\mathrm{RX}\mathrm{FOV}}]$. The angle ${\theta }_{\mathrm{RX},\mathrm{PD},\max }$ denotes the maximum coupling angle from the RX optics to the PD surface normal. In the next step, the irradiance $E_{\mathrm{TX}}(z,{\theta }_{\mathrm{TX}\mathrm{FOV}})$ is substituted by the product of the optical TX power $P_{\mathrm{TX}}$ and the TX efficiency ${\eta }_{\mathrm{TX}}$ divided by the illuminated spot area $A_{\mathrm{FOV}\mathrm{TX}}$.

$$P_{\mathrm{PD}}=\frac{P_{\mathrm{TX}}·{\eta }_{\mathrm{TX}}}{A_{\mathrm{FOV}\mathrm{TX}}}·{\left(\frac{n_1·sin{\theta }_{\mathrm{RX},\mathrm{PD}}}{n_{air}·sin{\theta }_{\mathrm{RX}\mathrm{FOV}}}\right)}^2·A_{\mathrm{PD}}·{\eta }_{\mathrm{RX}}\left({\theta }_{\mathrm{i}}\right)·{ϵ}_{\mathrm{c}}$$

(3)

$A_{\mathrm{FOV}\mathrm{TX}}$ is replaced by the corresponding triangular relationship, which includes the communication distance z and the tangent of the TX FOV ${\theta }_{\mathrm{TX}\mathrm{FOV}}$. Finally, Equation (4) is a useful expression for the most important geometric dependencies of $P_{\mathrm{PD}}$.

$$P_{\mathrm{PD}}=\frac{P_{\mathrm{TX}}·{\eta }_{\mathrm{TX}}}{\pi ·{(z·tan{\theta }_{\mathrm{TX}\mathrm{FOV}})}^2}·{\left(\frac{n_1}{sin{\theta }_{\mathrm{RX}\mathrm{FOV}}}\right)}^2·A_{\mathrm{PD}}·{\eta }_{\mathrm{RX}}·{ϵ}_{\mathrm{c}}$$

(4)

The performance merits from Equation (4) are ${\eta }_{\mathrm{TX}}$, ${\eta }_{\mathrm{RX}}$ and ${ϵ}_{\mathrm{c}}$. The transmitter efficiency ${\eta }_{\mathrm{TX}}$ specifies how much of the emitted power $P_{\mathrm{TX}}$ is concentrated into the FOV. Its loss mechanisms are ${\zeta }_{\mathrm{TX}\mathrm{M}}$ and ${\zeta }_{\mathrm{TX}\mathrm{F}}$. ${\zeta }_{\mathrm{TX}\mathrm{M}}$ describes rays that strike the target plane outside the FOV. ${\zeta }_{\mathrm{TX}\mathrm{F}}$ describes back-reflected rays due to Fresnel-reflections.

Since ${\eta }_{\mathrm{TX}}$ contains no information concerning irradiance homogeneity, we additionally introduce the effective transmitter efficiency ${\eta }_{\mathrm{TX}\mathrm{eff}}$. It is defined according to Equation (5) by using the minimum irradiance within the FOV $E_{\min }$ [28,31]. The difference between ${\eta }_{\mathrm{TX}}$ and ${\eta }_{\mathrm{TX}\mathrm{eff}}$ is called the inhomogeneity factor ${\zeta }_{\mathrm{TX}\mathrm{H}}$.

$${\eta }_{\mathrm{TX}\mathrm{eff}}=\frac{E_{\min }}{E_{\min ,\mathrm{ideal}}}=\frac{E_{\min }{A}_{\mathrm{FOV}\mathrm{TX}}}{P_{\mathrm{TX}}}$$

(5)

The sensitivity of RX is typically limited by the interaction of several internal noise mechanisms that exhibit a Gaussian probability function [26]. The signal can additionally be corrupted by optical crosstalk, i.e., adjacent communication channels. This nonGaussian noise has a limited range of variation. This effect introduces a power penalty $PP$ [26] that reduces the usable peak-to-peak amplitude of the signal. The optical power that falls onto the PD consists of a signal part $P_{\mathrm{PD}}$ and another part arising from crosstalk $P_{\mathrm{PD}}\mathrm{cross}$. The corrected power value $P_{\mathrm{PD}\mathrm{PP}}$ takes the eye-closure effect into account by subtracting the $P_{\mathrm{PD}\mathrm{cross}}$ from $P_{\mathrm{PD}}$ [26]. It holds $P_{\mathrm{PD}\mathrm{PP}}<P_{\mathrm{PD}}$ as soon as crosstalk is present. In this work, all optical power values are understood as average values to ensure comparability with literature. When dealing with the power penalty the signal strength is considered in a peak-to-peak manner. It is still valid to consider average values as long as the extinction ratio of the signal and the crosstalk signal is equal. This assumption applies to our case, since both link directions are designed equally and the crosstalk arises from the signal itself.

2.1.2. Ideal Arrangement

The ideal arrangement consists of TX and RX placed at the same position on the optical axis as shown in Figure 1. Both FOVs are equal in size and they fully overlap. The arrangement can be realized by using an LED. On the one hand, applying a forward-bias to the LED causes a forward-current and leads to photon generation. Biasing the PN-junction reversely enables fast photo-detection on the other hand. As a result, the transceiver only needs a single optoelectronic component for TX and RX. Data rates of up to 150 $\mathrm{Mbit}/\mathrm{s}$ were demonstrated for close distances [32]. However, the LED is a rather bad PD. It has a low responsivity, small area $A_{\mathrm{PD}}$ and low bandwidth [32]. Although a bidirectional link is realizable, data transfer is restricted to half-duplex mode, because the link direction is switched in time domain.

For high-speed bidirectional data transfer in full-duplex mode, TX and RX are separated. The optical arrangement becomes more challenging and design trade-offs have to be met. The following section introduces several geometrical arrangements for optical wireless rotary communication scenarios and compares their performance.

2.1.3. Geometrical Arrangements

Figure 2 shows three geometrical approaches for rotary OWC. In Figure 2a TX and RX are placed next to each other separated by a spacing d. In Figure 2b, TX and RX are radially separated regarding the optical axis. Third, both elements arranged along the optical axis and the front element are transparent as illustrated in Figure 2c.

Approach (a) is used by commercial low data rate IrDA transceivers like Vishays TFBS4711 [33] but also by a high-speed transceiver demonstrated by Faulwaßer et al. [34]. The axis of rotation is placed trough one of the elements or between them. This leads to misaligned FOVs. The designer tries to minimize the spacing d. Next, both FOVs are chosen large enough to ensure a sufficient ${ϵ}_{\mathrm{c}}$. As we learned from Equation (4), $P_{\mathrm{PD}}$ scales with $1/tan{\theta }_{\mathrm{TX}}^2{}_{\mathrm{FOV}}$ and $1/sin{\theta }_{\mathrm{RX}}^2{}_{\mathrm{FOV}}$. This penalty is typically significant. A numerical example is given in Section 3.

This penalty is not present for the radial separation from Figure 2b. The approach directly provides aligned FOVs and a high ${ϵ}_{\mathrm{c}}$. TX can be placed in the center surrounded by RX as shown in Figure 2b or vice-versa.

The approach shown in Figure 2c avoids shadowing of the back component by designing the front element transparent. If the front component is TX, it has to have separated emission and absorption bands, i.e., it must exhibit a Stokes shift similar to the fluorescence materials [35]. The issue here is the back plane of the emitter: on the one hand, it has to be reflective to direct the transmitted signal towards the other transceiver. On the other hand, it has to be transparent for the incoming signal. This contradiction does not seem to be easily resolved.

Clearly the radial separation from Figure 2b has superior performance over Figure 2a and has no fundamental concept issue like architecture Figure 2c. However, what is the best way of realizing radial separation? Figure 3 illustrates three principles.

Designing a transmitter element surrounded by one or more high-speed PDs or vice versa on a single chip, like it is shown in Figure 3a, comes with many design challenges including process compatibility. Thereby, it seems easier to produce the chips separately and stack them afterwards. A small emitter die is bonded onto a large area PD chip as it is depicted in Figure 3b. The emitter is contacted with bonds or directly trough the PD chip. The PD could be separated into multiple parts to ease the contacting and to reduce the transit time of the electrons and holes as they might limit the bandwidth [26]. The main issues of this approach include crosstalk between TX and RX, shadowing of the PD by the emitter and the fact that both chips are custom designs.

By using an optical system like it is shown in Figure 3c, conventional emitter and receiver chips can be used. Those dies are placed next to each other and a hybrid TX-RX lens is used to redirect the rays to achieve radial separation. The spatial separation of both chips is favorable to reduce electrical crosstalk. Injection molding allows the fabrication of the lens in high volume [36] and low cost. Ultra-precise drilling and milling [36] is used to produce the mold tool.

Since a part of the lens is used to direct the TX rays, the maximum theoretical gain $g_{\max }$ cannot be reached. From the PDs point of view, the solid angle of the TX lens part is not used for optical concentration. In Equation (4) this is expressed by a reduced ${\eta }_{\mathrm{RX}}$.

In summary, using a hybrid lens for radial separation is most promising: besides the potential for ${ϵ}_{\mathrm{c}}\approx 1$, commercial emitter and PD chips can be used. Since the hybrid lens can be fabricated at low cost by injection molding, the system costs are expected to be low.

2.2. Hybrid Lens

2.2.1. Concept

In order to achieve separation in radial direction, the lens is divided into a TX and RX part, i.e., a center part and a surrounding one. Generally, both parts consist of nonrotationally freeform surfaces at the top and bottom of the lens to form a constant irradiance pattern $E_{\mathrm{TX}}$ and provide a homogeneous gain g. In order to limit the lens thickness t, Fresnel-structures could be applied to the top and bottom surface. However, it is favorable to keep the top flat to improve reliability, since cavities tend to fill up with particles.

There are two possible arrangements of TX and RX:

• The emitter is placed centrally and the PD is positioned off-axis as depicted in Figure 4a. In this case only the emission profile of the emitter has to be adjusted. This includes the homogenization of the profile and an adjustment of the angle ${\theta }_{\mathrm{TX}}$. This can ideally be achieved with a single freeform interface. Therefore, the lower surface can be used for beam shaping and the top surface can be flat. The downside of this approach is a challenging RX lens design: the focus point of the RX lens part is off-axis.
• The PD is placed centrally and the emitter is positioned off-axis as shown in Figure 4b. The TX lens part fulfills two tasks: first, it compensates the displacement of the ray bundle regarding the optical axis. Second, it reshapes the ray bundle to the anticipated FOV. Because two surfaces are required, the top aperture of the lens cannot be flat. On the other hand, the design of the RX lens part is simplified, because the focal point is on the optical axis. If the TX part is neglected, the RX lens can be designed to be rotationally symmetrical. However, the shadowing effect of the TX part introduces a nonrotationally symmetric factor. Theoretically, this can be partly compensated by a nonrotationally symmetric RX lens part. The shadowing effect is also mitigated by minimizing the size of TX.

We choose Approach 2, due to the simplified RX lens design. In this configuration, the lens thickness t typically results from the vertical distance between both TX surfaces. A low t is desirable for size, weight and cost reduction. The costs are reduced, because less material is needed and due to faster processing [36,37]. Nevertheless, a certain thickness t is required to keep the refraction angles at the TX surfaces low in order to limit undesirable Fresnel-reflections. If the RX part determines t, the surface can be split into several Fresnel-grooves to reduce t.

2.2.2. Optic Design Methods

The hybrid lens is a nonimaging optical system. It can be designed by two fundamental approaches [38]: numerical optimization and direct calculation.

Numerical optimization is a straight-forward approach for designing complex optic modules. Modern optic simulation tools like Optic Studio Zemax enable forming and optically simulating arbitrarily shaped optics by overlapping parametric objects. Optimization algorithms like the Levenberg–Marquardt algorithm are used for adjusting parameters of those objects until a sufficiently good result is achieved. Due to the large amount of variables, the optimization is typically inefficient, because of many local minima in the merit function [38]. As a result, it is easy to find a solution, but its performance is very limited, especially if the systems become more complex.

In contrast, direct calculation algorithms follow well-defined design procedures and yield deterministic outcomes. Thereby, they provide better results than numerical optimization methods [38], especially if the systems are complex. A great variety of design methods are known, for example ray mapping [39,40,41,42,43,44,45,46], forming surfaces using Cartesian ovals [47], the simultaneous multiple surface method in 2D [48] and 3D [38,49] or the tailored freeform design method proposed by Ries and Muschaweck [50]. Nowadays ray mapping approach, i.e., the combination of energy mapping in conjunction with geometrical surface construction, is in the focus of illumination research [39,40,41,42,43,44,45,46]. Here, a mass-transfer problem is solved by transforming the source power irradiance $E_{\mathrm{s}}$ into the target power irradiance $E_{\mathrm{t}}$. This transformation is represented by a mapping $\varphi :{\mathsf{\Omega }}_{\mathrm{s}}\to {\mathsf{\Omega }}_{\mathrm{t}}$ from the source domain ${\mathsf{\Omega }}_{\mathrm{s}}$ to the target domain ${\mathsf{\Omega }}_{\mathrm{t}}$. Then, the laws of refraction and reflection are applied for $k\times l$ points to calculate a corresponding vector field $\mathit{N}$ containing the surface normals ${\mathit{n}}_{i,j}$ with $i\in [1,l]$, $j\in [1,k]$. The challenge is to find a mapping $\varphi$ which yields a vector field $\mathit{N}$ that satisfies the integrability condition for a continuous surface. This condition is shown in Equation (6) [41,42,50]. It states that $\mathit{N}$ has to be curl-free or exhibit at least minimum curl.

$$\mathit{N}·(\nabla \times \mathit{N})=0$$

(6)

Nowadays, parametrization and consecutive optimization are widely employed for generating a mapping $\varphi$ [28,31,39,40]. Circular shaped FOV are formed with the mapping shown in Figure 5a. An equi-flux grid around the source in spherical coordinates ${\theta }_{\mathrm{s}},\varphi$ is mapped onto a target grid in polar coordinates $\beta ,r$. In 3D, mappings like these typically lead to a normal vector field $\mathit{N}$ with substantial curl. Therefore, an optimization procedure purposely distorts the target grid, for instance by varying r, to improve the performance. It was shown that this approach works well for on-axis scenarios [28,39]. In case of an off-axis placed emitter and two optical surfaces, the scenario is more challenging due to the nonparaxial nature. Hence, the result will deviate from the anticipated irradiance pattern. An additional variable $\beta$ could be used in lateral direction as illustrated in Figure 5a. The downside is a slower optimization process.

For the TX lens part of the hybrid lens, inefficient optimization can be avoided by taking the curl of $\varphi$ directly into account. An initial curl-free mapping ${\varphi }_0$ is generated, assuming that the resulting vector field $\mathit{N}$ exhibits minimum curl [42]. Figure 5b illustrates the irradiance in front of the source $E_{\mathrm{s}}$ and the anticipated irradiance at the target plane $E_{\mathrm{t}}$. The mapping $\varphi$ has to ensure that the infinitesimal area elements at ${\sigma }_{\mathrm{s}}$ and ${\sigma }_{\mathrm{t}}$ are passed by the same flux. This is achieved by expansion or contraction of the area elements. Mathematically spoken, the mapping has to satisfy Equation (7) for every ${\sigma }_{\mathrm{s}}$ of the source grid [42,46].

$$det(\nabla \varphi \left({\sigma }_{\mathrm{s}}\right)){\rho }_{\mathrm{t}}\left(\varphi \left({\sigma }_{\mathrm{s}}\right)\right)={\rho }_{\mathrm{s}}\left({\sigma }_{\mathrm{s}}\right)$$

(7)

The term $det(\nabla \varphi (\sigma \left)\right)$ represents the expansion and contraction of the area element [46]. Although many mappings $\varphi$ might satisfy Equation (7), there is only a single one that minimizes transport cost [46]. Solving Equation (7) turns out to be nontrivial [42,46]. In this article, the algorithm proposed by Wu et al. [46] is used since it provides good convergence.

For the RX lens part, a rotationally symmetric concentrator lens is designed. It consists of two sections: the center is based on refraction, whereas the outer section works with total internal reflection (TIR). TIR is generally superior over refraction for large angles ${\theta }_{\mathrm{RX}\mathrm{PD}\mathrm{i}}$, since it limits the lens diameter and reduces Fresnel-losses [51]. Similar to the TX lens part, the surface is calculated from input and output vectors ${\mathit{v}}_{i,j}$ and ${\mathit{o}}_{i,j}$. In the first attempt, the edge ray principle [30] was applied for generating the input vectors ${\mathit{v}}_{i,j}$. Due to the TX lens part and the discontinuity between both RX lens sections, the edge ray principle is not valid. Therefore, the gain will vary over the FOV and may drop at certain alignments. In the second attempt, this issue is addressed by defining the input vectors for a range of angles of incidence rather than only for the maximum incidence angle. The output rays ${\mathit{o}}_{i,j}$ are derived from coupling angles ${\theta }_{\mathrm{RX}\mathrm{PD}\mathrm{i}}$ to the PD active area. Ideally, they cover the whole half-space in front of the PD.

The normal vector fields $\mathbf{N}$ of both lens parts and an initial point ${\mathbf{P}}_{1,1}$ for each surface is used to calculate a finite number of surface points ${\mathbf{P}}_{i,j}$. The result is only a point cloud representation for the optical surfaces. Nonuniform rational B-splines (NURBS) [52] are used for interpolation. NURBS are very flexible and they are supported in the most popular computer aided design formats, which makes them suitable for optical simulation in third-party software and for subsequent fabrication.

2.3. System with Separated TX and RX

In order to show the potential of the methodology and the concept of the hybrid TX-RX lens, a second optical wireless link based on the principle shown in Figure 2a is developed. A concave lens is chosen for TX to widen up the beam. A convex RX lens is used for optical concentration. The distance d between the TX and RX determines the maximum size of both lenses. The RX lens must not be too small to enable a sufficient optical gain g. Therefore, d will be larger compared to the hybrid lens approach. The exact optical parameters are determined by numerical optimization. The hybrid TX-RX lens and the fully separated TX-RX lens system are denoted by H TX-RX and S TX-RX respectively.

2.4. Simulation Parameters

For optical simulations, Monte Carlo ray tracing in Optic Studio Zemax 17 is used. For each simulation, $5·{10}^5$ rays are traced. Thereby, polycarbonate lenses are used with a refractive index of $n_1=1.57$ at $\lambda =940 \mathrm{n}\mathrm{m}$. We especially investigate Fresnel-reflections which lead to direct optical crosstalk $P_{\mathrm{PD}}\mathrm{cross}$ if the reflected rays reach the PD. The detector is a circular shaped PIN-PD with a radius of 100 $\mathsf{\mu }$$\mathrm{m}$. The LD has an output power of $P_{\mathrm{TX}}=4\mathrm{m}\mathrm{W}=6\mathrm{dBm}$ at $\lambda =940 \mathrm{n}\mathrm{m}$. First, the transmitter part is considered. Then, the full channel is characterized. In order to give a comparable result to other rotary communication technologies, the link distance is set to $12.5 \mathrm{m}\mathrm{m}$.

Conventionally, optical concentrators are characterized by estimating the gain over the angle of incidence by assuming parallel light [53]. However, Figure 6a shows a large divergence of the incident rays. Thus, the assumption of parallel rays does not apply for the present arrangement. The RX performance depends directly on the TX characteristics. Therefore, it is evaluated by the full-channel simulation.

3. Results

Figure 6a shows a render of a cross-section through the lens. It is $6.0$ mm in diameter and has a thickness of $t=2.0 \mathrm{mm}$ in the center and $t=2.7 \mathrm{mm}$ at the groove. The PD is placed on-axis at the origin. The LD position is $(x,y)=(-0.5 \mathrm{m}\mathrm{m},0 \mathrm{m}\mathrm{m})$.

For the second system, the numerical optimization leads to a separation between LD and PD of $d=1.9 \mathrm{m}\mathrm{m}$. The LD is placed at $x=-0.95 \mathrm{m}\mathrm{m}$ and the PD at $x=0.95 \mathrm{m}\mathrm{m}$. The TX and RX lens have a diameter of $1.3$ $\mathrm{m}$ and $2.2$ $\mathrm{m}\mathrm{m}$ and a focal length of $-9$ $\mathrm{m}\mathrm{m}$ and $2.7$ $\mathrm{m}\mathrm{m}$, respectively.

3.1. TX Performance

Figure 7 shows the irradiance at a distance of $12.5$ $\mathrm{m}\mathrm{m}$, measured from the TX aperture.

Table 2. TX merits for the laser diode (LD), the hybrid TX-RX system (H TX-RX) and the separated TX-RX lens system (S TX-RX).

Merit	Unit	LD	H TX-RX	S TX-RX
Emin	$\frac{\mathsf{\mu }\mathrm{W}}{{\mathrm{mm}}^2}$	14.6	117	25.9
${\eta }_{\mathrm{TX}}$	%	$89.5$	$63.2$	$51.4$
${\eta }_{\mathrm{TX}\mathrm{eff}}$	%	$7.2$	$57.6$	$12.8$
${\zeta }_{\mathrm{TX}\mathrm{M}}$	%	$10.5$	$24.7$	$36.0$
${\zeta }_{\mathrm{TX}\mathrm{F}}$	%	0	$12.1$	$12.6$
${\zeta }_{\mathrm{TX}\mathrm{H}}$	%	$82.3$	$5.6$	$38.6$

lists the detailed merits of both profiles. Figure 7a shows the donut-like shaped profile of the LD without optics. The off-axis placement of the LD is directly observable as profile displacement along the x-axis. Figure 7b shows that the hybrid TX-RX lens is able to remove the displacement and homogenize the profile. The effective transmitter efficiency ${\eta }_{\mathrm{TX}\mathrm{eff}}$ is $57.6\%$ with a minimum irradiance of $117\mathsf{\mu }\mathrm{W}/\mathrm{m}{\mathrm{m}}^2$ at $(x,y)=(1.5 \mathrm{m}\mathrm{m},1.5 \mathrm{m}\mathrm{m})$. ${\eta }_{\mathrm{TX}}$ and ${\eta }_{\mathrm{TX}\mathrm{eff}}$ are both reduced by ${\zeta }_{\mathrm{TX}\mathrm{M}}=24.7\%$ and ${\zeta }_{\mathrm{TX}\mathrm{F}}=12.1\%$. Moreover, ${\eta }_{\mathrm{TX}\mathrm{eff}}$ is lowered by another ${\zeta }_{\mathrm{TX}\mathrm{H}}=5.6\%$ due to inhomogeneity within the FOV.

Figure 7c shows the irradiance for the system with fully separated TX and RX as described in Section 2.3. Due to the profile displacement, ${\zeta }_{\mathrm{TX}\mathrm{M}}=36.0\%$ of the power misses the FOV. Moreover, the spherical lenses are not able to correct the donut-shaped profile resulting in a high ${\zeta }_{\mathrm{TX}\mathrm{H}}=38.6\%$. As a result, the irradiance is very inhomogeneous within the FOV and it drops down to $25.9 \mathsf{\mu }\mathrm{W}/\mathrm{m}{\mathrm{m}}^2$ at the right side. This corresponds to ${\eta }_{\mathrm{TX}\mathrm{eff}}=12.8\%$.

3.2. Full-Channel Performance

Figure 8a–c displays $P_{\mathrm{PD}}$, $P_{\mathrm{PD}\mathrm{cross}}$ and $P_{\mathrm{PD}\mathrm{PP}}$ at a lens-to-lens distance of $12.5 \mathrm{m}\mathrm{m}$ for the hybrid lens. The values are determined by misaligning the receiving transceiver, whereas the transmitting one is placed at the origin.

Table 3. Minimum $P_{\mathrm{PD}}$ depending on the misalignment in negative x-direction ($y=0$). Values are determined by choosing the minimum value $P_{\mathrm{PD}}$ on a circle around the center with a radius of the misalignment. Values lower than $40 \mathrm{dBm}$ are very noisy due to the finite number of simulated rays.

Misalignment/mm	−0	−0.5	−1	−1.5	−2	−2.5	−3	−3.5
H TX-RX: $P_{\mathrm{PD}}/\mathrm{dBm}$	$-10.1$	$-8.5$	$-9.5$	$-12.0$	$-14.1$	$-15.3$	$-16.0$	$-26.4$
H TX-RX: $P_{\mathrm{PD}\mathrm{cross}}/\mathrm{dBm}$	$-19.2$	$-22.7$	$-23.9$	$-22.9$	$-25.2$	$-27.4$	$-28.8$	$-30.4$
H TX-RX: $P_{\mathrm{PD}\mathrm{PP}}/\mathrm{dBm}$	$-10.7$	$-8.7$	$-9.7$	$-12.4$	$-14.5$	$-15.6$	$-16.2$	$-28.6$
S TX-RX: $P_{\mathrm{PD}}/\mathrm{dBm}$	$-13.7$	$-15.2$	$-20.6$	<$-40$	<$-40$	<$-40$	<$-40$	<$-40$

shows numerical values for different misalignments in negative x-direction. Figure 8a shows a rotationally symmetric performance in its fundamental structure. However, $P_{\mathrm{PD}}$ is not fully homogeneous within the FOV for a revolution. The largest variation $\Delta P_{\mathrm{PD}}$ over one revolution is reached if the misalignment is between 2 $\mathrm{m}\mathrm{m}$ and 3 $\mathrm{m}\mathrm{m}$. There, $\Delta P_{\mathrm{PD}}$ is in the range of $3 \mathrm{dB}$ to $3.29 \mathrm{dB}$. The crosstalk power $P_{\mathrm{PD}\mathrm{cross}}$ in Figure 8b is similarly distributed as $P_{\mathrm{PD}}$ with a power level which is about $10 \mathrm{dB}$ lower than $P_{\mathrm{PD}}$. As a result, the optical power with applied power penalty $P_{\mathrm{PD}\mathrm{PP}}$ in Figure 8c is similar to $P_{\mathrm{PD}}$. This can also be observed in Table 3: the difference between $P_{\mathrm{PD}}$ and $P_{\mathrm{PD}\mathrm{PP}}$ is in the range of $0.1$ $\mathrm{dB}$... $0.9$ $\mathrm{dB}$ (1.023...1.230). The crosstalk $P_{\mathrm{PD}\mathrm{cross}}$ consists of two parts: a constant part $P_{\mathrm{PD}\mathrm{cross}1}$ of $-32.4 \mathrm{dBm}$ and an alignment-depended part $P_{\mathrm{PD}\mathrm{cross}2}$.

The system with fully separated TX and RX is shown in Figure 8d. It suffers from a displacement of the communication-area in x-direction. The effect can be seen in Table 3; a communication is only possible for a misalignment of $0.5$ $\mathrm{m}\mathrm{m}$... 1 $\mathrm{m}\mathrm{m}$, depending on the data rate.

4. Discussion

4.1. TX Performance

Figure 7 proves the suitability of the ray mapping method based on curl-free mapping for the TX part. In order to assess the performance of the design, it is meaningful to have a closer look at the loss mechanism: the main loss is represented by ${\zeta }_{\mathrm{TX}\mathrm{M}}=24.7\%$. This magnitude is quite common. It results from the extended source effect from the LD with regards to the TX lens part and the remaining curl in $\mathbf{N}$. Furthermore, the overall ray mapping performance depends on the extent of the TX output aperture and the distance to the target plane [41]. The reason for this is that ray mapping is an optical far-field design method that neglects the rays’ position vector on the output aperture.

The inhomogeneity within the FOV is with ${\zeta }_{\mathrm{TX}\mathrm{H}}=5.6\%$ very low. Generally, a ${\zeta }_{\mathrm{TX}\mathrm{H}}$ below $10\%$ is a good result. The major part of the remaining inhomogeneity results from the FOV edge, where the irradiance starts to decrease. The Fresnel-loss ${\zeta }_{\mathrm{TX}\mathrm{F}}$ is with $12.1\%$ in a common region for two material interfaces. Although an anti-reflection (AR) coating might reduce this effect by a factor of 3 to 4, it is challenging to homogeneously coat the nonplanar TX surfaces.

4.2. Full-Channel Performance

$P_{\mathrm{PD}\mathrm{PP}}$ is in a sufficient range for data transfer, but how does it correspond to the data rate? For a bit error rate of ${10}^{-12}$ Säckinger calculates a sensitivity of $-26.5$ $\mathrm{dBm}$ for a $2.5$ $\mathrm{Gbit}/\mathrm{s}$ link and $-20.5$ $\mathrm{dBm}$ for a 10 $\mathrm{Gbit}/\mathrm{s}$ link [26]. Tzeng et al. [54] demonstrated a sensitivity of $-21.5$ $\mathrm{dBm}$ for a 10 $\mathrm{Gbit}/\mathrm{s}$ link. It can be concluded from Table 3 that $P_{\mathrm{PD}\mathrm{PP}}$ is sufficient for data transmission in the $\mathrm{Gbit}/\mathrm{s}$ data rate range and even provides a margin for ageing effects and other nonidealities. The optical crosstalk introduces a power penalty of up to $0.9 \mathrm{dB}$. The magnitude of $P_{\mathrm{PD}\mathrm{cross}}$ depends directly on $P_{\mathrm{TX}}$. A lower $P_{\mathrm{TX}}$ leads to reduced $P_{\mathrm{PD}\mathrm{cross}}$. The downside is that the SNR cannot be improved by increasing $P_{\mathrm{TX}}$ if the crosstalk is the dominant noise factor. The constant crosstalk part $P_{\mathrm{PD}\mathrm{cross}1}$ is alignment-independent. Therefore, it has to be the internal optical crosstalk. In contrast, the back-reflected signal $P_{\mathrm{PD}\mathrm{cross}2}$ from the opposite transceiver depends on the transceiver alignment. The crosstalk and thus the power penalty is reduced if z is increased. In the far field, $P_{\mathrm{PD}}$ is decreasing with approximately $P_{\mathrm{PD}}\propto z^{-2}$ and the crosstalk scales with $P_{\mathrm{PD}\mathrm{cross}2}\propto z^{-4}$, since the back-reflected rays have to travel twice the distance. Although this relation is not fully correct for the near field, the trend is still valid.

The crosstalk $P_{\mathrm{PD}\mathrm{cross}2}$ can be lowered by reducing the Fresnel-reflections ${\zeta }_{\mathrm{RX},\mathrm{F}}$. The planar top surface is well suited for an AR coating. The problem is that the nonplanar top surface of the TX lens part is also affected by the coating. If one is able to solve the TX coating problem, a single coating is twice as effective, because the ray passes the coating two times: first at TX and second at RX. The link-budget is improved by approx. 6% per coating instead of only approx. 3%. Moreover, the crosstalk $P_{\mathrm{PD}\mathrm{cross}2}$ is reduced. Alternatively, the top surface of the RX lens part could be designed nonplanar in a way that back-reflected rays miss the transmitting device.

Although TX exhibits a quite homogeneous performance, $P_{\mathrm{PD}\mathrm{PP}}$ varies about $6.9$ $\mathrm{dB}$ along radial direction for misalignment $\le 3 \mathrm{m}\mathrm{m}$ according to Figure 8a and Table 3. Thus, there is clearly some room for improvement for the RX lens part. The main issue results from the mismatch between near and far field in terms of ray optics. This can be seen in Figure 6a: the incident ray bundle (green) exhibits a large divergence. The situation is different at $z=50 \mathrm{m}\mathrm{m}$, for instance. The solid angle of incidence is much smaller and the incident ray bundle exhibits a smaller divergence. The larger the distance, the better the design approach with the edge-ray principle works. Another issue is the nonrotationally symmetric shadowing effect of the TX lens part that manifests itself as a local minimum around $(x,y)=(-2.5 \mathrm{m}\mathrm{m},0 \mathrm{m}\mathrm{m})$ in Figure 8a,c. The variation $\Delta P_{\mathrm{PD}}=3.29 \mathrm{dB}$ is not crucial for data transmission. Assuming a misalignment of $-2.5 \mathrm{m}\mathrm{m}$ and a rotation speed of $10000 \mathrm{rpm}$, the link moves $9.42 \mathsf{\mu }\mathrm{m}$ over the surface in Figure 8a during one bit-duration of $1 \mathrm{n}\mathrm{s}$ ($1 \mathrm{Gbit}/\mathrm{s}$). The change of $P_{\mathrm{PD}}$ over a sequence of bits is small enough and does not influence the transmission drastically. Ideally, the receiver circuit includes a decision-point control mechanism for continuous adaptation of the decision level to improve the BER [26].

4.3. Suitability for Rotary Scenarios

Faulwaßer et al. [19] already reported data rates of up to 10 $\mathrm{Gbit}/\mathrm{s}$. However, what data rates are generally possible compared to FORJ? From an electrical point of view, OWC is able to achieve similar data rates as single-FORJ. In contrast to FORJ, the PD has typically a larger area and thus a larger capacitance, which limits the bandwidth. Choosing a smaller PD, will reduce the maximum tolerable misalignment. The hybrid lens from this work exhibits $14.5$ dB link loss at $(x,y)=(0 \mathrm{m}\mathrm{m},0 \mathrm{m}\mathrm{m})$. This link loss is a part of the active link concept. The signal is directly recovered at RX by amplification and optimally by subsequent analog-to-digital conversion. The magnitude of the OWC link loss depends on the magnitude of tolerable misalignment. Thus, a higher data rate requires a smaller FOV. In summary, the data rate of OWC links might be slightly below single-fiber FORJ due to a higher link loss.

With regards to the communication distance, the hybrid lens is flexible and not restricted to $z=12.5 \mathrm{m}\mathrm{m}$. If the lens is designed for larger ranges, the TX beam exhibits a lower divergence and RX should be designed for smaller angles of incidence. Thereby, the hybrid lens can be tailored to the ideal distance of the rotary system.

As mentioned in Section 2.1.3, the hybrid lens approach has the potential to be very cost effective: in contrast to fully separated TX and RX, only a single lens has to be fabricated, potentially coated and assembled.

The proposed hybrid lens works in on-axis configuration like FORJs. Future work will deal with off-axis optical links to improve flexibility. Only data rates in the range of $\mathrm{kbit}/\mathrm{s}$ to a few $\mathrm{Mbit}/\mathrm{s}$ have been demonstrated [2,3], which cannot compete with modern capacitive links [6]. Another interesting field of research is the realization of multi-channel optical wireless links similar to multi-FORJs. In nonrotary scenarios, data rates of several hundreds of $\mathrm{Gbit}/\mathrm{s}$ have already been demonstrated [55].

5. Conclusions

This work has shown the suitability of OWC for full-duplex, high-speed data transfer in on-axis rotary scenarios. The proposed hybrid lens is able to provide a sufficient RX signal level $P_{\mathrm{PD}\mathrm{PP}}$ of more than $-16.2 \mathrm{dBm}$ even for misalignments of up to $3 \mathrm{m}\mathrm{m}$ at a communication distance of $z=12.5 \mathrm{m}\mathrm{m}$. OWC is therefore able to provide a robust data transfer without the strict mechanical tolerances compared to FORJs. The results show a maximum power penalty resulting from optical crosstalk of $0.9 \mathrm{dB}$ within the FOV. The approach is promising since it allows low-cost fabrication. Besides the electronical components, only a single optical component is required that can be fabricated by injection molding in high volume.

6. Patents

The hybrid TX-RX lens and its derivatives are covered by several patents including DE102018205559 B3 [56] (WO19197343A1 [57]). Further patents are submitted.