Total Internal Reﬂection Lens for Optical Wireless Receivers

: This work considers the use of a freeform total internal reﬂection (TIR) lens for optical concentration and provides for the ﬁrst time experimental results in the context of optical wireless communications (OWC). The lens is placed on a surface-mounted device (SMD) avalanche photodiode (APD) to minimize position tolerances and simplify assembly. The lens achieves a concentration ratio of g o = 44.7 (16.5 dB) within the FOV center and exhibits an acceptance angle of ± 5 ◦ . The TIR lens approach is validated by comparing eye diagrams and bit error ratios (BER) of a receiver with and without a TIR lens. For the measurements, non-return-to-zero (NRZ) on-off keying (OOK) signals are transmitted with a data rate of 1.289 Gbit/s.


Introduction
The growth of the mobile data traffic [1,2] challenges our wireless communication infrastructure. Moreover, new applications such as the tactile Internet [3,4] demand lowlatency data transfer. Spatially well-confined optical wireless communication (OWC) channels are able to mitigate interference issues and allow for real-time data transfer with high reliability. Therefore, for certain applications, OWC can be an alternative to radio frequency-based technologies.
Optical wireless transceivers use photodiodes (PDs) such as P-i-N photodiodes (PIN PDs) [5] or avalanche photodiodes (APDs) [6]. To improve range and coverage, additional receiver lenses are used for optical concentration. For the sake of simplicity and cost, simple spherical and aspherical lenses have been used in industry and science [5,[7][8][9][10][11][12][13]. A specific example is a hemispherical lens, which is well suited for large FOVs [11,13,14], [15] (pp. 103-104). The restricted shape of simple lenses limits design freedom and concentration ratio. For instance, it is barely possible to get a constant concentration ratio over the FOV with steep FOV edges. Moreover, large lenses are not suited for most practical applications due to their costly production. Freeform lenses have the potential to overcome these limitations of spherical lenses.
Different freeform lenses have been studied over the last decades in the context of OWC. The compound parabolic concentrator (CPC) exhibits high efficiency and has been widely employed in science in various configurations [13,14], [15] (pp. 104-106). A successor of the CPC is the dielectric totally internally reflecting concentrator (DTIRC), that was proposed in 1987 [24]. The main drawback of the CPC and the DTIRC is that their height and volume scale with increasing output aperture and decreasing acceptance angle [15] (pp. 104-122). Therefore, large PDs and small FOVs lead to bulky concentrators. Garcia Marquez et al. [25] and Valencia-Estrada et al. [26] described catadioptric lenses in 2015 and 2018, respectively. These lenses use at least one surface for refraction and TIR at the same time. Other works dealt with fisheye lenses [27] in 2014 and fly-eye lenses [28] in 2015. Moreover, there were practical and theoretical works on different hybrid transmitter-receiver lenses for low-range OWC [29,30] in 2018 and 2020, respectively. A TIR concentrator for OWC was proposed in 2019 [31]. However, the investigations were only theoretical and limited to simulations.
Although a growing interest in freeform lenses for OWC exists, there is a substantial lack of experimental data. Only a part of the works describes the concentrators in detail and only a few provide an experimental characterization. An experimental characterization is crucial for identifying relevant practical problems.
This paper considers freeform TIR lenses for OWC receivers. In this work, a lens prototype is fabricated and experimentally characterized. In contrast to previous works, we design the lens for a flat SMD package and mount it directly on top of it. For the first time, we show practical performance values in the context of OWC, as well as practical challenges and limits of the TIR lens approach.
The rest of the paper is organized as follows. Section 2 considers the concept of the TIR lens and the assembly. Section 3 presents the measurement setup and the results from the optical performance measurements and the data transfer experiment. The results are discussed in Section 4. Section 5 provides a summary.

Concept
The optical concentration ratio g o can be defined as the ratio of the detected power with the lens divided by the detected power without the lens. The maximum concentration ratio g o max due to the conservation of Étendue can be expressed according to Equation (1) [16] (pp. [18][19][20][21][22]. Therefore, n 1 and n 2 denote the refractive indices of the mediums at the concentrator's input and output. The angles θ in and θ out denote the maximum input and output angles, respectively. (1) Figure 1 shows a cross-sectional concept render of the TIR lens and sketches incident rays for −5 • , 0 • , and +5 • that reach the PD. The lens has a central refracting region and a surrounding TIR groove. TIR allows for larger deflection angles without large Fresnel losses in contrast to refraction [32]. Thus, it is suited to couple rays under larger angles θ out to the PD chip. Therefore, the TIR lens has the potential to achieve a high concentration ratio.
In our setup, an APD is employed. However, the TIR lens is also compatible to other types of PDs. The PD features a cuboidal surface-mounted device (SMD) package. The PD chip has a diameter of 0.5 mm, which corresponds to an active area of approximately 0.2 mm 2 . The chip is bonded onto a carrier and molded into a transparent polymer. The polymer top is flat. Because the polymer has no integrated lens, the receiver performance is prone to misalignments between the PD and the lens. Even a displacement of 200 µm takes a strong influence due to the small chip size. The problem is addressed by mounting the lens directly on the SMD package. For this purpose, the lens has mounting structures at the bottom that fit the SMD package. The mounting structures are located between the center and the TIR region. The size of the center region is determined by the SMD package. For handling and manufacturing, the lens features a cylindrical-shaped top. The flat top surface makes the lens robust against dirt particles. In our setup, an APD is employed. However, the TIR lens is also compatible to other types of PDs. The PD features a cuboidal surface-mounted device (SMD) package. The PD chip has a diameter of 0.5 mm, which corresponds to an active area of approximately 0.2 mm 2 . The chip is bonded onto a carrier and molded into a transparent polymer. The polymer top is flat. Because the polymer has no integrated lens, the receiver performance is prone to misalignments between the PD and the lens. Even a displacement of 200 µm takes a strong influence due to the small chip size. The problem is addressed by mounting the lens directly on the SMD package. For this purpose, the lens has mounting structures at the bottom that fit the SMD package. The mounting structures are located between the center and the TIR region. The size of the center region is determined by the SMD package. For handling and manufacturing, the lens features a cylindrical-shaped top. The flat top surface makes the lens robust against dirt particles. Figure 2 illustrates the geometrical surface construction. The lens is designed by using the edge-ray principle and subsequent optimization of the angle of incidence θi = [θa′, θa] to flatten go within the FOV. Therefore, θa denotes the acceptance angle and it holds θa′ < θa. The construction procedure takes the flat PD surface into account. The interface is modeled with the parametric curve . For this work, the curve is a simple straight line, where the parameter x refers to the x-coordinate. In general, the shape might be more complex and an arbitrary parameter might be chosen.   Figure 2 illustrates the geometrical surface construction. The lens is designed by using the edge-ray principle and subsequent optimization of the angle of incidence θ i = [θ a , θ a ] to flatten g o within the FOV. Therefore, θ a denotes the acceptance angle and it holds θ a < θ a . The construction procedure takes the flat PD surface into account. The interface is modeled with the parametric curve C(x). For this work, the curve is a simple straight line, where the parameter x refers to the x-coordinate. In general, the shape might be more complex and an arbitrary parameter might be chosen. In our setup, an APD is employed. However, the TIR lens is also compatible to other types of PDs. The PD features a cuboidal surface-mounted device (SMD) package. The PD chip has a diameter of 0.5 mm, which corresponds to an active area of approximately 0.2 mm 2 . The chip is bonded onto a carrier and molded into a transparent polymer. The polymer top is flat. Because the polymer has no integrated lens, the receiver performance is prone to misalignments between the PD and the lens. Even a displacement of 200 µm takes a strong influence due to the small chip size. The problem is addressed by mounting the lens directly on the SMD package. For this purpose, the lens has mounting structures at the bottom that fit the SMD package. The mounting structures are located between the center and the TIR region. The size of the center region is determined by the SMD package. For handling and manufacturing, the lens features a cylindrical-shaped top. The flat top surface makes the lens robust against dirt particles. Figure 2 illustrates the geometrical surface construction. The lens is designed by using the edge-ray principle and subsequent optimization of the angle of incidence θi = [θa′, θa] to flatten go within the FOV. Therefore, θa denotes the acceptance angle and it holds θa′ < θa. The construction procedure takes the flat PD surface into account. The interface is modeled with the parametric curve . For this work, the curve is a simple straight line, where the parameter x refers to the x-coordinate. In general, the shape might be more complex and an arbitrary parameter might be chosen.  Initially, the point P 0 1,j is defined. The input vector v 1,j is derived from the angle of incidence θ i . The vector v 1,j is calculated from v 1,j by the taking the refraction at the flat top surface into account. The point P 1,j is given by the intersection of v 1,j and a tangential vector t 1,j−1 . The tangential vector t 1,j−1 is derived by rotating the normal n 1,j−1 of the previous point P 1,j−1 around the y-axis by π/2. In general, it is not possible to directly calculate the normal vector n 1,j , because the ray does not target the final point P t directly. It is refracted at the PD package top at an unknown point P PD 1,j . The output vector o 1,j is unknown as well since it depends on P PD 1,j . The point P PD 1,j is determined as follows: We define the law of refraction at the point P PD 1,j according to Equation (2), where ∠ describes the angle between the vectors in the brackets. We assume n 1 = 1 and n 2 = 1.5. The problem is solved by expressing all variables as a function of C(x) and bringing the equation into the form C(x) = 0. In the next step, the Newton algorithm is applied to extract the coordinate x j . The point P PD 1,j corresponds to C x j . The vectors o 1,j and v 1,j are given by o 1,j = P t −C(x j ) and v 1,j = C x j − P 1,j . The normal vector n PD 1,j is the rotated tangential vector of the function C x = x j according to Equation (3).

Geometrical Construction
Thus, R α (π/2) denotes the rotation matrix. For a simple flat SMD package, n PD 1,j is parallel to the z-axis. After P PD 1,j is known, v 1,j and o 1,j are calculated. By using this procedure, a 2D curve of the lens is generated. The number of calculated points per curve N is 1200. The calculated curves are interpolated using cubic non-uniform rational basis splines (NURBS) [32]. A 3D representation is generated by rotating the curve around the z-axis. A render image of the design is shown in Figure 1. Figure 3 shows a lens prototype. It is fabricated with an injection molding process. The tool was formed by an ultra-precision diamond-turning/-milling process. The milling tip features a radius of 100 µm. The surface roughness is classified as SPI A-2, i.e., the optical interfaces have an average surface roughness R a of 25 nm to 50 nm. The lens material is black polycarbonate with a refractive index of n = 1.57. The material absorbs ambient light for λ < 700 nm and provides a high transmittance of T > 0.9 for λ > 800 nm. This makes the material an ideal candidate for near-infrared (NIR) signals. The lens has a height of 6.9 mm and an optical input aperture of Ø8.8 mm. The planarity of the top surface can be seen from the reflection. The diameter of the lens cavity is 3.1 mm. The inner sides of the cavity are inclined by 2 • , which is required for the diamond-turning.

Frabication and Assembly
to directly calculate the normal vector , , because the ray does not target the final point directly. It is refracted at the PD package top at an unknown point , . The output vector , is unknown as well since it depends on , . The point , is determined as follows: We define the law of refraction at the point , according to Equation (2), where ∠ describes the angle between the vectors in the brackets. We assume n 1 = 1 and n 2 = 1.5.
The problem is solved by expressing all variables as a function of and bringing the equation into the form = 0. In the next step, the Newton algorithm is applied to extract the coordinate xj. The point , corresponds to . The vectors , and , are given by , = C(xj) and , = , . The normal vector , is the rotated tangential vector of the function = according to Equation (3).
Thus, π/2 denotes the rotation matrix. For a simple flat SMD package, , is parallel to the z-axis. After , is known, , and , are calculated. By using this procedure, a 2D curve of the lens is generated. The number of calculated points per curve N is 1200. The calculated curves are interpolated using cubic non-uniform rational basis splines (NURBS) [32]. A 3D representation is generated by rotating the curve around the z-axis. A render image of the design is shown in Figure 1. Figure 3 shows a lens prototype. It is fabricated with an injection molding process. The tool was formed by an ultra-precision diamond-turning/-milling process. The milling tip features a radius of 100 µm. The surface roughness is classified as SPI A-2, i.e., the optical interfaces have an average surface roughness Ra of 25 nm to 50 nm. The lens material is black polycarbonate with a refractive index of n = 1.57. The material absorbs ambient light for λ < 700 nm and provides a high transmittance of T > 0.9 for λ > 800 nm. This makes the material an ideal candidate for near-infrared (NIR) signals. The lens has a height of 6.9 mm and an optical input aperture of Ø8.8 mm. The planarity of the top surface can be seen from the reflection. The diameter of the lens cavity is 3.1 mm. The inner sides of the cavity are inclined by 2°, which is required for the diamond-turning.  The lens is fixed to the PD by using a low-outgassing adhesive, which is transparent at the wavelength of communication. Because the glue is not applied to the top of the PD, the glue does not necessarily have to be transparent. However, it must be low-outgassing to avoid an undesired covering of the lenses' center cavity. When the lens is placed on the PD, it wiggles slightly, indicating that the mounting structures do not fit exactly. This issue can be eliminated by a selective revision of the tool.  Figure 4 shows a sketch of the measurement setup. A LED is placed at a distance of Z o = 600 mm in front of the receiver. The LED features a center wavelength of 850 nm and an optical output power of 260 mW. The optical receiver is rotated around the y-axis by an angle θ i . To remove the residual influence of the LED profile, we measure the photocurrent I PD over θ i with and without the TIR lens and define g o as the ratio of both results. For the I PD measurement, we measure the voltage drop over a resistor in series to the APD with the multimeter Keysight U1272A. We temporally deactivate the APD bias to avoid corruption by the APD's avalanche effect (M = 1). The optical power is derived from I PD by using the PD responsivity R = 0.5 A/W. The optical power with lens divided by center power without the lens is defined as the optical concentration ratio g o . The small angular error resulting from the distance between the lens surface and the axis of rotation is taken into account during the measurements.

Frabication and Assembly
at the wavelength of communication. Because the glue is not applied to the top of the PD, the glue does not necessarily have to be transparent. However, it must be low-outgassing to avoid an undesired covering of the lenses' center cavity. When the lens is placed on the PD, it wiggles slightly, indicating that the mounting structures do not fit exactly. This issue can be eliminated by a selective revision of the tool. Figure 4 shows a sketch of the measurement setup. A LED is placed at a distance of Zo = 600 mm in front of the receiver. The LED features a center wavelength of 850 nm and an optical output power of 260 mW. The optical receiver is rotated around the y-axis by an angle θ i . To remove the residual influence of the LED profile, we measure the photocurrent IPD over θ i with and without the TIR lens and define g o as the ratio of both results. For the IPD measurement, we measure the voltage drop over a resistor in series to the APD with the multimeter Keysight U1272A. We temporally deactivate the APD bias to avoid corruption by the APD's avalanche effect (M = 1). The optical power is derived from IPD by using the PD responsivity R = 0.5 A/W. The optical power with lens divided by center power without the lens is defined as the optical concentration ratio go. The small angular error resulting from the distance between the lens surface and the axis of rotation is taken into account during the measurements.  Figure 5a shows the results of the IPD measurement. The photodiode without the lens shows quite similar IPD values for the observed θ i . In contrast, the TIR lens shows a strong dependence from θ i . It increases IPD within the center from 303 nA without the lens to 13.55 µA. Figure 5b shows the simulated and measured optical concentration ratio go over the angle of incidence θ i . The simulation is based on Monte Carlo ray tracing within Zemax Optic Studio 17. For each simulation point, 10 6 rays are traced. The simulated lens features a center gain of up to 18.5 dB and enables data reception for θi < 7.5°. For larger angles, the gain drops rapidly. The simulated acceptance angle (−3 dB) is = 5.5°. From the experiment, we derive a concentration ratio of go = 10lg (44.7) = 16.5 dB at the center, which is 2 dB lower than the simulation. The measured curve has an acceptance angle of = ±5°. Moreover, it drops slower outside the FOV compared to the simulated curve.

Optical Results
Within our laboratory, the ambient light is 38 nA without the TIR lens and 0.8 nA with the TIR lens. This corresponds to an attenuation of ambient light by 16.8 dB.   Figure 6 shows the experimental setup for data transmission. For the validation of the TIR lens, we transmit a non-return-to-zero (NRZ) on-off keying (OOK) signal through an optical wireless channel with and without the TIR lens. This is close to the baud rate of 8B/10B-coded 1000BASE-T networks that are highly relevant for industrial real-time communication protocols. A Stratix V field-programmable gate array (FPGA) is used to generate a pseudo-random bit sequence (PRBS-7) at the transmitter. The transmitter converts the signal into the optical domain. Thus, a custom laser driver converts the voltage signal from the FPGA into a current and directly drives an edge-emitting laser diode (LD) to modulate the optical intensity. The center wavelength of the LD is 850 nm. The receiver printed circuit board (PCB) contains an APD for signal detection. The APD bias is varied  Within our laboratory, the ambient light is 38 nA without the TIR lens and 0.8 nA with the TIR lens. This corresponds to an attenuation of ambient light by 16.8 dB. Figure 6 shows the experimental setup for data transmission. For the validation of the TIR lens, we transmit a non-return-to-zero (NRZ) on-off keying (OOK) signal through an optical wireless channel with and without the TIR lens. This is close to the baud rate of 8B/10B-coded 1000BASE-T networks that are highly relevant for industrial real-time communication protocols. A Stratix V field-programmable gate array (FPGA) is used to generate a pseudo-random bit sequence (PRBS-7) at the transmitter. The transmitter converts the signal into the optical domain. Thus, a custom laser driver converts the voltage signal from the FPGA into a current and directly drives an edge-emitting laser diode (LD) to modulate the optical intensity. The center wavelength of the LD is 850 nm. The receiver printed circuit board (PCB) contains an APD for signal detection. The APD bias is varied within a control loop to adjust the APD gain depending on the incident optical power. For strong input signals, the bias and thus the APD gain is reduced to avoid saturation. The photocurrent is converted into a voltage using a transimpedance amplifier (TIA). The eye diagram is measured using a digital oscilloscope of type Lecroy 10-36Zi. The Stratix V FPGA is used to carry out a bit error rate test (BERT) at the receiver. The measurements are repeated for Z o = 50 cm, Z o = 100 cm and Z o = 150 cm with and without the TIR lens. Table 1 summarizes the optical wireless link parameters.  Figure 6 shows the experimental setup for data transmission. For the validation of the TIR lens, we transmit a non-return-to-zero (NRZ) on-off keying (OOK) signal through an optical wireless channel with and without the TIR lens. This is close to the baud rate of 8B/10B-coded 1000BASE-T networks that are highly relevant for industrial real-time communication protocols. A Stratix V field-programmable gate array (FPGA) is used to generate a pseudo-random bit sequence (PRBS-7) at the transmitter. The transmitter converts the signal into the optical domain. Thus, a custom laser driver converts the voltage signal from the FPGA into a current and directly drives an edge-emitting laser diode (LD) to modulate the optical intensity. The center wavelength of the LD is 850 nm. The receiver printed circuit board (PCB) contains an APD for signal detection. The APD bias is varied within a control loop to adjust the APD gain depending on the incident optical power. For strong input signals, the bias and thus the APD gain is reduced to avoid saturation. The photocurrent is converted into a voltage using a transimpedance amplifier (TIA). The eye diagram is measured using a digital oscilloscope of type Lecroy 10-36Zi. The Stratix V FPGA is used to carry out a bit error rate test (BERT) at the receiver. The measurements are repeated for Zo = 50 cm, Zo = 100 cm and Zo = 150 cm with and without the TIR lens. Table 1 summarizes the optical wireless link parameters.

Data Transmission Results
Figure 7a-c show the measured eye diagrams for the channel without the lens and Figure 7d-f for the channel with the lens. Table 2 lists the BERT data. The BER measurement is clipped at 10 −12 , when no bit errors occurred after 1000 s. For all measurements, the eye opening increases with a TIR lens. The improvement varies with the received signal power and thus the APD bias that is adjusted to avoid saturation. The improvement is clearly visible for rather high distances with a weak input signal. For instance, at Z o = 150 cm, there is practically an error-free transmission with the TIR lens, but no transmission at all without it. The avalanche effect leads to stronger noise at the high level compared to the low level [33] (pp. 58-60). This can be seen in particular for the weak signals in Figure 7b,c, where the APD gain is high.

Data Transmission Results
Figure 7a-c show the measured eye diagrams for the channel without the lens and Figure 7d-f for the channel with the lens. Table 2 lists the BERT data. The BER measurement is clipped at 10 −12 , when no bit errors occurred after 1000 s. For all measurements, the eye opening increases with a TIR lens. The improvement varies with the received signal power and thus the APD bias that is adjusted to avoid saturation. The improvement is clearly visible for rather high distances with a weak input signal. For instance, at Zo = 150 cm, there is practically an error-free transmission with the TIR lens, but no transmission at all without it. The avalanche effect leads to stronger noise at the high level compared to the low level [33] (pp. 58-60). This can be seen in particular for the weak signals in Figure 7b,c, where the APD gain is high.

Performance
The measurement from Figure 5b shows a high concentration ratio of 44.7 (16.5 dB) despite the small form factor of the lens. The eye diagram measurements and the BERT prove that the optical concentration improves the data transmission drastically. In our tests, the TIR lens increased the eye opening, thereby improving the BER. The eye opening is not directly proportional to the optical concentration ratio due to adjusted APD gain. The range measurements were limited by our optical bench to a maximum distance of 150 cm. At a first approximation, a concentration ratio of 44.7 corresponds to a range improvement by a factor of approximately √ 44.7 ≈ 6.7.
Nevertheless, the measured values are well below the simulation values according to Figure 5b. The shift of the profile and the fact that the mounting structures did not fit perfectly indicate that there is a misalignment between the lens and the PD chip. Because the gain falls slower outside the FOV than simulated, it is likely that the focus point of the lens is larger than intended, which could be caused by an additional z-displacement between lens and PD. A part of those deviations is caused by the finite shape accuracy of the tooling process and roundings at the edges of the surfaces. A precise analysis is challenging and costly, especially for the inner surfaces of the cavity. Therefore, such analyses are out of the scope of this work. A rework of the mounting structures is one way to reduce the position tolerances. However, the final position accuracy also depends on the spatial tolerances of the SMD packages. Those are typically in the range of a few hundredths of µm due to the sawing process that is used for device separation. Active positioning is another possibility to address the mechanical tolerances. However, this makes the fabrication more complex and thus is not desired for mass fabrication.
A significant drawback of our setup is the flat surface of the SMD package. In contrast to a package with an integrated convex lens, the optical concentration ratio is reduced, because the flat surface reduces the angular acceptance region and the TIR lens cannot fully profit from the higher refractive index n 2 of the PD package surrounding the active chip according to Equation (1). Moreover, an integrated lens would further reduce the influence of a displacement between the lens and the PD chip.

Comparison with CPC and DTIRC
The planar SMD package might be suited for a CPC or DTIRC. Compared to a DTIRC, the TIR lens covers the PD chip completely and reduces the ambient light photocurrent to a minimum by material absorption. The CPC and DTIRC most likely achieve a higher concentration ratio due to the direct contact to the PD package. Nevertheless, they feature challenges including precise alignment, the need for an index matching gel, and excess glue at the output aperture. Moreover, CPCs and DTIRC are higher and might not be suited for all applications. For the present setup, a realistic height is in the range of multiples of tens of millimeters depending on the realization. In comparison, the presented TIR lens is a compact concentrator with height of only 6.9 mm. It seems especially favorable for PD packages with non-flat surfaces, such as packages with integrated lenses that are not suited for the CPC or DTIRC. It allows for an easy assembly because the glue is not applied into the ray path and there is no issue with excess glue. The flat top of the TIR lens enables integration into a transceiver case without the need for an additional front plate.

Conclusions
This article considers the design and experimental characterization of a freeform TIR lens as an optical concentrator for OWC receivers. The experimental characterization of the lens prototype provides practical performance merits for the first time in the context of OWC and proves the general feasibility for data transmission. The proposed TIR lens is a compact concentrator with an entrance aperture of Ø8.8 mm and a height of only h = 6.9 mm. It fits on a cuboidal SMD package and requires no index-matching gel or glue within the optical path. Therefore, it simplifies the assembly compared to a corresponding CPC. The acceptance angle θ a of the prototype is ±5 • . It achieves a concentration ratio of g o = 44.7 (16.5 dB) within the FOV center. Small concentrators such as the proposed TIR lens require precise positioning. The investigations suggest that more precise mounting structures can improve the assembly. One drawback of the TIR lens is the complex lens structure that makes the characterization of the shape accuracy difficult, especially within the lens cavity.