LiDAR receiving three-element range-compensating lens optical design for automotive autonomy and advanced driver assist

Jason Mudge; Adam Phenis

doi:10.1117/1.OE.63.8.085101

5 August 2024 LiDAR receiving three-element range-compensating lens optical design for automotive autonomy and advanced driver assist

Jason Mudge, Adam Phenis

Author Affiliations +

Optical Engineering, Vol. 63, Issue 8, 085101 (August 2024). https://doi.org/10.1117/1.OE.63.8.085101

Abstract

A key sensor in the automotive stride for autonomy as well as improvement in advanced driver assist is the depth sensor light detection and ranging (LiDAR). However, one issue with LiDARs in general is the lack of dynamic range. Objects with high reflectivity and near range produce very high signals and low reflectivity far away targets yield a very low signal return. While the target reflectivity is generally not within the LiDAR designer’s control, the signal return as a function of range is when utilizing a range-compensating lens (RCL). This unique receiving lens disrupts the one-over-range-squared attenuation and provides a more constant return over range as initially demonstrated by Mudge [Appl. Opt., 58, 7921-7928, (2019)] using a two-element RCL. In this work, we provide an RCL design for automotive autonomy and/or advanced driver assist using a three-element RCL for improved range compensation over a two-element design while balancing complexity with greater than three elements. Additionally, we employ a more easily manufacturable variant RCL type given the anticipated high manufacturing volume.

1. Introduction

The genesis of a range-compensating lens (RCL) acting as the receiving lens is to alleviate the need for a very high dynamic range detector (pixel) and potentially provide a damage threshold safety mechanism, in light detection and ranging (LiDAR) sensors. Near objects with high reflectivity produce very high return signals, and far targets with low reflectivity yield relatively low return signals. Target reflectivity is generally not within the LiDAR designer’s control, but when utilizing an RCL, the signal return as a function of range is within the designer’s control. While ranges required for autonomous and advanced driver assist are under debate, far ranges often considered are on the order of 150 to 400 m for autonomy. Even in these relatively limited ranges, the one-over-range-squared attenuation still requires a high dynamic range detector. For example, if the LiDAR sensor (optics plus detector plus laser) has one unit of return energy due to the laser beam on a target at a range of 1 m, then at 100 m the energy will be one 10,000’th of a unit holding all other aspects fixed. There are two notable benefits of an RCL over other range adjusting methods, such as adjusting the detector gain, e.g., avalanche photodiode gain: (1) an RCL is passive, thus requiring no power nor an actuator nor a feedback loop and (2) it is instantaneous—focus or rather defocus at the detector plane is essentially instantaneous with a change in the target range.

In the initial RCL concept (circa $\sim 2019$ ¹^,²), a two-element RCL consisting of two lenses working in parallel with differing f-numbers and a mounting tube is developed and shown in Fig. 1. A first order set of RCL equations based on Fig. 1 are generated, and a set of simplified equations are developed by assuming the target range is much greater than the focal length. This simplified set of equations provides insight into the RCL mechanism and provides guidance in design given the particular application. Additionally, a variant RCL is provided, which is generally simpler to construct and more compact but is somewhat less efficient. A two-element variant RCL is shown in Fig. 2 and consists of one lens, one opaque ring, and one neutral density (ND) filter (as we will see, this variant RCL type is used in this work to develop the automotive three-element design). On the heels of this work, a verification of the equations developed in Ref. 2 is done by a comparing with a raytracing software and is detailed in Ref. 3 along with an example of an RCL over a long range on the order of kilometers. This yields an initial RCL theoretical foundation and leads to the design process of an n-element RCL—recognizing that the more elements provide a flatter return over range with the general drawback of a more complicated lens system with increased volume.

Fig. 1

A simplified rendition of a receiving two-element RCL. The object (target) is at infinity, and the outer lens is faster than the inner lens. The lens is shown in a cutaway view. Taken directly from Ref. 2.

Fig. 2

A receiving two-element variant RCL using one lens, an opaque ring, and an ND filter. The side view on left and down the optical axis on the right. Taken directly from Ref. 2.

Next, an RCL was incorporated in an incoherent time-of-flight LiDAR simulation showing the flattening of the signal-to-noise (SNR) as well as maintaining the minimum probability of detection (PD) over the range of 25 to 350 m—see Ref. 4. The general incoherent LiDAR design approach or method used in this particular design is provided in Ref. 5, which inverts the problem similarly to Albersheim’s relationship⁶ used in radar design where there is a relatively simple relationship between PD, probability of false alarm (PFA), and SNR. Additionally, Ref. 4 tackles the impacts of laser speckle on this particular design based on Goodman’s work from the mid-1960’s, as detailed in Ref. 7.

With significant modeling and simulation, or theoretical work, verified, and completed, a comparison between the theoretical results and experimental is the next logical step and often required by many as proof-of-concept. In Ref. 8, the experimental results for a two-element variant RCL are provided, which validates the theoretical results. The experimental RCL was designed to exercise each of the two-element RCL’s four (range) zones, as shown in Fig. 2 of Ref. 3 over a short 10 m range. The experiment was conducted in a 13 m long by 7.5 m wide climate-controlled environment. Interestingly, the experimental RCL’s design aligns with potential applications in gesture-based interface systems, which aim to supplant traditional input devices like keyboards and mice. This experimental work provides the needed comparison with Refs. 2 and 3 for community satisfaction. Additionally, the resulting data provide insight into the RCL mechanism by providing full images at the detector focus position for various ranges—see in Fig. 8 of Ref. 8. These images show the four zones over range for a two-element RCL as theoretically given in Fig. 2 of Ref. 3.

Up to this point, these analyses have assumed the return is perfectly diffuse (Lambertian) or at least diffuse. However, the work done in Ref. 9 examines a specular return to determine if there are any RCL advantages or benefits with this type of return and/or if any rejection of signal from other nearby LiDARs exists. The results show that since the light exiting the LiDAR is assumed effectively collimated with a slight divergence over distance, there is little benefit an RCL brings in reducing the potentially large specular returns and/or rejecting nearby LiDARs collimated light emission.

With a two-element RCL resting on a robust theoretical and experimental foundation, this current work uses these new found, verified, and validated concepts to design a three-element variant RCL. The design is nominally for an incoherent LiDAR with applications to automotive autonomy and/or advanced driver assist. This work utilizes the top-level requirements provided in Ref. 4, which are reproduced in the following section, to give continuity and focus to the work. This new three-element variant design shows a limited 15% projected solid angle variation over the prescribed range, whereas the two-element RCL in Ref. 4 has a 30% variation, which is a factor of two improvement.

2. Requirements

This RCL prescription stems from the need for automotive autonomy and/or advanced driver assist and is aligned with Ref. 4. However, there are two deviations with respect to the Ref. 4 prescription, which are: (1) the RCL is a three-element version, not a two-element, and (2) the variant RCL is invoked keeping it more compact and cost effective for this mass-produced application. We will not consider the operational wavelength selection at this juncture, only that the lenses must perform well at that particular wavelength, the detector (pixel) performance should be on par with what is used in Ref. 4, which is a silicon detector operating with at a laser wavelength of 905 nm, and the atmospheric transmission losses at 905 nm should be similar (for operational wavelength selection see Ref. 10).

2.1.

Top- or Sensor-Level Requirements

To provide continuity, the top- or sensor-level requirements are from Ref. 4 and reproduced in Table 1. The far range requirement is 350 m, which is a reasonable range for automotive autonomy, and the near range requirement is 25 m, recognizing that the perfectly diffuse return (assumed in Ref. 4 simulations) is coupled with some specular return, which tends to increase the signal in the near range, meaning the minimum range would generally be further reduced. The last two entries in Table 1, SNR and the projected solid angle ( $Ω$ ), are derived using Refs. 4 and 5 and based on required PD and PFA values shown in Table 1.

Table 1

Top- or sensor-level requirements.

PD min	0.99
PFA max	$10^{- 5}$
Range min	25 m
Range max	350 m
SNR min – derived	6.5
$Ω$ min – derived	$5.5 \times 10^{- 8} sr$

The last requirement listed in Table 1 comes from examining Fig. 11 of Ref. 4. This figure shows the $Ω$ versus range, which encompasses the magnification change in each of the elements and considers only those optical rays that impinge on the detector area. To meet the required PD, the $Ω$ must be at or above the minimum value shown in Fig. 11 of Ref. 4 or $5.5 \times 10^{- 8} sr$ favoring the low target reflectivity of 7% in place of 15% to give the LiDAR more robustness over the required range.

For further insight, the projected solid angle of the RCL is the cumulative sum of the solid angles of each lens, adjusted by the square of their respective magnifications. The projected solid angle is

Eq. (1)

Ω = m_{i}^{2} Ω_{i}^{'} + m_{m}^{2} Ω_{m}^{'} + m_{o}^{2} Ω_{o}^{'},

where

m_{i}

and

Ω_{i}^{'}

are the magnification of the inner element and its projected solid angle, respectively, as detailed in Eq. (1) of Ref. 2 or the approximation Eq. (10) in that same reference. This

Ω_{i}^{'}

is the commonly appreciated projected solid angle with the twist that the value is physically capped when very defocused as described in Ref. 2 and depicted in Fig. 2 of Ref. 3. The other two terms in Eq. (1) above are for the middle and outer lenses, respectively, with the added recognition that in a very defocused state they will provide little to no radiance on the detector – see Eqs. (2) and (11) of Ref. 2. The variant RCL differs from the non-variant in that the magnifications are equal (

m_{i} = m_{m} = m_{o}

), but the performance is similar. Again, the overarching design objective is for the RCL projected solid angle to be above the needed

5.5 \times 10^{- 8} sr

over the entire operating range.

2.2.

Three-Element Variant RCL

The primary deviation in the requirements from Ref. 4 is the lens/detector combination of a three-element variant RCL, which consists of one lens, two opaque rings, and two differing ND filters as opposed to a two-element (non-variant) RCL. In this three-element design, shown in Fig. 3, the outer ND filter is weaker than the inner ND filter. This essentially replaces the one main peak in the two-element RCL $Ω$ curve for two smaller peaks, which improves the flatness particularly over the middle ranges.

Fig. 3

A receiving three-element variant RCL using one lens, two opaque rings, and two ND filters. The side view on left and down the optical axis on the right.

3. Prescription and Performance

The resulting paraxial three-element variant RCL design or prescription is given in Table 2 and provides the derived f-number for the outer element for a single receive channel. The prescription for the two-element (non-variant) RCL detailed in Ref. 4 is reproduced in Table 3 primarily for comparison purposes.

Table 2

Three-element variant RCL prescription.

$f$ (lens focal length)	120 mm
$D_{l}$ (lens diameter)	97 mm
$D_{ond}$ (outer ND filter diameter)	55.7 mm
$D_{ind}$ (inner ND filter diameter)	32.5 mm
$T_{o}$ (outer ND filter transmission)	0.90 (OD 0.046)
$T_{i}$ (inner ND filter transmission)	0.80 (OD 0.097)
$t_{ir}$ (inner ring thickness)	0.5 mm
$t_{or}$ (outer ring thickness)	1.6 mm
$\sqrt{A_{D}}$ (detector characteristic size)	$40 μ m$
f-number (outer element) – derived	1.24

Table 3

Two-element (non-variant) RCL prescription from Ref. 4.

$f_{o}$ (outer lens focal length)	120 mm
$f_{i}$ (inner lens focal length)	150 mm
$D_{o}$ (outer lens diameter)	100 mm
$D_{i}$ (inner lens diameter)	40 mm
$t_{r}$ (ring thickness)	6 mm
$\sqrt{A_{D}}$ (detector characteristic size)	$40 μ m$
f-number (outer element) – derived	1.20

The RCL $Ω$ represents the performance and is plotted against the ordinate ( $A Ω$ product $\div A$ ) This ordinate is chosen to allow all the RCL components to be on the same plot as described in Sec. 4 of Ref. 2. The three-element RCL performance and the individual elements that make up the RCL performance are provided in Fig. 4. Additionally, the two-element RCL performance is shown in Fig. 5 and is determined using raytracing to generated data as opposed to the analytical version, as in Fig. 11 of Ref. 4. According to Table 1, the projected solid angle must be at or above the needed $5.5 \times 10^{- 8} sr$ over the prescribed range to meet requirements and is shown in Figs. 4 and 5 by the dashed line.

Fig. 4

The three-element variant RCL design performance.

Fig. 5

The two-element RCL performance.

For comparison purposes, both the RCL performance curves without the respective components are provided in Fig. 6 with the $y$ -axis expanded to provide more fidelity. From this figure, it is clear that there is a minute dip below the requirement at approximately 170 m for the two-element RCL. This is not the case for the three-element variant RCL. Additionally, the three-element variant is significantly flatter over range and is closer the requirement over the full range—improved performance. The three-element variant RCL has a peak of $6.5 \times 10^{- 8} sr$ at 300 m and a minimum or valley at $5.5 \times 10^{- 8} sr$ at 350 m giving a $\sim 15 %$ variation over the range, and the two-element has a peak at $7.6 \times 10^{- 8} sr$ at 310 m and a valley of $5.4 \times 10^{- 8} sr$ at 170 m with a range variation of $\sim 30 %$ . This limited $\sim 15 %$ projected solid angle variation over range for the three-element RCL allows the detector dynamic range to better to handle large target reflectivity variations, which generally cannot be controlled by the designer.

Fig. 6

The three-element variant RCL performance over range with the two-element RCL comparison from Ref. 4.

4. Design Method

In Zemax, the ray trace performance curves are generated in a relative radiometric sense and not in an absolute sense. To obtain the absolute position of the curve or height, as shown in Figs. 4 Fig. 5–6, we extend Eqs. (10), (11), and (12) from Ref. 2 to three elements but only for the far range where all the elements act in unison—essentially the last line in Eqs. (10) and (11). The equation represents the incoherently summed projected solid angles due to each element in the far range and is

Eq. (2)

Ω |_{z = z_{\max}} = T_{i} (\frac{f}{z_{\max}})^{2} \frac{π}{4} (\frac{D_{ind}}{f})^{2} + T_{o} (\frac{f}{z_{\max}})^{2} \frac{π}{4} ((\frac{D_{ond}}{f})^{2} - (\frac{D_{ind} + 2 t_{ir}}{f})^{2}) + (\frac{f}{z_{\max}})^{2} \frac{π}{4} ((\frac{D_{l}}{f})^{2} - (\frac{D_{ond} + 2 t_{or}}{f})^{2}) .

This provides a radiometric anchor to the ray trace data requires. For this design, we set the far range as given in Table 2 to the max range, which is $z = z_{\max} = 350 m$ (notice range squared in the dominator of Eq. (2) when all elements act in unison), and using Eq. (2), we design the RCL to provide the needed solid angle of $5.5 \times 10^{- 8} sr$ at this maximum range. This is an iterative design in that we are driving for a relatively flat curve using the ray trace data results but must also have the correct overall radiometry as provided by Eq. (2). When the iteration converges, the performance curves are generated as in Figs. 4 Fig. 5–6.

The opaque ring thickness is designed small relative to the two-element (non-variant) RCL and is an aspect that differs between the variant and non-variant RCLs. This is because the non-variant has a faster lens on the outside, which pulls the rays further toward the center and onto the detector (pixel), which has the effect of reducing the dip(s), whereas the variant does not have this effect. Additionally, the (non-variant) RCL rings act as a mounting tube, which provides mechanical structure to the RCL limiting the ring thinness—see Fig. 1. Therefore, to have an acceptable dip given the requirement in the three-element variant RCL, the opaque rings are reduced in size relative to the two-element (non-variant) RCL. For this design, we have set the opaque ring thickness to 0.5 mm for the inner ring and 1.6 mm for the outer ring, which is much less than 6 mm for the two-element (non-variant) RCL. Despite their reduced size, the opaque rings will adequately cover the ND coating transition regions with either a coating or a lithographic masking process. Meaning, the ND coating transition region’s light is blocked with an opaque ring, which assures a consistently reliable design implementation. Finally, the lens volume is reduced in the three-element variant RCL due, in part, to the ring thickness. The volume for the two-element (non-variant) RCL is $3.5 \times 10^{5} {mm}^{3}$ , whereas the volume for the three-element variant RCL is $3.0 \times 10^{5} {mm}^{3}$ , which is a volume drop of approximately 15%. This volume reduction tends to be beneficial in mass produced item—another positive for the variant RCL. However, an RCL volume is increased over a non-RCL having the same far range, as described in Ref. 2, since it has no ND filters and/or rings obstructing the return signal.

The lens’ optical design is initially carried out in a sequential manner and subsequently transformed into a non-sequential analysis model. Within the non-sequential framework, a detector conforming to the design parameters, as provided in Tables 2 and 3, is modeled. The relative number of rays on the detector is then represented as a function of the target range for each distinct element, relying solely on non-sequential raytracing. To streamline the process, the sources in the non-sequential model are configured for their respective elements, where each source’s power is relatively determined by the ratio of its elements area to the total area of the RCL diameter. Additionally, in the case of the non-variant type RCL, the areas are scaled based on the respective ND filters.

Non-sequential ray tracing is employed because it disregards rays that do not reach the detector, as illustrated in Fig. 2 of Ref. 3 and Fig. 8 of Ref. 8 (the Ref. 8 appendix provides the Zemax implementation steps). The total flux default merit function as generated with the optimization wizard is calculated as a function range for each element’s source.¹¹ Sources not intended for tracing are assigned zero rays. It is possible to trace all sources simultaneously; however, this approach obscures the individual contribution of each element to the overall response and may not allow the designer the ability to determine what can be adjusted to better flatten the response. The summed rays reaching the detector do not reflect the one-over-range-squared attenuation, but this is addressed in post processing by multiplying the projected solid angles relative amounts from the ray trace by the respective magnifications squared and summing—see Eq. (1). These then anchored and range attenuated values represent the final data for each element. Each of these values is plotted in the figures above, and their incoherent summation yields the RCL’s performance.

5. Summary

The LiDAR receiving three-element variant RCL design provided is directed at automotive autonomy and/or advanced driver assist incoherent LiDARs. This design shows the performance to be significantly flatter over than the two-element RCL for the required range, thus leaving more dynamic range for large target reflectivity variations. Furthermore, this three-element variant design is more compact and easier to manufacture in high volumes over the two-element (non-variant) RCL design.

Disclosures

The author declares no conflicts of interest.

Code and Data Availability

There is no code or data publicly available.

Acknowledgments

Parts of this research was supported by the optics consulting company Golden Gate Light Optimization, LLC, and AMP Optics, LLC, which is an optical design and analysis consultancy. The components used in the RCL construction should be properly recycled when its life cycle has passed. In memory of Eugene Cross.

References

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J. Mudge, “Range correcting radiometric lens, method of optical design, and range finding system using same,” (2024). Google Scholar

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Biography

Jason Mudge is a principal at the optics consulting firm Golden Gate Light Optimization, LLC. He received his BS and PhD degrees in mechanical engineering from the University of California at Davis and his initial MS degree in mechanical engineering from Stanford University and a second MS degree in optical sciences from University of Arizona. He is the author of more than 30 technical publications. His current research interests include imaging systems and image quality, LiDAR, range-compensating lens, interferometry, polarimetry, radiometry, and sampling/aliasing. He is a senior member of SPIE and a member of Optical Society of Southern California.

Adam Phenis, of AMP Optics, is an optical engineering consultant. His expertise spans from extreme ultraviolet to very long-wavelength infrared, excelling in practical design, delivering on-time, on-budget solutions for electro-optical systems, high-power laser material processing systems, and more. His systems operate in diverse environments—from high-power laser labs to harsh conditions, e.g., space and airborne. As a senior SPIE member and OEOSC board member, he also contributes to the optical industry and ISO optics standards. He holds a BS degree in optics from the University of California at Davis and an MS degree from the University of Arizona under Professor James C. Wyant.

CC BY: © The Authors. Published by SPIE under a Creative Commons Attribution 4.0 International License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation Download Citation

Jason Mudge and Adam Phenis "LiDAR receiving three-element range-compensating lens optical design for automotive autonomy and advanced driver assist," Optical Engineering 63(8), 085101 (5 August 2024). https://doi.org/10.1117/1.OE.63.8.085101

Received: 4 April 2024; Accepted: 12 July 2024; Published: 5 August 2024

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KEYWORDS

LIDAR

Design

Sensors

Solids

Optical filtering

Reflectivity

Ray tracing

1.

Introduction