The Perfect LiDAR – does it exist?
As a consultant, I have often been approached by unrealistic requirements from customers, and the LiDAR business is particularly prone to this. This is perhaps due to that the technology is new. A customer can ask for:
1. Large field-of-view
2. Long Range
3. High Resolution
4. Small form factor
5. Low price
6. Off-the-shelf (or max 2-3 weeks from production)
If a customer was to write a specification for a camera system, this would not happen, as there are known and better understood trade-offs, for example between a large field-of-view and high resolution. My impression is that this is not yet the case in the LiDAR industry. One example is the trade-off between range and field-of-view. A scanning LiDAR system does not necessarily have this problem (which could be part of it), but mechanical scanning systems are becoming, to be honest, old news (at least for automotive LiDAR). Reliability and manufacturing yields are too big issues. For a true solid-state LiDAR with a lens and a detector array, the range will depend on the lens entrance pupil – this is what collects light. And this needs to have a certain size to achieve a certain required range. This is well known in Radar where range ~ √A [1] (A being the antenna area).
For that reason, a miniaturized long-range LiDAR is very difficult to achieve. Even if you make the electronics and detector small, you will still need an optical system with a large (5-15 mm?) entrance pupil. It doesn’t matter if you are using FMCW (a type of coherent LiDAR – see my previous blogpost on LiDAR detection techniques) or not, you simply need the photons.
Furthermore, adding a large field-of-view (FOV) to this makes it even more difficult. A large field of view requires a short focal length (a strong lens), and a short focal length means a smaller aperture – you cannot have both. To understand why, it helps if you understand Etendue.
Etendue means that the product of the range of positions (area) and the range of angles will not decrease, for a lossless system. This is also known as “You cannot make a laser out of a flashlight” or “You cannot focus the sun into an infinitely small point”. It is one of the most important concepts in non-imaging optics and something that an optical designer faces in almost every project.
In imaging optics, the term etendue is not used but instead the LaGrange Invariant or the Optical invariant is used (read about the difference here: https://en.wikipedia.org/wiki/Lagrange_invariant) The difference is that these entities remain constant throught the system (thereby the name invariants). Let us compare the detector plane and the entrance pupil plane. The size of the detector and the size of the entrance pupil are constants. The angular range at the detector is given by the F#, which is not realistically smaller than 1.2. So, through the Lagrange Invariant, the FOV (or the range of angles at the entrance pupil) is given from these three constants.
We can formulate this in an equation. Let the collecting area (entrance pupil), be A, and a given size of the sensor D. We then have the maximum FOV as
The focal length, f, is related to the F# and the entrance pupil diameter as
Let’s say the link budget (the budget on how many photons are needed for a specific detection) requires a 5 mm entrance pupil and the sensor size is 10 mm, this means that you have a maximum field-of-view of 90 degrees. Fig 1 shows a plot of FOV vs aperture for a 10 mm sensor.
Figure 1. FOV vs aperture size for a 10 mm sensor and F# 1.2.
In addition, there is also distortion. By reducing the entrance pupil at higher angles, the range of angles can be increased. For example, 100% distortion (50 o to 100o) horizontally will give a 50% reduction in pupil area (corresponds to a pupil diameter reduction by 50% in one direction). Since range scales with the square root of the area, the range is reduced by √0.5 = 0.7, i.e 30% shorter range (in most cases). Anyway, most non-scanning LiDARs have roll-off of range for high angles, although it is not usually specified. To finish off the list I started with, the system form factor will depend on cost (number of aspheres or diffractives), resolution (number of surfaces in the system), sensor size (sets the lens diameter and limits FOV!), F# and how good your designer is. Let me know if you need help in finding someone really good 😊! Delivery time is never three weeks! Be realistic!
Furthermore, do not forget eye safety! This may limit your form factor, range or other key performance factors and is best to consider early in your design process.
Figure 2. Roll off for an f-theta lens (fisheye), following the distortion (right graph)
References
[1] The Radar Range Equation, https://www.radartutorial.eu/01.basics/The%20Radar%20Range%20Equation.en.html
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