You Say Radar, I Say Lidar

We’re going to return to land this week and talk about how police use the principles of radar to measure a vehicle’s speed. A policeman can use a radar gun to send an electromagnetic pulse  at a specific frequency from his radar’s transmitting antenna towards a target vehicle. In the U.S., the most common radar frequency is in the 33.4 GHz to 36 GHz range. The wave bounces off the target and returns a short time later to the radar’s receiving antenna. This information can be used to calculate how fast the vehicle is going. If the car is exceeding the speed limit, the officer can issue a ticket.

We learned a couple weeks ago that we can use the Doppler shift to determine the speed of an object. Let the frequency of the radar pulse leaving the policeman’s radar detector equal f_p and let the radar return frequency be f_s. We assume that the policeman is traveling at a speed of v_p and gaining on the speeder, who is going v_s. The constant c represents the speed of light. Then the Doppler equation told us that:

f_s = f_p (c + v_s) / (c – v_p)

or solving for v_s we get

f_s

— (c – v_p) – c = v_s

f_p

Unfortunately, there are some serious limitations for radar in this application. The beam of the radar gun spreads out from its point of origin. By the time it has traveled 1000 feet, it has spread to a width of 250 feet, kind of like a beam spreads from a flashlight. Eventually it reaches a limit, and so its actual field of view looks like an ellipsoid or lobe, as shown in the graphic. Now suppose that you’re trying to measure the speed of a vehicle in heavy traffic. If the object isn’t very close to your radar gun, then the width of the beam may have spread enough to include other vehicles. You will be unable to issue a valid speeding ticket if the measurement is confounded by radar returns from multiple vehicles.

One way around this problem is to use a different kind of wave. Instead of a radio wave, one can use an EM wave at an optical frequency using a laser. LASER stands for Light Amplification by Stimulated Emission of Radiation. A laser is an optical device that emits a high-intensity narrow beam of light. Laser light is special. Unlike ordinary light, laser light emits photons at a single wavelength (color) and are phase-aligned in space and time. They all also travel in just one direction as a high intensity beam. While people may think lasers exist in only the visible part of the spectrum (430 THz to 750 THz), they have actually been created in a much broader range of frequencies (28 THz to 1554 THz).

Lasers are used to create LIDAR systems. LIDAR stands for Light Detection And Ranging, or Light Imaging Detection and Ranging. A lidar system uses light in very much the same way that radar uses radio waves. It measures the roundtrip time and frequency shift of a pulse of laser light reflected back from an object. This allows the lidar system to calculate the distance to and relative speed of the object. Lidar is used in many applications, like geography, geology, and law enforcement.

Why do police use lidar? The light beam from a lidar system remains narrow and doesn’t spread out like a radar lobe. It’s much more likely that a policeman using lidar will only have one vehicle in its field of view, as shown in the diagram. This makes it easier to clock speeders, even in heavy traffic. Police lidar commonly uses a frequency of around 332 THz, which is invisible to the human eye.

There is, however, another problem with radar and lidar systems, which is known as the cosine effect. In the previous case, the police car was moving in the same direction as the speeder, along the x-axis, and we found an expression for its velocity. The same speeder in our picture is traveling diagonally away from stationary police car 2. The equipment in this cruiser will estimate the velocity of the vehicle along the x-axis, in the direction of the highway:

v_x = v cos theta < v

That means that the radar (or lidar) will consistently underestimate the velocity of the speeder along the highway! Police have to comply with some very complicated instructions about using such systems before issuing tickets.  Of course, the most precise measures will be obtained by aiming one’s radar or lidar system at the vehicle at an angle as close to zero as possible.

Next week, we will talk about another application of radar-like technology that many automobile commuters take advantage of every day. Yes, I’m talking about paying tolls on highways electronically using things like E-ZPass. I’ll tell you what life was like before E-ZPass, and why we all consider ourselves fortunate to have this invention available today.