LIDAR Speed Guns and the Frequency of a Photon 

The radio is playing some forgotten song… 
Golden Earing 

There seems to be a widespread misconception (in some venues) that LIDAR speed guns do not use the Doppler effect to measure speed. This may be traced to a popular online source, which gives the following ‘explanation’ for how a LIDAR speed gun measures speed, contrasting it with how a radar gun measures speed: 

A normal radar set sends out a radio pulse and waits for the reflection. Then it measures the Doppler shift in the signal and uses the shift to determine the speed. Laser (or LIDAR, for light detection and ranging) speed guns use a more direct method that relies on the reflection time of light rather than Doppler shift. 

This ‘explanation’ actually highlights two misconceptions. First, it is obviously wrong to suggest that LIDAR speed guns don’t rely on the Doppler effect to determine speeds. It’s true that a LIDAR gun measures the distance to the target by evaluating the difference between the time t_{1} of the emission of the pulse and the time t_{2} of the reception of the reflected pulse, and then computing the distance D_{1} = (t_{2} – t_{1})c/2. However, this by itself is not sufficient to determine the speed of the target. To determine the speed we need to know how the distance is changing with time. So we must emit another pulse at later time t_{3}, and receive the reflected pulse at time t_{4}, from which we compute the distance D_{2} = (t_{4} – t_{3})c/2. 

Now, working in the frame of the LIDAR gun, it’s obvious that D_{1} was the distance of the target at the time of reflection of the first pulse, which was T_{1} = (t_{1} + t_{2})/2, halfway between emission and reception. Likewise D_{2} was the distance of the target at time T_{2} = (t_{3} + t_{4})/2. Therefore, the speed v of the target based on these two measurements is 



The difference t_{3} – t_{1} is the time between emissions, so the emission frequency is ν_{e} = 2π /(t_{3}–t_{1}), and the difference t_{4} – t_{2} is the time between receptions, so the reception frequency is ν_{r} = 2π /(t_{4}–t_{2}). Thus we can write the above expression as 



Recall that the familiar relativistic Doppler formula for a oneway signal is 



and for a roundtrip signal the Doppler frequency shift is the square of the righthand expression, so we have 



Solving this for v/c gives 


which is identical to the formula (1) used in a LIDAR speed gun to determine the speed. Needless to say, the roundtrip Doppler shift formula given by equation (2) is also the equation used in radar speed guns. Therefore, the popular online reference is wrong to imply that LIDAR does not use the Doppler effect to determine the speed of the target. 

In view of this, why does the online reference claim that radar uses the Doppler effect but LIDAR does not? The answer highlights another widespread misconception. Note that the reference said radar sends out a radio pulse, waits for the reflection, and then measures the Doppler shift. Since the Doppler effect involves frequency, the reference is tacitly assuming that an individual “pulse” has some identifiable frequency. Classically this would imply that the “pulse” actually consists of a short burst with a certain frequency, i.e., several wave crests of an electromagnetic wave. The period (i.e., time interval between consecutive wave crests) of the transmitted wave is then compared with the period of the received wave, exactly as in the LIDAR gun. Expressed in terms of frequency, the radar gun uses the Doppler equation (2), just as does the LIDAR gun. The only difference is that a radar gun deals with the frequency of the carrier wave, whereas LIDAR deals with the frequency of a signal wave, as depicted below. 



The radar gun in effect measures the period of the waves that comprise a burst, whereas the LIDAR gun measures the period between bursts. But in either case the ratio of the transmitted and received periods is obviously the same, given by the (squared) Doppler factor. Naturally there are different methods of measuring the period or frequency or wavelength of a signal. We could use a tuning analog detector that is excited by a range of frequencies, or a differential circuit that directly responds to the difference between emitted and received frequencies. Alternatively we could measure the time intervals between consecutive wavecrests or pulses. Radar guns and LIDAR guns ordinarily use different methods for measuring the frequencies, but they both rely on determining the ratio of transmitted to received frequencies (or periods) using the standard Doppler formula. 

Indeed, aside from parallax, it’s difficult to imagine any way of inferring the speed of an object from reflected signals other than by the Doppler effect, i.e., by exploiting the fact that the distance is changing so the rate of reflected pulses is different than the rate of transmitted pulses, because the distance each pulse must travel is changing. So how could anyone even entertain for a moment the possibility that LIDAR guns do not rely on the Doppler effect? 

The underlying misconception seems to be the idea that an electromagnetic wave possesses and exhibits a frequency that is somehow different from the frequency of arrival of successive wave crests. Recall that the online reference describing radar spoke of the frequency of a radio pulse, but it would be more accurately described as a burst comprised of a small number of cycles with a characteristic period, as depicted in the figure above. The word pulse tends to suggest a single entity with no internal structure (like a Dirac delta function impulse), and it seems to be vaguely associated in the minds of some people with the concept of a photon in quantum theory. Indeed the wellknown expression E = hν gives the relationship between the frequency of light and the energy of the photons comprising that light. Since an individual photon has a certain energy, this leads some people to the idea that an individual photon has a frequency. It is true that quantum interference effects are exhibited by individual photons (such as in a twoslit experiment with extremely ow intensity source), but this does not imply that an individual photon has a frequency in the temporal sense. 

Needless to say, radar guns do not examine individual photons or quantum interference effects for a sequence of photons, they work on the classical level of electromagnetic waves following Maxwell’s equations. It would be possible in principle to pass a large number of individual photons through a twoslit apparatus until the interference pattern became evident, and then deduce the frequency of the light from that pattern, but this still would not enable us to derive a frequency from timing of a single photon. Alternatively we could measure the energy of a photon and infer the associated frequency from the relation E = hν, but we would find that the ratio of transmitted and received energies is the same as the ratio of frequencies of a classical wave. 

In some ways it is more useful to regard a photon as a discrete transfer of energy along a null interval. The phase of the quantum wave function of a pulse of light or photon does not advance at all during transit. (In general, the advance of the quantum wave function of any system passing between two events is proportional to the proper time interval along the path, which is zero for massless energy.) Rather than saying a photon has a frequency it is better to say the source has a frequency. The source (atom) is in null contact with the receiving system (atom), possibly via multiple paths through spacetime, and the amplitude for a transfer to occur from any given emitter’s worldline to a given reception point is the superposition of the amplitudes for all possible piecewise null paths. Thus the frequency over a range of events at the source is imposed on the receiving event, and this accounts for the interference effects. 

Quantum electrodynamics is an interesting subject but, again, this is not relevant to how radar and LIDAR guns actually measure the speed of the target. Both of these devices measure the frequency of the emitted signals (at the classical level) and compare with the frequency of the reflected signals, and both devices infer the speed of the target from the ratio of the frequencies using the squared Doppler equation. 

The misconception about intrinsic temporal frequencies of individual photons arises in other contexts as well. For example, in discussions of gravitational redshift we sometimes see claims that the difference in frequency is not due to time dilation but rather due to “something happening” to the signals (photons) in transit from one gravitational potential to another. The classical image is of a ballistic particle losing speed as it rises in a gravitational field, and some people vaguely associate this with a drop in frequency, but of course in a stationary situation each particle would slow by the same amount, so this would not affect the frequency of arrival. Likewise the wavecrests of a light wave would take the same time to climb, so we cannot attribute the difference in arrival frequency to something that happens to the pulses (or bullets, or whatever). The difference can only be due to time dilation between the locations of different gravitational potential. 
