Problem : Laser sensor for illumination and ranging of the moon

Number of emitted photons :

The number of emitted laser photons emitted during a pulse is the ratio between the pulse and the photon energies, i.e. :

Peak laser irradiance of the moon surface :

According to Bouguer's law, this irradiance is Ep,moon = Ip moon /d 2 où d where d is the distance between the earth and the moon (380 000 km), hence :

Peak radiance and intensity of the moon in return :

Since the moon surface is considered as lambertian, its peak radiance is :

and the peak intensity of the backscattered laser beam is:

Number of photons de returning onto the detector :

the reception field of view of the sensor is :

Hence it is twice as large as the divergence of the transmitted laser beam, which means that theis sensor is a flux collector with respect to the laser spot on the moon. The laser flux being backscattered into the entrance pupil of the sensor, and hence onto the detector, if one considers the telescope as being perfectly transparent, is the following :

The number of photons N1 incident upon the detector is then :

That means that in the average, there is one photon every ten shots.

If the existing detector is being replaced by another one that is smaller, the flux on the detector remains unchanged as long as the field of view of the receiver is larger than the laser spot on the moon. This is the case here, because with a detector diameter of 60 mm, the receiver field of view is 12µrad, till slightly larger than the total divergence of the emitted laser beam. Admitting that there is no misalignment between laser axis and telescope axis, one finds that :

The on-axis laser intensity of the set of cube corners (or retro-reflectors RR) is the product of its radiance by its area :

It is is hence about 30 times larger than that of the ground of the moon. This advantage is the same in the number of photons returning from the cube corners with respect with those returning from the ground, But the largest advantage of the photons coming from the cubes as compared those coming from the ground is that they all come from the same tiny area (1m2 instead of 10 km2). So they all come back at the same instant onto the telescope, which is not the case for the other photons. In this application, where the number of “useful”photons est extremely small, signal processing is based upon photon counting techniques implying signal integration over very short periods of time (corresponding to “range windows”) case distance per case distance, and correlation from pulse to pulse.