# MID-TERM EXAM The problem has three parts that are independent; however, reading and understanding the premises of a previous part may…

## MID-TERM EXAM

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**The problem has three parts that are independent; however, reading and understanding the premises of a previous part may be needed to address the following part. Good presentation, correct style and detailed written argumentation are part of the overall grade.**

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*The purpose of this problem is to examine various aspects of a laser telemetry system implemented by NASA during the Apollo lunar missions, which aimed at obtaining precise measurements of the Earth-Moon distance, based on the time of flight of some “lucky” photons.*

## PART I

A trihedral mirror is an assembly of three planar mirrors that are perpendicular to one another. We will examine such a device, where the mirrors are in the planes respectively. The vectors represent the direction of a light ray, and are all unit vectors.

1. If an incident ray has a direction ** **when hitting a reflective surface of normal , show that the reflected ray has a direction that fulfills the two conditions:

2. Use the relations above to calculate the components of , directional vector of the ray reflected by , in term of the components of the incident ray’s directional vector, which is in this case:

3. Do the same for vectors and which result from reflections of by and by respectively.

4. Establish a direct relation between and to justify the term “retro-reflector” used to describe trihedral mirrors.

5. Would this also hold true if only one or two reflections occurred (briefly justify)?

## PART II

A ruby laser emits pulses of coherent monochromatic light of wavelength, with a power of

. Each pulse lasts millisecond and can be seen as a beam in diameter, which fulfills Gauss’s condition. It is aimed at the surface of the Moon, which, from the Earth’s surface, is between

354,994 km and 397,586 km away.

1. If we want the area illuminated at the Moon to be a disc of at least 6,500 meters in diameter, what should be the focal length of the diverging lens used?

2. Draw a ray-tracing diagram of the system with one diverging lens (not to scale!)

3. Assuming a planar-concave lens is used, what would be the radius of curvature of the concave side (assume flint glass of)? Comment on the result.

4. Instead of a single lens, a pair of convex and concave lenses are placed very close to one another. Calculate the spacing required to achieve the focal length of question 1 using a pair of lenses.

5. What is the approximate number of photons received by an object that is about 1 m^{2} in area within the region illuminated by the laser, if only 1.00% of the photons emitted travel past the Earth’s atmosphere towards the moon?

## PART III

An array of trihedral mirrors is illuminated under the conditions described in

Part II. The width of the array is and its height is larger, but not to be considered in this problem. The array is equivalent to a continuous distribution of secondary light point-sources, emitting light back in the direction it came from:

1. Verify that the setup fulfills Fraunhofer’s condition, if the reflected rays are expected to be received on Earth.

2. Calculate the path difference, then the phase difference between the reflected ray at and the reflected ray at any position along the retro-reflector (*hint: extra distance is traveled in both directions)*

3. Calling the overall amplitude of the incident wave, calculate the amplitude contribution of an element at a position along the array.

4. Integrate over the width of the array to prove that reflected amplitude in the direction is, with .

5. Calculate the angular span of the central maximum to find the minimum width of the brightest fringe received back on Earth.

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