MID-TERM EXAMThe problem has three parts that areindependent

MID-TERM EXAMThe problem has three parts that areindependent; however, reading and understanding the premises of a previous partmay be needed to address the following part. Good presentation, correct style and detailed written argumentation arepart of the overall grade.The purpose of this problem is to examinevarious aspects of a laser telemetry system implemented by NASA during theApollo lunar missions, which aimed at obtaining precise measurements of theEarth-Moon distance, based on the time of flight of some “lucky” photons.PART IA trihedral mirror is an assembly of three planar mirrorsthat 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, andare all unit vectors.1.  Ifan incident ray has a direction  whenhitting a reflective surface of normal  , show that the reflected ray hasa direction  that fulfills the two conditions:2.  Usethe relations above to calculate the components of  , directional vector of the rayreflected by , in term of the components of theincident ray’s directional vector, which is in this case:  3.  Dothe same for vectors and which result from reflections of  by and  by respectively.4.  Establisha direct relation between  and  to justify the term “retro-reflector” used todescribe trihedral mirrors.5.  Wouldthis also hold true if only one or two reflections occurred (briefly justify)?PART IIA ruby laser emits pulses ofcoherent monochromatic light of wavelength, with a power of. Each pulse lasts millisecond and can be seen as a beam in diameter, which fulfills Gauss’scondition.  It is aimed at the surface ofthe Moon, which, from the Earth’s surface, is between 354,994 km and 397,586 km away.1.  Ifwe want the area illuminated at the Moon to be a disc of at least 6,500 metersin diameter, what should be the focal length of the diverging lens used?2.  Drawa ray-tracing diagram of the system with one diverging lens (not to scale!)3. Assuming a planar-concave lens is used, what wouldbe the radius of curvature of the concave side (assume flint glass of)? Comment on the result.4. Instead of a single lens, a pair ofconvex and concave lenses are placed very close to one another.  Calculate the spacing required to achievethe focal length of question 1 using a pair of lenses.5. What is the approximate number of photonsreceived by an object that is about 1 m2 in area within the regionilluminated by the laser, if only 1.00% of the photons emitted travel past theEarth’s atmosphere towards the moon?PART IIIAn array of trihedral mirrorsis illuminated under the conditions described in Part II.  The width of the array is [img src=’file:///C:UsersLab-PCAppDataLocalTempmsohtmlclip1?1clip_image055.png’ height=’20’ width=’82’>and its height is larger, but not to beconsidered in this problem.  The array isequivalent to a continuous distribution of secondary light point-sources,emitting light back in the direction it came from:1.  Verifythat the setup fulfills Fraunhofer’s condition, if the reflected rays areexpected to be received on Earth.2.  Calculatethe 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 bothdirections)3.  Callingthe overall amplitude of the incident wave,calculate the amplitude contribution of an element at a position along the array.4.  Integrateover the width of the array to prove that reflected amplitude in the direction is, with.5.  Calculatethe angular span of the central maximum to find the minimum width of thebrightest fringe received back on Earth.