7) (25 points) You are tasked with designing part of a laser guide star projecto

Question

7) (25 points) You are tasked with designing part of a laser guide star projector for an adaptive optics system for the UH 2.2-m telescope. The projector needs to focus the light from the laser to a distance of 10 km. The UV laser produces collimated light that is 1 mm in (Assume this is coming from a source at infinity) Lens 1 has an effective focal lengt UV diamete the optical elements have the following separations: ti 300 mm, t2 t 150 mm, t4 1400 mm. ( Assume this is a paraxial optical system and that the lens thicknesses are negligible.) Assuming that Lens 2 is made from high-purity fused silica (n-1.47) and is a plano-convex lens, what is the radius of curvature required to focus the laser light at the desired distance? Laser projector assembly Mirror UV Laser Lens 1 Lens 2 Mirror

Explanation / Answer

form the given data
focal laenght of lens 1 = f = 14.4 mm
now, assuming that the incoming light is coming from infinity, object distance u = infinity
image distance = v
then
1/v - 1/u = 1/f
1/v = 1/f
v = 14.4 mm
so, for the plano convex lens
object distance u = -(t4 - v)
t4 = 1400 mm
hence u = -(1400 - 14.4) = -1385.6 mm = -1.3856 m
given image distance, v = 10 km = 10,000 m
hence the focal lenght of the plano convex lens = F
then
1/v - 1/u = 1/F
1/10,000 + 1/1.3856 = 1/F
F = 1.38540803 m .. (1)

now refractive index of the material of the plano convex lens, n = 1.47
let radius of curvature be R
then
for a lens with refractive index n in air
1/F = (n - 1)[ 1/R1 - 1/R2]
R1 = infinity
R2= -R
hence
1/F = (1.47 - 1)[1/R]
hence R = F*0.47 = 0.65114177 m

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