The figure shows a beam expander made with two coaxial converging lenses of foca

Question

The figure shows a beam expander made with two coaxial converging lenses of focal lengths f_1 and f_2 and separation d = f_1 + f_2. The device can expand a laser beam while keeping the light rays in the beam parallel to the central axis through the tenses. Suppose a uniform laser beam of width W_f = 3.5 mm and intensity I_f = 9.9 kW/m^2 enters a beam expander for which f_1 = 12.7 cm and f_2 = 38.7 cm. What are (a) W_f and (b) I_f of the beam leaving the expander? (c) mat value of d is needed for the beam expander if lens 1 is replaced with a diverging lens of focal length f_1 = -27 cm? (a) Number Units b) Number Units c) Number Units

Explanation / Answer

a)

Parallel rays are bent by positive-f lenses to their focal points F1, and rays that come from the focal point positions F2 in front of positive-f lenses are made to emerge parallel. The key, then, to this type of beam expander is to have the rear focal point F1 of the first lens coincide with the front focal point F2 of the second lens. Since the triangles
that meet at the coincident focal point are similar (they share the same angle; they are vertex angles), then

Wf/f2= Wi/f1

Wf=(f2/f1)Wi=(38.7/12.7)*3.5

Wf=10.67 cm

b)

The area is proportional to W2. Since intensity is defined as power P divided by area,we have

If/Ii=(P/Wf2)/(P/Wi2)=Wi2/Wf2=f12/f22

If=(12.72/38.72)*9.9=1.066 KW/m2

c)

The previous argument can be adapted to the first lens in the expanding pair being of the diverging type, by ensuring that the front focal point of the first lens coincides with the front focal point of the second lens. The distance between the lenses in this case is

f2-|f1|=38.7-27=11.7 cm

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