Some identical thin convex lenses are arranged at edge of a full circle of radius R, according to the picture below. The lenses have equal angle spacing of θ (θ<<1). Each lens has aperture length of D (D<<R).Find the longest focus distance, f, of the lenses so that it is possible to trap a ray of light inside the circle!


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A very thin lens has diameter D and focal length f. It has a weird equation:
            (1/f)^n = (1/s)^n + (1/s')^n
where s is an object's position and s' is the image position measured from the lens, and n is an integer where n > 0 (a common lens has n = 1).
Then ray of light come to the lens with a small angle θ from axis of the lens (θ<<1). And a screen is placed behind the lens at focus.Determine radius of light appear on the screen!
Use (1+x)^n = 1 + nx + 1/2*n(n-1)x^2 for x << 1.

Consider two mirrors and a light source. The mirrors form an angle φ and -φ with a horizontal axis, and the light source is located at horizontal axis and near junction of the two mirrors (see the black dot below).The light source emits light of total power P with uniform distribution. Assume the mirrors are perfect reflectors. Also assume that this system is only 2 dimensional. Find the distribution of power, dP/dθ, as a function of θ (the angle formed by light and horizontal axis after the light is out from the system)!