Question

2) Thin lens imaging formulas for Gaussian beams. A Gaussian beam waist of radius w1, located a distance Li to the left of a thin lens of focal length f, is imaged by that lens into a Gaussian beam waist of radius w2 located a distance L2 to the right of the lens. a) Write an expression relating the “object distance” L1, the “image distance” L2, and the focal length f (this expression may also include dependence on wi and w2). b) Write an expression for the linear magnification M = w2/w1 in terms of L1, L2, and f. c) In the limit as w1 +0, show that these expressions reduce to the results for geometric optics. Hint: Use the ABCD transformation law between the object and image points, and recognize that the respec- tive q-parameters will be purely imaginary at those waists (the q's are more conveniently expressed in terms of ZR1 and 2 R2 instead of wi and w2). The results for parts (a) and (b) will then be obtained by demanding that the real and imaginary parts (respectively) of the transformation law be satisfied separately.

Answer 1

TOOi Zoj Zo1n 62 Zor F'Auy 9,=11+1201 FaY lens A 1 A9, + B - 91 /4+121)f 4-2 f-1,-1201 (Fl12afF-L)+ izu fL)221)2 PL)IF-L) Zo 4 equohiy ed imagine PL)CE-L-7.7P P-4904 (79)2 Z.) FI, 4(f-) Zof PLI)24 (20)2 Z0 3-

f a (4) - Xf] Ay Pl, (A42-2Af FLi(P)-f2 LO62 Zo (146) - fl,(F)-( 1+f) exOE + 1 A, + fif-) Zo2 f4If--(2 -f) Zol XW (Af)fF-L L)-( f2(14f) 2 (F-1) -)t, - (F1) + 2 (Jeiled F-1)F -f) + (1+ f) =f f-L)6-)(- +f 1+f) M - f Lineat magni Acrkon

(fA.)CFL) (F-14 (F4) L (f-4) fLr-422f - 2-1) tp-LI) fLI fl, f -4-