Solutions To Fresnel Diffraction Problems

 1) The innermost zone of a zone plate has a diameter 0.425mm.  

(a)Find the focal length of the plate when it is used with parallel incident light of wavelength 4471 Å from a helium lamp.

(b) Find the first subsidiary focal length for the zone plate.

Solns.

a)  The focal length f =  s _m ² / m l

Where in this case: s _m  = s ₁   =  0. 0425 cm   (so m = 1)

l   =   4471 Å  = 4. 471   x  10 ^-5  cm

Then:  f =  (0. 0425 cm) ² / (4. 471   x  10 ^-5  cm)  =  40. 39 cm

b) The first subsidiary focal length will correspond to f/ 3,  so:

f /3  =  40. 39 cm/  3   =  13. 46 cm

2) A zone plate is to be set up on an optical bench where it will be used as an enlarging lens.  Its innermost zone will have a diameter of 0.2250 mm with monochromatic green light of wavelength 4800 Å  to be used.  If all enlargements are to be eightfold in diameter find the focal length f of the zone plate.

Soln.

An eight-fold enlargement means we need to find f/ 16 for the 8th subsidiary maximum of the plate.  Then:  

 f =  (0.0225 cm)/  (4. 8   x  10 ^-5  cm) =  10. 54 cm

And: f/ 16 = 10. 54 cm/  16  =  0.66 cm