1 Electromagnetic waves: Multiple beam Interference Friday November 8, 2002

2 Anti-Reflection coatings A. Determine thickness of film film glass air n1n1n1n1 n3n3n3n3 n2n2n2n2 n 1 < n 2 < n 3 Thus both rays (1 and 2) are shifted in phase by  on reflection. 1 2 For destructive interference (near normal incidence) 2n 2 t=(m+1/2) 2n 2 t=(m+1/2) Determines the thickness of the film (usually use m=0 for minimum t)

3 Anti-Reflection coatings B. Determine refractive index of film Near normal incidence Amplitude at A film glass air n1n1n1n1 n3n3n3n3 n2n2n2n2 1 2 A A’ Since   ’ ~ 1

4 Anti-reflection coating Amplitude at A’ B. Determine refractive index of film To get perfect cancellation, we would like E A = E A’ should be index of AR film

5 Multiple Beam interference Thus far in looking at reflectivity from a dielectric layer we have assumed that the reflectivity is small The problem then reduces to two beam interference Now consider a dielectric layer of uniform thickness d and assume that the reflectivity is large e.g. |  | > 0.8 This is usually obtained by coating the surface of the layer with a thin metallic coating – or several dielectric coatings to give high reflectivity Or, one can put coatings on glass plates, then consider space between plates

6 Multiple beam interference Let  12 =   21 =  ’  12 =   21 =  ’ n1n1n1n1 n2n2n2n2 n1n1n1n1  ’’’’ ABCD  EoEoEoEo  ’  ’ E o  ’ E o (  ’) 3  ’E o (  ’) 2  ’E o (  ’) 6  ’E o (  ’) 4  ’E o (  ’) 5  ’E o (  ’) 7  ’E o

7 Multiple Beam Interference Assume a (for the time being) a monochromatic source ,  ’ small ( < 30 o ) usually Now |  | = |  ’| >> ,  ’ Thus reflected beams decrease rapidly in amplitude (from first to second) But amplitude of adjacent transmitted beam is about the same amplitude Amplitude of successfully reflected beams decreases slowly (from the second) Thus treat in transmission where contrast should be somewhat higher The latter is the configuration of most applications

8 Multiple Beam Interference Assume phase of transmitted beam at A is such that, Now let  ’be the phase shift in travelling across and back once, i.e.

9 Multiple Beam interference At B: At C: At D: etc…

10 Multiple beam interference Consider N beams which interfere at infinity We can use a lens and then beams shown interfere in focal plane of lens The phase difference between adjacent rays outside is, ΔS1ΔS1 N N-3 N-2 N-1 N-4 ΔS1ΔS1 ’’  d n2n2 n1n1 n1n1

11 Multiple beam interference Amplitude at point P,

12 Multiple beam interference Amplitude at point P, Let

13 Multiple beam interference This is just a geometric series with r < 1. Thus,

14 Multiple beam interference Thus,

15 Multiple beam interference Evaluate Thus,

16 Multiple beam interference Now,

17 Multiple beam interference Now recall the definition of the intensity of an electromagnetic wave Thus, is the intensity distribution in the focal plane of the lens.

18 Multiple beam interference Fringe pattern

19 Multiple beam interference Maximum intensity when,