1. 1 Electromagnetic waves: Multiple beam Interference Wed. Nov. 13, 2002

2. 2 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

3. 3 Multiple beam interference Evaluate Thus,

4. 4 Multiple beam interference Now,

5. 5 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.

6. 6 Multiple beam interference Fringe pattern

7. 7 Multiple beam interference Maximum intensity when,

8. 8 Multiple beam interference Thus maxima are circles in focal plane of lens – or rings The maximum intensity And intensity distribution is,

9. 9 Multiple beam interference Minima intensity when, Intensity distribution, If R > 0.9 I min <<I max

10. 10 Multiple beam interference Let the contrast, or co-efficient of finesse be defined by, Then the transmitted light is described by and Airy function, The same analysis for the reflected light gives

11. 11 Multiple beam interference: Transmission curves R=0.87 R=0.046 R=0.18 F=200 F=1 F=0.2

12. 12 Multiple beam interference: Reflectivity curves R=0.87 R=0.046 R=0.18 F=200 F=1 F=0.2

13. 13 Observation of fringe patterns source f1f1 f2f2 Bright circles Screen or eye

14. 14 Typical parameters in an experiment Consider a quartz slab (n 2 =1.5) of thickness d ~ 0.5 cm The condition for constructive interference requires, For light of wavelength o = 500 nm, incident at a small angle , i.e.  ’ also small, and m is large:

15. 15 Spectroscopy applications: Fabry- Perot Interferometer Assume we have a monochromatic light source and we obtain a fringe pattern in the focal plane of a lens Now plot I T along any radial direction Let I MAX =I M The fringes have a finite width as we scan mm-1m-2 order m-3

16. 16 Fabry-Perot Interferometer Full width at half maximum = FWHM, is defined as the width of the fringe at I=(½)I M Now we need to specify units for our application Let us first find  such that I = ½ I M

17. 17 Fabry-Perot Interferometer We want, This gives,

18. 18 Fabry-Perot Interferometer m m-1  = 2m  = 2(m-Δm)  I= ½ I M  = 2(m-1) 

19. 19 Fabry Perot Interferometer Thus at I = ½ I M sin(  /2) = sin (m – Δm)    sin Δm  Assume Δm is small and sin Δm   Δm  Thus FWHM ~ Fraction of an order occupied by fringe

20. 20 Fabry-Perot Interferometer The inverse of the FWHM is a measure of the quality of the instrument. This quality index is called the finesse It is the ratio of the separation between the fringes to the fringe width

21. 21 Fabry-Perot Interferometer Not that  is determined by the reflectivity If R ~ 0.90  = 30 R ~ 0.95  = 60 R ~ 0.97  = 100 In practive, can’t get much better than 100 since the reflectivity is limited by the flatness of the plates (and other factors of course)

22. 22 Fabry-Perot Interferometer Now consider the case of two wavelengths ( 1, 2 ) present in the beam Assume 1  2 and 1 < 2 Increase 2, dashed lines shrink e.g. order m-1 of 2 moves toward mth order of 1 Eventually (m-1) 2 =m 1 This defines the free spectral range m m-1 m-2 2 1 2 1 2 1

23. 23 Fabry-Perot Interferometer m( 2 - 1 )= 2 or mΔ = Δ FSR = /m Now since We have, e.g. =500 nm, d = 5mm, n=1  Δ FSR = 25(10 -2 )mm = 0.25Å