Chap. 4 Multiple-Beam Interference
4.1 Interference with Multiple Beam
- Consider two semi-reflecting mirrors that are considered to be thin and identical.
- The geometrical path difference any two successive transmitted rays is cos.




The corresponding phase difference   cos   cos   cos

  
  

- Then,             ⋅⋅⋅ → 
  

- The intensity    of the transmitted light is    
   
 
In general, r is a complex number;    
where  is the phase change for one reflection.

- The phase change for a dielectric is either 0 or , depending on the relative index
of refraction (see Sec. 2.7); but in the case of a metal film, the phase change can be
any value (see Sec. 6.5).

Let           , then    
with     
  
                cos   

 


        sin   


   




- The formula for the intensity      ;

    
  sin 




Airy function  and the coefficient of finesse   
   

   sin 

-    sin   has the maxima at   .
- As → (small), F is small and the interference fringes are broad and indistinct.
- The condition for a fringe maximum    is equivalent to

   cos  with N = the order of interference.



- If two reflecting surfaces are NOT identical,        ,
    
 
    
 
     
- For non-zero absorption at each reflection (A), we must have       


The maximum intensity max  



4.2 The Fabry-Perot Interferometer
- 1899, C. Fabry and A. Perot: to measure wavelengths with high precision.
- Two optically flat, partially reflecting plates of glass or quartz with their reflecting surfaces
held accurately parallel (see Fig. 4.4).
The plate spacing can be mechanically varied (interferometer) or
held fixed by spacers (etalon).
A flatness of the order of 1/20 to 1/100 wavelength is required!
- Free spectral range:        →

       cos →


      

for small 
4.3 Resolution of Fabry-Perot Instruments
- Suppose a spectrum consisting of two closely spaced frequencies  and ′ to be analyzed.
- Assuming that two components are of equal intensities. the fringe pattern is given by the sum
of two Airy functions,




 ′
     sin  with    and
      sin 


   

′
 ≈        , ′ ≈   ′     for small .






- If there is a dip in the intensity curve, the two frequencies ( ′) can be resolved.
☛ Taylor Criterion: two equal lines are considered to be resolved if the individual curves
cross at the half-intensity point, so that the total intensity at the saddle
point is equal to the maximum intensity of each line alone.


   ′ 

  →
      sin 






  ′

 sin    





For small   ′,   ′      


 
 
In terms of the angular frequency,     ′       

 





Resolving power (RP) =      ;



4.4 Theory of Multilayer Films


RP =   




- Optical surfaces having any desired reflectance and transmission characteristics; optical filters,
anti-reflecting coatings, etc.
- Boundary conditions: the electric and magnetic fields should be continuous at each interface.
For normal incidence,
(1) First interface;
Electric field
    ′       ′
Magnetic field     ′      ′
or
  ′    ′
(2) Second interface;
Electric field
  ′   

Magnetic field     ′    
or
- Eliminating  ′,
  ′
 
 ′


     cos    sin  



 
 ′

      sin  cos  



 ′
or, in matrix form,      

 
 



sin
 


 

 sin cos 
cos
 
  
′




  
  and transmission   
with
reflection










sin
cos 

transfer matrix  
where      
 sin cos 
- Simplified form
 
 
- Then, for N layers having indices of refraction    and thicknesses       ,
 
 
 




    





 
Let    
 

, then solve the above equation for r and t;

● Antireflecting
         
 
         
→
   and

 
         
→
  
Films: for a single film of index  and thickness l, placed on
a glass substrate of index 
      cos        sin
With   ,   
      cos        sin
- If the optical thickness of the film is 1/4 wavelength, then    ;
 











 

     
→
 if   

With magnesium fluoride (MgF)   , glass  ≈ , for a quarter-wave film of MgF
on a glass, about 1 % of reflectance (1/4 of the uncoated glass)
- Use of two layers, one of high index and the other of low index; nearly zero reflectance
available for given wavelength → more layers for zero reflectance at broader wavelengths
●
High-Reflectance Films: a stack of alternating layers of high index ( ) and low index ( )
with the thickness of each layer being 1/4 wavelength

  
 


- For two adjacent layers, transfer matrix is
    
  

   


 
 
 
 
 

     
- For the stack consisting of 2N layers,  

  




 




 



 
  










 
 




Assuming that    and   , the reflectance of a multilayer stack is given by
                  
 
    



        
        

[Homework]
 

Prob. #4.4, #4.9