Question

Consider pulse propagation in a dispersive, isotropic medium with a refractive index given as a function of vacuum wavelength lmda0 by: n(lmda0) = nc + a (lmda0/lmdac )^2

Here nc, a, and lmdac are fixed parameters describing the medium.

a) Derive an expression for the phase velocity of radiation in this medium, and evaluate your answer for vacuum wavelength lmda0 = lmdac .

b) Derive an expression for the velocity of a pulse propagating in this medium, and evaluate your answer when the pulse’s central vacuum wavelength is lmda0 = lmdac. For a transform‐limited optical pulse of initial duration t0, and a central vacuum wavelength of lmda0 = lmdac, write an expression for the pulse broadening after propagation over a distance d.

Answer 1

Let us assume the speed in the vacuum is c.
  And so, the phase velocity in the medium is
  
Now given
  
And so, the phase velocity in terms of the vacuum wavelength is
  
And so, the phase velocity at is


b)

Group velocity is defined as
     
     
     
  

  
Now as we have already got
   
So,
  
Ans so,

  
     
And so, at
   

Let the initial duration is .
   And the pulse has propagated a distance d. Let us assume that as the pulse propagates it broadens to a width .
   This amount of spreading occurs because the different frequency components move at a different group velocity. And so, if we assume the two ferquency components
   
Now as the initial duration is
     
So, we have

  
   
   
   
     
Now we have
  
  
And so,
   
  
Now at
  
And so, at