Redshift Academy


Double Slit Experiment
----------------------



δ = path difference = dsinθ' 

For D>>a, θ' ~ θ therefore δ = dsinθ

For a maximum we need dsinθ = mλ

Now if θ is small, sinθ ~ tanθ = y/D.  Therefore,

dy/D = mλ

Intensity: 

I = I₀cos²(π.d.sinθ/λ) 

  = I₀cos²(π.d.y/λ.D) for small θ

It is worth bearing in mind that these results are an idealization assuming an 
infinitesimally small width of the slits. A full treatment of the situation, 
accounting for the finite width of the slits requires us to take diffraction 
into account. 

Single Slit Diffraction
-----------------------
Diffraction manifests itself in the apparent bending of waves around small 
obstacles and the spreading out of waves past small openings. There are
two types - Fraunhoher and Fresnel.  Fraunhofer diffraction deals with the 
limiting cases where the light appoaching the diffracting object is parallel 
and monochromatic, and where the image plane is at a distance large compared 
to the size of the diffracting object. The more general case where these 
restrictions are relaxed is called Fresnel diffraction.



Condition for a maximum: 

Using the same geometry and reasoning as the double slit, the condition is,

asinθ = mλ

 ay/D = mλ 

Intensity: I = I₀[sin(π.a.sinθ/λ)/(π.a.sinθ/λ)]²

             = I₀[sin(π.a.y/λD)/(π.a.y/λ.D)² for small θ

Diffraction and Interference Combined
-------------------------------------
Combined diffraction and interference is obtain by multiplying diffraction 
intensity for single slit by interference for double slit.  Thus,

I = I₀[cos²(π.d.sinθ/λ)][sin(π.a.sinθ/λ)/(π.a.sinθ/λ)]²

The first and the second terms in the above equation are referred to as 
the "interference factor" and the "diffraction factor". The former yields 
the interference substructure, the latter acts as an envelope which sets 
limits on the the number of interference peaks. 



Diffraction Grating
-------------------
A diffraction grating is a large number of close, parallel equidistant slits 
ruled on glass or metal.   

The condition for maximum intensity is the same as that for the double 
slit.  However, as more parallel equidistant slits are added, the intensity 
and sharpness of the principal maxima increase and those of the subsiduary 
maxima decrease.



Bright or principal maxima are obtained when,

dsinθ = mλ

d can be calculated from the number of slits per metre.

Spectra
-------
Diffraction gratings provides a valuable way of studying spectra. If white 
light is incident normally on a diffraction grating, several colored spectra 
are observed on either side of the normal.  This occurs because the wavelength 
of violet light is 3.8 x 10^-7 m and the wavelength of red light is 7.0 x 10^-7 m.  
Therefore, θ is less for violet than red. 



Now,
  
dsinθ = 2 x 3.8 x 10^-7 for violet the 2nd order spectrum

      = 7.6 x 10^-7

and,

dsinθ = 7.0 x 10^-7 for red the 1st order spectrum

Thus, the red in the 1st order spectrum does not overlap the violet in the 
2nd order spectrum. However, overlapping of colors does occur in higher order 
spectra.

Ex:  Two slits separated by a distance d = 1cm allow light of wavelength 
     500nm to pass through and interfere. A screen is situated 10m from the slits, 
     where a pattern of bright and dark fringes are observed. As measured from 
     the center 0th order fringe, what is the distance to the first dark fringe 
     in μm?


y = mDλ/d => 1*10*500 x 10^-9/10^-2 => 5*10^-4 => 500μm

This is the distance between 2 peaks.  The distance from the center is this 
value/2.

Bragg's Law
-----------



When x-rays are scattered from a crystal lattice, a diffraction pattern is 
observed. A plot of the intensity of the scattered waves as a function of 
scattering angle shows peaks in the pattern when the following condition 
is satisfied.

mλ = 2dsinθ ... Bragg's Law

The condition for maximum intensity allows us to calculate details about 
the crystal structure, or if the crystal structure is known, to determine 
the wavelength of the x-rays incident upon the crystal.