Lecture 35: Diffraction: Single slit

Diffraction: bending of light around the obstacles

• Diffraction is a spreading of light around the edges of obstacles.
• Diffraction is a manifestation of the wave nature of light
• Clearest explanation of diffraction is using viewing wave propagation according to Huygen's principle: each point disturbed by the advancing wave front can be viewed as a source of a spherical wave, new front is enveloped of these secondary spherical wave fronts
  
• We have already used Huygens principle to describe phenomenon of bending of the light ray on interface between media with different refractive index but it also tells that the light will propagate behind the obstacles - diffract
  

  Refraction	Diffraction

• We have used diffraction before to create two coherent sources of light by a wave passing through two holes.

Diffraction and interference

• Outcome of diffraction is not simple. Diffraction leads to intersecting rays of light, which are coherent and interfere.
• Interference pattern formed by the diffracted light is called, for short diffraction pattern
• Sometimes it is emphasized that we speak about diffraction picture when we consider interference from many sources - as for example continous distribution of sources between the barriers in the Figure above.
• Note: Diffraction and interference are different phenomena (despite what textbook says :) )

Single slit diffraction

• One of the most important examples of diffraction effects is the passage of light through a hole of finite size.
• If light passes through narrow, point like hole, almost exact spherical wave forms. However if the hole has a finite size, an interference pattern forms behind the hole
• We shall consider narrow (but finite width) slits, rather than round holes. The wavefronts that are formed by the slit are cyllindical rather than spherical
  Single Slit Diffraction Applet
• Note that the interference patern of diffracted light from a wide slit although has geometrical similarity to two narrow slits, is guite different from as far as intensity distribution is concerned (see for example much brighter central fringe)
• We shall compute the distribution of the intensity at the screen beyond the slit.
• The pattern near the slit, where exact wavefront profiles are important are called near-field, or Fresnel, diffraction . The pattern at the screen far away, where we can use geometrical optics rays and approximations we have done in intereference studies is called far-field, or Fraunhofer diffraction
• We shall consider exclusively the far-field, Fraunhofer, diffraction pattern.

Intensity in Fraunhofer diffraction pattern from a single slit

• I will replace several pages of discussion in the text book (but read them) which avoids mathematics with one line calculation using a simple integration.
• Just to illustrate that often mastering advanced mathematics makes things easier - thats what for it is introduced in the first place, actually.
• Resulting pattern is the sum of the waves from every point in a slit.
  
• With continous distribution of source points, it is an integral over position of a point in a slit y If the center of a slit has the coordinate y = 0 and the width of the slit is a, then y varies from -a/2 to a/2
• So we have for the amplitude
  
• and, finally, for the intensity
• Here is how it looks like

Minima of Intensity in Fraunhofer diffraction pattern from a single slit

• This is simple, minima is achived at observation angles θ where sin(π(a/λ) sinθ) vanishes, i.e sin(θ_min) = ±m λ/a
• but not for m = 0 , only for m = 1, 2 ...
• The separation between minima widens when the wavelengths increases or the slit width decreases
• This is why when slit is very narrow a ≈ 0 , the first minima is far away and one has continous distribution of light.
• sinθ varies from -1 to 1 with zero achieved straight ahead of the slit. Thus, the number of minima observed is n_min = 2 int(a/λ)
• If slit is narrower than the wavelength, a < λ, no minima are observed.

Maxima of Intensity in Fraunhofer diffraction pattern from a single slit

• The main maxima of intensity is at θ = 0 (yes, zero divide by zero gives one here ! )
• Because main maximum is where minimum is expected to be for m=0, it is twice wider than other maxima, occuping space between minima at ± λ/a . It's width in terms of sinθ is 2 λ/a
• Positions of other maxima are only approximately halfway between the minima sin(θ_max) ≈ ± (m + ½) λ/a
• Depending on m we speak about first order (m=±1), second order (m=±2) etc. maxima
• The approximate value of peak intensity at the maximum m > 0 is then I_m ≈ I₀ / ( π² (m + ½)²)