### Fresnel Diffraction

#### Overview of General Diffraction

What is Diffraction?

Diffraction is the deviation of a wave from its straight ray propagation.  This occurs when part of the wave is obstructed by a boundary.  Diffraction is a wave phenomenon so light undergoes diffraction because of its wave nature.

Why does it occur?

In accordance with Huygen’s Principle as a wave from a single source propagates through space each wavefront can be thought of as consisting of numerous individual secondary wavefronts.  Diffraction occurs because part of the wave (some of the secondary wavelets) that is obstructed changes phase or amplitude and interferes with the rest of the unaltered wave (the rest of the secondary wavelets).  This causes multiple interference patterns resulting in the diffraction of the wave.

What affects the diffractive behavior of a light source at a boundary?

-The wavelength of the light source

-The distance from light source to the boundary

-The distance from the boundary to the screen where the diffraction pattern is projected

-The size, shape, and nature of the boundary or the obstruction

##### Fresnel Diffraction

What are the conditions for Fresnel Diffraction?

Fresnel diffraction occurs when either the distance from the source to the obstruction or the distance from the obstruction to the screen is comparable to the size of the obstruction.  These comparable distances and sizes lead to unique diffractive behavior.

Why is Fresnel Diffraction different than other types of diffraction?

The approximations made for Fraunhofer Diffraction are no longer valid.  The light source can no longer be considered a planar wavefront at the aperture because it can longer be approximated to originate at infinity.  It must be considered a spherical wavefront.

-The relative phase difference for a curved wavefront is not constant.

-Amplitudes of the individual light waves (secondary wavefronts) at the observation point are not equal because the distances traveled by each element, or wavefront, can no longer be considered approximately equal.  Therefore, the of light on the screen varies from point to point.

How is Fresnel Diffraction dealt with mathematically?

-All parameters (lengths, distances, widths, etc.) must be considered in the mathematical interpretation of Fresnel diffraction because of their comparable sizes.

-One can determine the diffraction pattern caused by Fresnel diffraction by determining the intensity of light at each point on the viewing screen.

s

light source                                                                      q

 a
 b

The intensity at each point on the screen can be determined by a parameter V

V = v1 - v2

where

v1 = [s/2 + k][2(a +b)/ab(wavelength))]1/2

v2 = [-s/2 + k][2(a +b)/ab(wavelength))]1/2

and

# k = q[a/(a+b)]

k depends on q (the distance from the center of the screen to the point (x,y) on the screen where you wish to determine the intensity)

v1 and v2 depend on k so for each distance q there is a different v1 and v2

therefore there is a unique V for every point on the screen and its value determines the intensity at that point

How is Fresnel Diffraction represented graphically?

This parameter V and, therefore the resultant Fresnel diffraction pattern, can be represented graphically as well.

The resultant amplitude (intensity) of light at an observation point can be determined by vectorially adding (head to tail) each of the individual amplitude components.

For Fraunhofer diffraction, where the relative phase of these elements is constant, the vector addition simply gives an arc length of a circle and the intensity is the cord length of that arc.

Fresnel diffraction introduces another phase shift due to the curvature of the wave.  Therefore, each consecutive amplitude is longer displaced the same amount (forming a circular arc length) rather these elements bend into a spiral curve.

What is a Cornu Spiral?

The Cornu Spiral is the geometric depiction of the Fresnel diffraction patterns due to different barriers.  Each point on the spiral has a value that corresponds to certain v1(k) or v2(k). Therefore, the cord length between any two points v1 and v2 on the spiral, corresponding to a certain k value gives V for that value k (and thus q).   V determines the intensity at that point q (which has both an x- and y-coordinate) on the screen.

How do different barriers and apertures affect the diffraction pattern?

Opaque Barrier

One can see from the graph above that the cord length between two points determines the intensity of the pattern produced on the screen.  Because the bottom half of the barrier extends to infinity, v1 always has an infinite value.  Since the Cornu spiral spirals inward infinitely, an infinite value is found at the center of the spiral.  The points 1,2,3,4,5 correspond to different v2 values.  The smaller the length of the cord the less intense the light is at that point on the screen.  The zero point on the graph (point 3) represents the center of the screen (where the barrier begins in this case).  One can see that some light extends past this barrier into the shadow produced by the barrier.

Single Slit

The Fresnel diffraction pattern for a single slit is similar to that of Fraunhofer diffraction except the areas of minimum intensity do not go to zero (i.e. there is never complete destructive interference)

 The diffraction pattern due to an opaque circle is identical to that of a circular aperture except for the bright spot in the center of the pattern.  The mathematics of Fresnel's theory and the Cornu Spiral predict this bright spot but it is often very hard to see.  Poisson actually tried to argue that Fresnel's theory was wrong because it predicted such a spot that had not been found experimentally.  This spot is now often referred to as Poisson's Spot.

References

Dauger, Dean. Fresnel Diffraction. http://www.dauger.com/fresnel/

Weisstein, Eric. Fresnel Diffraction - from Eric's Wonderful World of Physics. http://scienceworld.wolfram.com/physics/FresnelDiffraction.html

Fresnel Diffraction. http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/fresnelcon.html

Hecht, Eugene. Optics 4ed. Addison Wesley, San Francisco : 2002.