1. Fraunhofer Diffraction
Lecture 19
What is Diffraction?
Diffraction manifests itself in the apparent bending of waves around small obstacles and the spreading out of waves past small openings.
Diffraction
Fraunhofer Diffraction
Single Slit
Rectangular & Circular Apertures
Double Slit
Multiple Slits
Diffraction Grating
Rayleigh Criterion
Fresnel Diffraction
Cornu Spiral
Fresnel Zone plate
As screen is moved away from aperture, image of aperture passes through forms predicted in turn by
Geometrical optics  Near-field diffraction  Far-field diffraction
(no edge effects) (Fresnel diffraction) (Fraunhofer diffraction)

2. Spreading of waves past apertures
What is diffraction?
Spreading of waves past apertures
Diffraction pattern from a circular aperture
Diffraction pattern from a single-slit
Apparent bending of waves around obstacles
Diffraction pattern from a corner of a double-edged razor blade
Diffraction pattern from edge of metal

3. Fraunhofer Diffraction (far-field diffraction)
It deals with the limiting case where the light appoaching the diffracting object is parallel (i.e. wavefronts arriving at aperture and observation screen are considered plane) and monochromatic, and where the image plane is at a distance large compared to the size of the diffracting object
Diffraction pattern changes uniformly in size only as the viewing screen is moved relative to the aperture
Fresnel Diffraction (near-field diffraction)
Fresnel diffraction refers to the general case where the restrictions for Fraunhofer diffraction above are relaxed.
Curvature of wavefront must be taken into account.
Shape and size of diffraction pattern depend on distance between aperture and screen

4. Fraunhofer Diffraction from a Single Slit :
Single slit  rectangular aperture (length >> width)
Source far enough so that wavefronts reaching slit are plane (by using positive lens with source at focal point)
Observation screen is also very far, near infinity (a positive lens may be used to focus onto screen)  s ds f y P b  
Construction to determine irradiance on screen due to Fraunhofer diffraction by single-slit

5. because inverse square law for irradiance (E2  1/r2)
Fraunhofer Diffraction from a Single Slit :
Light that reaches point P is due to parallel rays of light from different portions of the wavefront at the slit (dashed pink line)
From Huygens-Fresnel principle, spherical wavelets are considered to propagate from a continuous array of wavelet sources on this wavefront
Waves arriving at P may not be in-phase (for example, having optical path length difference of  as shown), they are added according to principle of superposition to give the resultant field at P
Considering interval of wavefront ds as source, the resultant of all such sources is obtained by integrating over the entire slit width b
The differential field of the spherical wavelets reaching P originating from each interval ds at height s may be written as:
(19-1)
because inverse square law for irradiance (E2  1/r2)
r = optical path length from interval ds to point P; dE0 = amplitude at unit distance from source

6. Fraunhofer Diffraction from a Single Slit :
Rewritting Eqn. (19-1) in terms of path difference  and taking r = r0 when source is at s = 0, we have:
(19-2)
Path difference  being << r0 can be neglected in the amplitude term dE0/(r0+) ; however, the phase is very sensitive to small differences and  is therefore included
But and
EL = amplitude per unit width of slit at unit distance from source.
Eqn. (19-2) is rewritten as:
(19-3)
Integrating over width of slit, we have:
(19-4)

7. Therefore, Fraunhofer Diffraction from a Single Slit :
Considering only the bracketed term (giving the amplitude ER):
(19-5)
Substituting   ½ kb sin  ; we have
Therefore,
(19-6)
  phase difference between waves from the center and either endpoints of the slit; and its associated path difference is:
(19-7)

8. Irradiance at P  ER2 given by:
Fraunhofer Diffraction from a Single Slit :
Irradiance at P  ER2 given by:
(19-8) or I I0
sinc 
sinc2 
 = (kb/2) sin 
3
2   2 3
Irradiance function for single-slit Fraunhofer diffraction, normalized to I0 at center of pattern
Note:
sinc  1 when  0 as
Zeroes occur when sin  = 0, i.e., when
(19-9)
(m = 0 excluded)

9. Primary maximum irradiance occurs at  = 0 or y = 0
Fraunhofer Diffraction from a Single Slit :
But k = 2/ and condition for minimum irradiance of Fraunhofer diffraction pattern from single-slit is:
(19-10)
On the screen at distance f (see figure in Slide 5), minimum irradiance occurs at
(19-11)
(as  is small, sin   y/f )
Primary maximum irradiance occurs at  = 0 or y = 0
Secondary maxima occur when sinc function has maxima at condition
yielding

10. Fraunhofer Diffraction from a Single Slit :2  3 y 
y = tan 
y = 
Intersections of the curves y =  and y = tan  determines the angles  at which the sinc function is a maximum.
Points of intersection do not occur at exactly midway between successive m.
1.43
2.46
3.47
However, most of the energy of the diffraction pattern falls under the central maximum, which is much larger than the secondary maxima on either side.

11. Central maximum essentially represents image of the slit on screen.
Quantitative example for Fraunhofer Diffraction from a Single Slit :
What is the ratio of irradiances at central peak maximum to the first of the secondary maxima?
Solution:
The required ratio is
In other words, the irradiance of the 1st secondary maxima is only (1/21.2)x100% = 4.7% of the irradiance of central peak.
Central maximum essentially represents image of the slit on screen.
Edge has series of maxima & minima that tail off - blurring due to diffraction

12. Beam spreading due to diffraction
Spread of central maximum in far-field diffraction pattern of single-slit
 between 1st maxima on either side = Angular spread of central maximum
From (19-10), i.e. with m = 1 and approximating sin  by ,
(20-12)

13. Spread  is wider when slit width b is smaller or when  is larger.
Beam spreading due to diffraction
Spread  is wider when slit width b is smaller or when  is larger.
Angular spread independent of distance between aperture and screen, and thus linear dimensions of diffraction patterns increase with distance L such that width W of central maximum on screen is
(19-13)
This beam spreading is valid for rectangular apertures. (For circular apertures, Eqn. (19-13) is different.)
And we have assumed plane wavefront of uniform irradiance.
In fact, even a collimated beam (as if it emerged from slit-opening) has diffraction spreading.
Example: Consider a parallel beam of 546 nm of width b = 0.5 mm propagating across the room, a distance of 10 m.
The final width of the beam due to diffraction spreading is:

14. It is concluded that it is in the far-field when
Condition for far-field diffraction from consideration of beam spreading:
Fraunhofer diffraction requires L to be much larger than some minimum value Lmin at which the beam width W = b
Thus,
It is concluded that it is in the far-field when
A general approach yields the criterion for far-field diffraction:
(19-14)
(19-15)

15. Rectangular apertures:
Slit aperture
Screen
Rectangularaperture a b
Single slit diffraction. Only small dimension b of narrow slit causes appreciable spreading of light along the x-axis
Single slit diffraction. Both dimensions of square aperture are small and a 2-D diffraction pattern is discerned.

16. For aperture of width a, the irradiance will be (analogous to (19-8)):
Rectangular apertures:
For aperture of width a, the irradiance will be (analogous to (19-8)):
(19-16)
Therefore, the 2-D diffraction pattern from rectangular aperture of height & width, a  b, will have zero irradiance at
Irradiance over the screen is the product of the irradiance functions in each dimension, i.e.,
(19-17)
- involved double integration over both dimensions (area) of the aperture