# Diffraction Theory 0 Diffraction Theory

```Diffraction Theory
Diffraction Theory 0
Irradiance pattern (“Fraunhofer Diffraction Pattern” ) on an observation
screen far from a square aperture illuminated by an incident plane wave.
The dimensions of the aperture are on the order of the wavelength.
Diffraction Theory
“Interference” and “Diffraction” are arbitrarily distinguished from each
other.
Diffraction Theory 1
Interference  irradiance due to a collection of discrete sources.
Diffraction  irradiance due to a continuous distribution of sources.
“Fraunhofer Diffraction”  the observation point P is far away from the
distribution of sources or the irradiance in the focal plane of a lens is
observed.
Diffraction Theory
We will analyze “Fraunhofer Diffraction” from
apertures (illuminated by incident plane waves)
whose dimensions are on the order of the
wavelength. In our discussion of N slit interference,
we assumed the slit width to be infinitesimally small.
Diffraction Theory 2
Diffraction Theory
The superposition field at the observation point is generally
determined by an integral over the distribution of Huygens
emitters in the aperture.
Diffraction Theory 3
Diffraction Theory
Fraunhofer Diffraction also describes the irradiance pattern
in the focal plane of a lens.
Diffraction Theory 4
Single Slit Diffraction
Rather than evaluating an integral, we will determine the field from a
single slit (of width b on the order of the wavelength  ) by filling the
slit with an array of Huygens emitters (N slits of infinitesimal width)
and let N   as the separation between emitters  0.
Single Slit Diffraction 0
The slit: L>>  and b   .
The N emitters we place
inside this (vertical) slit are
infinitesimally wide vertical
slits.
Single Slit Diffraction
Rather than evaluating an integral, we will determine the field from a
single slit (of width b on the order of the wavelength  ) by filling the
slit with an array of Huygens emitters (N slits of infinitesimal width)
and let N   as the separation between emitters  0.
Single Slit Diffraction 1
S1 is at the edge of the
slit and the distance
Huygens slit emitters is
“a” (which we will shrink
to zero as N   ).
Point C is the centre of
the slit.
To point P,
far away (or
in lens focal
plane).
Single Slit Diffraction
The field at P due to the N Huygens slit emitters is:
Single Slit Diffraction 2
Our earlier
grating result.
is the phase change undergone by a
field propagating from C  P.
is the field amplitude at each
Huygens emitter.
Single Slit Diffraction
The field at P due to the N Huygens slit emitters is:
Single Slit Diffraction 3
As we let N   and a  0 :
Single Slit Diffraction
So, as N   and a  0 :
Single Slit Diffraction 4
where the last step is true by the small angle approx.
Single Slit Diffraction
So, as N   and a  0 , the field becomes:
Single Slit Diffraction 5
with
Physical realism requires that, as
such that
This has to be true in order for the overall field passing through the slit
to remain finite as the number of Huygens emitters becomes infinite!
Thus:
Single Slit Diffraction
The irradiance at observation point P is:
Single Slit Diffraction 6
2
with
We can put this in a more useful form by noting that
as
as
Single Slit Diffraction
We define the irradiance in the  = 0 (forward) direction as
Single Slit Diffraction 7
and we can express the irradiance in some general direction,  , in
terms of this quantity:
This is a practical formula as we are able to calculate the irradiance in
some general direction,  , relative to the irradiance in the forward
direction. This type of (relative) quantity can be measured easily in an
experiment.
Single Slit Diffraction
We can express the single slit irradiance function in a more common
way by introducing the “sinc” function:
Single Slit Diffraction 8
with
with
Properties of the “sinc” function:
Zeroes:
Maxima:
Single Slit Diffraction
The sinc function.
Single Slit Diffraction 9
sinc() vs 
sinc2() vs 
Single Slit Diffraction
The single slit diffraction pattern.
Single Slit Diffraction 10
Single Slit Diffraction
The single slit diffraction pattern: Variation with slit width.
Single Slit Diffraction 10a
Single Slit Diffraction
The single slit diffraction pattern: Limiting Cases.
Narrow Slit.
Single Slit Diffraction 11
sin 
=0
Single Slit Diffraction
The single slit diffraction pattern: Limiting Cases.
Wide Slit.
Single Slit Diffraction 12
Except near  = 0 !
sin 
=0
Single Slit Diffraction
Geometric optics limit:
(wide slit)
Single Slit Diffraction 13
Lens
focal
plane.
P
P


wide slit and a point source for
s   . This approximates the
point image expected in
geometrical optics.
For a narrow slit, the irradiance
function (diffraction pattern) is
the point image: “diffraction
limited optics”.
N Slit Diffraction
Expand our previous discussion of diffraction from a single slit to an
array of N slits with each slit having width “b” (b   ) and the
separation between slits (ie centre to centre distance) is “a”.
N Slit Diffraction 1
We consider N identical slits,
illuminated by an incident plane
wave and determine the
at an observation point P located
far from the slits (or lying in a lens
focal plane).
To observation
point P.
N Slit Diffraction
Expand our previous discussion of diffraction from a single slit to an
array of N slits with each slit having width “b” (b   ) and the
separation between slits (ie centre to centre distance) is “a”.
N Slit Diffraction 2
Each slit gives rise to a field
component at P:
where 0 is the phase change
undergone by the field in travelling
from the slit centre to P. The
phase change 0 is different for
each slit!
To observation
point P.
N Slit Diffraction
N Slit Diffraction 3
The overall superposition field at P
is the sum of field components
arising from each of the N slits.
Identifying 0i as the phase change
undergone by the field in travelling
from the centre of slit “i” to P we can
write this superposition field as:
N Slit Diffraction
N Slit Diffraction 4
Having identified 0i as the phase
change undergone by the field in
travelling from the centre of slit “i” to
P we can define the quantity  as
the difference in this phase change
arising from neighbouring slits.
N Slit Diffraction
With  defined, the superposition field can be written as:
N Slit Diffraction 5
We can sum the series (as we did for the grating) using the formula:
with
N Slit Diffraction
The result:
N Slit Diffraction 6
Where:
(b = slit width)
(a = slit separation)
is the phase change undergone
by a field travelling from the centre of the array to P.
N Slit Diffraction
The important quantity: Find the irradiance at P !
N Slit Diffraction 7
As before, for the single slit, we can write this in a more useful form by
seeing what happens as   0.
As   0 :
N Slit Diffraction
Defining the irradiance in the  = 0 (forward) direction as:
N Slit Diffraction 8
we can express the irradiance in some arbitrary direction,  , in terms
N Slit Diffraction
N
Slit
Diffraction
9
So we have the irradiance function:
N Slit Diffraction
Example: N=4
N Slit Diffraction 10
15
The black curve is the irradiance pattern.
N Slit Diffraction
Example: N=4
N Slit Diffraction 11
Interference
Fringes
Diffraction
“Envelope”
15
Interference fringes correspond to the principal maxima of the N slit
value of the single slit irradiance pattern: the “diffraction envelope”.
N Slit Diffraction
N Slit Diffraction 11a
Interference Fringes
(Principal Maxima)
Diffraction
“Envelope”
Interference fringes correspond to the principal maxima of the N slit
value of the single slit irradiance pattern: the “diffraction envelope”.
N Slit Diffraction
N
Slit
Diffraction
12
Interference fringes (principal maxima):
Diffraction envelope:
Zeroes:
Central maximum:
Subsidiary maxima:
(usually not concerned with these.)
N Slit Diffraction
N Slit Diffraction 12a
N Slit Diffraction 12a
N Slit Diffraction
Example: N=2 (Young’s Double Slit with finite width slits)
N Slit Diffraction 15
1. Double slit pattern for slit width b  
2. Double slit pattern for slit width b << 
3. Single slit pattern for slit width b in 1.
N Slit Diffraction
Example: N=2 (Young’s Double Slit with finite width slits)
N Slit Diffraction 15a
15
N Slit Diffraction
Example: N=2 (smaller slit width)
As b  0 , we approach our earlier double slit result.
N Slit Diffraction 16
15
N Slit Diffraction
Example: N=2 (smaller slit width)
As b  0 , we approach our earlier double slit result.
N Slit Diffraction 17
15
N Slit Diffraction
“Missing Order” in the irradiance pattern.
N Slit Diffraction 13
If the 1st order diffraction zero overlaps a principal maximum location,
then we have a missing order. Example:
15
The 3rd order
maximum is
missing here.
N Slit Diffraction
Calculating the missing order:
N Slit Diffraction 14
Overlap of the 1st diffraction zero with a principal maximum  they
occur at the same angle  . The order, m, of the principal maximum
for which this occurs is given by:
1st diff zero:
princ. max:
N Slit Diffraction
“Missing Order” in the irradiance pattern.
N Slit Diffraction 14a
For this example: a/b = 3  the 3rd order fringe is missing.
15
The 3rd order
maximum is
missing here.
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