Diffraction Theory 0Diffraction Theory

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 1“Interference” and “Diffraction” are arbitrarily distinguished from each other.

Interference irradiance due to a collection of discrete sources.Diffraction irradiance due to a continuous distribution of sources.

Diffraction Theory

“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 2Diffraction 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 3Diffraction 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 4Diffraction Theory

Fraunhofer Diffraction also describes the irradiance pattern in the focal plane of a lens.

Single Slit Diffraction 0Single Slit DiffractionRather 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.

The slit: L>> and b .

The N emitters we place inside this (vertical) slit are infinitesimally wide vertical slits.

Single Slit Diffraction 1Single Slit DiffractionRather 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.

S1 is at the edge of the slit and the distance between adjacent 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 2Single Slit DiffractionThe field at P due to the N Huygens slit emitters is:

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 3Single Slit DiffractionThe field at P due to the N Huygens slit emitters is:

As we let N and a 0 :

Single Slit Diffraction 4Single Slit DiffractionSo, as N and a 0 :

where the last step is true by the small angle approx.

Single Slit Diffraction 5Single Slit DiffractionSo, as N and a 0 , the field becomes:

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 6Single Slit DiffractionThe irradiance at observation point P is:

We can put this in a more useful form by noting that

as

as

with

2

Single Slit Diffraction 7Single Slit DiffractionWe define the irradiance in the = 0 (forward) direction as

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 8Single Slit DiffractionWe can express the single slit irradiance function in a more common way by introducing the “sinc” function:

with

with

Properties of the “sinc” function:

Zeroes:

Maxima:

Single Slit Diffraction 9Single Slit DiffractionThe sinc function.

sinc() vs

sinc2() vs

Single Slit Diffraction 10Single Slit DiffractionThe single slit diffraction pattern.

Single Slit Diffraction 10aSingle Slit DiffractionThe single slit diffraction pattern: Variation with slit width.

Single Slit Diffraction 11Single Slit DiffractionThe single slit diffraction pattern: Limiting Cases.Narrow Slit.

sin = 0

Single Slit Diffraction 12Single Slit DiffractionThe single slit diffraction pattern: Limiting Cases.Wide Slit.

Except near = 0 !

sin = 0

Single Slit Diffraction 13Single Slit DiffractionGeometric optics limit: (wide slit)

The irradiance function for a 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 “spread out” in comparison to the point image: “diffraction limited optics”.

Lens focal plane.

P

P

N Slit Diffraction 1N Slit DiffractionExpand 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”.

We consider N identical slits, illuminated by an incident plane wave and determine the superposition field and irradiance at an observation point P located far from the slits (or lying in a lens focal plane).

To observation point P.

N Slit Diffraction 2N Slit DiffractionExpand 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”.

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 3N Slit Diffraction

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 4N Slit Diffraction

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 5N Slit DiffractionWith defined, the superposition field can be written as:

We can sum the series (as we did for the grating) using the formula:

with

N Slit Diffraction 6N Slit DiffractionThe result:

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 7N Slit DiffractionThe important quantity: Find the irradiance at P !

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 8N Slit DiffractionDefining the irradiance in the = 0 (forward) direction as:

we can express the irradiance in some arbitrary direction, , in terms of the “forward” irradiance:

N Slit Diffraction 9N Slit Diffraction

So we have the irradiance function:

N Slit Diffraction 10N Slit DiffractionExample: N=4

The black curve is the irradiance pattern.

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N Slit Diffraction 11N Slit DiffractionExample: N=4

Diffraction “Envelope”

Interference Fringes

Interference fringes correspond to the principal maxima of the N slit irradiance pattern. The irradiance in a given fringe depends on the value of the single slit irradiance pattern: the “diffraction envelope”.

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N Slit Diffraction 11aN Slit Diffraction

Diffraction “Envelope”

Interference Fringes (Principal Maxima)

Interference fringes correspond to the principal maxima of the N slit irradiance pattern. The irradiance in a given fringe depends on the value of the single slit irradiance pattern: the “diffraction envelope”.

N Slit Diffraction 12N Slit DiffractionProperties of the irradiance function.

Interference fringes (principal maxima):

Diffraction envelope:

Zeroes:

Central maximum:

Subsidiary maxima: (usually not concerned with these.)

N Slit Diffraction 12aN Slit Diffraction

N Slit Diffraction 12a

N Slit Diffraction 15N Slit Diffraction

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.

Example: N=2 (Young’s Double Slit with finite width slits)

N Slit Diffraction 15aN Slit DiffractionExample: N=2 (Young’s Double Slit with finite width slits)

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N Slit Diffraction 16N Slit DiffractionExample: N=2 (smaller slit width)As b 0 , we approach our earlier double slit result.

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N Slit Diffraction 17N Slit DiffractionExample: N=2 (smaller slit width)As b 0 , we approach our earlier double slit result.

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N Slit Diffraction 13N Slit Diffraction“Missing Order” in the irradiance pattern.

If the 1st order diffraction zero overlaps a principal maximum location, then we have a missing order. Example:

The 3rd order maximum is missing here.

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N Slit Diffraction 14N Slit DiffractionCalculating the missing order:

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 14aN Slit Diffraction“Missing Order” in the irradiance pattern.

For this example: a/b = 3 the 3rd order fringe is missing.

The 3rd order maximum is missing here.

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