Chapter 2.8 - Diffraction

2.8 DIFFRACTION

Consider a screen having a small hole with sharp edges at the periphery and dimensions
comparable to the wavelength of light (this may be an aperture, a slit, or a
grating) and a second screen behind it at distance d (Figure 1.18). Consider a parallel
(collimated) and monochromatic beam, known as light from a source at infinity,
directed to the small hole. Since light travels in free space in a straight path, a small
round projection is expected, D_EXP. However, a wider projection is seen instead,
D_ACT, with concentric rings surrounding it. The smaller the hole diameter, the wider
the projection.

This phenomenon is attributed to interference at each edge point of the hole,
which acts like a secondary source resulting in diffraction; this is also known as the
phenomenon of Fresnel. Moreover, the intensity of rings, J, decreases with distance
from the center of the projection (Figure 2.5). This pattern is also known as Fraunhofer
diffraction.

According to Huygens’s principle, the incident wave excites coherent secondary
waves at each point of the wavefront. All these waves interfere with each other and
cause the diffraction pattern. The pure mathematical analysis of diffraction is studied
by approximating the scalar Helmholz equation in which the wavefront is a
spherical function. However, to illustrate this point, if a plane wave is incident on

$Figure 2.5. Intensity distribution of diffracted light passing through a round hole.$

an aperture, then the radiant intensity per unit of projected area of the source viewed
from that direction (that is the radiance, L), is described by

where α, β, and γ are propagation vectors in the x, y, and z directions, respectively,
A is the area of the diffracting aperture, λ is the wavelength, F is the Fourier transform,
and U is the complex amplitude distribution emerging from the aperture
(hence, z = 0).

Integrating the above, invoking the conservation of energy theorem, and using
Parceval’s (or Rayleigh’s) theorem, which converts the integral of the squared modulus
of a function f(x) to the integral of the squared modulus of its Fourier transform
f(ω),

where the integrals (int) are from – to + infinity, the radiance is expressed by

and K is a normalization constant.

Now, let us consider an aperture that is a narrow rectangular slit with height h and
width w. In this case, the monochromatic collimated beam passes through it and, due
to refraction, the projection on a screen is a rectangle rotated by 90°. That is, the refracted
pattern is narrow in the direction in which the aperture is wide. Moreover, because
of two-dimensional Fourier expansion, the refracted light forms many secondary
rectangles in the X–Y plane of the screen, with intensity fading away from the
axis of symmetry (Figure 2.6). This is also known as Fraunhofer diffraction of a rectangular
aperture. The condition for these rectangles on the screen is

and

R(x, y) = 0 elsewhere

Theoretically speaking, when a diffracting aperture is illuminated by a uniform
monochromatic plane wave, λ, the total radiant power, P_TR, emanating from the diffracting
aperture is calculated using Rayleigh’s theorem and Fourier transforms:

where U₀ is the function of the complex amplitude distribution emerging from the
diffracting aperture, β is the propagating vector and {x, y} are the Cartesian coordinates.

The above discussion applies to monochromatic light. If light is polychromatic,
then each color (frequency) component would be diffracted differently; that is, at a
different angle. Then, fringes on the screen would be of different colors but overlapping.
The corollary is that the angle of diffraction depends on frequency (color)
of the light.

If in the above experiment, instead of one slit there is an array of parallel slits
then, because of interference among the slits, light is diffracted at certain angles.
This diffraction of light is known as diffraction by transmission. Interesting
phenomena occur when slits are positioned in a matrix configuration. Diffraction
then takes place in a two-dimensional plane and the fringe pattern depends
on size, shape, density, wavelength, and topological arrangement of slits. Thus,
the accuracy of diffraction-based devices may impact the spectral content of an
optical signal, which may be manifested as signal-to-noise ratio and BER degradation.

$Figure 2.6. Diffraction of light passing through a rectangular hole (diagram and photo).$

Now, if in the above discussion, instead of slits in the diaphragm there are parallel
grooves each at a width comparable to the wavelength, then the same theory applies.
The difference is that light diffracts back at specified angles. This is known as
diffraction by reflection.

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