Fraunhofer diffraction

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In optics, Fraunhofer diffraction or far-field diffraction is diffraction of light through an aperture, or slit, for small values of the Fresnel number, $F \ll 1$ . It is also known as parallel beam diffraction. This corresponds to the observation of diffraction on a system (aperture, grating, slit, etc.) which is at large distance both from the source and the screen (where the interference pattern is observed). Essentially the same situation takes place when the optical system is sandwiched by a pair of positive lenses with the source and the screen at their foci.

1 Aperture diffraction
2 Slit diffraction
3 Theory
4 Examples
- 4.1 Diffraction from a slit
- 4.2 Gaussian profile
5 See also
6 External links
7 References

[edit] Aperture diffraction

A far-field pattern exists at distances that are large compared with s ²/λ, where s is a characteristic dimension of the source and λ is the wavelength. For example, if the source is a uniformly illuminated circle, then s is the radius of the circle.

The far-field diffraction pattern of a source may also be observed (except for scale) in the focal plane of a well-corrected lens. The far-field pattern of a diffracting screen illuminated by a point source may be observed in the image plane of the source.

If a light source and an observation screen are effectively far enough from a diffraction aperture (for example a slit), then the wavefronts arriving at the aperture and the screen can be considered to be collimated, or plane. Fresnel diffraction, or near-field diffraction occurs when this is not the case and the curvature of the incident wavefronts is taken into account.

In far-field diffraction, if the observation screen is moved relative to the aperture, the diffraction pattern produced changes uniformly in size. This is not the case in near-field diffraction, where the diffraction pattern changes both in size and shape.

[edit] Slit diffraction

Fraunhofer diffraction through a slit can be achieved with two lenses and a screen. Using a point-like source for light and a collimating lens it is possible to make parallel light, which will then be passed through the slit. After the slit there is another lens that will focus the parallel light onto a screen for observation. The same setup with multiple slits can also be used, creating a different diffraction pattern.

Since this type of diffraction is mathematically simple, this experimental setup can be used to find the wavelength of the incident monochromatic light with high accuracy.

[edit] Theory

For an aperture with an amplitude transmittance $f (x)$ on which plane waves are incident, the diffracted far-field amplitude as a function of the angle $θ$ with the propagation direction of the incident waves is

$\psi(\theta) \propto \int_{-\infty}^{\infty} f(x) e^{i(k\theta)x} dx,$

where $k = 2π / λ$ is the circular wavenumber of the incident waves. The expression is the Fourier transform of the aperture function against the variable $k θ$ . This approximation is only valid for small values of $θ$ . Note that the aperture function acts on the amplitude, not on the intensity (amplitude squared) of the waves. It can be complex-valued to indicate a phase shift.

[edit] Examples

[edit] Diffraction from a slit

The simplest example is Fraunhofer diffraction from a slit, i.e. $f (x) = 1$ for $- a / 2 < x < a / 2$ and $f (x) = 0$ elsewhere. In this case,

ψ(θ) = sinc(π a θ / λ),

The unnormalized sinc function peaks at $θ = 0$ and has zeroes at $\theta=\pm n\lambda/a$ , where $n=1, 2, \ldots$ .

[edit] Gaussian profile

An aperture with a Gaussian profile $f (x) = exp( - a x 2)$ results in

$\psi(\theta) = \exp\left(\frac{-k^2\theta^2}{4a}\right) .$

For example, if a laser beam has an intensity profile with a full width at half maximum (FWHM) $W$ , then $a = 2ln2 / W 2$ . At a wavelength $λ$ , the amplitude profile is

$\psi(\theta) = \exp\left(-\frac{\pi^2W^2}{2 \lambda^2\ln 2}\theta^2\right),$