In optics, the Fraunhofer diffraction equation is used to model the
diffraction of waves when the diffraction pattern is viewed at a long
distance from the diffracting object, and also when it is viewed at the focal
plane of an imaging lens. In contrast, the diffraction pattern created near the
object, in the near field region, is given by the Fresnel diffraction
equation. The equation was named in honor of
Joseph von Fraunhofer although he was not actually involved in the development
of the theory. This article explains where the
Fraunhofer equation can be applied, and shows the form of the Fraunhofer
diffraction pattern for various apertures. A detailed mathematical
treatment of Fraunhofer diffraction is given Fraunhofer diffraction.
The Fraunhofer diffraction equation When a beam of light is partly blocked
by an obstacle, some of the light is scattered around the object, and light
and dark bands are often seen at the edge of the shadow – this effect is
known as diffraction. These effects can be modelled using the Huygens–Fresnel
principle. Huygens postulated that every point on a primary wavefront acts as a
source of spherical secondary wavelets and the sum of these secondary waves
determines the form of the wave at any subsequent time. Fresnel developed an
equation using the Huygens wavelets together with the principle of
superposition of waves, which models these diffraction effects quite well.
It is not a straightforward matter to calculate the displacement given by the
sum of the secondary wavelets, each of which has its own amplitude and phase,
since this involves addition of many waves of varying phase and amplitude.
When two waves are added together, the total displacement depends on both the
amplitude and the phase of the individual waves: two waves of equal
amplitude which are in phase give a displacement whose amplitude is double
the individual wave amplitudes, while two waves which are in opposite phases
give a zero displacement. Generally, a two-dimensional integral over complex
variables has to be solved and in many cases, an analytic solution is not
available. The Fraunhofer diffraction equation is a
simplified version of the Kirchhoff's diffraction formula and it can be used
to model the light diffracted when both the light source and the viewing plane
are effectively at infinity with respect to the diffracting aperture. In this
case, the incident light is a plane wave so that the phase of the light at each
point in the aperture is the same. The phase of the contributions of the
individual wavelets in the aperture varies linearly with position in the
aperture, making the calculation of the sum of the contributions relatively
straightforward in many cases. Strictly speaking, the Fraunhofer
approximation only applies when the diffracted pattern is viewed at
infinity, but in practice it can be applied in the far field, and also in
the focal plane of a positive lens. = Far field =
When the distance between the aperture and the plane in which the pattern is
observed is large enough that the difference in the optical path length
between the light from the extremes of the aperture is much less than the
wavelength, then individual contributions can be treated as though
they are parallel. This is often known as the far field and is defined as being
located at a distance which is significantly greater than W2/λ, where λ
is the wavelength and W is the largest dimension in the aperture. The
Fraunhofer equation can be used to model the diffraction in this case.
For example, if a 0.5mm diameter circular hole is illuminated by a laser
with 0.6μm wavelength, the Fraunhofer diffraction equation can be employed if
the viewing distance is greater than 1000mm.
= Focal plane of a positive lens = A plane wave incident on a positive lens
is focused at a point by the lens; all the 'rays' have the same phase at the
point of focus, so that this is equivalent to viewing the plane wave at
infinity. Thus, if the diffracted light is focused with a lens, the observed
diffraction pattern can be modelled using Fraunhofer diffraction. The
diffracted light can be considered to be made up of a set of plane waves of
varying orientation. When a lens is located in front of the diffracting
aperture, each plane wave is brought to a focus at a different point in the
focal plane with the point of focus being proportional to the x- and
y-direction cosines, so that the variation in intensity as a function of
direction is mapped into a positional variation in intensity.
Examples of Fraunhofer diffraction In each of these examples, the aperture
is illuminated by a monochromatic plane wave at normal incidence.
= Diffraction by a slit of infinite depth =
The width of the slit is W. The Fraunhofer diffraction pattern is shown
in the image together with a plot of the intensity vs. angle θ. The pattern has
maximum intensity at θ = 0, and a series of peaks of decreasing intensity. Most
of the diffracted light falls between the first minima. The angle, α,
subtended by these two minima is given by:
Thus, the smaller the aperture, the larger the angle, α subtended by the
diffraction bands. The size of the central band at a distance z is given by
For example, when a slit of width 0.5 mm is illuminated by light of wavelength
0.6 µm, and viewed at a distance of 1000 mm, the width of the central band in the
diffraction pattern is 2.4 mm. The fringes extend to infinity in the y
direction since the slit and illumination also extend to infinity.
If W