Fraunhofer diffraction
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.[1][2] 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.[3]
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 (mathematics).
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.[4] 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.[5]
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.[6] 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
Fraunhofer diffraction occurs when: |
– aperture or slit size,
– wavelength, – distance from the aperture |
When the distance between the aperture and the plane in which the pattern is observed is large enough that the difference in phase 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.[7]
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
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.[9] 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:[10]
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 < λ, the intensity of the diffracted light does not fall to zero, and if D << λ, the diffracted wave is cylindrical.
Semi-quantitative analysis of single slit diffraction
The angle subtended by the first minima on either side of the centre is then, as above:
There is no such simple argument to enable us to find the maxima of the diffraction pattern.
Diffraction by a rectangular aperture
If the illuminating beam does not illuminate the whole length of the slit, the spacing of the vertical fringes is determined by the dimensions of the illuminating beam. Close examination of the double-slit diffraction pattern below shows that there are very fine horizontal diffraction fringes above and below the main spot, as well as the more obvious horizontal fringes.
Diffraction by a circular aperture
where W is the diameter of the aperture.
The Airy disk can be an important parameter in limiting the ability of an imaging system to resolve closely located objects.
Diffraction by an aperture with a Gaussian profile
The output profile of single mode laser beam may have a Gaussian intensity profile and the diffraction equation can be used to show that it maintains that profile however far away it propagates from the source.[14]
Diffraction by a double slit
The angular spacing of the fringes is given by
The spacing of the fringes at a distance z from the slits is given by [16]
where d is the separation of the slits.
The fringes in the picture were obtained using the yellow light from a sodium light (wavelength = 589 nm), with slits separated by 0.25 mm, and projected directly onto the image plane of a digital camera.
Double slit interference fringes can be observed by cutting two slits in a piece of card, illuminating with a laser pointer, and observing the diffracted light at a distance of 1 m. If the slit separation is 0.5 mm, and the wavelength of the laser is 600 nm the spacing of the fringes viewed at a distance of 1 m would be 1.2 mm.
Semi quantitative explanation of double-slit fringes
The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves.
When the two waves are in phase, i.e. the path difference is equal to an integral number of wavelengths, the summed amplitude, and therefore the summed intensity is maximum, and when they are in anti-phase, i.e. the path difference is equal to half a wavelength, one and a half wavelengths, etc., then the two waves cancel and the summed intensity is zero. This effect is known as interference.
The interference fringe maxima occur at angles
where λ is the wavelength of the light. The angular spacing of the fringes is θf is given by
When the distance between the slits and the viewing plane is z, the spacing of the fringes is equal to zθ and is the same as above:
Diffraction by a grating
A grating whose elements are separated by S diffracts a normally incident beam of light into a set of beams, at angles θn given by:[17]
This is known as the grating equation The finer the grating spacing, the greater the angular separation of the diffracted beams.
If the light is incident at an angle θ0, the grating equation is:
The detailed structure of the repeating pattern determines the form of the individual diffracted beams, as well as their relative intensity while the grating spacing always determines the angles of the diffracted beams.
The image on the right shows a laser beam diffracted by a grating into n = 0, and ±1 beams. The angles of the first order beams is about 20°; if we assume the wavelength of the laser beam is 600 nm, we can infer that the grating spacing is about 1.8 μm.
Semi-quantitative explanation
This is the same relationship that given above.
See also
- Fraunhofer diffraction (mathematics)
- Diffraction
- Huygens-Fresnel principle
- Kirchhoff's diffraction formula
- Fresnel diffraction
- Airy disc
- Fourier optics
References
- ↑ Born & Wolf, 1999, p. 427.
- ↑ Jenkins & White, 1957, p288
- ↑ http://scienceworld.wolfram.com/biography/Fraunhofer.html
- ↑ Heavens and Ditchburn, 1996, p. 62
- ↑ Born & Wolf, 1999, p. 425
- ↑ Jenkins & White, 1957, Section 15.1, p. 288
- ↑ Lipson, Lipson and Lipson, 2011, p. 203
- ↑ Hecht, 2002, p. 448
- ↑ Hecht, 2002, Figures 10.6(b) and 10.7(e)
- ↑ Jenkins & White, 1957, p. 297
- ↑ Born & Wolf, 1999, Figure 8.10
- ↑ Born & Wolf, 1999, Figure 8.12
- ↑ Hecht, 2002, Figure 11.33
- ↑ Hecht, 2002, Figure 13.14
- ↑ Born & Wolf, 1999, Figure 7.4
- ↑ Hecht, 2002, eq. (9.30)
- ↑ Longhurst, 1957, eq.(12.1)
Reference sources
- Born M & Wolf E, Principles of Optics, 1999, 7th Edition, Cambridge University Press, ISBN 978-0-521-64222-4
- Heavens OS and Ditchburn W, Insight into Optics, 1991, Longman and Sons, Chichester ISBN 978-0-471-92769-3
- Hecht Eugene, Optics, 2002, Addison Wesley, ISBN 0-321-18878-0
- Jenkins FA & White HE, Fundamentals of Optics, 1957, 3rd Edition, McGraw Hill, New York
- Lipson A., Lipson SG, Lipson H, Optical Physics, 4th ed., 2011, Cambridge University Press, ISBN 978-0-521-49345-1
- Longhurst RS, Geometrical and Physical Optics,1967, 2nd Edition, Longmans, London