Maxima and minima in the diffraction pattern arise due to differences in the phase of the beams scattered from each point along the object. We need to take this into account to calculate the diffraction pattern.

What is diffractionin physics

If the central (zero order) spot (undiffracted straight-through beam) is solely used, the resulting image is known as the bright field image. If a non-zero order diffraction spot is solely used then a dark field image.

where n is the refractive index of the medium (n = 1 for air) and sinα is known as the numerical aperture, N.A. (commonly printed on the side of a lens). If a microscope can just resolve a "400 lines per mm" grating, what would the N.A. of the lens be?

$${\rm{/(S) = (A(S)}}{{\rm{)}}^2}{\rm{ = }}{{\rm{A}}_0}^2\left( {{{{{\sin }^2}\left( {{{\pi w\sin 2\theta } \over \lambda }} \right)} \over {{{\left( {{{\pi \sin 2\theta } \over \lambda }} \right)}^2}}}} \right)$$

Consider a one-dimensional object extending in the x direction. A beam of coherent monochromatic light is incident on the object and it is diffracted away from its original trajectory to produce a scattered beam. We can use unit vectors to record the direction of the incident beam and the scattered beam. The incident beam is parallel to the vector s0 and the scattered beam is parallel to the vector s. Two points on the object at O and X are joined by the vector x. At each point, scattering of the incident beam occurs isotropically (i.e. through all angles). We will consider the beam scattered through an angle 2θ from the original beam direction.

Diagram ofdiffractionof light

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Diffraction patterns can be calculated by from a function representing the mask. The symmetry of the pattern can reveal useful information on the symmetry of the mask. For a periodic object, the pattern is equivalent to the reciprocal lattice of the object.

Multi-beam images (composed of various spots, and known as 'high resolution images') are commonly used to study dislocations. Dark field imaging can be used to highlight the dislocation lines, and by tilting the electron beam, the Burgers vector can be determined. These techniques are common in Transmission Electron Microscopy.

The observed diffraction pattern is neither the sum nor the product of the original patterns of the individual gratings, but the separate patterns are repeated to form a two-dimensional array.

The diffraction pattern of a mask without a centre of symmetry will still be symmetrical. This can be seen in the mathematics of calculating the pattern. The non-centrosymmetric nature of the mask will however cause non-centrosymmetric variations in the phase.

As in the previous questions, a convex lens has a focal length of 150 mm and is placed 180 mm from a mask on an optical bench, giving an image distance of 900 mm. If the above lens is 56 mm in diameter, what is the finest grating size that could be resolved theoretically using light of wavelength 0.6328 μm?

where d is the distance from the source to mask, L is the distance from mask to image, w is the width of the slit and λ is the wavelength of the radiation. If the condition is not met, then curvature of the waves is involved and the resulting effect is known as Fresnel diffraction.

Diffractiondiagram

$$\delta {\rm{A}} = {{\rm{A}}_0}\rho ({\rm{x}})\delta {\rm{x}}\exp \left( {2\pi {\rm{i}}{{{\bf{x}} \cdot {\bf{S}}} \over \lambda }} \right)$$

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In optics, the basic shape of the mask is preserved in the bright field image, and some fine detail is lost. In electron diffraction, the contrast of the bright field image is due entirely to thickness and density variations in the sample. A convex glass lens is typically used to focus laser light, but magnetic fields are required to focus electron beams. By selecting individual diffraction spots, dark field images can be used in electron microscopy to distinguish phases (such as characterising two phase intergrowths in crystals).

Diffraction patterns can be calculated mathematically. The operation that directly predicts the amplitude of the diffraction pattern from the mask is known as a Fourier Transform (provided the conditions for Fraunhofer Diffraction are satisfied). The derivation of some simple patterns can be found here.

The theoretical resolution of a microscope is given bywhere n is the refractive index of the medium (n = 1 for air) and sinα is known as the numerical aperture, N.A. (commonly printed on the side of a lens). If a microscope can just resolve a "400 lines per mm" grating, what would the N.A. of the lens be?

As light, or any wave, passes a barrier, the waveform is distorted at the boundary edge. If the wave passes through a gap, more obvious distortion can be seen. As the gap width approaches the wavelength of the wave, the distortion becomes even more obvious. This process is known as diffraction. If the diffracted light is projected onto a screen some distance away, then interference between the light waves create a distinctive pattern (the diffraction pattern ) on the screen. The nature of the diffraction pattern depends on the nature of the gap (or mask) which diffracts the original light wave.

In conventional image formation, a lens focuses the diffracted waves into an image. Since the individual sections (spots) of the diffraction pattern each contain information, by forming an image from only particular parts of the diffraction pattern, the resulting image can be used to enhance particular features. This is used in bright and dark field imaging.

The nature of diffraction from a single slit allows macro-scale measurements to be used to calculate micro-scale dimensions. This has important implications - for example, allowing microscopes resolve to very fine scale (nanometre scale).

The angle γ* in the diffraction pattern is complementary to the angle between the grating axes, γ. i.e. γ + γ* = 180º This can be seen by rotating the gratings with respect to each other.

Diffractionexamples

Whilst a very basic knowledge of the physics of waves and optics is assumed, this teaching and learning package covers the fundamentals of diffraction and imaging.

The nature of the diffraction pattern (shape, symmetry, dimensions, etc.) is determined by the nature of the mask that diffracts the light. A lens can recombine the (accessible) diffracted light to generate a magnified image of the mask. However, by forming the image from a limited proportion of the pattern, then elements of the mask can be enhanced.

$${\rm{A(}}{\bf{S}}{\rm{)}} = {{\rm{A}}_0}\int\limits_{ - \infty }^\infty {\rho ({\rm{x}})\exp \left( {2\pi {\rm{i}}{{{\bf{x}} \cdot {\bf{S}}} \over \lambda }} \right){\rm{d}}x} = {{\rm{A}}_0}\int\limits_{ - w/2}^{w/2} {\exp \left( {2\pi {\rm{i}}{{{\bf{x}} \cdot {\bf{S}}} \over \lambda }} \right){\rm{d}}x} $$

Laser diffraction experiments can be conducted using an optical bench, as shown below. Light from the laser (of wavelength λ) is diffracted by a mask (usually a small aperture or grating) and projected onto the screen, located at a large distance away, such that Fraunhofer geometry applies. The light on the screen is known as the diffraction pattern.

The diffraction pattern from a single slit is projected onto a screen 0.90 m from the slit. Seven spots are visible, with a bright central spot. The maxima of the outer spots are a distance of 15 cm apart. A He-Ne laser is used, which produces light of a wavelength of 0.6328 μm. What is the width of the slit?

Fresneldiffraction

When the diffraction pattern from a particular grating is projected onto a screen, the even diffraction spots are missing, i.e. the second, fourth, etc. What is the relationship between the width of the slits (w) of the grating and the distance between each slit (s)?

A diffraction pattern shows just two-fold symmetry. Which one of these apertures could not have produced such a pattern?

The form of this relationship is a sinc2 function. The variation of I(S) with the variable (sin 2θ / λ) is shown below. The central maximum is much more intense than the diffracted maxima - this central maximum corresponds to the undiffracted beam.

By considering diffraction from a grating, the reciprocal nature of the pattern can be derived. This relationship can be seen in the diffraction patterns of the slits: small features of the diffracting object give wide spacings in the diffraction pattern

If only one diffraction spot is allowed through the back focal plane then no information about the spacing of the slits is passed on to the image and individual slits will not be resolved. Note, however, that each diffraction spot is made up of beams scattered from all parts of the object. Therefore, information about the size and shape of the object as a whole is passed on to the image through a single diffraction spot.

A convex lens has a focal length of 150 mm. If it is placed 180 mm from a mask on an optical bench, where must the screen be placed in order to focus the diffracted light into a sharp image?

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A lens will focus light from infinity to the 'focal point', at a distance from the lens known as the focal length, f. Located at the focal point, is the back focal plane of the lens where the diffraction pattern is visible (by using a screen). The diffraction pattern acts as a source of light that propagates to the screen where the image is formed. This theory was first described by Ernst Abbe in 1872.

What is diffractiongrating

The resulting diffraction pattern of a complex mask can be predicted by considering the individual diffraction patterns associated with the components that make up the shape of the mask. This can be seen in the diffraction pattern of a row of apertures.

Diffractionof waves

This form is familiar: the scattered amplitude A(S) is the Fourier transform of the scattering density ρ(x). Note that the information about the angle 2θ is contained within the scattering vector S.

The observable pattern projected onto the screen (a distance L away) has an intensity pattern as follows, where x is the distance from the straight-through position:

The basic features of diffraction and imaging have been presented in this package. When a wave, such as light, passes through a small aperture, it will be distorted. It will form a distinctive pattern on a screen, known as the diffraction pattern. This pattern contains information on the diffracting aperture (such as a mask or grating), with an inverse relationship in dimensions. The form of the intensity pattern can be predicted mathematically.

The following heart-shaped aperture produces the adjacent diffraction pattern.Which of the following masks should be placed in the back focal plane in order to best study the horizontal stripes of the aperture in the image? Dashed lines are shown to identify location of central diffraction spot with respect to mask.

The symmetry of a diffraction pattern can reveal useful information on the symmetry of the mask. This is exploited in the electron diffraction of crystals, where the pattern can reveal the nature of the crystallographic symmetry, e.g. the periodicity of the structure; the distribution of atoms in the unit cell; and the shape of the crystal. X-ray diffraction patterns are used to measure spacing between layers or rows of atoms, and to determine crystal orientations and structures.

$${\rm{A(}}{\bf{S}}{\rm{) = }}{{\rm{A}}_0}\int\limits_{ - w/2}^{w/2} {\exp \left( {2\pi {\rm{i}}{{{\bf{x}} \cdot {\bf{S}}} \over \lambda }} \right){\rm{d}}x} = {{\rm{A}}_0}\int\limits_{ - w/2}^{w/2} {\exp \left( {2\pi {\rm{i}}{{{\bf{x}}\sin 2\theta } \over \lambda }} \right){\rm{d}}x} $$ $${\rm{ = }}{{\rm{A}}_0}\left[ {{{\exp \left( {{{2\pi {\rm{i}}{\bf{x}}\sin 2\theta } \over \lambda }} \right)} \over {{{2\pi {\rm{i}}\sin 2\theta } \over \lambda }}}} \right]_{ - w/2}^{w/2}$$ $${\rm{ = }}{{\rm{A}}_0}\left( {{{\sin \left( {{{\pi w\sin 2\theta } \over \lambda }} \right)} \over {\left( {{{\pi \sin 2\theta } \over \lambda }} \right)}}} \right)$$

Increasing the spread of the diffraction pattern allows for more accurate measurement of the spot spacing. Which one of these will achieve this?

As in the previous questions, a convex lens has a focal length of 150 mm and is placed 180 mm from a mask on an optical bench, giving an image distance of 900 mm. What is the magnification of the object in this set-up?

A lens can be used to form an image of the mask onto the screen. The diffraction pattern of the mask can be seen in the back focal plane of the lens. By forming the image from selected portions of the diffraction pattern in the back focal plane, particular information present in the image can be enhanced.

Fraunhofer diffraction occurs when the distances from the object to the source and the object to the image are so great that the incoming and outgoing waves are effectively planar. It is named after Joseph von Fraunhofer, who studied diffraction, and even built the first diffraction grating. The condition is given by

More complicated masks, for example a periodic row of apertures, will show more intricate diffraction patterns, but still follow the same basic inverse relationship.

Which of the following masks should be placed in the back focal plane in order to best study the horizontal stripes of the aperture in the image? Dashed lines are shown to identify location of central diffraction spot with respect to mask.

As in the previous question, a convex lens has a focal length of 150 mm and is placed 180 mm from a mask on an optical bench. Where must the screen be placed in order to observe the diffraction pattern?

When a convex lens is placed between the mask and the screen, the optical bench can form magnified images of the mask onto the screen. Use of a mirror can simply extend the effective screen distance. Note: caution should be taken when using a mirror to reflect laser light.

A diffraction grating is effectively a multitude of equally-spaced slits. The diffraction pattern from a complex mask such as a grating can be constructed from simplier patterns via the convolution theorem. The observed diffraction pattern is composed of repeated "sinc-squared" functions. Their positions from the central spot are determined by s (the spacing between slits) and their relative intensity is dependent on w (the width of individual slits).

$${\rm{A(}}{\bf{S}}{\rm{) = }}{{\rm{A}}_0}\int\limits_{ - \infty }^\infty {\rho ({\rm{x}})\exp \left( {2\pi {\rm{i}}{{{\bf{x}} \cdot {\bf{S}}} \over \lambda }} \right){\rm{d}}x} $$ $$\eqalign{ & {\rm{ = }}{{\rm{A}}_0}\int\limits_{ - b/2}^{b/2} {\left( {\delta \left( {x - {b \over 2}} \right) + \delta \left( {x + {b \over 2}} \right)} \right)\exp \left( {2\pi {\rm{i}}{{{\bf{x}}\sin 2\theta } \over \lambda }} \right){\rm{d}}x} \cr & {\rm{ = 2}}{{\rm{A}}_0}\cos \left( {{{\pi b \sin 2\theta } \over \lambda }} \right) \cr} $$

A variable aperture can be placed at the back focal plane. Thus the aperture can be adjusted to limit the region of the diffraction pattern that goes on to form the image. The minimum area of the pattern necessary to form a “full” image of the zebra (with overall shape and stripes visible) contains the undiffracted beam and one of the first diffraction spots. In order to properly resolve the features of the mask, both first order diffracted spots should be included.

This gives rise to a sinusoidal variation of amplitude with (sin 2θ) / λ. The intensity of the diffraction pattern is shown in the diagram below.

Fraunhoferdiffraction

A diffraction pattern shows just two-fold symmetry. Which one of these apertures could not have produced such a pattern?

Now consider the diffracted amplitude at angle 2θ. The contribution from the scattering point at X is proportional to A0ρ(x)δx, as we saw before, but we must include a term that takes into account the phase difference between the scattered rays from O and X. This second term must go to zero when the rays are π radians out-of-phase, i.e. when the path difference is nλ/2 (n integer), and it should be a maximum when the rays are in phase, i.e. when the path difference is a whole number of wavelengths. A complex exponential function fulfils these criteria. The full expression for the contribution to the diffracted amplitude at angle 2θ from the element at X is:

The distance between the object and lens (u), the distance between the image and lens (v) and the focal length of the lens (f) are related by the equation

The physics of diffraction is reasonably complicated, and the algebra required to calculate diffraction patterns is lengthy. Despite this, the derivations that follow should not be beyond an interested student who is confident with mathematics.

$${\rm{A(}}{\bf{S}}{\rm{)}} = {{\rm{A}}_0}\int\limits_{ - \infty }^\infty {\rho ({\rm{x}})\exp \left( {2\pi {\rm{i}}{{{\bf{x}} \cdot {\bf{S}}} \over \lambda }} \right){\rm{d}}x} $$

The principles of optical diffraction and image formation are equally valid for other waves: for example, neutrons, electron beams and X-rays. The similarity in the diffraction behaviour means that the theory presented here is applicable to them as well.

The phenomenon of diffraction was first documented in 1665 by the Italian Francesco Maria Grimaldi. The use of lasers has only become common in the last few decades. The laser's ability to produce a narrow beam of coherent monochromatic radiation in the visible light range makes it ideal for use in diffraction experiments: the diffracted light forms a clear pattern that is easily measured.

This diffraction pattern of a two-dimensional array of apertures is analogous to the reciprocal lattice of the array, and can be labelled (indexed) as such. Inverse axes are therefore created (where x* is perpendicular to y, and y* is perpendicular to x). In a reciprocal lattice, the magnitude of the reciprocal lattice vector is inversely proportional to the magnitude of the original vector. This inverse relationship is evident between the pattern and the mask (the x-axis repeat is smaller than in the y-axis, whereas the x*-axis repeat is larger than in the y*-axis). See the X-ray Diffraction and Reciprocal Space TLPs for an explanation of the reciprocal lattice in terms of diffraction.

This TLP was prepared when DoITPoMS was funded by the Higher Education Funding Council for England (HEFCE) and the Department for Employment and Learning (DEL) under the Fund for the Development of Teaching and Learning (FDTL).

The diagram above shows that the path difference between two rays scattered at an angle 2θ from O and X is x•s - x•s0. It is useful to define the scattering vector S = s - s0, so that the path difference is simply x•S.

Electron diffraction patterns are two-dimensional sections of the reciprocal lattice of the diffracting crystal. X-ray diffraction patterns are simply 3-dimensional extensions of Fraunhofer diffraction. With X-rays, the crystal only diffracts in a few directions.

We can describe the object in terms of its scattering density. The scattering density is represented by a function ρ(x) which varies along the dimension of the object. If the amplitude of the incident beam is A0, then the amplitude scattered from the infinitesimal point at X is proportional to A0ρ(x)δx, where x is the magnitude of the vector x and δx represents the infinitesimal length of the point.

To find the total diffracted amplitude at angle 2θ we must sum the contributions from all elements making up the object. This is done by integrating: