Diffraction

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Diffraction is a particular type of wave interference, caused by the partial obstruction or lateral restriction of a wave. The interference is undergone by electromagnetic waves such as light and radio waves. Diffraction also occurs when any group of waves of a finite size is propagating; for example, a narrow beam of light waves from a laser must, because of diffraction of the beam, eventually diverge into a wider beam at a sufficient distance from the laser. As a simple example of diffraction, if you speak into one end of a cardboard tube, the sound waves emerging from the other end spread out in all directions, rather than propagating in a straight line like a stream of water from a garden hose.

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Introduction

It is important to understand that not all interference is diffraction; for example, sound waves emitted by two stereo speakers will interfere with each other if they are of the same frequency and have a definite phase relationship, but this is not diffraction. Diffraction will not occur if the wave is not coherent, and diffraction effects become weaker (and ultimately undetectable) as the size of obstruction is made larger and larger compared to the wavelength. In well-defined cases, a diffraction pattern may be observed.

Diffraction is not the same as refraction, although both are phenomena in which a wave does not propagate in a single direction. Refraction is not an interference phenomenon, and, e.g., can occur without coherence.

It is the diffraction of "particles," such as electrons, which stood as one of the powerful arguments in favor of quantum mechanics. It is possible to observe diffraction of particles such as neutrons or electrons and hence we are able to infer the existence of wave-particle duality. Indeed, this diffraction is a useful tool; the wavelengths of these particle-waves are small enough that they are used as probes of the atomic structure of crystals. See electron diffraction and neutron diffraction.

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Double-slit diffraction



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Double-slit diffraction
(red laser light)

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2-slit and 5-slit diffraction

The most conceptually simple example of diffraction is double-slit diffraction in which both slits have relatively narrow widths compared to the wavelength of the wave. Suppose, for the sake of visualization, that these are water waves. After passing through the slits, two overlapping patterns of semicircular ripples are formed, as shown in the first figure. Where a crest overlaps with a crest, a double-height crest will be formed; this is constructive interference. Constructive interference also occurs where a trough overlaps another trough. However, when a trough and a crest overlap, they cancel out; the interference is destructive. The second figure shows the result of this process with light waves of a single wavelength originating from a laser. The constructive-interference locations are called maxima, because they have maximum brightness. The destructive-interference locations are the minima. Historically, the first proof that light was a wave phenomenon came from the double-slit experiment of Thomas Young.

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General facts about diffraction

Several qualitative observations can be made:

  • The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction, in other words: the smaller the diffracting object the 'wider' the resulting diffraction pattern and vice versa. (More precisely, this is true of the sines of the angles.)
  • The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to a dimension, a, of the diffracting object.
  • When the diffracting object is repeated, the effect is to narrow each maximum, concentrating its energy within a narrower range of angles. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, a, between the center of one slit and the next.
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Mathematical description

It is mathematically easier to consider the case of far-field or Fraunhofer diffraction, where the diffracting obstruction is far from the point at which the wave is measured. The more general case is known as near-field or Fresnel diffraction, and involves more complex mathematics. As the observation distance is increased the results predicted by the Fresnel theory converge towards those predicted by the simpler Fraunhofer theory. This article considers far-field diffraction, which is commonly observed in nature.

Quantitatively, the angular positions of the minima in multiple-slit diffraction are given by the equation

<math> \sin \theta = \frac{\lambda}{a} m </math>     

where

m is an integer that labels the order of each minimum,
<math>\lambda</math> is the wavelength,
<math>a</math> is the distance between the slits
and θ is the angle for destructive interference

The central maximum is two orders wide, however, so m = 0, θ = 0 is the absolute maximum of the distribution and intensity functions. This is a form of Bragg's law (see below).

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Graph and image
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Quantitative analysis of single-slit diffraction

As an example, an exact equation can now be derived for the intensity of the diffraction pattern as a function of angle in the case of single-slit diffraction.

A mathematical representation of Huygens' principle can be used to start an equation.

Consider a monochromatic complex plane wave <math>\Psi^\prime</math> of wavelength λ incident on a slit of width a.

If the slit lies in the x′-y′ plane, with its center at the origin, then it can be assumed that diffraction generates a complex wave ψ, traveling radially in the r direction away from the slit, and this is given by:

<math>\Psi = \int_{slit} \frac{i}{r\lambda} \Psi^\prime e^{-ikr}\,dslit</math>

let (x′,y′,0) be a point inside the slit over which it is being integrated. If (x,0,z) is the location at which the intensity of the diffraction pattern is being computed, the slit extends from <math>x^\prime=-a/2</math> to <math>+a/2\,</math>, and from <math>y'=-\infty</math> to <math>\infty</math>.

The distance r from the slot is:

<math>r = \sqrt{\left(x - x^\prime\right)^2 + y^{\prime2} + z^2}</math>
<math>r = z \left(1 + \frac{\left(x - x^\prime\right)^2 + y^{\prime2}}{z^2}\right)^\frac{1}{2}</math>

Assuming Fraunhofer diffraction will result in the conclusion <math>z >> \big|\left(x - x^\prime\right)\big|</math>. In other words, the distance to the target is much larger than the diffraction width on the target. By the binomial expansion rule, ignoring terms quadratic and higher, the quantity on the right can be estimated to be:

<math>r \approx z \left( 1 + \frac{1}{2} \frac{\left(x - x^\prime \right)^2 + y^{\prime 2}}{z^2} \right)</math>
<math>r \approx z + \frac{\left(x - x^\prime\right)^2 + y^{\prime 2}}{2z}</math>

It can be seen that 1/r in front of the equation is non-oscillatory, i.e. its contribution to the magnitude of the intensity is small compared to our exponential factors. Therefore, we will lose little accuracy by approximating it as z.

<math>\Psi \,</math> <math>= \frac{i \Psi^\prime}{z \lambda} \int_{-\frac{a}{2}}^{\frac{a}{2}}\int_{-\infty}^{\infty} e^{-ik\left[z+\frac{ \left(x - x^\prime \right)^2 + y^{\prime 2}}{2z}\right]} \,dx^\prime \,dy^\prime</math>
<math>= \frac{i \Psi^\prime}{z \lambda} e^{-ikz} \int_{-\frac{a}{2}}^{\frac{a}{2}}e^{-ik\left[\frac{\left(x - x^\prime \right)^2}{2z}\right]} \,dx^\prime \int_{-\infty}^{\infty} e^{-ik\left[\frac{y^{\prime 2}}{2z}\right]} \,dy^\prime</math>
<math>=\Psi^\prime \sqrt{\frac{i}{z\lambda}} e^\frac{-ikx^2}{2z} \int_{-\frac{a}{2}}^{\frac{a}{2}}e^\frac{ikxx^\prime}{z} e^\frac{-ikx^{\prime 2}}{2z} \,dx^\prime</math>

To make things cleaner, a placeholder 'C' is used to denote constants in the equation. It is important to keep in mind that C can contain imaginary numbers, thus the wave function will be complex, however at the end, the ψ will be bracketed, which will eliminate any imaginary components.

Now, in Fraunhoffer diffraction, <math>kx^{\prime 2}/z</math> is small, so <math>e^\frac{-ikx^{\prime 2}}{2z} \approx 1</math>. The same approximation holds for <math>e^\frac{-ikx^2}{2z}</math>. Thus, taking <math>C = \Psi^\prime \sqrt{\frac{i}{z\lambda}}</math>, this results in:

<math>\Psi\, </math> <math>= C \int_{-\frac{a}{2}}^{\frac{a}{2}}e^\frac{ikxx^\prime}{z} \,dx^\prime</math>
<math>=C \frac{\left(e^\frac{ikax}{2z} - e^\frac{-ikax}{2z}\right)}{\frac{2ikax}{2z}}</math>

It can be noted through Euler's formula and its derivatives that <math>\sin x = \frac{e^{ix} - e^{-ix}}{2i}</math> and <math>\sin \theta = \frac{x}{z}</math>.

<math>\Psi = C \frac{\sin\frac{ka\sin\theta}{2}}{\frac{ika\sin\theta}{2}}</math>

Now, substituting in <math>\frac{2\pi}{\lambda} = k</math>, the intensity I of the diffracted waves at an angle θ is given by:

<math>I(\theta)\, </math> \Psi(\theta)\big|^2 </math>
\Psi \rangle \,</math>
<math>= I_0 {\left[ \operatorname{sinc} \left( \frac{\pi a}{\lambda} \sin \theta \right) \right] }^2 </math>

where the sinc function is given by sinc(x) = sin(x)/x.

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Quantitative analysis of n-slit diffraction

Let us again start with the mathematical representation of Huygens' principle.

<math>\Psi = \int_{slit} \frac{i}{r\lambda} \Psi^\prime e^{-ikr}\,dslit</math>

Consider n slits in the prime plane of the equal size (a, <math>\infty</math>, 0) and spacing d spread along the x′ axis. As above, the distance r from the slit 1 is:

<math>r = z \left(1 + \frac{\left(x - x^\prime\right)^2 + y^{\prime2}}{z^2}\right)^\frac{1}{2}</math>

To generalize this to n slits, we make the observation that while z and y remain constant, x′ shifts by

<math>x_{j=0 \cdots n-1}^{\prime} = x_0^\prime - j d </math>

Thus

<math>r_j = z \left(1 + \frac{\left(x - x^\prime - j d \right)^2 + y^{\prime2}}{z^2}\right)^\frac{1}{2}</math>

and the sum of all n contributions to the wave function is:

<math>\Psi = \sum_{j=0}^{N-1} C \int_{-\frac{a}{2}}^{\frac{a}{2}} e^\frac{ikx\left(x^\prime - jd\right)}{z} e^\frac{-ik\left(x^\prime - jd\right)^2}{2z} \,dx^\prime</math>

Again noting that <math>\frac{k\left(x^\prime -jd\right)^2}{z}</math> is small, so <math>e^\frac{-ik\left(x^\prime -jd\right)^2}{2z} \approx 1</math>, we have:

<math>\Psi\, </math> <math>= C\sum_{j=0}^{N-1} \int_{-\frac{a}{2}}^{\frac{a}{2}} e^\frac{ikx\left(x^\prime - jd\right)}{z} \,dx^\prime</math>
<math>= C \sum_{j=0}^{N-1} \frac{\left(e^{\frac{ikax}{2z} - \frac{ijkxd}{z}} - e^{\frac{-ikax}{2z}-\frac{ijkxd}{z}}\right)}{\frac{2ikax}{2z}}</math>
<math>= C \sum_{j=0}^{N-1} e^\frac{ijkxd}{z} \frac{\left(e^\frac{ikax}{2z} - e^\frac{-ikax}{2z}\right)}{\frac{2ikax}{2z}}</math>
<math>= C \frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}} \sum_{j=1}^{N-1} e^{ijkd\sin\theta}</math>

Now, we can use the following identity

<math>\sum_{j=0}^{N-1} e^{x j} = \frac{1 - e^{Nx}}{1 - e^x}.</math>

Substituting into our equation, we find:

<math>\Psi\, </math> <math>= C \frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}}\left(\frac{1 - e^{iNkd\sin\theta}}{1 - e^{ikd\sin\theta}}\right)</math>
<math>= C \frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}}\left(\frac{e^{-iNkd\frac{\sin\theta}{2}}-e^{iNkd\frac{\sin\theta}{2}}}{e^{-ikd\frac{\sin\theta}{2}}-e^{ikd\frac{\sin\theta}{2}}}\right)\left(\frac{e^{iNkd\frac{\sin\theta}{2}}}{e^{ikd\frac{\sin\theta}{2}}}\right)</math>
<math>= C \frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}}\frac{\frac{e^{-iNkd \frac{\sin\theta}{2}} - e^{iNkd\frac{\sin\theta}{2}}}{2i}}{\frac{e^{-ikd\frac{\sin\theta}{2}} - e^{ikd\frac{\sin\theta}{2}}}{2i}} \left(e^{i(N-1)kd\frac{\sin\theta}{2}}\right)</math>
<math>= C \frac{\sin\left(\frac{ka\sin\theta}{2}\right)}{\frac{ka\sin\theta}{2}} \frac{\sin\left(\frac{Nkd\sin\theta}{2}\right)} {\sin\left(\frac{kd\sin\theta}{2}\right)}e^{i\left(N-1\right)kd\frac{\sin\theta}{2}} </math>

We now make our k substitution as before and represent all non-oscillating constants by the <math>I_0</math> variable as in the 1-slit diffraction and bracket the result. Remember that

<math>\langle e^{ix} \Big| e^{ix}\rangle\ = e^0 = 1</math>

This allows us to discard the tailing exponent and we have our answer:

<math>I\left(\theta\right)\, </math>
<math>=I_0 \left[ \operatorname{sinc} \left( \frac{\pi a}{\lambda} \sin \theta \right) \right]^2 </math>
<math>\left[\frac{\sin\left(\frac{N\pi d}{\lambda}\sin\theta\right)}{\sin\left(\frac{\pi d}{\lambda}\sin\theta\right)}\right]^2</math>
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Other cases

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Bragg diffraction

Diffraction from multiple slits, as described above, is similar to what occurs when waves are scattered from a periodic structure, such as atoms in a crystal or rulings on a diffraction grating. Each scattering center (e.g., each atom) acts as a point source of spherical wavefronts; these wavefronts undergo constructive interference to form a number of diffracted beams. The direction of these beams is described by Bragg's law:

<math> m \lambda = 2 d \sin \theta,\, </math>

where

λ is the wavelength,
d is the distance between scattering centers,
θ is the angle of diffraction
and m is an integer known as the order of the diffracted beam.

Bragg diffraction is used in X-ray crystallography to deduce the structure of a crystal from the angles at which X-rays are diffracted from it. Since the diffraction angle θ is dependent on the wavelength λ, diffaction gratings impart angular dispersion on a beam of light.

The most common demonstration of Bragg diffraction is the spectrum of colors seen reflected from a compact disc: the closely-spaced tracks on the surface of the disc form a diffraction grating, and the individual wavelengths of white light are diffracted at different angles from it, in accordance with Bragg's law.

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Diffraction limit of telescopes

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For diffraction through a circular aperture, there is a series of concentric rings surrounding a central Airy disc. The mathematical result is similar to a radially symmetric version of the equation given above in the case of single-slit diffraction.

A wave does not have to pass through an aperture to diffract; for example, a beam of light of a finite size also undergoes diffraction and spreads in diameter. This effect limits the minimum size d of spot of light formed at the focus of a lens, known as the diffraction limit:

<math> d = 2.44 \lambda \frac{f}{a},\, </math>

where λ is the wavelength of the light, f is the focal length of the lens, and a is the diameter of the beam of light, or (if the beam is filling the lens) the diameter of the lens. (See Rayleigh criterion).

By use of Huygens' principle, it is possible to compute the diffraction pattern of a wave from any arbitrarily shaped aperture. If the pattern is observed at a sufficient distance from the aperture, it will appear as the two-dimensional Fourier transform of the function representing the aperture.

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See also

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External links

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