Diffraction

Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves emerging from an aperture. It occurs with any type of wave, including sound waves, water waves, electromagnetic waves such as light and radio waves, and matter displaying wave-like properties according to the wave–particle duality. While diffraction always occurs, its effects are generally only noticeable for waves where the wavelength is on the order of the feature size of the diffracting objects or apertures.

Explanation

Photograph of single-slit diffraction in a circular ripple tank

The most conceptually simple example of diffraction is single-slit diffraction in which the slit is narrow, that is, significantly smaller than a wavelength of the wave. After the wave passes through the slit, a pattern of semicircular ripples is formed, approximately equally strong in all directions, as if there were a simple wave source at the position of the slit. This semicircular wave is a diffraction pattern.

When the slit is significantly more than a wavelength wide, the wave propagates more nearly straight through, but a diffraction pattern at the edges of the wave can be seen. The center part of the wave travels through largely unaffected at short distances, but the wave forms a stable diffraction pattern at longer distances. This pattern is most easily understood and calculated as the interference pattern of a large number of simple sources spaced closely and evenly across the width of the slit.

In multiple-slit experiments, narrow enough slits can be analyzed as simple wave sources.

A slit is an opening that is infinitely extended in one dimension, which has the effect of reducing a wave problem in 3-space to a simpler problem in 2-space. All the same effects can be seen and analyzed for small round holes and other shapes, in 3D, but they're harder to describe, compute, and illustrate.

Diffraction of particles

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.

History

The effects of diffraction were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction. The results of Grimaldi's observations were published in 1665. Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, effectively the first diffraction grating. Thomas Young observed two-slit diffraction in 1803 and deduced that light must propagate as waves. Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christian Huygens and reinvigorated by Thomas Young, against Newton's theories.

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, for example in a diffraction grating the effect is to create narrower maximum on the interference fringes, 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.

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, corresponding to path length differences of an odd number of half wavelengths, are given by the equation

{a}\sin \theta ={\frac {\lambda }{2}}(2m+1)

where

m is an integer that labels the order of each minimum,

\lambda

is the wavelength,

a

is the distance between the slits

and θ is the angle for destructive interference

And the corresponding maxima are at path differences of an integer number of wavelengths:

{a}\sin \theta =\lambda m\,

Quantitative analysis of single-slit diffraction

Graph and image 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 $\Psi ^{\prime }$ 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:

\Psi =\int _{slit}{\frac {i}{r\lambda }}\Psi ^{\prime }e^{-ikr}\,dslit

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 $x^{\prime }=-a/2$ to $+a/2\,$ , and from $y'=-\infty$ to $\infty$ .

The distance r from the slot is:

r={\sqrt {\left(x-x^{\prime }\right)^{2}+y^{\prime 2}+z^{2}}}

r=z\left(1+{\frac {\left(x-x^{\prime }\right)^{2}+y^{\prime 2}}{z^{2}}}\right)^{\frac {1}{2}}

Assuming Fraunhofer diffraction will result in the conclusion $z\gg {\big |}\left(x-x^{\prime }\right){\big |}$ . 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:

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

r\approx z+{\frac {\left(x-x^{\prime }\right)^{2}+y^{\prime 2}}{2z}}

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.

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

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

$\Psi \,$	$=C\int _{-{\frac {a}{2}}}^{\frac {a}{2}}e^{\frac {ikxx^{\prime }}{z}}\,dx^{\prime }$
	$=C{\frac {\left(e^{\frac {ikax}{2z}}-e^{\frac {-ikax}{2z}}\right)}{\frac {ikx}{z}}}$

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

$\Psi =aC{\frac {\sin {\frac {ka\sin \theta }{2}}}{\frac {ka\sin \theta }{2}}}=aC\left[\operatorname {sinc} \left({\frac {ka\sin \theta }{2}}\right)\right]$

where the (unnormalized) sinc function is defined by $\operatorname {sinc} (x)\equiv {\frac {\operatorname {sin} (x)}{x}}$ .

Now, substituting in ${\frac {2\pi }{\lambda }}=k$ , the intensity (squared amplitude) $I$ of the diffracted waves at an angle θ is given by:

I(\theta )\,

=I_{0}{\left[\operatorname {sinc} \left({\frac {\pi a}{\lambda }}\sin \theta \right)\right]}^{2}

Quantitative analysis of N-slit diffraction

Double-slit diffraction of red laser light

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

\Psi =\int _{slit}{\frac {i}{r\lambda }}\Psi ^{\prime }e^{-ikr}\,dslit

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

r=z\left(1+{\frac {\left(x-x^{\prime }\right)^{2}+y^{\prime 2}}{z^{2}}}\right)^{\frac {1}{2}}

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

x_{j=0\cdots n-1}^{\prime }=x_{0}^{\prime }-jd

Thus

r_{j}=z\left(1+{\frac {\left(x-x^{\prime }-jd\right)^{2}+y^{\prime 2}}{z^{2}}}\right)^{\frac {1}{2}}

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

\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 }

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

$\Psi \,$	$=C\sum _{j=0}^{N-1}\int _{-{\frac {a}{2}}}^{\frac {a}{2}}e^{\frac {ikx\left(x^{\prime }-jd\right)}{z}}\,dx^{\prime }$
	$=C\sum _{j=0}^{N-1}{\frac {\left(e^{{\frac {ikax}{2z}}-{\frac {ijkxd}{z}}}-e^{{\frac {-ikax}{2z}}-{\frac {ijkxd}{z}}}\right)}{\frac {2ikax}{2z}}}$
	$=C\sum _{j=0}^{N-1}e^{\frac {ijkxd}{z}}{\frac {\left(e^{\frac {ikax}{2z}}-e^{\frac {-ikax}{2z}}\right)}{\frac {2ikax}{2z}}}$
	$=C{\frac {\sin {\frac {ka\sin \theta }{2}}}{\frac {ka\sin \theta }{2}}}\sum _{j=1}^{N-1}e^{ijkd\sin \theta }$

Now, we can use the following identity

$\sum _{j=0}^{N-1}e^{xj}={\frac {1-e^{Nx}}{1-e^{x}}}.$

Substituting into our equation, we find:

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

We now make our k substitution as before and represent all non-oscillating constants by the $I_{0}$ variable as in the 1-slit diffraction and bracket the result. Remember that

\langle e^{ix}{\Big |}e^{ix}\rangle \ =e^{0}=1

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

I\left(\theta \right)=I_{0}\left[\operatorname {sinc} \left({\frac {\pi a}{\lambda }}\sin \theta \right)\right]^{2}\cdot \left[{\frac {\sin \left({\frac {N\pi d}{\lambda }}\sin \theta \right)}{\sin \left({\frac {\pi d}{\lambda }}\sin \theta \right)}}\right]^{2}

Other cases

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:

m\lambda =2d\sin \theta

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, diffraction 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.

Diffraction limit of telescopes

The Airy disc around each of the stars from the 2.56m telescope aperture can be seen in this *lucky image* of the binary star zeta Boötis.

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:

d=1.22\lambda {\frac {f}{a}},\,

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. The diameter given is enough to contain about 70% of the light energy; it is the radius to the first null of the Airy disk, in approximate agreement with the Rayleigh criterion. Twice that diameter, the diameter to the first null of the Airy disk, within which 83.8% of the light energy is contained, is also sometimes given as the diffraction spot diameter.

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.

External links

Wave Optics - a chapter of an online textbook
2-D wave java applet displays diffraction patterns of various slit configurations.
Diffraction java applet displays diffraction patterns of various 2-D apertures.
Diffraction approximations illustrated MIT site that illustrates the various approximations in diffraction and intuitively explains the Fraunhofer regime from the perspective of linear system theory.
Diffraction Limited Photography understanding how airy disks, lens aperture and pixel size limit the absolute resolution of any camera.