11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Diffraction | 4/5 | https://en.wikipedia.org/wiki/Diffraction | reference | science, encyclopedia | 2026-05-05T10:54:57.453682+00:00 | kb-cron |
where
θ
i
{\displaystyle \theta _{i}}
is the angle at which the light is incident,
d
{\displaystyle d}
is the separation of grating elements, and
m
{\displaystyle m}
is an integer which can be positive or negative. The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns. The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.
=== General case for far field ===
A more mathematical approach involves treating the problem as a summation over spherical waves derived from the relevant wave equation; see for instance Born and Wolf for details. The wave that emerges from a point source has an amplitude
ψ
{\displaystyle \psi }
at location
r
{\displaystyle \mathbf {r} }
that is given by the solution of the frequency domain wave equation for a point source (the Helmholtz equation),
∇
2
ψ
+
k
2
ψ
=
δ
(
r
)
,
{\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r} ),}
where
δ
(
r
)
{\displaystyle \delta (\mathbf {r} )}
is the 3-dimensional delta function. By direct substitution, the solution to this equation can be shown to be the scalar Green's function, which in the spherical coordinate system (and using the physics time convention
e
−
i
ω
t
{\displaystyle e^{-i\omega t}}
) is
ψ
(
r
)
=
e
i
k
r
4
π
r
.
{\displaystyle \psi (r)={\frac {e^{ikr}}{4\pi r}}.}
which is a spherical wave emanating from the origin, the mathematical form of Huygen"s wavelets in the Huygens-Fresnel appriach. This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector
r
′
{\displaystyle \mathbf {r} '}
and the field point is located at the point
r
{\displaystyle \mathbf {r} }
, then we may represent the scalar Green's function (for arbitrary source location) as
ψ
(
r
|
r
′
)
=
e
i
k
|
r
−
r
′
|
4
π
|
r
−
r
′
|
.
{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}
In the far field, where
r
{\displaystyle r}
is large the Green's function simplifies to
ψ
(
r
|
r
′
)
=
e
i
k
r
4
π
r
e
−
i
k
(
r
′
⋅
r
^
)
{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ikr}}{4\pi r}}e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}}
The expression for the far (Fraunhofer region) wave then becomes
Ψ
(
r
)
∝
e
i
k
r
4
π
r
∬
a
p
e
r
t
u
r
e
E
i
n
c
(
x
′
,
y
′
)
e
−
i
k
(
r
′
⋅
r
^
)
d
x
′
d
y
′
.
{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}\,dx'\,dy'.}
with an electric field
E
i
n
c
(
x
,
y
)
{\displaystyle E_{\mathrm {inc} }(x,y)}
incident on the aperture for the case of an electromagnetic wave. In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see Fourier optics). In the far field, where r is essentially constant, then the equation:
Ψ
=
∫
a
p
e
r
t
u
r
e
i
r
λ
Ψ
′
e
−
i
k
r
d
a
p
e
r
t
u
r
e
{\displaystyle \Psi =\int _{\mathrm {aperture} }{\frac {i}{r\lambda }}\Psi ^{\prime }e^{-ikr}\,d\mathrm {aperture} }
is equivalent to doing a Fourier transform on the gaps in the barrier. Similar forms can be derived for matter and other types of waves. For instance, with electron diffraction the aperture would be replaced by the electrostatic potential, while for x-ray diffraction it would be the electron charge density.
=== Dynamical diffraction ===
In the cases discussed above it is implicitly assumed that the wave encounters some single barrier or obstruction and is then diffraction by it. In reality it may encounter a number of barriers along the direction that it is travelling. The wave diffraction by the first one it encounters can be diffraction by the next, and so forth. The case when only single diffraction occurs is called kinematical diffraction, the more general case is called dynamical diffraction. Dynamical diffraction is quite well developed for x-rays, and also for electrons. As discussed extensively in the existing literature and reviews the results with dynamical diffraction can be quite different from those when only single scattering is considered. It can also occur with light, one example being opals and photonic crystals.
== Matter wave diffraction ==
According to quantum theory every particle exhibits wave properties and can therefore diffract. Diffraction of electrons and neutrons is one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a non-relativistic particle is the de Broglie wavelength
λ
=
h
p
,
{\displaystyle \lambda ={\frac {h}{p}}\,,}
where
h
{\displaystyle h}
is the Planck constant and
p
{\displaystyle p}
is the momentum of the particle (mass × velocity for slow-moving particles). For example, a sodium atom traveling at about 300 m/s would have a de Broglie wavelength of about 50 picometres. Diffraction of matter waves has been observed for small particles, like electrons, neutrons, atoms, and even large molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic structure of solids, molecules and proteins.