## Maxwell2D: Animations of electromagnetic waves

This page contains animations of numerical solutions to Maxwell's equations in a 2D domain (using the Finite-Difference Time-Domain method, FDTD). A wave is generally excited at the bottom of the domain and propagates upwards. The interaction with the various distributions of complex dielectric constant (the square of the refractive index) demonstrates many phenomena that occur both in nature and in scientific instruments. The plots below show the field of dielectric constant - click on them to see the animation of the component of electric field perpendicular to the plane of the domain.
Wikipedia links: Maxwell's equations, Finite-difference time-domain (FDTD) method

### Documents and presentations

• Radiation parametrization and clouds
Hogan, R. J., and J. K. P. Shonk. Proc. ECMWF Seminar, 1-4 Sept 2008: PDF (includes a description of the numerical method employed); PPT

### Quick links to animation categories

Propagation | Reflection and refraction | Particle scattering | Clear-air scattering
Antennas | Waveguides | Diffraction grating | Michelson interferometer | Miscellaneous

### Propagation

 Propagation of electromagnetic waves in a vacuum. Propagation of electromagnetic waves in the presence of a gradient in refractive index. The bending of the wave is what happens in mirages, shimmering of the light over a hot surface and anomalous propagation of radar beams. Wikipedia link: Refraction

### Reflection and refraction

 A sharp change in refractive index results in specular reflection from the surface and refraction into the medium according to Snell's law, with a reduction of the wave speed. The animation shows two panels: the left shows the complete wave field, while the right shows just the component that is scattered (calculated simply by subtracting the equivalent field in a vacuum). The bending of the light explains why the bottom of a pool of water appears is nearer than it really is. This kind of reflection is occasionally seen by clear-air radars as Fresnel scattering from sharp horizontal gradients in atmospheric refractive index. Wikipedia link: Snell's law Consider light propagating in a medium of higher refractive index incident upon an interface to a medium with lower refractive index. If the angle beween the direction of propagation and the normal to the interface is greater than the critical angle then total internal reflection will occur. This property is used in the prisms of bicycle reflectors and is easy to observe when looking through water into air. Wikipedia links: Total internal reflection, Evanescent wave Consider the same set-up as with total internal reflection, but introducing a new medium a distance of order a wavelength away from the original interface. In this case, there is still some propagation across the gap, a phenomenon known as frustrated total internal reflection. Wikipedia links: Total internal reflection, Evanescent wave

### Refractive index inhomogeneities

 When there are fluctuations in refractive index on the scale of half the wavelength of the radiation, then the scattered waves constructively interfere in the backward direction (note that the apparent scattered wave travelling in the forward direction in this example is simply the energy extracted from the incident wave. This is how clear-air radars work, for which it is often referred to misleadingly as Bragg scattering. When the fluctuations in refractive index are not on the scale of half the wavelength (such as the example here where they are on the scale of the wavelength itself) then destructive interference means that no significant backscatter is detected.

### Waveguides

 A microwave waveguide is normally a hollow metallic conductor, and must be more than half a wavelength in diameter to support electromagnetic waves without strong losses. In this animation the same frequency wave is passed along waveguides with different diameters. Wikipedia link: Waveguide (electromagnetism) An optic fibre has a core with a slightly higher refractive index than the cladding layer surrounding it, and can act as a waveguide for light. This animation shows a single-mode optic fibre, in which the diameter of the core is of the same order as the wavelength of the light, such that only one mode of wave can be supported. Wikipedia link: Optic fiber A fibre Bragg grating is an optic fibre with fluctuations of refractive index in the core. This leads to full transmission of all wavelengths except those close to the Bragg wavelength. In this example a wave with this wavelength is propagated along such a fibre. Wikipedia link: Fiber Bragg grating The fibre is as in the previous example, but this time the wave has a slightly shorter wavelength than the Bragg wavelength leading to complete transmission. Wikipedia link: Fiber Bragg grating

### Diffraction grating

 A diffraction grating diffracts light by an angle that depends on its wavelength. It consists of a series of equally spaced groves of a similar spacing as the wavelength of the radiation. The simplest way to understand how it works is using Huygens' Principle; each grove acts like a separate source of electromagnetic waves, but the phase of each is offset from that of its neighbours to a degree that depends on the angle of the grating with respect to the incident ratiation. The diffraction angle is that which results in a constructive interference between all the different sources. The sources also combine constructively in the direction of the incident radiation. Wikipedia link: Diffraction grating This example uses exactly the same grating as above, but the frequency has been increased by 50%. It can be seen that the diffraction angle is smaller, and there is a weaker second-order diffraction towards the right. Wikipedia link: Diffraction grating This example again uses the same grating as above, but two frequencies are incident on it simultaneously. This demonstrates the ability of the grating to separate the different frequencies by different angles. Wikipedia link: Diffraction grating A reflection grating is a type of diffraction grating that exploits the frequency dependence of the angle of reflection from a device with groves that have a spacing of the same order as the wavelength of the light. In this case two frequencies are incident on a reflection grating. Both are reflected significantly in the direction one would expect if the grating were a mirror, but the light is also diffracted into beams at an angle that is frequency dependent. Wikipedia link: Diffraction grating A grism is a prism on which has been etched the grooves of a diffraction grating. It is used in astronomical instruments to filter light into different frequencies. In this example, two frequencies enter the prism from the base, and on leaving the other side, a significant fraction of the radiation is refracted in the direction that would be expected for a usual prism. However, there is an additional diffracted component, the angle of which is frequency dependent. Wikipedia link: Grism

### Michelson interferometer

 A Michelson interferometer was used in the famous Michelson-Morley experiment to demonstrate the non-existence of the luminiferous aether, a hypothetical medium in which light was believed to travel. In the configuration to the left, coherent light is incident on a 45° half-silvered mirror (in this case simply a thin bar with a refractive index chosen such that half the light is transmitted and half is reflected). The transmitted and reflected beams travel equal paths to ordinary mirrors (in this case simply bars with a very high refractive index), and are reflected back and recombine at the 45° mirror. This results in constructive interference for the outgoing beam to the right. If the luminiferous aether existed and the earth was moving with respect to it, then light would be expected to travel at different speeds along the two orthogonal paths, resulting in a change in the interference behaviour depending on the orientation of the instrument and the time of year. The fact that this behaviour was not observed demonstrated that this aether did not exist. Wikipedia links: Michelson interferometer, Michelson-Morley experiment A Fourier transform spectrometer consists simply of a Michelson interferometer but with with one of the two mirrors movable. When one of the path lengths is changed by a quarter of a wavelength (as in this example), the radiation from the two paths destructively interferes in the rightward direction. Instead, the light constructively interferes in the downward direction (unlike in the previous case). The frequency spectrum of incoming light may be obtained by measuring the intensity of the outgoing radiation to the right as a function of the path length, then performing a Fourier Transform on it. This principle is used by a number of meteorological satellites instruments, such as AIRS and IASI. Wikipedia link: Fourier transform spectroscopy The same mirror configuration as in the previous example, but now two frequencies are used simultaneously. It can be seen that only one of the two frequencies emerges to the right. Again, the frequency dependence of the transmission is key to the workings of a Fourier Transform spectrometer.

### Limitations

• A sponge layer absorbs waves leaving the domain, but this is not perfect so there are sometimes odd effects at the edge of the domain.
• The simulations are two-dimensional, so effectively we have assumed the dielectric constant to be constant in the third dimension. This means that in the "Particle scattering" section we are not strictly simulating Rayleigh and Mie scattering, but actually the scattering for wires of different diameters. Nonetheless, most of the same phenomena are observed.

### Future ideas

• Hexagonal crystals
• Explanation for the extinction paradox
• Slotted waveguide

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