Explain Light: Dispersion, Phase Velocity, and More

[arkain101]

The purpose of this topic is to investigate the dynamics of light, beginning with an example of light as it passes through a prism. The inclusions below are the areas of research that led me to asking some questions about these dynamics.

 

 

 

 

 

 

 

 

Energy Frequency Proportionality

Albert Einstein's mathematical description in 1905 of how the photoelectric effect was caused by absorption of quanta of light (now called photons), was in the paper named "On a Heuristic Viewpoint Concerning the Production and Transformation of Light". This paper proposed the simple description of "light quanta", or photons, and showed how they explained such phenomena as the photoelectric effect. His simple explanation in terms of absorption of discrete quanta of light explained the features of the phenomenon and the characteristic frequency. Einstein's explanation of the photoelectric effect won him the Nobel Prize in Physics in 1921.

 

The idea of light quanta began with Max Planck's published law of black-body radiation ("On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik 4 (1901)) by assuming that Hertzian oscillators could only exist at energies E proportional to the frequency f of the oscillator by E = hf, where h is Planck's constant. By assuming that light actually consisted of discrete energy packets, Einstein wrote an equation for the photoelectric effect that fitted experiments. It explained why the energy of photoelectrons were dependent only on the frequency of the incident light and not on its intensity: a low-intensity, high-frequency source could supply a few high energy photons, whereas a high-intensity, low-frequency source would supply no photons of sufficient individual energy to dislodge any electrons.[

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Dispersion

The most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrum by a prism. From Snell's law it can be seen that the angle of refraction of light in a prism depends on the refractive index of the prism material. Since that refractive index varies with wavelength, it follows that the angle that the light is refracted by will also vary with wavelength, causing an angular separation of the colors known as angular dispersion.[

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An image illustrating the separation of white light.

 

 

 

 

 

 

Phase Velocity

The phase velocity (or phase speed) of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase speed is given in terms of the wavelength λ (lambda) and period T as:

 

[math]v_\mathrm{p} = \frac{\lambda}{T}[/math]

 

In a dispersive medium, the phase velocity varies with frequency and is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate. [

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Matter Wave Phase

In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis [

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[math]v_\mathrm{p} = \frac{\omega}{k} = \frac{E/\hbar}{p/\hbar} = \frac{E}{p}[/math]

 

 

Using relativistic relations for energy and momentum, we have:

 

[math]v_\mathrm{p} = \frac{E}{p} = \frac{\gamma m c^2}{\gamma m v} = \frac{c^2}{v} = \frac{c}{\beta}[/math]

 

 

 

 

 

 

 

Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

 

[math]p=\frac{nE}{c}[/math]

 

In 1909, Max Abraham proposed

 

[math] p=\frac{E}{nc}[/math]

 

Rudolf Peierls raises this in his book More Surprises in Theoretical Physics.[8] Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews, has discussed this, including experiments to resolve it.[9][

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Investigation

 

This effect, known as chromatic dispersion

, results from the fact that the speed of light in glass depends on the wavelength of the light. The speed slightly decreases with decreasing wavelength; this means that the index of refraction, which is inversely proportional to the speed, slightly increases with decreasing wavelength. For glass, the index of refraction for red light (the longest visible wavelength) is about 1 percent less than that for violet light (the shortest visible wavelength).

 

 

I want to look into the explanations as to why the velocity of light through a medium is proportional to its frequency. I noticed that light of shorter wavelengths / higher frequencies has a different angled path then that of a longer wavelength / lower frequency. Why does light do this? Furthermore, what is the physics involved in the explanation to the overall mechanics the causes this frequency dispersion.

 

 

 

 

 

 

 

 

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[Qfwfq]

I want to look into the explanations as to why the velocity of light through a medium is proportional to its frequency.

Actually this is not so. It would imply the wavelength being fixed, which is false.

 

A medium being dispersive means that there is a dependency but it isn't proportionality. The exact causes of dispersivity are complex and depend on the specific medium; basically it's the polarizability [imath]\epsilon[/imath] which varies and intuitively you can think of the polarization as lagging somewhat behind cuz it can't quite keep up with such fast oscillations, so its amplitude isn't the same as it is statically. ;)

[arkain101]

Actually this is not so. It would imply the wavelength being fixed, which is false.

 

A medium being dispersive means that there is a dependency but it isn't proportionality. The exact causes of dispersivity are complex and depend on the specific medium; basically it's the polarizability [imath]\epsilon[/imath] which varies and intuitively you can think of the polarization as lagging somewhat behind cuz it can't quite keep up with such fast oscillations, so its amplitude isn't the same as it is statically. ;)

 

I agree, that was probably the wrong use of language to use for the information I am referring to understanding.

 

What I should have used was what I included above my choice of words:

 

This effect, known as chromatic dispersion, results from the fact that the speed of light in glass depends on the wavelength of the light.

 

 

I regret to tell you that I don't understand your answer on face value. I am most specifically interested in the mechanics of this process while at the same time including an understandable conceptual...hmmm.. approach. Not only would I like to gather understanding personally, but I would be pleased to see a topic cover a full range of explanations of light and/or energy. IE, I encourage participation! :)

 

So far I have read through some wave-like explanations and theorems, but if light also has particle theory, then particle theory should also be capable to provide an explanation.

 

My intuition is that there is some very fundamental basic laws hidden within these effects. There is also personal interests as to why I would like to develop a rich knowledge related to the behavior of light.

 

 

 

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Anyway, universal theories aside :hihi:, as mentioned, unless a thread covering the mechanics and dynamics of light in general already exists, let's contribute and put one together.