Phase lag of light in materials

[hacivat]

There are 5 fantastic videos in this website: http://www.alfredleitner.com/
He is a very good educator and it is also very good to see those authentic experiments and aparatus.

Anyway, in the following one at exacly 8:00 minutes he says that the phase lag induced by the dipole is always 90 degrees. That really didn't make sense from his "time of arrival" argument since the dipoles would constitute a continuum. I would like to know if there is a better explanation for this 90 degrees phase lag.

 

[Philip Koeck]

There are 5 fantastic videos in this website: http://www.alfredleitner.com/
He is a very good educator and it is also very good to see those authentic experiments and aparatus.

Anyway, in the following one at exacly 8:00 minutes he says that the phase lag induced by the dipole is always 90 degrees. That really didn't make sense from his "time of arrival" argument since the dipoles would constitute a continuum. I would like to know if there is a better explanation for this 90 degrees phase lag.

I can't make sense of his explanation either.

The way I see it it just results from a series expansion.
Start by writing a plane wave like this: ψ = A e^{i(kx - ωt)}
Bold type letters are vectors. k is the wave vector giving the wavelength and direction of the wave. ω is the angular frequency.
If this wave travels through some material it will experience a phase shift given by a phase factor of the following form: e^iφ.
The phase shift φ depends on the material and the distance that the wave travels through it.
You can develop this factor into a series: e^iφ ≈ 1 + i φ + ... = 1 + e^{i π/2} φ + ...
There you have the 90 degree phase shift between the singly scattered and the unscattered wave.

 

[tech99]

Imagine we reduce the problem to a single block of the dielectric somewhat smaller than the wavelength. A pure capacitor shunting the incoming E-field would have its current flowing 90 degrees ahead of the voltage, giving the re-radiation a 90 deg phase advance.
However, it looks as if re-radiation opposes the accelerating potential, so we have an additional 180 deg shift, creating a lag in the re-radiation. Regarding the exact phase of the re-radiation, the capacitor is impure because the electron has radiation resistance, so the lag will possibly be greater than 90 degrees.

 

[Philip Koeck]

I can't make sense of his explanation either.

The way I see it it just results from a series expansion.
Start by writing a plane wave like this: ψ = A e^{i(kx - ωt)}
Bold type letters are vectors. k is the wave vector giving the wavelength and direction of the wave. ω is the angular frequency.
If this wave travels through some material it will experience a phase shift given by a phase factor of the following form: e^iφ.
The phase shift φ depends on the material and the distance that the wave travels through it.
You can develop this factor into a series: e^iφ ≈ 1 + i φ + ... = 1 + e^{i π/2} φ + ...
There you have the 90 degree phase shift between the singly scattered and the unscattered wave.

I have to point out that what I write doesn't explain why light slows down in a dielectric (with normal dispersion).
I only say that the effect on a light wave of an area with n that differs from the surrounding (for example a small blob with n > 1 surrounded by air) is a phase shift because the wavelength in this area is shorter than in the surroundings.
The exponential function describing this phase shift can be expanded into a series and the first term of this series, corresponding to single scattering, is phase shifted by exactly 90 degrees compared to the unscattered wave. This is always true, no matter how big or small the phase shift is.

I'd also like to see a good explanation why the phase velocity of light usually decreases in a dielectric.

For electrons it's easy to understand, on the other hand.

 

[tech99]

You say for electrons it is easy to understand. Well the dielectric action involves electrons, so I was wondering if you could elaborate on your thinking?

 

[Philip Koeck]

You say for electrons it is easy to understand. Well the dielectric action involves electrons, so I was wondering if you could elaborate on your thinking?

I meant for an electron wave going through matter.

The electrons of the electron wave gain momentum when they "fall" into the electrostatic potential wells of the specimen atoms. That means they get a shorter wavelength.
When the electron wave leaves the specimen on the other side it lags behind in phase compared to a (possibly hypothetical) wave that hasn't gone through the specimen.

It's easy to understand because there is an actual force between electrons and the atoms of the specimen.
Photons are different in that respect.

 

[hutchphd]

In matter the speed of light slows by n. But the frequency cannot change at any interface so the wavelength shortens according to $$\omega k=\frac cn $$ and the phase kx increases with n-

Lordy! OOPS: $$\frac \omega k =\frac cn $$

 

[Philip Koeck]

In matter the speed of light slows by n. But the frequency cannot change at any interface so the wavelength shortens according to $$\omega k=\frac cn $$ and the phase kx increases with n

Completely agree, but I see two question in the original post:
Why is the phase velocity of light usually smaller in a dielectric and where do the 90 degrees come from?

 

[vanhees71]

You mean $\omega=c k/n$. I find the explanation not so good in this movie.

The argument is not that difficult though: First what is discussed here are plane waves, i.e., there's an em. wave with a harmonic time dependence present within the material for a very long time. A classical picture, which leads to an amazingly good qualitative model for the refractive index, is that the dielectric consists of a lattice of positive charged very heavy ions and electrons bound to them. As long as the external electromagnetic plane wave is weak compared to the typical fields binding the electrons you can assume that the corresponding disturbance leads to a restoring force linear to the displacement of the bound electrons from their equilibrium positions. In addition there's also some friction, i.e., these electrons behave like a damped harmonic oscillator when displaced a little bit from the equilibrium positions. Without external force the motion will be damped, and the electrons relax to their static equilibrium positions after some relaxation time $\tau$ related to the friction coefficient.

Now the plane wave is present for a long time and its electric field for the bound electrons at any place is just a harmonic disturbance which has acted for a considerable time, so that you can assume that all the transient motion has been damped out and they thus perform only the enforced harmonic oscillation with the frequency of the external em. field.

This model is worked out nicely in the Feynman lectures:

https://www.feynmanlectures.caltech.edu/II_32.html