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Re: speed of light of different colours



The question is:
> >why is the speed of light for red bigger that , say, blue, when both are
> >in an optcally denser medium?

So, at 08:31 PM 5/10/01 -0700, Geoff Nunes wrote:

Light in a medium propagates by excitation and re-radiation.

OK.

So why different speeds for different wavelengths? The harmonic
oscillator is being driven at a frequency below its resonant frequency.
...
> The bluer the light, the closer it is to the resonant frequency

Sorry, that's not right. In particular:

1) In a real optical medium, we are quite likely to have some resonators
for which the drive is below their resonant frequency, *and* some for which
the same drive is above their resonant frequency. For example, look at the
absorption spectrum of water. Here is a graph
http://www.deas.harvard.edu/~jones/es151/pages/gallery/images/water_spec.
http://www.deas.harvard.edu/~jones/es151/pages/gallery/images/water_spec.html
that somebody lifted from page 291 of Jackson, _Classical
Electrodynamics_. We see that throughout most of the optical band, the
absorption is dominated by the shoulder of resonators that lie at !lower!
frequencies.

So the motion of the oscillator is not in phase with the incoming wave,
but reaches it's maximum a little bit after the EM wave does. That means
the radiation that it puts out is also behind.

2) Even in cases where that happens to be true, it is irrelevant, for
several reasons given below.

The cumulative effect of all these little delays is a propagation speed
less than c. The bluer the light, the closer it is to the resonant
frequency (which is in the ultraviolet for most glasses). The closer to
the resonance, the bigger the "phase lag," and the slower the wave travels.

2.1) No, it is not the "phase lag" that plays the dominant role. There
is no such thing as absolute phase; relative phase is the only thing that
matters. And for calculating the intensity of a wave, the sign of the
relative phase doesn't matter; that is, given two waves A and B, it
doesn't matter whether A is _ahead_ of B by ten degrees of phase, or
_behind_ B by ten degrees of phase.

2.2) The "closer-to-resonance" argument predicts that for
drive-frequencies above the material's resonant frequency, we should see
anomalous dispersion, i.e. red being more strongly refracted than
blue. This is not what is observed; see for instance figure 7.8 on page
286 of Jackson.

In fact we observe
-- normal dispersion below (but not too close) to resonance,
-- normal dispersion above (but not too close) to resonance, and
-- anomalous dispersion close to resonance.

We don't often notice anomalous dispersion, because near resonance the
_absorption_ is large, and we don't usually build refractors out of
strongly absorbing materials.

=========================

People's intuition about discrete harmonic oscillators often leads them
astray when thinking about resonances in optical materials. Consider the
following contrast:
-- For discrete oscillators, we usually observe the _magnitude_ of the
response. The magnitude has a strong peak at the resonant frequency.
-- For optical materials, we must distinguish the _real part_ of the
response versus the _imaginary part_ of the response. Combining those two
parts to form the magnitude would be very unhelpful. The real part is zero
when the magnitude is at its maximum.

To say the same thing in terms that are slightly more physical and less
mathematical: We can resolve the re-radiated wave into components:
* One component is in phase with the main wave.
Call this the "real part" or the "cosine" component.
* One component is 90 degrees out of phase.
Call this the "imaginary part" or the "sine" component.

(The term "cosine component" is predicated on choosing WLoG the phase of
the main wave so that it is a pure cosine.)

Anyway, it is the cosine component of the re-radiated wave that is
responsible for the index of refraction. (The sine component is
responsible for absorption). And... The behavior of the cosine component
is radically different from the behavior of the more-familiar magnitude of
the response function.

The cosine component is an increasing function of frequency, for all
frequencies below resonance (but not too close). It then falls
precipitously. It is ZERO right at resonance, unlike (very unlike!) the
magnitude. It falls steeply through zero, and then starts rising
again. It is again a rising function of frequency for all frequencies
above resonance (but not too close).

Rather than delving into the mathematics, let's see if we can shed some
light on this using a hands-on demo of a resonator. Hint: you have to be
careful to consider only the IN-PHASE component of the response.

handle-----------spring--------mass
(position = (position =
excitation) response)

So, let's go: If you shake the handle below resonance, the mass responds
in phase to the excitation. As you get moderately close to resonance, the
in-phase response gets bigger and bigger. But right at resonance, there is
*no* in-phase response. The total response is huge, but it is exactly 90
degrees out of phase. Well above resonance, the response is nearly 180
degrees out of phase; we can consider this an in-phase (cosine-phase)
response with a minus sign. This goes to zero as the frequency increases
-- but that is to say, it is negative and becoming closer to zero, which is
again an increasing function of frequency.

It's not easy to draw a good graph of the response function using ASCII
art; so I will refer you to figure 7.8 on page 286 of Jackson, or the
figures on pages A2-2 and A2-3 of
http://www.bioc.rice.edu/~graham/BIOS481/chapters/Apndx_2_harm_osc.pdf