problems like this were unknown at thermotron

OPTI 447,

Spring 2012 Problem Sets 5 Prof. Ewan M.Wright


Due: Beginning of class, Wednesday 2/29/12 (21 points)

1. This problem examines the formula for the optical properties of a gas of Rubidium atoms
according to the Lorentz oscillator model
N2(!) = 1 +ÃNe2m²0!1(!20 ¡ !2 ¡ i°!) ; (1)

where N is the density of atoms, e and m are the charge and mass of the electron, !0 is the transition or resonant frequency, and ° the linewidth of the transition.

Consider the case of Rubidium vapor which has a resonance at frequency !

0 = 2:4 £ 1015 rad s¡1 with linewidth ° = 36 £ 106 rad s¡1, and

assume an atomic density N = 1016 m¡3.

(a - 1pt) Consider a plane-wave propagating along the z-axis E(z) = E(0)ei( !

c )N(!)z. Show by substi-tuting N(!) = n(!) + i·(!) into this plane-wave that the ¯eld intensity evolves according to Beer's law I(z) = I(0)e¡®(!)z with absorption coe±cient ®(!) = 2!·(!)c .

Thus, we see that the extinction index ·(!) is related to absorption due to the medium described by the Lorentz oscillator model.

(b - 1pt) Following on from part (a) show that the ¯eld evolves as E(z) = E(0)eiÁ(z)e¡®(!)z=2, where the phase-shift accumulated over the distance z is given by Á(z) = (!c )n(!)z. Thus, we see that n(!)is related to the phase-shift accumulated by the propagating ¯eld.

(c - 3pts) Write a Matlab code to plot the refractive-index di®erence (n(!) ¡ 1) and the absorp-tion ®(!) both as functions of scaled frequency detuning ­ = (! ¡ !0)=° for ¡5 < ­ < 5. Note that = 0 corresponds to resonance ! = !0, and, for example, ­ = 2 corresponds to a frequency detuning
of two linewidths ! = !0 + 2°.

Please attach a copy of your code along with your plots.

(d - 2pts) Your plot of the absorption ®(!) from part (c) should be largest at resonance ­ = 0 or ! = !0. Based on your plot from part (c) show that the full-width at half-maximum ¢! of the absorption curve is ¢! = °, and this is the origin of the name linewidth for °. Thus, absorp-tion is most relevant at resonance but becomes negligible far from resonance j­j >> 1 or j!¡!0j >> °.(e - 2pts) Based on your plot from part (c) demonstrate that the refractive-index n(!) displays anomalous dispersion over the spectral region (!0 ¡ °=2) < ! < (!0 + °=2) around resonance, and
normal dispersion for all other detunings. Thus, for large detunings j­j >> 1 the refractive-index displays normal dispersion as we discussed in class.
(f - 2pts) As for the example of Rubidium vapor above, it is often the case that the linewidth °is much smaller than the optical and transition frequencies !; !0 >> °, with ! ¼ !0 of comparable size.

Assuming this to be the case prove that far o® resonance j(! ¡ !0)j >> °, or j­j >> 1, the complex refractive-index is real to a very good approximation given by
n2(!) ¼ 1 +Ã Ne2 m²0!1(!20 ¡ !2) :

This form for the refractive-index is valid o®-resonance in the normal dispersion region. (Hint:To do this problem you will want to show that under the conditions stated (!20 ¡ !2 ¡ i°!) ¼(!0 + !)[(!0 ¡ !) ¡ i°=2] and proceed from there.)
12.

So far our treatment of optical dispersion in class has tacitly dealt with dilute media for which the density N is relatively low, eg. atomic gases. For a single Lorentz oscillator a more general expression for the complex refractive-index valid for dense media is (see p. 71 of Hecht for a discussion)
(N2(!) ¡ 1)(N2(!) + 2)=ÃNe23²0m!1(!20 ¡ !2 ¡ i°!) ; (2)with N(!) = (n(!) + i·(!)).

Note that for zero density N = 0 the right-hand-side is zero and the
solution is N(!) = n(!) = 1 as expected physically.
(a - 2pts) A dilute medium is de¯ned by the fact that the density N is low enough that the magnitude
of the right-hand-side of Eq. (2) is small compared to unity. Show that for a dilute medium the general result in Eq. (2) may be safely approximated by the result obtained in class

You should plot these both on the same ¯gure and clearly distinguish them.


Please include your code with your solution.