Section 31–3 Dispersion

 (Normal dispersion / Anomalous dispersion / Refining dispersion equation)

 

The three interesting concepts in this section are normal dispersion, anomalous dispersion, and the refinements for the dispersion equation.

 

1. Normal dispersion:

“The phenomenon that the index depends upon the frequency is called the phenomenon of dispersion, because it is the basis of the fact that light is “dispersed” by a prism into a spectrum (Feynman et al., 1963, p. 31–6).”

 

Feynman explains that dispersion is a phenomenon in which the refractive index depends upon the frequency and it is based on the fact that light is dispersed by a prism into a spectrum. However, one may define dispersion as a spreading of light waves into their full spectrum of wavelengths in a dispersive medium. Furthermore, we may state the condition of dispersive medium whereby the speed of a light wave is dependent on its frequency, i.e., all material media are dispersive except vacuum. Specifically, normal dispersion refers to the increase in refractive index provided the frequency (w) increases toward a resonant frequency (w₀) and (w₀² - w²) decreases in accordance with the equation (31.19). In other words, the slope dn/dw is positive as long as w is not too close to one of the resonant frequencies of the dispersive medium.

 

Setting ω₀ = 0 in our dispersion equation yields the correct formula for the index of refraction for radiowaves in the stratosphere, where N is now to represent the density of free electrons (number per unit volume) in the stratosphere. But let us look again at the equation, if we beam x-rays on matter, or radiowaves (or any electric waves) on free electrons the term (ω₀²−ω²) becomes negative, and we obtain the result that n is less than one (Feynman et al., 1963, p. 31–6).”

 

According to Feynman, if we beam x-rays or radiowaves (or any electric waves) on free electrons in the stratosphere, the term (ω₀²−ω²) becomes negative and it means n is less than one. This is not quite correct because the term (ω₀²−ω²) would be positive for frequencies of radiowaves that are lesser than ω₀. In addition, there are several resonant frequencies (ω_0i) depending on various materials as shown in the equation (31.20). For example, the stratosphere also contains ozone molecules, which protect life on earth by reducing the harmful ultraviolet radiation because ozone molecules have stronger absorption in the ultraviolet region. Note that British geophysicists, Farman, Gardiner, and Shanklin (1985) reported a significant decrease in stratospheric ozone levels over the Antarctic stations.

 

In spite of the fact that it is said that you cannot send signals any faster than the speed of light, it is nevertheless true that the index of refraction of materials at a particular frequency can be either greater or less than 1. This just means that the phase shift which is produced by the scattered light can be either positive or negative… It is this advance in phase which is meant when we say that the ‘phase velocity’ or velocity of the nodes is greater than c (Feynman et al., 1963, p. 31–6-7).”

 

According to Feynman, the refractive index can be greater or less than 1 means that the phase shift which is due to the scattered light can be either positive or negative, and specifically, advance in phase means the “phase velocity” is greater than c. Importantly, the phase velocity is the velocity of any point within the wave, i.e., it is not necessarily the nodes. Perhaps Feynman could have proved this fact by using kx - wt = constant, which may refer to any point (or constant phase) in the wave. In short, d(kx - wt)/dt = 0 Þ dx/dt = w/k = c, which is independent of w and k for a sinusoidal wave. On the other hand, it is not strictly correct to say that the refractive index can be greater or less than 1 because it may be a complex number or less than zero. (In 1968, Veselago hypothesized the possibility of a negative refractive index.)

 

2. Anomalous dispersion:

“Very near the resonant frequencies, however, there is a small range of ω’s for which the slope is negative. Such a negative slope is often referred to as ‘anomalous’ (meaning abnormal) dispersion, because it seemed unusual when it was first observed, long before anyone even knew there were such things as electrons. From our point of view both slopes are quite ‘normal’! (Feynman et al., 1963, p. 31–8).”

 

Feynman explains that it seemed unusual to have a negative slope (the rate of change of refractive index with respect to frequency) when it was first observed without the knowledge of electrons. In short, the dispersion is normal if dn/dω is greater than zero (dn/dω > 0 or dn/dl < 0), i.e., the dispersion is anomalous if dn/dω is less than zero. In other words, anomalous dispersion of light refers to the refraction spectra in which the normal order of the separation of component waves is reversed in the region near the resonant frequency. In this frequency range, the material is almost opaque and it may be described as “regions of resonant absorption” (Jackson, 1999, p. 130). The anomalous dispersion regions may not be observed because the medium strongly absorbs energy from the light waves within the absorption band.

 

During Weiner’s interview on March 5, 1966, Feynman says: “They asked me which color was on top of a rainbow. I said I didn’t know, but I could figure it out. They said, all right, figure it out, and I drew a drop of water and said, let’s see, now, the red rays are bent more than the blue — And the professor said, ‘Would you draw a curve of index refraction against wavelength?’ And I thought: aha, I got it wrong. So when I drew the index versus wavelength I turned it around so that the index was higher for the blue end than for the red end — the opposite of what I just said. When I looked at my curve I just drew, I said, ‘Oh yes, excuse me, it’s the other way around.’ But I drew the curve because I knew from the question it was backward, and I thought, I’m fooling him. Of course, he’d just given me a hint that I was wrong on the other.”

 

Historically, Leroux discovered the phenomenon of anomalous dispersion involving iodine vapor whose absorption bands fall within the visible region. In Leroux’s (1862) words, “Iodine vapor disperses light in different direction to any substance yet studied; that is, a prism full of iodine vapor refracts red rays to a greater extent than blue rays (p. 246).” That is, a prism formed using iodine vapor will deviate the red rays more than the violet, giving a different spectrum as compared to a substance having normal dispersion. When it was later discovered that transparent substances (e.g., glass) also possess absorption regions, such as ultraviolet and infrared radiation, the term anomalous is no longer appropriate. Although Leroux adopted the term abnormal dispersion, it could be better to replace it with resonance dispersion.

 

3. Refining dispersion equation:

“Let us now look again at our dispersion equation... To be completely accurate we must add some refinements. First, we should expect that our model of the atomic oscillator should have some damping force (otherwise once started it would oscillate forever, and we do not expect that to happen). We have worked out before (Eq. 23.8) the motion of a damped oscillator and the result is that the denominator in Eq. (31.16), and therefore in (31.19), is changed from (ω₀²−ω²) to (ω₀²−ω²+iγω), where γ is the damping coefficient (Feynman et al., 1963, p. 31–7).”

 

Feynman says that our model of the atomic oscillator should have some damping force so that it would not oscillate forever. The presence of damping force means that the amplitude of oscillations would not approach infinity when the driving frequency of light waves is equal to the resonant frequency of electrons. Some may explain that it is difficult to observe the region of anomalous dispersion because the electrons strongly absorb energy from the light waves whose frequencies are close to their resonant frequency. For example, metals are opaque because the electrons in the metals absorb the light waves and re-emit them during their passage through a metal. However, Lorentz oscillator model has already included damping force and thus another refinement could be including free and bound electrons (Hecht, 2002).

 

“We need a second modification to take into account the fact that there are several resonant frequencies for a particular kind of atom. It is easy to fix up our dispersion equation by imagining that there are several different kinds of oscillators, but that each oscillator acts separately, and so we simply add the contributions of all the oscillators. Let us say that there are N_k electrons per unit of volume, whose natural frequency is ω_k and whose damping factor is γ_k. We would then have for our dispersion equation n = 1 + (q_e²/2ϵ₀m)[∑kN_k/(ω²k−ω²+iγ_kω)] (Feynman et al., 1963, p. 31–7-8).”

 

In an endnote of chapter 31, Feynman adds that the dispersion equation (31.20) n = 1 + (q_e²/2ϵ₀m)[∑kN_k/(ω_k²−ω²+iγ_kω)] is still valid in quantum mechanics, but its interpretation is somewhat different. To be more accurate, an atom with one electron, like hydrogen, has several resonant frequencies. However, in the summary of the lecture (as shown on the blackboard), Feynman has also included the equation n – 1 = (Nq_e²/2ϵ₀m)[∑f_i/(ω_k²−ω²+iγ_kω)] in which f_i = fractional strength of oscillator i. Currently, we may explain that the f_i terms, which satisfy the requirement that å_if_i = 1, are weighting factors known as oscillator strengths (Hecht, 2002). Alternatively, the f_i terms are known as transition probabilities, and perhaps Feynman could have cited the Thomas-Reiche-Kuhn sum rule.

 

Review Questions:

1. Do you agree with Feynman that the refractive index n is definitely less than one if we beam radiowaves (or any electric waves) on free electrons?

2. Would you adopt the term anomalous dispersion?

3. How would you refine and interpret the new dispersion equation?

 

The moral of the lesson: Anomalous dispersion (or resonance dispersion) occurs in the region near a resonant frequency, whereas normal dispersion occurs below the resonant frequency; these two phenomena repeat in other resonant frequencies as shown below.

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics,
radiation, and heat. Reading, MA: Addison-Wesley.

2. Farman, J. C., Gardner, B. G. & Shanklin, J. D. (1985). Large losses of total ozone in Antarctica reveal seasonal Cl0x/NOx interaction. Nature, 315, 207–210.

3. Hecht, E. (2002). Optics (4th edition). San Francisco: Addison Wesley.

4. Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons, New York.

5. Leroux, M. F.-P. (1862). Researches on the refractive indices of bodies which only assume the gaseous condition at high temperatures. Abnormal dispersion of iodine vapour. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 24(160), 245-247.