Anomalous Dispersion and Lorentz theory

Lorentz Theory Of Dispersion Of Light

The theory of dispersion based on the electromagnetic theory based on the following assumptions;

• There is no appreciable interaction between the atoms or between the molecules.
• The electric field of the electromagnetic wave induces a dipole moment in the gas molecules.
• Electrons are bound to the nucleus by linear restoring force.
• Over a molecules or an atom E is constant in space, i.e.

$$E=E_0e^{-i(\omega t-kr)}$$

$$E=E_0e^{-i\omega t}$$

According to Lorentz theory, the dispersion formula is; $$\mu^2=1+\frac{1}{4\pi\epsilon_0} \frac{4\pi N e^2}{m}\sum_k\frac{f_k}{(\omega^2_k-\omega^2-j\gamma_k\omega)}$$

Anomalous Dispersion

The dispersion in which the refractive index increase with increase in wavelength and vice-versa is called anomalous dispersion. When light is passed through the prism, containing iodine vapour, anomalous dispersion is observed. That means, in this case, the red color has the greatest wavelength and also the refractive index. In the reverse order, violet color shows the least wavelength and the refractive index. For such a dispersion, a graph of refractive index versus the wavelength is straight line having positive upward slopes. The variation of refractive index with wavelength is shown in the diagram below.

FIGURE HERE;

In the figure, AB, CD & EF represents the normal dispersion and BC & DE represents the anomalous dispersion. For small wavelength range, the refractive index increase with increase in and the dispersion is anomalous dispersion. The substance which exhibits the anomalous dispersion follows Sellmeier’s formula ,which is given by

$$\mu=1+\sum_k \frac{A_K\lambda^2}{\lambda^2-\lambda^2_k}$$

Where \(A_K\) is a proportionality constant to the number of particles per unit volume that vibrate with natural frequency corresponding to the wavelength \(\lambda_k\). For two wavelength \(\lambda_1\) and \(\lambda_2\),

$$\mu^2=1+\frac{A^2_1\lambda^2}{\lambda^2-\lambda^2_1}+\frac{A^2_2\lambda^2}{\lambda^2-\lambda^2_2}$$

Mathematical derivations

We have dispersion formula as;

$$\mu^2=1+\frac{1}{4\pi\epsilon_0} \frac{4\pi N e^2}{m}\sum_k\frac{f_k}{(\omega^2_k-\omega^2-j\gamma_k\omega)}\dotsm(1)$$

If the impressed frequencies do not include a natural frequency of the electrons, then the dispersion is always normal. But, when the impressed frequency contain the natural frequency from the so many natural frequencies of the electron, the normal course of dispersion is disturbed. For simplicity, let us assume that there is one natural frequency \(\omega_k = \omega_0\) . Then the equation (1) becomes;

$$\mu^*^2=1+\frac{1}{4\pi\epsilon_0}\frac{4\pi Ne^2}{m}\frac{1}{(\omega^2_k-\omega^2-i\gamma\omega)}$$

$$\mu^*^2=1+\biggl[\frac{1}{4\pi\epsilon_0}\frac{4\pi Ne^2}{m}\frac{1}{(\omega^2_0-\omega^2-i\gamma\omega)}\biggr]^{\frac{1}{2}}\dotsm(2)$$

$$\mu^*=1+\frac{1}{4\pi\epsilon_0}\frac{2\pi Ne^2}{m}\frac{1}{(\omega^2_0-\omega^2-i\gamma\omega)}+\dotsm\;\;\;\dotsm(3)$$

Now, multiplying to the numerator and denominator of right hand side by (\omega^2_0-\omega^2+i\gamma\omega), we have,

$$\mu^*=1+\frac{2\pi Ne^2}{4\pi\epsilon_0 m}\biggl{\frac{(\omega^2_0-\omega^2+i\gamma\omega)}{(\omega^2_0-\omega^2-i\gamma\omega)(\omega^2_0-\omega^2+i\gamma\omega)}\biggr}$$

$$or,\;\;\;\mu^*=1+\frac{2\pi Ne^2}{4\pi\epsilon_0m}\biggl{\frac{\omega^2_0-\omega^2+i\gamma\omega}{(\omega^2_0-\omega^2)^2+\gamma^2\omega^2}\biggr}$$

$$or,\;\;\;\mu^*=1+\frac{2\pi Ne^2}{4\pi\epsilon_0m}\biggl{\frac{(\omega^2_0-\omega^2)}{(\omega^2_0-\omega^2)^2+\gamma^2\omega^2}+\frac{i\gamma\omega}{(\omega^2_0-\omega^2)^2+\gamma^2\omega^2}\biggr}\dotsm(4)$$

$$or,\;\;\;\mu^*=1+\frac{2\pi Ne^2}{4\pi\epsilon_0m}\biggl{\frac{(\omega^2_0-\omega^2)}{(\omega^2_0-\omega^2)^2+\gamma^2\omega^2}\biggr}+i\frac{2\pi Ne^2}{4\pi\epsilon_0m}\biggl{\frac{\gamma\omega}{\omega^2_0-\omega^2+\gamma^2\omega^2}\biggr}$$

Where \(\mu\) = n + ik

Where, n = 1 + \(\frac{2\pi Ne^2}{4\pi\epsilon_0m}\biggl{\frac{(\omega^2_0-\omega^2)}{(\omega^2_0-\omega^2)^2 + \gamma^2\omega^2}\biggr}\dotsm(5)\)

And , K = \(\frac{2\pi Ne^2}{4\pi\epsilon_0m}\biggl{\frac{\gamma\omega}{(\omega^2_0 - \omega + \gamma^2\omega^2)}\biggr}\dotsm\;\;\;\dotsm(6)\)

The figure given below shows the real and imaginary parts of \(\mu\) plotted as a function of frequency \(\omega\). The figure shows that at low frequencies n is slightly greater than unity. It increases as \(\omega\) increases. It reaches a maximum for \(\omega_1 =\omega_0 - \frac{\gamma}{2} and then decreases rapidly until \(\omega_2 = \biggl(\omega_0 + \frac{\gamma}{2}\biggr)\) when \(\omega = \omega_0\) then n = 1. After \(\omega_2\), the curve again increases and approaches unity asymptotically.

The imaginary part corresponds to the absorption of electromagnetic wave waves propagating through the gas. K is maximum when \(\omega = \omega_0\), where n is unity. It has a width aat half maximum approximately equal to \(\gamma\). Thus, in the region where n changes rapidly, the gas is relatively highly absorbing.

In case of real gas, there exist many resonant frequencies and corresponding damping coefficients. The complex refractive index for real gas is given by,

$$\mu*=1+\frac{1}{4\pi\epsilon_0}\frac{2\pi Ne^2}{4\pi\epsilon_0m}\sum_k\frac{f_k(\omega^2_k-\omega^2-i\gamma\omega)}{(\omega^2_k-\omega^2)^2+\gamma^2\omega^2}\dotsm\;\;\dotsm(7)$$

Similarly, we can consider the behaviour of real and imaginary parts.

FIGURE: Plot of n and K as a FUNCTION of \(\omega\)

References:

Adhikari, P.B, Daya Nidhi Chhatkuli and Iswar Prasad Koirala. A Textbook of Physics. Vol. II. Kathmandu: Sukunda Pustak Bhawan, 2012.

Jenkins, F.A and H.E White. Fundamental of optics. New York (USA): McGraw-Hill Book Co, 1976.

wood, R.W. Physical Optics. New York (USA): Dover Publication , 1934.