Normal Dispersion

DISPERSION

A beam of light when it passes through the prism is splits into its constituent colors. Different colors suffers different amount of deviation. The spread of color is called dispersion of light. The medium which produces dispersion is called dispersive medium.

Or simply, the flow of light energy through a medium causes a electrical disturbances that varies sinusodially with time. When the refractive index of medium varies with frequency, the medium is said to be dispersive and the variation of refractive index \(\mu\) of the medium with wavelength \(\lambda\), i.e. frequency constitutes the phenomena of dispersion. The dispersive power of the medium is represented by \(\frac{d\mu}{d\lambda}\). It is generally observed that the refractive index decreases with the increase in wavelength. Hence the variation of refractive index with wavelength is called dispersion. It is divided into two categories; normal and anomalous dispersion.

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

Normal Dispersion

The dispersion in which refractive index decreases with increase in wavelength and vice-versa is called normal dispersion. Most of the transparent crystals exhibits normal dispersion. The dispersion of light through a glass prism is normal dispersion, in which the red color has greatest wavelength and least refractive index whereas the violet color has the greatest refractive index and least wavelength. This dispersion (normal dispersion ) satisfy the Cauchy dispersion relation, which is given as,

$$\mu=A+\frac{B}{\lambda^2}$$

Where A & B are the Cauchy constant.

The variation of refractive index with wavelength is shown in the diagram below.

The graph shows that the refractive index decrease with increase in wavelength. There is rapid decrease in refractive index in the short wavelength region.

Mathematical calculations:

We have dispersion formula as;

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

In the region remote from the natural frequency of oscillators, that is , the absorption frequencies of the medium, the term \(\gamma k^{\omega}\) of the equation is very small. Therefore, it can be neglected in comparison to \(\omega^2_k - \omega^2\). Thus, $$\mu^2=1+\frac{1}{4\pi\epsilon_0}\frac{4\pi Ne^2}{m}\sum_k\frac{f_k}{\omega^2_k-\omega^2}\dotsm(1)$$

This expression shows that the refractive index is real and increase with frequency of the incident wave. This is normal dispersion. $$\omega=2\pi f=\frac{2\pi c}{\lambda}$$

Substituting the value of \(\omega\) in equation (1), we get

$$\mu^2=1+\frac{1}{4\pi\epsilon_0}\frac{4\pi Ne^2}{m}\sum\frac{f_k}{\biggr(\frac{4\pi^2 c^2}{\lambda_k^2}-\frac{4\pi^2 c^2}{\lambda^2}\biggl)}$$

$$=1+\frac{N e^2}{\epsilon_0 m4\pi^2 c^2}\sum_k\frac{f_k \lambda^2\lambda^2_k}{(\lambda^2-\lambda^2_k)}$$

$$\mu^2=1+\sum_k\frac{A_K \lambda^2}{(\lambda^2-\lambda_k^2)}\dotsm(2)$$

Where, \(A_K = \frac{Ne^2\lambda^2_k f_k}{\epsilon_0 m 4\pi^2 c^2}\)

Equation (2) is known as Sellmeier’s equation.

if we suppose that \(\lambda\) less than or equal to \(\lambda_k\), then equation (2) can be written as $$\mu^2=1+\sum_k A_K\biggl(\frac{\lambda^2-\lambda_k^2}{\lambda^2}\biggr)^{-1}$$

$$=1+\sum_K A_k\biggr(1-\frac{\lambda_k^2}{\lambda^2}\biggl)^{-1}$$

$$=1+\sum_k A_K \biggr(1+\frac{\lambda_k^2}{\lambda^2}\biggl)+\;\dotsm$$

[expanding binomially and neglecting the higher order terms.]

$$=1+\sum_k A_K+\sum_K A_K\frac{\lambda_k^2}{\lambda^2}+\sum_k A_K \frac{\lambda_k^4}{\lambda^4}$$

$$\mu^2=A+\frac{B}{\lambda^2}+\frac{C}{\lambda^4}\dotsm(3)$$

Equation (3) represents the Cauchy’s dispersion formula. Where, A & B are called Cauchy’s constants. The values of A & B depend on the medium. From equation (3), it is clear that the refractive index of the medium decreases with increase in wavelength of light.

If a graph is plotted between \(\mu\) and \(\frac{1}{\lambda^2}\), it will be a straight line. The intercept OM on the y-axis gives the value of A. The slope of the line MN gives the value of B.

Where, A = 1 + \(\sum_k A_K \) ;

$$B=\sum_k A_K \lambda_k^2$$

$$C=\sum_k A_K \lambda_k^4$$

The equation (3) is called the Cauchy’s dispersion formula. A is the coefficient of refraction and B is the coefficient of dispersion.

Differentiating on both sides of the equation (3), we get

$$2\mu \frac{d\mu}{d\lambda}=-\frac{2B}{\lambda^3}-\frac{4C}{\lambda^5}$$

$$or,\;\;\; 2\mu \frac{d\mu}{d\lambda}=-\frac{2}{\lambda^3}\biggl(B+\frac{2C}{\lambda^2}+\dotsm\biggr)$$

$$or,\;\;\; \frac{d\mu}{d\lambda}=-\frac{1}{\lambda^3}\biggl(B+\frac{2C}{\lambda^2}+\dotsm\biggr)\frac{1}{\mu}$$

$$or,\;\;\; \frac{d\mu}{d\lambda}=-\frac{1}{\lambda^3}\biggl(B+\frac{2C}{\lambda^2}\biggr) \biggl(A+\frac{B}{\lambda^2}\biggl)^{-\frac{1}{2}}$$

$$or,\;\;\; \frac{d\mu}{d\lambda}=-\frac{BA}{\lambda^3}=-\frac{K}{\lambda^3}$$

So, \(\frac{d\mu}{d\lambda}=-\frac{K}{\lambda^3}\), where K = BA \(\dotsm(4)\)

The equation (4) gives the expression for the dispersive power. From this expression, it is clear that the dispersive power varies inversely proportional to the cube of the wavelength. The minus sign indicates that the slope of the dispersive curve is negative.

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.