Keywords: Normal dispersion, Anomalous dispersion, Absorption. Ref: M. Born and E. Wolf: Principles of Optics; R.S. Longhurst: Geometrical

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1 28 Dispersion Contents 28.1 Normal dispersion 28.2 Anomalous Dispersion 28.3 Elementary theory of dispersion Keywords: Normal dispersion, Anomalous dispersion, Absorption. Ref: M. Born and E. Wolf: Principles of Optics; R.S. Longhurst: Geometrical And Physical Optics; A. Sommerfeld: Optics Normal dispersion When a white light (eg. sunlight, or light from an incandescent lamp) is passed through a prism we observe colour separation. Light being electromagnetic oscillations, the different colours have different wavelengths and different frequencies in vacuum for all of them move with the same speed c. Our eyes sense(map) different wavelengths of visible spectrum with different colours. The separation happens because the different wavelengths have different refractive indices. Whenever light enters a dielectric medium this separation happens and the phenomenon is known as dispersion. In simple term it the variation of refractive index with the wavelength. In other words

2 2 28 Dispersion the variation of the frequency with the wavelength in a medium is dispersion. Cauchy studied dispersion and gave a formula which described the dispersion in the visible range quite well. The following formula is known as Cauchy s dispersion formula, n(λ) = A+ B λ + C 2 λ4, (28.1) where A, B and C are constants which depend on the medium. Experimentally the constants can be determined be measuring the the refractive index for three wavelengths. In usual condition the first two terms would suffice to give an accurate value of n. The derivative of the refractive index is given by dn dλ = B λ 3 (28.2) to a good accuracy. Since A and B both are positive the refractive index decreases increasing the wavelength. Problem 1: Suppose n violet = 1.60, n green = 1.53, and n red = 1.50, draw the dispersion curve in the visible range, (i.e. wavelength in a range 4000Å to 7000Å), [ λ violet = 4000Å, λ green 5500Å and λ red 7000Å] Anomalous Dispersion For material transparent to visible region Cauchy s formula works very well but if one further increases the wavelength say to the infra-red, one finds the refractive index suddenly decreases very fast and does not obey the Cauchy s

3 28.2 Anomalous Dispersion 3 law. One now approaches the absorption region. Further increasing the wavelength once again refractive index becomes large. Again the behaviour isquitesimilartothevisibleregionfortheincreaseinwavelength. Iftherange is increased further one again observes another absorption band as shown in the figure (28.1) below. The pattern may repeat further as shown, giving many absorption bands. This dispersion is known as anomalous dispersion. Refractive Index (n) Α Visible Absoption Bands Wavelength ( λ) Fig. 28.1: Normal and anomalous dispersion The first theory of it came from Sellmeier who assumed that all elastically bound particles in the medium oscillate with a natural frequency ω 0 which correspond to a wavelength λ 0 in the vacuum. Sellmeir, formula gave, n 2 = 1+ Aλ2 λ 2 λ 2, (28.3) 0 where A is a constant. If one is away from λ 0 it can be expanded in powers of λ 0 /λ and one would get a formula of the Cauchy type (28.1).

4 4 28 Dispersion Problem 2: Obtain Cauchy constants A, B and C in terms of Sellmeier constants A and λ 2 0. To explain many absorption bands one has to assume different species of electrons with different natural frequencies ω j corresponding to wavelengths λ j in the substance and then n 2 = 1+ j A j λ 2 λ 2 λ 2. (28.4) j 28.3 Elementary theory of dispersion We will now see the effect of dispersion when an electromagnetic field incident on a dielectric. In chapter 3 we have seen that for the LIH we can write, D = ǫe, (28.5) The equation (28.5) is further written in the following way D = ǫ 0 E = ǫ 0 E+P = ǫ 0 (1+χ)E. (28.6) The vector P(= ǫ 0 χe) is called the Polarisation vector and is assumed to be linearly proportional to the electric field. The factor χ is called the electric susceptibility of the substance and the factor (1+χ) = ǫ r is nothing but the relative permittivity or the dielectric constant introduced in chapter 3 earlier. When an electric field is applied to a substance electrons of the molecules of that substance are displaced from there mean position as shown in the fig.

5 28.3 Elementary theory of dispersion 5 (28.2). The electron does not leave the molecule as it is bound to it by some force. In this situation we say the substance is polarised. The polarisation is measured by the quantity P, the net dipole moment per unit volume. If one has N such electrons per unit volume of that substance, polarisation is given by P = Nex (28.7) where x is the displacement of the electron from its mean position as shown in the figure. Where we are assuming implicitly that the field applied is along the positive-x direction. The negative sign comes due to the negative charge of the electron and hence the displacement is along the negative-x direction. If the electron is little bit displaced from its mean position and left it would e E x Fig. 28.2: Polarisation of a molecule oscillate with a natural frequency ω 0. The positive ions do not move much due to their large masses. Now if the substance is under time varying electric field, E, the displacement would change with time, i.e. it will undergo a forced oscillation, the equation of which will be given by, ẍ+rẋ+ω0x 2 = e E, (28.8) m

6 6 28 Dispersion where a natural damping term is introduced proportional to the velocity. We can recast the above equation in terms of P, P+rṖ+ω2 0P = Ne2 E. (28.9) m Now we use the other two Maxwell equations, viz. E = Ḃ and H = Ḋ to elemi nate P from the equation (28.9). Ḃ = µ 0 D ( E) = µ 0 ǫ 0 Ë µ 0 P 2 E = µ 0 ǫ 0 Ë+µ 0 P (28.10) whereweusedthemaxwellequation E = 0andhaveassumedpermeability of the substance, µ µ 0, i.e. B = µ 0 H. So, P = 1 ( 2 E 1 µ 0 c2ë). (28.11) Differentiating the equation (28.9) with respect to time we get ( 2 t 2 +r t +ω2 0 ) P = Ne2 Ë. (28.12) m Using (28.11) we can now elemi nate P in (28.12) to obtain, ( 2 t +r ) 2 t +ω2 0 ( 2 E 1 c 2Ë) = µ Ne 2 0 Ë. (28.13) m Now when a plane electromagnetic wave polarised along the x direction, E = E x = E 0 e i(kz ωt) and E y = E z = 0, is incident on the substance,

7 28.3 Elementary theory of dispersion 7 with frequency ω, we have on substituting in (28.13) the required dispersion relation. ( ω 2 irω +ω 2 0)( k 2 + ω c 2) = µ 0Ne 2 After some re-arrangement of terms we get [ k 2 = ω2 c 2 1+ (µ 0c 2 Ne 2 /m) ω0 2 iω ω2 m ω2. (28.14) ]. (28.15) Now ω/k is the phase velocity v, and refractive index n = c/v, so we have the dependence of refractive index on frequency as n 2 = 1+ (µ 0c 2 Ne 2 /m) ω0 2 (28.16) iω ω2. So we find that in general the refractive index can be imaginary depending on the frequency and this would lead to absorption. Now if ω 2 0 ω 2 >> rω that is damping is small and absorption is neglected, the refractive index is real, n 2 = 1+ (µ 0c 2 Ne 2 /m) ω 2 0 ω2. (28.17) In vacuum the wavelength λ = 2πc/ω and if λ 0 = 2πc/ω 0, we get the Sellmeier equation (28.3) n 2 = 1+ (µ 0Ne 2 λ 2 0λ 2 /4π 2 m) λ 2 λ 2 0 (28.18) with an estimate of A as µ 0 Ne 2 λ 2 0/(4π 2 m). For the case of electrons of different restoring forces, we would write the polarisation as, P = N j f j x j (28.19)

8 8 28 Dispersion wherex j isthedisplacementofthej th typeofelectron, thef j isthefractionof that type and the total electrons available per unit volume is N. Proceeding as earlier this would lead to the general Sellmeier type formula (28.4) n 2 = 1+ µ 0Ne 2 4π 2 m f j λ 2 j λ 2 λ 2. (28.20) j j

INTERACTION OF LIGHT WITH MATTER

INTERACTION OF LIGHT WITH MATTER Already.the speed of light can be related to the permittivity, ε and the magnetic permeability, µ of the material by Rememberε = ε r ε 0 and µ = µ r µ 0 where ε 0 = 8.85