17. Electromagnetic waves From the Maxwell’s work it follows that optics is a branch of electromagnetism. A beam of light is a traveling wave of electric and magnetic fields – EM wave. The spectrum of light as a part of electromagnetic spectrum From HRW 4

17.1. Generation of EM waves Emission of EM waves is effectively realised when we use an electric dipole (antenna). Starting from the simple resonance LC circuit and „opening” the circuit with increasing frequency, one can schematically show how the radiating dipole is formed. Realistic shape of the EM wave at a distant point Generation of EM wave in the shortwave radio region

17.2. Equation of EM wave From the Maxwell’s equations it is easy to obtain for both electric and magnetic fields the expressions having the shape of wave equations (11.5) (Chapter 11) (17.1) (17.2) Comparing (17.1) and (17.2) with the wave equation (11.5) it can be concluded that in the case of an EM wave or (17.3) where c is the speed of EM wave in vacuum (all EM waves, including light, have the same speed in vacuum).

Equation of EM wave, cont. Similarly to (17.3) the speed of light in a medium (with neglected absorption) is given by (17.4) where ε, μ are electric permittivity and magnetic permeability of a given material, respectively. The solutions of equations (17.1) and (17.2) are simple for the case of plane waves. Assuming for electric and magnetic vectors the following components: equations (17.1) and (17.2) can be written as (17.5) (17.6) It is easy to prove that solutions of above equations are (17.7) where

Equation of EM wave, cont. The sinusoidal electric and magnetic fields are perpendicular to each other and perpendicular to the direction of propagation indicated by the wave vector k (EM waves are transverse). Configuration of fields E and B for a plane wave propagating in x direction at speed c. The instantaneous values of fields E and B as functions of x. The wave components E and B are in phase. The two fields continuously create each other via induction.

17.3. Polarization The wave for which the E vector oscillates only in one plane is polarized (plane-polarized). Polarized waves are emitted by e.g. radio and TV transmitters, microwave antennas. The light waves emitted by natural light sources (the Sun) or common sources as a bulb are unpolarized. This is connected with the mechanism of radiation. Vector E of EM wave is parallel to the dipols axes oriented vertically. Turning the receving dipole by 90o makes the received signal disappear. Natural unpolarized light is polarized by the polarization sheet (a Polaroid filter) and passes through another filter – an analyser.

Polarization, cont. The intensity of polarized light transmitted by the filter can be found from the analysis of the transmitted component of electric field Ey θ – angle between E and Ey (17.8) As the intensity of light is proportional to the square of electric field amplitude one obtains from (17.8) (17.9) From (17.9) it follows that I = Im for θ = 0 or θ = π and I = 0 when polarizer and analyzer are crossed. When the light incident on the filter is unpolarized, the angle θ varies randomly and in this case the intensity of transmitted light is (17.10) which follows from the averaging: This is called the one-half rule. The vector of electric field can be resolved into two components. Component Ey parallel to the polarizing direction is transmitted by the sheet, component Ez is attenuated.

17.4. Reflection and refraction of light In geometrical optics we consider the light waves as straight lines (rays). Law of reflection: The reflected ray, incident ray and normal to the surface lie in one plane. The angle of reflection is equal to the angle of incidence Law of refraction (Snell’s law): An angle of refraction θ2 is related to the angle of incidence θ1 by n2,1 – relative index of refraction The index of refraction for a given medium is defined as v – speed of light in a medium c – speed of light in vacuum The incident ray of light is reflected from the interface separating two media and refracted. All angles are measured relative to the normal. medium n water 1.33 fused quartz 1.46 crown glass 1.52 flint glass 1.65 diamond 2.42

Chromatic dispersion The index of refraction depends on the wavelength. For the light consisting of different wavelengths one observes different angles of refraction for these wavelengths what is called chromatic dispersion. Figures from HRW 4 prism A prism separates white light into component colors

Total internal reflection When light travels from a medium of larger index of refraction to a medium with a smaller index of refraction, then the total internal reflection may occur (when the angles of incidence are greater than the critical angle). From HRW 4 θc – critical angle The refraction angle for ray e is 900. Rays f and g only reflect. Total internal reflection is used to guide light in optical fibres

Polarization by reflection The reflected light in general is partially polarized because the electric fields along one direction have greater amplitudes than those oscillating along other directions. However at particular angle of incidence, called Brewster angle θB , the reflected light has only perpendicular components. The reflected light is then fully polarized. Experimentally it was found that in this case the reflected and refracted rays are perpendicular to each other. In this case θB + θr = 90o . From the law of refraction n1 sin θB = n2 sin θr and then n1 sin θB = n2 sin (90o- θB ) tan θB = n2/n1. Finally one obtains: From HRW 4

17.5. Interference from thin films Rays 1 and 2 are the result of reflections by the front and back sides of the film, respectively. These waves interfere and the result of interference depends on their phase shift. The phase shift depends not only on the thickness d and refractive index n2 of the thin film but also it has to be taken into account that during reflection at the interface the change in phase depends on the refractive index of the medium from which the ray reflects.. If the ray reflects from the medium of higher refractive index, it undergoes a phase shift of p rad ( half of a wavelength). If the ray reflects from the medium of lower refractive index it does not undergoe a change in phase. For nearly perpendicular incidence of the ray shown in the figure above and for the condition n3 > n2 > n1 in order to observe fully constructive interference of rays 1 and 2, the following condition must be fulfilled 2d= m ln2 , where m – integer number, ln2 - wavelength in medium n2.

Interference from thin films, cont. Antireflective coatings The structure shown in the figure is an example of suppression of unwanted reflections from glass by deposition of thin film of magnesium fluoride with adequate thickness. In this case the waves of selected wavelengths reflected from the two film interfaces should be exactly out of phase l - wavelength in air. If we want the least thickness of the coating, one selects m = 0. In this case one obtains for l = 550 nm (the middle of the visible spectrum) Professional antireflective coatings consist of many layers with proper thicknesses and indices of refraction to reduce the reflection in a desired region on wavelength.