Lecture 5: Polarisation of light 2 Lecture aims to explain: 1. Circularly and elliptically polarised light 2. Optical retarders - Birefringence - Quarter-wave plate, half-wave plate
Circularly and elliptically polarised light
Circularly polarised light Both Ex and Ey components have equal amplitude and phase difference ε=±π/2+πm y E x ( t) = E0 sin ( ωt) E y E y t) π = E sin ωt ± + πm 2 ( 0 E x x The shape traced out in a fixed plane by the electric field vector is a description of the polarization state Note direction of rotation: clock-wise rotation for rightcircularly polarised, anti-clock-wise rotation for leftcircularly polarised (wave moves towards the observer)
Elliptically polarised light Ex and Ey components have different amplitudes and a fixed phase difference ε E x ( t) = E0 x sin ( ωt) E y ( t) 0 = E sin t y ( ω + ε ) Both circularly and linearly polarised light are specific cases of elliptically polarised light
Optical retarders
Properties of birefringent materials Crystalline materials may have different indices of refraction associated with different crystallographic directions. A common situation is birefringence i.e. two distinct indices of refraction for light with polarisation along (extraordinary, e-wave) and perpendicular (ordinary, o-wave) to the so-called "optical axis" of the crystal. Typical material calcite (CaCO 3 ). At a wavelength of ~590 nm n o =1.658 and n e =1.486, respectively. Between 190 and 1700 nm, n o varies roughly between 1.6 and 1.4, while n e varies between 1.9 and 1.5. The birefringent effect (using calcite) was first described by the Danish scientist Rasmus Bartholin in 1669
Optical retarders: quarter- and half-wave plates Crystal is cut so that the front and back surfaces are parallel to the so called optical axis Optical path difference d( n ) e no 2πd ε = λ Phase difference (λ- wavelength in vacuum): ( ) n e n o The quarter-wave plate (QWP) introduces a phase shift of π/2 (optical path difference of λ/4) between the o-wave and e-wave The half-wave plate (HWP) introduces a phase shift of π (optical path difference of λ/2)
EXAMPLE 5.1 Linearly polarised light (polarisation in horizontal direction) is incident on a a) quarter-wave plate b) half-wave plate The plate is rotated around its optical axis. Describe how the polarisation of the transmitted light will change c) In (a) and (b) light then passes through a linear polariser with a horizontal transmission axis. Calculate the intensity of light transmitted through the polariser as a function of the angle of rotation of the QWP/HWP.
EXAMPLE 5.2 A quarter-wave plate is inserted between two ideal linear polarisers whose transmission axes differ by 90 o. Light is transmitted through this optical system. Calculate how the intensity of the transmitted light will change when the quarter-wave plate is rotated around its axis.
EXAMPLE 5.3 (a) Describe how circularly polarised light can be converted into linearly polarised light using a quarter-wave plate. (b) Using a photo-detector, light of intensity I 0 is measured from a specimen emitting a mixture of left and right circularly polarised light. Using the result of (a) describe how the individual intensities of left and right circularly polarised light can be measured using a quarter-wave plate and a linear polariser.
Summary Circularly polarised light: Ex and Ey components have equal amplitudes and a phase difference ε=±π/2+πm Clock-wise rotation for right-circularly polarised, anti-clock-wise rotation for left-circularly polarised (wave moves towards observer) Elliptically polarised light: the shape traced by the electric field vector is an ellipse (linear and circular light are specific cases of elliptical light) Retarders are made of crystalline materials birefringence i.e. two distinct indices of refraction for light with polarisation along (extraordinary, e-wave) and perpendicular (ordinary, o-wave) to the so-called "optical axis" of the crystal. The quarter-wave plate (QWP) introduces a phase shift of π/2 (optical path difference of λ/4) between the o-wave and e-wave The half-wave plate (HWP) introduces a phase shift of π (optical path difference of λ/2) See Hecht pp 325-329, 336-344, 352-357