Polarization Jones vector & matrices Phys 375 1 Matrix treatment of polarization Consider a light ray with an instantaneous Evector as shown E z, t iˆEx z, t ˆjE y z, t y Ey x E x Eoxe i kz t x Ex E y Eoy e i kz t y 2 Matrix treatment of polarization Combining the components i kz t y i kz t x ˆ ˆ E i Eox e jEoy e i y i x ˆ ˆ E i Eox e jEoy e e i kz t ~ i kz t E Eo e The terms in brackets represents the complex amplitude of the plane wave 3 Jones Vectors The state of polarization of light is determined by the relative amplitudes (Eox, Eoy) and, the relative phases ( = y - x ) of these components The complex amplitude is written as a twoelement matrix, the Jones vector ~ i x Eox Eox e Eox ~ i x Eo ~ e i i y Eoy Eoy e Eoy e 4 Jones vector: Horizontally polarized light The electric field oscillations are only along the x-axis The Jones vector is then written, ~ Eox Eox e i x A 1 ~ Eo ~ A Eoy 0 0 0 where we have set the phase x = 0, for convenience The arrows indicate the sense of movement as the beam approaches you y E x The normalized form is 1 0 5 Jones vector: Vertically polarized light The electric field oscillations are only along the y-axis The Jones vector is then written, ~ 0 0 E 0 ~ ox Eo ~ i y A Eoy Eoy e A 1 Where we have set the phase y = 0, for convenience y E x The normalized form is 0 1 6 Jones vector: Linearly polarized light at an arbitrary angle If the phases are such that = m for m = 0, 1, 2, 3, … Then we must have, Ex m Eox 1 Ey Eoy y x and the Jones vector is simply a line inclined at an angle = tan-1(Eoy/Eox) since we can write ~ E ~ m cos ox Eo ~ A 1 sin E oy 7 Circular polarization y Suppose Eox = Eoy = A and Ex leads Ey by 90o=/2 At the instant Ex reaches its maximum displacement (+A), Ey is zero A fourth of a period later, Ex is zero and Ey=+A x t=0, Ey = 0, Ex = +A t=T/8, Ey = +Asin 45o, Ex = Acos45o t=T/4, Ey = +A, Ex = 0 8 Circular polarization The Jones vector for this case – where Ex leads Ey is i ~ Eoxe x A Eo i i y 2 Eoy e Ae 1 A i The normalized form is, This vector represents circularly polarized light, where E rotates counterclockwise, viewed head-on This mode is called left-circularly polarized light What is the corresponding vector for right-circularly polarized light? 1 2 1 i Replace /2 with -/2 to get 1 2 1 i 9 Elliptically polarized light If Eox Eoy , e.g. if Eox=A and Eoy = B The Jones vector can be written A iB counterclockwise A iB clockwise Here A>B 10 Jones vector and polarization In general, the Jones vector for the arbitrary case is an ellipse ( m; (m+1/2)) y Eoy Eox A ~ Eo i Bcos i sin E e oy b a tan 2 2 Eox Eoy cos E ox2 E oy2 x Eox 11 Optical elements: Linear polarizer Selectively removes all or most of the Evibrations except in a given direction TA y x Linear polarizer 12 Jones matrix for a linear polarizer Consider a linear polarizer with transmission axis along the vertical (y). Let a 2X2 matrix represent the polarizer operating on vertically polarized light. The transmitted light must also be vertically polarized. Thus, a b 0 0 c d 1 1 Operating on horizontally polarized light, a b 1 0 c d 0 0 Thus, 0 0 M 0 1 Linear polarizer with TA vertical. 13 Jones matrix for a linear polarizer For a linear polarizer with a transmission axis at cos M sin cos 2 sin cos 2 sin 14 Optical elements: Phase retarder Introduces a phase difference (Δ) between orthogonal components The fast axis(FA) and slow axis (SA) are shown FA y x SA Retardation plate 15 Jones matrix of a phase retarder We wish to find a matrix which will transform the elements as follows: Eoxe i x int o Eoxei x x i y i y y Eoy e int o Eoy e It is easy to show by inspection that, ei x M 0 0 i y e Here x and y represent the advance in phase of the components 16 Jones matrix of a Quarter Wave Plate Consider a quarter wave plate for which |Δ| = /2 For y - x = /2 (Slow axis vertical) Let x = -/4 and y = /4 The matrix representing a Quarter wave plate, with its slow axis vertical is, e i 4 M 0 i 1 0 4 e 0 i 4 e 0 i 17 Jones matrices: Half-wave Plate For |Δ| = e i 2 M 0 e i 2 M 0 i 1 0 2 e 0 i e 2 i 1 0 2 e 0 i 2 e 0 1 0 1 HWP, SA vertical HWP, SA horizontal 18 Optical elements: Quarter/Half wave plate When the net phase difference Δ = /2 : Quarter-wave plate Δ = : Half-wave plate /2 19 Optical elements: Rotator Rotates the direction of linearly polarized light by a particular angle y x SA Rotator 20 Jones matrix for a rotator An E-vector oscillating linearly at is rotated by an angle Thus, the light must be converted to one that oscillates linearly at ( + ) a c b cos cos d sin sin cos One then finds M sin sin cos 21