Propagation of waves - UMD Physics

Propagation of waves - UMD Physics

Polarization Jones vector & matrices Phys 375 1 Matrix treatment of polarization Consider a light ray with an instantaneous Evector as shown E z, t iE x z , t jE y z , t y Ey Ex x E x Eox e i kz t x E y Eoy e i kz t y

2 Matrix treatment of polarization Combining the components i kz t y i kz t x E i Eox e jEoy e i y i x i kz t E i Eox e jEoy e e ~ 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 E o ~ 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 0 E

oy 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, ~ Eox 0 0 0 ~ E o ~ A i y 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 1 ~ Eox e x A Eo i A i y 2 Eoy e Ae 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 B cos 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 2 cos M sin cos sin cos

2 sin 14 Optical elements: Phase retarder Introduces a phase difference () 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: E e i x int o E e i x x ox

Eoy e ox i y int o Eoy e i y y It is easy to show by inspection that, Here x and y components e i x 0 M i y 0 e

represent the advance in phase of the 16 Jones matrix of a Quarter Wave Plate Consider a quarter wave plate for which || = / 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 || = 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 e 2 0 1 0 1 HWP, SA vertical HWP, SA horizontal 18

Optical elements: Quarter/Half wave plate When the net phase difference = /2 : Quarter-wave plate = : 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

Recently Viewed Presentations

  • Utility Axioms - UToledo Engineering

    Utility Axioms - UToledo Engineering

    Utility Axioms Axiom: something obvious, cannot be proven Utility axioms (rules for clear thinking) The seven axioms (rules of clear thinking) Ordering and transitivity Reduction of compound uncertain events Continuity Substitutability Monotonicity Invariance Finiteness 1.
  • Canada Business Products - ACN Compass Canada

    Canada Business Products - ACN Compass Canada

    Bundle High Speed Internet in Ontario and Quebec with ACN DigitalTalk®Express Customers get one bill, one service provider and one number to call. Competitively priced and can save business more
  • Education in Healthcare

    Education in Healthcare

    Social Learning Theory . Key Reference: Wenger, E. (2000). Communities of practice and social learning systems. Organization, 7(2), 225-246. Learning is a social process according to Wenger (1999). Wenger explains that part of being human is the willingness or desire...
  • Kathleen Quinlan Director of Communications Office of Tobacco

    Kathleen Quinlan Director of Communications Office of Tobacco

    Arial MS Pゴシック Times New Roman ApexSansBookT Wingdings Courier New Default Design OTC Microsoft PowerPoint Presentation Paint Shop Pro 7 Image Microsoft Excel Chart Kathleen Quinlan Going Smoke-free: The Irish Experience Background to Smoke Free Workplace Legislation Tobacco Free Policy...
  • Los pronombres - Woodland Hills School District

    Los pronombres - Woodland Hills School District

    te (to, for you) le, les (se) (to, for him, her, you, them) nos (to, for us) os (to, for you all; used in Spain) Los pronombresdirectos. Direct object pronouns are the person/object that receives the action of the verb....
  • Star Wars Franchise

    Star Wars Franchise

    Products. The success of the Star Wars films led the franchise to become one of the most merchandised franchises in the world.In 1977, while filming the original film, George Lucas decided to take a 500,000-dollar pay-cut to his own salary...
  • Titel der Präsentation -

    Titel der Präsentation -

    Don´t know answers that are optional for personal interviews are coded as missing values. Score is a mean score that is calculated if at least 5 of the 6 Items are completed.((SUM of Answers/Number of Items))
  • Romanticism


    Unit 6 CBHS Mr. Buttell