APPLIED OPTICS POLARIZATION
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1 A. La Rosa Lecture Notes APPLIED OPTICS POLARIZATION Linearly-polarized light Description of linearly polarized light (using Real variables) Alternative description of linearly polarized light using phasors Left circularly-polarized light Description of left-circularly polarized light (using Real variables) Alternative description of left -circularly polarized light using phases Right circularly-polarized light Description of left-circularly polarized light (using Real variables) Alternative description of left -circularly polarized light using phasors
2 Pattern of electromagnetic radiation from a dipole L AM radio tower AM radio wave f = 1 MHz = 300 m (Works better when the antenna's length L= (1/2)λ. Half of the antenna buried in the ground allows building the device to be (1/4)λ ~ 75 m tall.)
3 Linearly polarized light c Speed Oscillation of the electric field remains parallel to a fixed line in space (a) (b) (c) The figure shows three different cases of linearly polarized light (light advances in the z-direction.)
4 Description of linearly polarized light (using Real variables) Z E y E E x Given the following harmonic components (of frequency ω) of the electric field that exists at a given position z =z o, E x = E m cos (ωt) E = + Electric field E y = E m cos (ωt) how does the magnitude and orientation of the electric field changes as a function of time t? E m E The graph shows the electric field E at the time t=0 E m
5 Notice E x and E y are in phase E m E x -E m time E m E y time -E m E m E(t) -E m E m E changes with time, but always points along this line -E m
6 The first step is to create two phasors Linearly polarized light Notice, the phasors are referenced to two different axes, and respectively phasor E y phasor E x Real - space E x E y E(t) Linearly polarized light
7 Alternative description of linearly polarized light using phasors First, let's recall that given a real field E x = E m cos (ωt) it can be represented as the real component of a phasor Im E x phasor for E x The rotating radius and its associated angle together constitute a phasor Real Mathematical complex plane E x will be drawn as a horizontal component in the physical plane.
8 A given a real field Ey = Em cos (ωt) can be represented as the real component of the phasor Im phasor for Ey The rotating radius and its associated angle together constitute a phasor Ey Real Mathematical complex plane Ey will be drawn as a vertical component in the physical plane.
9 E x = E m = cos (ωt) = E x (t) E y = E= m cos (ωt ) = E y (t) (1 ) E = (E x, E y ) The first step is to create two phasors, and. Notice, the angle of the phasors and are referenced to the axes and respectively. E phasor E y E x phasor Phasor Phasor E(t) Linearly polarized light (in complex variables) Linearly polarized light
10 Circularly polarized light Real - space E y E Z E x Left-circularly polarized light Task: Given E x = E m cos (ωt ) E y = E m cos (ωt - π/2) Electric field vector E at a fixed position z=z o indicate how does the orientation and magnitude of the electric of the field vector change as a function of time Real - space At t = 0
11 time time Real - space Left circularly polarized light ω Electric field vector rotates counter-clockwise with angular velocity ω (when looking from the side in which the wave propagates and into the source). Left circularly polarized light
12 Alternative description of left-circularly polarized light using phasors Im Phasor of magnitude A E x = A cos (θ) A θ E x Real E y = B cos (γ) Im Real B γ E x = E m cos (ωt) E y = E m cos (ωt - π/2) E y Left circularly polarized light E x
13 Right-circularly polarized light Task: Given E x = E m cos (ωt ) E y = E m cos (ωt + π/2) Electric field vector E at a fixed position z=z o indicate how does the orientation and magnitude of the electric indicate field how vector the orientation change as and a function of time At t = 0
14 Cos (ωt ) time Cos (ωt + π/2) time Electric field ω Electric field vector rotates clockwise with angular velocity ω when looking along the direction in which the wave propagates and into the source Right circularly polarized light
15 Alternative description of right-circularly polarized light using phasors We create two phasors ω E x ω E y ω Electric field Right circularly polarized light An extra simplification step is possible. Indeed, since COS is a symmetric function, the mathematical expressions for E x and E y given above can be re-written as:
16 Right circularly polarized light Right circularly polarized light Right circularly polarized light Both phasors and the electric field rotate together in the clockwise direction
17 Summary Linearly polarized light E = E x + E y E x = E 1 cos (ωt ) E y = E 2 cos (ωt ) E
18 Summary Left circularly polarized light E = E x + E y E x = E m cos (ωt ) E y = E m cos (ωt - π/2) E Right circularly polarized light E x = E m cos (ωt ) E y = E m cos (ωt + π/2) E = E x + E y E x = E m cos (- ωt ) E y = E m cos (- ωt - π/2) E Rule of thumb for the complex variable description: Notice in the expressions above that The y-component phasor is written as lagging π/2 with respect to the x-component phasor ; the right or left circularly polarization character is given by the sign in front of of ω.
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APPLIED OPTICS POLARIZATION
A. La Rosa Lecture Notes APPLIED OPTICS POLARIZATION Linearly-polarized light Description of linearly polarized light (using Real variables) Alternative description of linearly polarized light using phasors
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