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
Pattern of electromagnetic radiation from a dipole
Linearly polarized light The figure shows three different cases of linearly polarized light
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) 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
E x E y E m E(t) -E m E m E changes with time, but always points along this line -E m
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
Alternative description of linearly polarized light using phasors First, let's recall that given a real field Ex = Em cos (ωt) it can be represented as the horizontal components of a phasor Im phasor for E x The rotating radius and its associated angle together constitute a phasor E x Real Mathematical complex plane E x will be drawn as a horizontal component in the physical plane.
A given a real field Ey = Em cos (ωt) can be represented as the horizontal components of a 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.
= = E x (t) = = E y (t) (1 ) E = (E x, E y ) The first step is to create two phasors. 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
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
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
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
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
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
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:
Right circularly polarized light Right circularly polarized light Right circularly polarized light Both phasors and the electric field rotate together in the clockwise direction
Summary Left circularly polarized light E x = E m cos (ωt ) E y = E m cos (ωt - π/2) Right circularly polarized light E x = E m cos (ωt ) E y = E m cos (ωt + π/2) E x = E m cos (- ωt ) E y = E m cos (- ωt - π/2) 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.