Optics 430/530, week VIII Polarization Superposition of plane waves This class notes freely use material from http://optics.byu.edu/BYUOpticsBook_2015.pdf P. Piot, PHYS430-530, NIU FA2018 1 Polarization: definition Polarization refer to the direction of the E field (this is a convention).

If the direction is unpredictable the wave is said to be unpolarized If the E-field direction is well define the wave is said to be polarized Starting with axis we can decompose E as and taking the z axis as propagation The relationship between the two transverse component describes the polarization P. Piot, PHYS430-530, NIU FA2018 2

Polarization: examples Linearly-polarized waves = Elliptically-polarized waves = with the special case of circularly polarized = P. Piot, PHYS430-530, NIU FA2018

3 Jones formalism (I) Consider Then P. Piot, PHYS430-530, NIU FA2018 4 Jones formalism (II) The strength is unimportant for polarization considerations it only

enters in the intensity as In Jones formalism the polarization is represented by the vector P. Piot, PHYS430-530, NIU FA2018 5 Example of special cases P. Piot, PHYS430-530, NIU FA2018

6 Linear polarizers and Jones matrices In Jones formalism the evolution of the polarization can be described by a 2x2 matrix (referred to as Jones matrix) A simple example regards the representation of a polarizer: an optical element which only let one polarization component to pass. In such a case we have P. Piot, PHYS430-530, NIU FA2018

7 Jones matrix Generally Note that the intensity does not remain the same as So one always renormalized the final Jones vector as P. Piot, PHYS430-530, NIU FA2018 8

Jones matrix of an arbitrarydirection polarizer (I) Consider an incoming wave Decompose in the. basis as So we have where P. Piot, PHYS430-530, NIU FA2018 9

Jones matrix of an arbitrarydirection polarizer (II) The effect of the polarizer is (a perfect polarizer would have ) Expliciting the basis vectors Gives P. Piot, PHYS430-530, NIU FA2018 10 Perfect polarizer with transmission angle Considering the the Jones matrix for a polarizer with transmission at

angle is Note that if we consider the case of a polarized wave along x we obtained Malus law: ) So that . (Maluss law LAB#2) P. Piot, PHYS430-530, NIU FA2018 11 Waveplates

We now consider a birefringent material with its index of refraction dependent on the direction of the polarization A waveplate is cut so that the slow and fast axis are 90 deg apart The phase difference between the two axis is P. Piot, PHYS430-530, NIU FA2018 12 Waveplates

Quarter waveplate Ca co n be n to pol vert use cir ari lin d t o cu ze lar d w earl y ly

po ave lar ize d Half waveplate P. Piot, PHYS430-530, NIU FA2018 13 Superposition of plane waves

(chapt. 7) To date we focus on a single plane wave Any type of wave can in principle be written as a sum of plane waves (with different and ). What is the total intensity of such a superimposed wave? Start with and The we can compute the Poynting vector P. Piot, PHYS430-530, NIU FA2018 14

Intensity of superimposed plane waves The Poynting vector is So we finally get =0 is the plane waves are moving along the same direction P. Piot, PHYS430-530, NIU FA2018 15

Intensity of superimposed plane waves (II) Gathering some term we finally have So the optical intensity is P. Piot, PHYS430-530, NIU FA2018 16 Sum of two waves

We now specialize to the case of two wave with equal amplitudes: The phase velocities is given by The superimposed field is So that the optical intensity is P. Piot, PHYS430-530, NIU FA2018 17 Group velocity Consider the previous equation From the argument of the cosine we can

define a velocity as this is the group velocity which describes the velocity of the wave envelope Note that the phase velocity of the superimposed wave is P. Piot, PHYS430-530, NIU FA2018 18 Frequency spectrum of light In Physics it is common to decompose a temporal signal over the Frequency domain.

Such a decomposition of the E field writes The function is referred to as the Fourier transform of . The previous operation is actually called inverse Fourier transform. The Fourier transform is defined as P. Piot, PHYS430-530, NIU FA2018 19 Power spectrum We saw that the optical intensity

We can also write this intensity in term of Fourier transform is such a case it is called power spectrum Note that is not the Fourier transform of . P. Piot, PHYS430-530, NIU FA2018 20 Fourier transforms P. Piot, PHYS430-530, NIU FA2018

21 Parsevals theorem The Parseval theorem is a general theorem that states Consider the example of a modulated Gaussian pulse We have for the Fourier transform So that both the time integral and frequency integral give P. Piot, PHYS430-530, NIU FA2018

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