Polarization Superposition of plane waves 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

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 and taking the z axis as propagation axis we can decompose E as The relationship between the two transverse component describes the polarization P. Piot, PHYS430-530, NIU FA2018

Polarization: examples Linearly-polarized waves Elliptically-polarized waves with the special case of circularly polarized 𝐸 𝑥 =±𝑎 𝐸 𝑦 𝐸 𝑥 =𝑎 𝑒 𝑖𝛿 𝐸 𝑦 𝐸 𝑥 =𝑖 𝐸 𝑦 P. Piot, PHYS430-530, NIU FA2018

Jones’ formalism (I) Consider Then P. Piot, PHYS430-530, NIU FA2018

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

Example of special cases P. Piot, PHYS430-530, NIU FA2018

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

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

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

Jones matrix of an arbitrary-direction polarizer (II) The effect of the polarizer is (a perfect polarizer would have 𝜉=0) Expliciting the basis vectors Gives P. Piot, PHYS430-530, NIU FA2018

Perfect polarizer with transmission angle 𝜃 Considering 𝜉=0 the the Jones matrix for a polarizer with transmission at angle 𝜃 is Note that if we consider the case of a polarized wave along x we obtained Malus’ law: 𝑬 𝑎𝑓𝑡𝑒𝑟 = E x (cos 2 𝜃 𝒙 +𝑠𝑖𝑛𝜃cos𝜃 𝒚 ) So that 𝑰 𝑎𝑓𝑡𝑒𝑟 = I before cos 2 𝜃. (Malus’s law LAB#2) P. Piot, PHYS430-530, NIU FA2018

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

Can be used to convert linearly polarized wave to circularly polarized Waveplates Quarter waveplate Half waveplate Can be used to convert linearly polarized wave to circularly polarized P. Piot, PHYS430-530, NIU FA2018

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 𝑘 𝑖 and 𝜔 𝑖 ). What is the total intensity of such a superimposed wave? Start with and The we can compute the Poynting vector 𝑺 P. Piot, PHYS430-530, NIU FA2018

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

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

Sum of two waves We now specialize to the case of two wave with equal amplitudes: The phase velocities is given by 𝑣 𝑝𝑖 = 𝜔 𝑖 𝑘 𝑖 The superimposed field is So that the optical intensity is P. Piot, PHYS430-530, NIU FA2018

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

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 𝑬 𝒓,𝜔 is referred to as the “Fourier transform” of 𝑬 𝒓,𝑡 . The previous operation is actually called inverse Fourier transform. The Fourier transform is defined as P. Piot, PHYS430-530, NIU FA2018

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 𝐼(𝒓,𝜔) is not the Fourier transform of 𝐼 𝒓,𝑡 . P. Piot, PHYS430-530, NIU FA2018

Fourier transforms P. Piot, PHYS430-530, NIU FA2018

Parseval’s 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