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Polarization Superposition of plane waves
Optics 430/530, week VIII Polarization Superposition of plane waves This class notes freely use material from P. Piot, PHYS , NIU FA2018
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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, PHYS , NIU FA2018
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Polarization: examples
Linearly-polarized waves Elliptically-polarized waves with the special case of circularly polarized πΈ π₯ =Β±π πΈ π¦ πΈ π₯ =π π ππΏ πΈ π¦ πΈ π₯ =π πΈ π¦ P. Piot, PHYS , NIU FA2018
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Jonesβ formalism (I) Consider Then P. Piot, PHYS , NIU FA2018
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Jonesβ formalism (II) The strength is unimportant for polarization considerations it only enters in the intensity as In Jonesβ formalism the polarization is represented by the vector P. Piot, PHYS , NIU FA2018
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Example of special cases
P. Piot, PHYS , NIU FA2018
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Linear polarizers and Jones matrices
In Jones formalism the evolution of the polarization can be described by a 2x2 matrix (referred to as Jonesβ 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, PHYS , NIU FA2018
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Jones matrix Generally
Note that the intensity does not remain the same as So one always renormalized the final Jones vector as P. Piot, PHYS , NIU FA2018
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Jones matrix of an arbitrary-direction polarizer (I)
Consider an incoming wave Decompose in the basis as So we have where P. Piot, PHYS , NIU FA2018
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Jones matrix of an arbitrary-direction polarizer (II)
The effect of the polarizer is (a perfect polarizer would have π=0) Expliciting the basis vectors Gives P. Piot, PHYS , NIU FA2018
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Perfect polarizer with transmission angle π
Considering π=0 the the Jones matrix for a polarizer with transmission at angle π is Note that if we consider the case of a polarized wave along x we obtained Malusβ law: π¬ πππ‘ππ = E x (cos 2 π π +π πππcosπ π ) So that π° πππ‘ππ = I before cos 2 π. (Malusβs law LAB#2) P. Piot, PHYS , NIU FA2018
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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, PHYS , NIU FA2018
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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, PHYS , NIU FA2018
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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 π π and π π ). What is the total intensity of such a superimposed wave? Start with and The we can compute the Poynting vector πΊ P. Piot, PHYS , NIU FA2018
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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, PHYS , NIU FA2018
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Intensity of superimposed plane waves (II)
Gathering some term we finally have So the optical intensity is P. Piot, PHYS , NIU FA2018
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Sum of two waves We now specialize to the case of two wave with equal amplitudes: The phase velocities is given by π£ ππ = π π π π The superimposed field is So that the optical intensity is P. Piot, PHYS , NIU FA2018
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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, PHYS , NIU FA2018
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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 π¬ π,π is referred to as the βFourier transformβ of π¬ π,π‘ . The previous operation is actually called inverse Fourier transform. The Fourier transform is defined as P. Piot, PHYS , NIU FA2018
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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 πΌ(π,π) is not the Fourier transform of πΌ π,π‘ . P. Piot, PHYS , NIU FA2018
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Fourier transforms P. Piot, PHYS , NIU FA2018
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Parsevalβs theorem The Parseval theorem is a general theorem that states Consider the example of a modulated Gaussian pulseβ We have for the Fourier transform So that both the time integral and frequency integral give P. Piot, PHYS , NIU FA2018
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