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I am reading the Sakurai's book "Modern Quantum Mechanics". It starts from analogy between electron spin and classical light polarization. So far so good, but I have no idea how and why classical circular light polarization is represented by complex numbers.

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Circular polarization is the sum of two equal amplitude linear polarizations that are orthogonal to each other and whose relative phases is exactly $90^{o} $ apart. Analytically this means that, say, the E-field is propagating in the $\hat z $ direction and is of the form $\mathbf E(x,y,z,t) = A \cos(\omega t - kz+ \phi)\hat x + A \cos(\omega t - kz+ \phi+\pi/2)\hat y$. We can rewrite this in complex form as: $$\mathbf E = A \cos(\omega t - kz+ \phi)\hat x + A \cos(\omega t - kz+ \phi+\pi/2)\hat y \\ =A \cos(\omega t - kz+ \phi)\hat x - A \sin(\omega t - kz+ \phi)\hat y \\ =A\Re[ e^{\mathfrak j (\omega t - kz+ \phi)})(\hat x + \mathfrak j \hat y)]$$ so $\mathbf E = \Re [\mathbf {\tilde E}e^{\mathfrak j (\omega t - kz}]$ where the complex vector amplitude that propagates as $e^{\mathfrak j (\omega t - kz)}$ is defined by $$\mathbf {\tilde E} = Ae^{\mathfrak j \phi}(\hat x + \mathfrak j \hat y).$$ Geometrically we can view this as the constant complex phasor $Ae^{\mathfrak j \phi}$ is rotating in the plane $\hat x + \mathfrak j \hat y$ whose rate of rotation is $\omega $, a circle.


At least in antenna practice, the wave represented by $$\mathbf E_1(x,y,z,t) = A \cos(\omega t - kz+ \phi)\hat x - A \sin(\omega t - kz+ \phi)\hat y$$ or by its equivalent complex amplitude $\mathbf {\tilde E_1} = Ae^{\mathfrak j \phi}(\hat x + \mathfrak j \hat y)$ is called left-handed circular polarization. On the contrary, if the wave is $$\mathbf E_2(x,y,z,t) = A \cos(\omega t - kz+ \phi)\hat x + A \sin(\omega t - kz+ \phi)\hat y,$$ or $\mathbf {\tilde E_2} = Ae^{\mathfrak j \phi}(\hat x - \mathfrak j \hat y)$ is called right-handed circular polarization.

An antenna that is designed to receive, say right-handed polarization will receive almost none of the left-handed polarized wave, and vice versa. To avoid confusing one's left hand with his own right hand, a quite common occurrence among system engineers, the prudent thing to do is always to check with the antenna designer what he meant by handedness...

(Thank you @naturallyInconsistent.)

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  • $\begingroup$ You should make it clear which handedness this is, in which convention. $\endgroup$ Jul 6, 2023 at 8:46

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