# Meaning of complex-number representation of circular polarization

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.

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.