Optics

Textbook: Max Born, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light

Optics(Max Born, CUP, 1999)

Ch1 Basic Properties of the Electromagnetic Field

1. The electromagnetic field.
2. The wave equation and the velocity of light:
   1. When $j=0,\rho =0$ and the medium is homogeneous, we have ${\nabla }^{2}E-\frac{\epsilon \mu }{c^2}\ddot{E}=0,{\nabla }^{2}H-\frac{\epsilon \mu }{c^2}\ddot{H}=0$. The “velocity” can be determined as $v=\frac{c}{\sqrt{\epsilon \mu }}$ by comparing ${\nabla }^{2}V-\frac{1}{v^2}\frac{{\partial }^{2}V}{\partial t^2}=0$.
      1. Note this holds only for plane waves. Thus $v=\frac{c}{\sqrt{\epsilon \mu }}$ should better be conceived as a “characteristic velocity of propagation”.
   2. The law of refraction: $\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{v_1}{v_2}=:n_{12}$(refractive index). An absolute refractive index $n:=\frac{c}{v}$ then satisfies $n_{12}=\frac{n_2}{n_1}$.
      1. Equivalent to that the wave-front is continuous, or the intersection of the incident and refracted wave moves at a same speed on both sides of the boundary, $v_1=v^{\prime }\sin {\theta }_1,v_2=v^{\prime }\sin {\theta }_2$.
      2. Maxwell’s formula: $n=\sqrt{\epsilon \mu }$. In fact $\mu$ is close to unity for most of nonmagnetic substances.
3. Scalar waves: Solving ${\nabla }^{2}V-\frac{1}{v^2}\frac{{\partial }^{2}V}{\partial t^2}=0$.
   1. Plane waves: $V=V(r\cdot s,t)$ where $r=r(x,y,z)$ is a position vector and $s=s(s_x,s_y,s_z)$ is a unit vector in a fixed direction. The term “plane wave” is due to that $V$ holds constant over planes $r\cdot s=const$ which are perpendicular to $s$. We may take $\zeta =r\cdot s$ and find ${\nabla }^{2}V=\frac{{\partial }^{2}V}{\partial {\zeta }^2}$, simplifying the equation to $\frac{{\partial }^{2}V}{\partial (\zeta -vt)\partial (\zeta +vt)}=0$. The general solution is $V=V_1(\zeta -vt)+V_2(\zeta +vt)$. $V_1$ holds constant for $(\zeta ,t)\to (\zeta +v\tau ,t+\tau )$. Therefore it represents a disturbance propagated with $v$ in the $+\zeta$ direction. Likewise $V_2$ -> $-\zeta$ direction.