I am currently studying the textbook Introduction to Optical Metrology by R. S. Sirohi. Section 1.3 Law of Refraction says the following:
Refraction takes place when the light travels from one medium to the other. An isotropic and homogeneous medium is characterized by an index of refraction or refractive index $\mu$, which is the ratio of the velocity of light in vacuum to that in the medium. The law of refraction can also be stated into two parts as.
- $\mu_1 \sin(\theta_i) = \mu_2 \sin(\theta_r)$
- The incident ray, the normal to the surface separating the media at the point of incidence, and the refracted ray lie in one plane.
Both the parts of the law of refraction can also be cast in a single vectorial equation as $$\vec{n}_2 = \vec{n}_1 - (\vec{n}_1 \cdot \vec{s}) \vec{s} + \left( \sqrt{(\mu_2)^2 - (\mu_1)^2 + (\vec{n}_1 \cdot \vec{s})^2} \right) \vec{s}, \tag{1.2}$$ where $\theta_i$ and $\theta_r$ are the angles of incidence and refraction, respectively. The refractive indices of the two media are $\mu_1$ and $\mu_2$, and the incident ray is in the medium of refractive index $\mu_1$.
Under paraxial approximation, the angle $\theta_i$ is small and hence $\vec{n}_1 \cdot \vec{s} \approx \mu_1$. Therefore, the laws of refraction can be expressed as $$\vec{n}_2 = \vec{n}_1 + (\mu_2 - \mu_1)\vec{s} \tag{1.3}$$
I recently asked how this was derived. The user 'JEB' answered with a correct step-by-step derivation, resulting in
$$\hat n_r = \frac{\mu_1}{\mu_2}[\hat n_i-(\hat n_i\cdot \hat s)\hat s] + \sqrt{1- \left\Vert\frac{\mu_1}{\mu_2}[\hat n_i-(\hat n_i\cdot \hat s)\hat s] \right\Vert^2}\hat s$$
But, continuing with the textbook explanation, what is the justification for having $\vec{n}_1 \cdot \vec{s} \approx \mu_1$ under the paraxial approximation, and how does it tie in with JEB's result? The textbook author says that this paraxial approximation allows the law of refraction to be expressed as $\vec{n}_2 = \vec{n}_1 + (\mu_2 - \mu_1)\vec{s}$, so I'm not how we get here.
