# Rays and optical pathlength minimization

My textbook, Fundamentals of Photonics, Third Edition, by Saleh and Teich, says the following in a section on Planar Boundaries:

The relation between the angles of refraction and incidence, $$\theta_2$$ and $$\theta_1$$, at a planar boundary between two media of refractive indices $$n_1$$ and $$n_2$$ is governed by Snell's law (1.1-3). This relation is plotted in Fig. 1.2-8 for two cases:

$$\blacksquare$$ External Refraction ($$n_1 < n_2$$). When the ray is incident from the medium of smaller refractive index, $$\theta_2 < \theta_1$$ and the refracted ray bends away from the boundary.

$$\blacksquare$$ Internal Refraction ($$n_1 > n_2$$). If the incident ray is in a medium of higher refractive index, $$\theta_2 > \theta_1$$ and the refracted ray bends toward the boundary.

The refracted rays bend in such a way as to minimize the optical pathlength, i.e., to increase the pathlength in the lower-index medium at the expense of pathlength in the higher-index medium. In both cases, when the angles are small (i.e., the rays are paraxial), the relation between $$\theta_2$$ and $$\theta_1$$ is approximately linear, $$n_1 \theta_1 \approx n_2 \theta_2$$, or $$\theta_2 \approx (n_1/n_2)\theta_1$$.

Reading what the authors have said here, and comparing it to figure 1.2-8, when they say that the optical pathlength is minimized, they're referring to the pathlength from the origin of the ray to some imaginary vertical line on the $$n_2$$ medium side, right? is this a correct way to conceptualize the minimization of optical pathlength?

You know that ray propagation in a homogeneous medium is straight line. Now assume that you are given both the source point in medium 1, say P, and a point Q in medium 2 that lies on the true refracted ray and let $$X_0$$ be the point on the interface of the two media where the ray refracts. For any point $$X$$ on the interface form the sum $$L[X]=n_1 PX + n_2 XQ$$ where $$PX$$ and $$XQ$$ are the lengths of the straight line segments in hte respective medium. Fermat' principle states that as $$X$$ is infinitesimally varied around $$X_0$$ the optical path length $$L[X] \ge L[X_0]$$
Note too that one can replace the fixed point $$Q$$ in medium 2 with a given fixed direction and minimize a different optical path length. Fix an arbitrary reference $$Z$$ and for every straight line in medium 2 project $$Z$$ unto the line, say $$T$$ and now measure the optical path length (distance time refractive index) from the fixed $$P$$ to $$T$$: $$L_1[X] = n_1 PX + n_2 XT$$ where now the segment $$XT$$ is now the fixed direction. Then again the minimum of $$L_1[X]$$ as $$X$$ is varied within an infinitesimal neighborhood of the actual $$X_0$$ at the interface takes its minimum value with Snell's law! This modification of Fermat's principle is due to Hamilton.
• Thanks for the answer. So the ray through the point $X$ on the interface does not refract, and the ray through the point $X_0$ on the interface does refract? – The Pointer Jan 25 '20 at 16:29
• It is not quite that a ray through $X$ does no refract, instead that the ray connecting $P$ and $Q$ refracts through $X_0$ and there is another ray that may refract through $X$ but then it does not go through $Q$. You can have any point on the interface refract some ray but the one going through from $P$ to $Q$ is when $X=X_0$ and then $L[X]=L[X_0+\delta X]$ is minimum when $\delta X=0$ with $X=X_0 + \delta X$ and $\delta X$ is an arbitrary infinitesimal. – hyportnex Jan 25 '20 at 18:22
• By the "true" refracted ray I mean the two straight line segments $PX$ and $XQ$ (or $XT$) along which the ray from $P$ to $Q$ actually propagates, that is when $X=X_0$; for any other $X=X_0+\delta X$ the segments are not true rays, for there is only one ray between two fixed points $P, Q$. Your question was about two semi-infinite homogeneous media joined at a plane and am ignoring now the more general case that you can also have an infinity of rays between a pair of points when properly focused. – hyportnex Jan 25 '20 at 21:21