A Simple Variational Monte Carlo Method

A ray of light follows the path between two points that requires the least amount of time.Pierre de Fermat

Our everyday experience with light naturally leads to the concept of light rays. The propagation of light rays can be formulated in terms of a variational principle known as Fermat's principle of least time, which is the basis of classical or geometrical optics. As shown in the Notes, this principle states that the path taken between two points A and B by a light ray is the path that can be traversed in the least amount of time. Put another way: because the speed of light $v$ is constant (expressed as the ratio of the speed of light in vacuum $c$ to the index of refraction $n$) along any path within a given homogeneous medium, the path of shortest time is a straight line. This program uses Fermat's principle and a simple Monte Carlo method to find the path of a light ray.

Interestingly, the principle of least time—or, more broadly, least action—has inspired mathematicians to formulate new methods in number theory related to the rational solutions of Diophantine equations: Secret Link Uncovered Between Pure Math and Physics.

The Law of Refraction

This program applies a simple variational Monte Carlo method to Fermat's principle of least time in geometrical optics.

Index of refraction $n_1$ in the left half of the system:

Index of refraction $n_2$ in the right half of the system:

Notes

gfortran -g -std=f2008 -Wall -Wextra -O2 -fall-intrinsics -o fermat fermat.f08

A transmitted ray is bent, or refracted, as it enters region 2 (right-hand side above). Snell's law—which this program has no knowledge of—helps us understand how Nature "chooses" a path that minimizes travel time. Conceptually, this tendency to minimize travel time can be understood is terms of the index of refraction and the angle as measured from the normal:

$n_1 \gt n_2$ requires $\theta_1 \lt \theta_2 \to$ refraction away from the normal in region 2;
$n_1 \lt n_2$ requires $\theta_1 \gt \theta_2 \to$ refraction toward the normal in region 2.

How does program fermat work? The strategy here is to begin with an arbitrary path and make changes to the path at random. These changes are accepted only if they reduce the travel time of the light ray. Because light propagates in a straight line in each medium, the path of the light ray is given by the position $y(k)$ at each boundary $k$. The interface between the medium on the left and the one on the right is located at the boundary $k$ = 5.

Snell's Law from Fermat's Principle of Least Time

The bending of a light ray near an interface can be calculated using the known refractive indices of two materials using Snell's law of refraction, named after Willebrord Snellius, who described it in 1621, but discovered first by the Arabian mathematician Ibn Sahl in 984. In 1662, Pierre de Fermat produced an ingenious method of deriving Snell's law on the basis of the principle of least time.

A light ray originating at point A is refracted upon crossing a vertical interface. Angles are measured relative to a normal line that is oriented horizontally at the contact point.

Let's start with $d = v t$. Rearranging, we get $t = d / v$ for the travel time. Using the diagram above, the travel times for regions 1 (left) and 2 (right) are:

$\displaystyle{ t_1 = \frac{ \sqrt{a^2 + x^2} }{v_1} \text{;} }$

(N.

$\displaystyle{ t_2 = \frac{ \sqrt{b^2 + \left( d - x \right)^2} }{v_2} \text{.} }$

(N.

This would make the total travel time for a beam of light $t = t_1 + t_2$. Fermat's principle of least time gives

$\displaystyle{ \begin{align*} \frac{\mathrm{d} t}{\mathrm{d} x} &= \frac{x}{v_1 \sqrt{a^2 + x^2}} \\ & - \frac{d - x}{v_2 \sqrt{b^2 + \left( d - x \right)^2}} = 0 \text{.} \end{align*} }$

(N.

Again referring to the above diagram, we see that:

$\displaystyle{ \sin \theta_1 = \frac{x}{\sqrt{a^2 + x^2}} \text{;} }$

(N.

$\displaystyle{ \sin \theta_2 = \frac{d - x}{\sqrt{b^2 + \left( d - x \right)^2}} \text{.} }$

(N.

Equation (N.3) now reads

$\displaystyle{ \frac{ \sin \theta_1 }{v_1} - \frac{ \sin \theta_2 }{v_2} = 0 \text{.} }$

(N.

Recall that the index of refraction is given as $n = c / v$, which implies $1 / v = n / c$. Now rewrite Equation (N.6) as

$\displaystyle{ \frac{ n_1 \sin \theta_1 }{c} - \frac{ n_2 \sin \theta_2 }{c} = 0 \text{.} }$

(N.

The factors of $c$ cancel to give Snell's law, or $n_1 \sin \theta_1 = n_2 \sin \theta_2$.