This article is part of a set of three; the common factor is Calculus of Variations. In classical physics Calculus of Variations is applied in three areas: Optics, Statics, and Dynamics. Each article in the set is written as a standalone article, resulting in some degree of overlap.
The other two articles:

Statics: The Catenary
Dynamics: Energy Position Equation


Fermat's stationary time

When light transitions from one medium to another there is refraction. This refraction is described by Snell's law. Snell's law is inferred from the experimental data.

Christiaan Huygens proposed a way of understanding Snell's law in terms of reconstitution of a wavefront. It is common to refer to this idea as ‘Huygens' Principle’. I actually prefer to use the designation ‘wavefront hypothesis’.

(In my opinion the qualification ‘Principle’ is used too often. If everything is a principle then the word ‘principle’ is rendered meaningless.)

It is assumed that the wavefront is always perpendicular to the direction of propagation.

Picture 1. Image
Refraction by wavefront reconstitution

In the diagram the length of the line segment 'd' is not important; the value of that length is necessary for calculation, but in the course of that calculation the value of 'd' drops out.

In the time interval 't' the light travels a shorter distance in the denser medium, in the proportion of v2/v1. So: the wavefront hypothesis gives the following expression that is equivalent to Snell's law:

\frac{\sin\alpha_2}{\sin\alpha_1} = \frac{v_2}{v_1}
(1)

Time and Space

As we know, light propagates so fast that for practical purposes we can think of the path of the light as something that has a static form. We can express the length of the path in terms of spatial length, or in terms of duration, the interconversion is straightforward.

Snell's law equates state of motion in n1 to state of motion in n2. That is: Snell's law can be interpreted as stating that in the process of refraction there is a quality that is conserved.

While it is the case that the light propagates slower in the denser medium the frequency remains the same; we can think of that as a conserved quantity. To express Snell's law in variational form we need to formulate a criterion that identifies the point in variation space such that in the process of refraction the frequency of the light is conserved.


Rectangular triangles

To set things up for discussing Fermat's stationary time I must first discuss a geometric property of rectangular triangles.

Picture 2. Graphlet

We will need an expression for the rate of change of the length of line segment C as the line segment A is shortened and lengthened. So we set up differentiation of C with respect to A:

\frac{dc}{da} = \frac{d(\sqrt{a^2 + b^2})}{da} = \frac{a}{\sqrt{a^2+b^2}} = \frac{a}{c}
(2)

With the intermediate steps removed:

\frac{dC}{dA} = \frac{A}{C}
(3)

(3) is a geometric property that is not specific to optics or even physics, it is a mathematical property.



Picture 3. Image
Fermat's stationary time

In Image 3 the letter 'S' stands for ‘Snell's point’. We will take as our starting point that there is a fixed point from where the light is transmitted, point 'T', and that there is a fixed point 'R' where the light is received. (T and R not shown in the image; T and R can be arbitrarily far away.)

If it is granted that the wavefront is perpendicular to the direction of propagation it follows that the angle β1 is equal to the angle α1, and that the angle β2 is equal to the angle α2.

The variation of the path of the light consists of moving point S along the refraction line. We want to find the criterion that identifies the location of point S in the variation space such that Snell's law is satisfied.

Repeating the statement that is equivalent to Snell's law:

\frac{\sin\alpha_2}{\sin\alpha_1} = \frac{v_2}{v_1}
(4)

Before including the division by velocity, as expressed in (4), we first set up the following two relations that follow from (3):

\frac{dC_1}{dA_1} = \sin \alpha_1  \qquad \frac{dC_2}{dA_2} = \sin \alpha_2
(5)

The next step is to accomodate the division by the velocity: let T1 be the time that it takes to traverse the length C1, T2 the time to traverse C2.

\frac{C_1}{v_1} = T_1 \qquad \frac{C_2}{v_2} = T_2
(6)

So the way to accommodate the division by velocity is to substitute the distance C with the time T:

\frac{dT_1}{dA_1} = \sin \alpha_1  \qquad \frac{dT_2}{dA_2} = \sin \alpha_2
(7)

Hence, in order to satisfy Snell's law:

\frac{dT_1}{dA_1} = \frac{dT_2}{dA_2}
(8)

A1 + A2 is constant, hence dA1 and -dA2 are equal. We can restate (8) as derivatives with respect to variation of Snell's point S. The variation space is a hypothetical space; I will refer to the position in this hypothetical space as Sh:

\frac{dT_1}{dS_h} = \frac{dT_2}{d(-S_h)}
(9)

Finally, we move the minus sign outside the differentiation:

\frac{dT_1}{dS_h} = -\frac{dT_2}{d(S_h)}
(10)



Picture 4. Graphlet
Snell's point is where the derivatives of T1 and T2 have the same magnitude.

Graphlet 4 displays exploration of the hypothetical variation space in accordance with (10). The value displayed in the slider knob is the position of the refraction point in the variation space.

In the righthand sub-panel: the curve labeled T1 represents the duration for the light to move from the point of emission to the refraction point, the curve labeled T2 represents the duration for the light to move from the refraction point to the point of reception.

Comparing slopes

The total time is not relevant here. The thing that is being evaluated is the derivative of the time. The variation arrives at Snell's point when the curves T1 and T2 have the same slope, with opposite sign.

Mathematics of comparing derivatives

Of course: when the hypothetical refraction point is at Snell's point the derivative with respect to Sh of (T1 + T2) is zero: the function (T1 + T2) is at an extremum. Arriving at this extremum follows from (10) automatically; it cannot not occur.

This is a structural pattern that is common in all forms of application of Calculus of Variations. In the case of refraction of light: you take the derivative with respect to the variation in order to identify the point in variation space where d(T1)/dS and d(T2)/dS match each other.

On the question of optimization

William Rowan Hamilton pointed out that a notion of economy and/or optimization is problematic:

In optics, for example, though the sum of the incident and reflected portions of the path of light, in a single ordinary reflexion at a plane, is always the shortest of any, yet in reflexion at a curved mirror this economy is often violated. If an eye be placed in the interior but not at the centre of a reflecting hollow sphere, it may see itself reflected in two opposite points, of which one indeed is the nearest to it, but the other on the contrary is the furthest; so that of the two different paths of light, corresponding to these two opposite points, the one indeed is the shortest, but the other is the longest of any.

Source:
On a general Method of expressing the Paths of Light, and of the Planets, by the Coefficients of a Characteristic Function
Transcription by Dr. David R. Wilkins

More generally, when a setup is created where there are multiple paths available, all with different duration, light will take all of those paths.

I think it is crucially important to be aware of the ramifications of that: when there are multiple paths available, all with different duration, light will take all of those paths.



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