This article is part of a set of four; 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 three articles:

Foundation: Calculus of Variations
Statics: The Catenary
Dynamics: Energy Position Equation

Fermat's stationary time

Refraction

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.


Right-angled triangles

To set things up for discussing Fermat's stationary time I must first discuss a geometric property of right-angled 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

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 brings us to Fermat's stationary time. Here 'stationary' refers to the point in variation space where the derivative of (T1 + T2) is zero.

Discussion

At this point let's look back on what has been established.

Snell's law is formulated in terms of the sines of the angle of incidence and angle of refraction respectively. This is because for the phenomenon of refraction it is the angle that counts.

For emphasis let me state explicitly: for refraction the distance traveled is not relevant. Only the angle counts.

An angle is 2-dimensional in the following sense: in order to have an angle at all the space must have at least 2 spatial dimensions. To specify an angle we give a ratio of two numbers. We construct a right-angled triangle with the angle as one of its corners, and then the angle is expressed in terms of sine, cosine, tangent.

The geometric property expressed in (2) is what allows Snell's law to be restated in terms of differentials.

In (5) the sines are restated in terms of a derivative of spatial lengths: the derivative of the length of the hypotenuse with respect to the length of the opposing side.

With (5) established we arrive at the point where a physics hypothesis is introduced: the hypothesis that in different media the speed of light is different, and that the index of refraction of a particular medium is correlated with the speed of light in that medium.

From (5) to (7) the substitution of spatial length (C1,C2) with duration (T1,T2) is performed.

This brings us to (10): although visually (10) looks to be an equation in terms of derivatives, it is still about angles. (10) is equivalent to Snell's law, and Snell's law is about angles.


Multiple paths

Picture 5. Diagram
Light taking multiple paths from a single start point to a single end point

Diagram 5 shows a setup with a stack of three prisms. This represents the geometry of a Fresnel lens in a schematic way. To see how often Fresnel lenses are used my recommendation is to google with the search terms:
"magnifier" "fresnel lens"

The diagram represents that light emitted from a single start point is refracted onto a single end point. We have for the three paths that for each the total duration is different.

Switching to the wavefront hypothesis: we observe that all of the paths that the light is taking are accounted for in terms of the wavefront hypothesis.

In fact: our general experience is that all the properties of propagation of light can be accounted for in terms of the wavefront hypothesis.

In diagram 5 each of the three paths has the following property: compared to closely adjacent paths the duration of the path is stationary.

Refraction at a curved interface

Picture 5. Graphlet
Refraction of light, curved surface

In graphlet 5 The egg shaped region represents a chunk of glass. For simplicity the index of refraction is assigned a value of 1.5 That is: to traverse 1 unit of distance through the glass takes 1.5 units of time.
The top slider changes the radius of the upper circle. The radius of the lower circle is decreased such that the total transit time remains 2.5 units of time.

The point of intersection of the two circles traces out a shape that has the property that all the rays that travel from point A to point B have the same transit time.

The top slider sweeps out the variation space.

The second slider modifies the egg shape. The number displayed in the lower right corner represents the transit time of light when it interacts with the modified shape.

When the curvature of the air-glass interface is increased the light from point A that reaches point B has traveled the path of most time.

This graphlet is designed for the purpose of demonstrating that Fermat time is not about least time. Instead the criterion is: light travels along a path such that the derivative of the transit time is zero.

That derivative-of-the-transit-time is the derivative with respect to variation. The variation moves the point of refraction along the air-glass interface. In this graphlet the radius of the circle around point A is used to generate the motion of sweeping out the variation space. So in order to define the derivative of the variation sweep: express the variation as a function of the radius of the circle.

Reflection

Picture 6. Graphlet
Reflection of light in the case of a flat surface

In graphlet 6: the variation space is the length of the straight line where the light will reflect. This particulr variation space is a 1-dimensional space. As the slider is moved the exploratory point sweeps out variation. As we know: if we grant Huygens' wavefront hypothesis then it follows that the true reflection point is the point such that the angle of reflection is equal to the angle of incidence.

In the section right-angled triangles a geometric property of the hypotenuse of a right-angled triangle was demonstrated: the derivative of the length of the hypotenuse (with respect to the triangle width) is equal to the sine of the hypotenuse's angle.

(In the case of reflection the velocity of the light is the same along the entire path, therefore using length-of-the-path or duration-of-the-path is equivalent.)

At the true reflection point the angles are equal.
Therefore: the derivatives of the lengths will be equal in magnitude (while being opposite in sign).
Therefore: the derivative of the sum of the lengths will be zero.

In graphlet 5: the number underneath the exploratory reflection point gives the sum of the two lengths. When the exploratory reflection point is at the true reflection point the derivative of the sum of the lengths is zero.



 
Picture 7. Graphlet
Reflection of light in the case of a concave surface

In graphlet 7 the variation space is the length of the concave surface. The concave surface is in the shape of an ellipse. In the starting configuration the ratio of major axis and minor axis is such that the two focal points of the ellipse coincide with the point of emission and the point of reception. As a consequence, the derivative of the total pathlength (with respect to variation) is zero everywhere.

Moving the lower slider modifies the major axis of the ellipse. Decreasing the major axis makes the reflecting surface more concave. With a more concave surface the true reflection point is at a maximum in the variation space.


All of the above combined demonstrates that Fermat's stationary time is not about minimizing time. The criterion is: identify the point in variation space such that the derivative of the time (with respect to variation) is zero.

The total time is not a relevant factor here. The relevant factor is the derivative of the time; taking the derivative of the length of the path recovers the angle.



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