Huygens on Descartes Awareness of Snell's Law

  In his Dioptrica, 1703, Huygens indicates that Descartes had come into contact with Snell's manuscript containing the ratio of sines for the angles involved in diffraction. Snell is most likely to have used the particle impulse model to arrive at his results. Huygens states the following,

"Hæc autem omnia, quæ de refractionis inquisitione volumine integro Snellius exposuerat‚ inedita mansere; quae & nos vidimus aliquando, & Cartesium quoque vidisse accepimus, ut hinc fortasse mensuram illam, quae in sinibús consistit, elicuerit; quá in explicanda iride & vitrorum figuris investigandis felicissimè est usus.

Cujusmodi vero sit illa Refractionis in sinubus proportio, cum radius ex aere in aquam, vitrumve, aut alia corpora diaphana desertur, id vel prismate, ut Cartesius praecipit, inquiri potest, vel aliis modis; quos, qui praecedentia intellexerit, non difficulter inveniet."


"All these things, about the refraction question in the work Snell completely explained, but it remained unpublished; what both we've seen for some time and Descartes had seen and heard so here possibly is the measure for it; which he concluded depends on the sines; which in explaining the rainbow of glass figures investigated is most successfully employed.

However with the preceding having been understood, it will not be difficult to discover that the Refraction is in the proportion of sines when a ray has been conveyed from air into water; glass; or any other transparent body; thus or for a prism or in other ways as Descartes has informed us, it is possible to inquire into."

The Law of Refraction From Wave Vectors

  The impulse acting on a particle analogy gives an incorrect value for the index of refraction so this theory has to be rejected. The direction unit vectors were not the source of the error. If we assume that ê₂ can be decompose in terms of ê₁ and n̂ it is not too difficult to solve for their coefficients.

We should have used the photon's momentum which is proportional to the wave vector, the gradient of the phase. ħ=h/(2π) is the reduced Plank's constant.

Thus we deduce the speed of light will be lower in denser mediums. The ratio of the angles of refraction and reflection equal a constant since the speed of light in the various media is due to the homogeneous nature of the media and so the ratio of the speeds is therefore independent of the angle of incidence. This turned out to be true for the particle impulse theory but we get the inverse of the index of refraction.

Regarding Descartes' Description of the Law of Refraction

  One finds in Descartes' La Dioptrique (1637) a description of the law of refraction with the proportion between the sines of the angles of incidence and those of refraction being the same for all angles of incidence. His discussion is very general and focuses on a ball struck by a tennis racket and bouncing off of the ground for reflection and penetrating it for refraction. For refraction he speculates (see figure on p. 20 in the link above),

 "...In the end, as long as the action of light follows the same laws as the movement of this ball, it must be said that when its rays pass obliquely from one transparent body into another, as the receiption is more or less easier than the first, they turn away in such a manner, that they are all less inclined to the surface of these bodies, from the side where the reception is easier, than that where it's the contrary: and this adjustment is in proportion to that which receives them more easily than the other. But we should take care that this inclination is measured by the quantity of straight lines, such as CB or AH, EB or IG, and the like, compared to each other; not by that of the angles, such as are ABH, or GBI, much less by those similar to DBI, which are called the angles of Refraction. For the ratio or proportion between these angles varies by all the various inclinations of the rays, whereas that which is between the lines AH & IG or the like, remains the same in all the refractions which are caused by the same bodies..."

One can clean this argument up some by assuming a particle of light receives an impulse as it enters a transparent body in the direction of the surface normal altering its trajectory. Let's assume that it is drawn into a denser body increasing its speed in the direction of the normal and so turning it in that direction as is observed.

Before and after the interaction with the surface we find uniform rectilinear motion. The interaction with the surface can be treated as an impulse similar to that of a collision by adding the velocity change to the initial motion.

One finds the proportion between the ratio of the sines is in inverse proportion to the speed as Huygens pointed out. If Δv is positive, the impulse being attractive, the speed increases and the angle of refraction is reduced. If it is negative, as with repulsion, the speed after entering the body will decrease and the angle of refraction increases. One gets a better match with observations if one assumes higher speeds in denser bodies. A complication is that the speed within the body isn't always the same but depends on the angle incidence. Later observations showed that the speed of light decreases in denser bodies.

Maupertuis' Three Laws of Motion for Light and Action as a Universal Principle

  In his Accord de différentes lois de la nature of 1744 Maupertuis gives Three Laws for the motion of light.

"Here are the laws that light follows.

The first is that, in a uniform medium, it moves in a straight line.

The second, that when light meets a body which it cannot penetrate, it is reflected; and the angle of its reflection is equal to the angle of its incidence; that is to say, after its reflection, it makes an angle with the surface of the body equal to that under which it had met it.

The third is, that when the light passes from one diaphanous medium to another, its path after the meeting of the new medium, makes an angle with that which it held in the first; & the sine of the angle of refraction is always in the same ratio to the sine of the angle of incidence. If, for example, a ray of light passing from the air into the water is broken so that the sine of the angle of its refraction is three quarters of the sine of its angle of incidence; under some other obliquity that it meets the surface of the water, the sine of its refraction will always be three quarters of the sine of its new incidence."

He also contrasts the term, quantity of action, with Descartes' quantity of motion in Letter X found in his Works and describes action as a universal principle governing all motion.

"There is a truly universal principle, from which these Laws follow, in regards to the movement of hard bodies, elastic bodies, light, and all bodily substances: That is, in all the changes which occur in the Universe, the sum of the products of each body multiplied by the space which it traverses, and by the speed with which it traverses it, what is called the quantity of action, is always the smallest that is possible."

Maupertuis On The Quantity Of Action (1744)

  Here's a rough translation of a 1744 passage by Maupertuis containing his statement of the quantity of action.

 "It was by this principle that Fermat solved the problem, by this principle so plausible, that light which, in its propagation & in its reflection always goes in the shortest possible time, still follows that same law in its refraction; & he did not hesitate to consider that light does not move with greater ease and more swiftly in rarer mediums than in those where, for the same space, there is found a greater quantity of matter: in fact, couldn't one assume at the onset that light traverses with greater ease & more swiftly in crystal and water than in air and the void?
  Also many of the most famous mathematicians are known to embrace the feelings of Fermat; Leibnitz is the one who has made the most use of it, and by his name and by a more elegant analysis which he has given of this problem: he was so charmed by the metaphysical principle, & here to find its final causes to which he was considerably attached, that he regarded, as an unmistakable fact, that light moves faster in air than in water or glass.
  It is however the opposite. Descartes had advanced the first, that the light moves most rapidly in the densest mediums, and although the explanation of the refraction which he had deduced from it was insufficient, his fault did not come from the supposition that he had made. All systems that give some plausible explanation for the phenomena of refection, assume the paradox, or confirm it.
  If one now supposes, that light moves most rapidly in the densest mediums, the whole edifice which Fermat & Leibnitz had built, is destroyed: the light, when it crosses different mediums, neither by the shortest way, or by the quickest way; the ray which passes from the air into the water making the greater part of its path in the air, arrives later than if it did not make the slightest difference. We can see in the Memoir that Mr. de Mairan gave on Reflection and Refraction, the history of the dispute between Fermat & Descartes, & the difficulty & impotence to which we have so far been able to grant the law of refection with metaphysical principle.
  While meditating deeply on this matter, I thought that the light, when it passes from one medium to another, already abandoning the shortest path, which is that of the straight line, could well also not follow that at the most rapid time: indeed, what preference should there be here of time over space? the light being unable to go at once by the shortest way, and by that of the quickest time, why should it go by one of these paths rather than by the other? so it does not follow either of the two, it takes a path that has a more real assertion: the path it takes is that by which the amount of action is a minimum.
  Now I have to explain what I mean by the quantity of action. When a body is carried from one point to another, it requires a certain action, this action depends on the speed of the body and the space it travels, but it is neither the speed nor the space taken separately. The quantity of action is all the greater as the speed of the body is greater, and the path which it traverses is longer, it is proportional to the sum of the spaces multiplied each by the speed with which the body travels. It is this, that quantity of ation which is here the true expenditure of Nature, and which it spares as much as possible in the movement of light.
  Let two different media, separated by a common surface be represented by the line CD, such that the speed of light in the medium which is above, set = V, & the speed in the medium which is below, set = W. Let a ray of light AR, which from a given point A must reach the given point B.

  To find the point R where it must break, I search for the point where the ray breaks, the amount of action is the least, & I have V.AR + W.RB which must be a minimum, or V.√(AC²+CR²) + W.√(BD²+CD²-2CDxCR+CR²) = min. So AC, BD & CD being constant, I have V.CR.dCR/√(AC²+CR²)-W.(CD-CR).dCR/√(BD²+DR²)=0, or V.CR/AR=W.DR/BR. CR/AR:DR/BR::W:V, that is, the sine of incidence to the sine of refraction is in inverse ratio to the speed that light has in each medium.
  All the phenomena of refraction now agree with the great principle, that Nature in the production of its effects always acts in the simplest ways. From this principle follows that when the light passes from one medium to another, the sine of its angle of refraction is to the sine of its angle of incidence in inverse ratio to the velocities of light in each medium."

Ptolemy, Kepler, Newton & Huygens on Refraction

  Some references on the history of refraction:

160s -
  Ptolemy - Optics  Lejeune  Smith

1611 -
  Kepler - Dioptrice - refringuntur & refracti

1672 -
  Newton - Philos. Trans., concave mirror vs convex lens (aberration)
  Newton - use of the word aberration in a response

1687 -
  Newton - Principia Mathematica, 1st Law, uniform rectilinear motion

1690 -
  Huygens - Traité de la lumière, refraction, proportion of sines

1703 -
  Huygens - Dioptrica, Snell, refraction, proportion of sines

1704 -
  Newton - Opticks, ratio of sines, refrangible rays (aberration)

1727-8² -
  Bradley - Philos. Trans. stellar aberration

1747¹ -
  Bradley - Philos. Trans., stellar aberration
_____________
¹ edit (Jul 13): Bradley 2nd ltr
² edit (Jul 13): Bradley 1st ltr

References for Hamilton's Theory of Systems of Rays, 1st Part

  Here are some references that may give some insight into Hamilton's terminology and sources regarding the presentation in his theory of optics and rays.

Malus - Treaté de l'Optique 1807


Least Action

Maupertuis - quantité d'action 1744

Laplace - Mechanics: Maupertuis least action footnote 1814


pencil

Hamilton -  pencils 1824

Coddington - A System of Optics, pencil 1829


Characteristic

cnrtl - caractéristique

Laplace - Mécanique céleste, caractéristiques 1799

Monge - Application de l'analyse à la géometrie, caractéristique 1807

Lagrange - caractéristique (symbol δ for variation) 1811

Lagrange (Eng) caractéristique -> symbol trans.


caustic*

Coddington - An Elementary Treatise on Optics, caustics 1825


arétes de rebroussement (cusp edges)

Hamilton - arétes de rebroussement

cnrtl - aréte de rebroussement

Monge - Géométrie descriptive, arétes de rebroussement (see also pages 18, 208)


Aberration

Hamilton - aberrations

Optical aberration wiki


*edit (Jul 11): Found an earlier Coddington book in my notes. Hamilton was associated with Trinity College, Dublin while Coddington was associated with Trinity College, Cambridge. They were contemporaries.

Focal Surface

  The impression of the caustic that we got in the last blog needs to be qualified to some extent. First the curves of constant ℓ and x are projections onto the x,y-plane but they are more like spirals in three dimensions. A photograph of a caustic would result from the flow of the sets of these evolving spirals through the frame of the photographic film during the time of the exposure. Hamilton chose a more limited definition of the surface of constant action based on a ray and those in its immediate vicinity which he called a pencil. However, we can use the prior results to play with statistics a little. The value of ℓ=2 did not give the the most compact image as it did for the focal point on the axis of the mirror but was close to it.

For both of the plots above the x,y-grid varied from (x,y)=(-1,-1) to (1,1) with steps of 0.1 in x and 0.2 in y.  As one increases the value of ℓ the focal point becomes more spread out, more blurred.

When one uses a smaller section of the mirror one gets a more well defined surface of constant action. The rays reflecting off the mirror centered at x=y=0 and with steps of 0.0001 in either direction produce a surface grid which gives following projection onto the x,y-plane. The red marker indicates the point that reflected off the mirror at x=y=-0.001. This appears to be what Hamilton referred to as a rectangular system. The horizontal lines are lines of constant x on the mirror. The vertical rows of markers indicate the lines of constant y on the mirror. The data is plotted relative to the mean values of x and y and ℓ was chosen to minimize the rms error.

So to study the system of rays and the focal properties of a mirror we need to think in terms of a flow field for the propagation of light.

edit (Jul 10): redid the last plot which also required some cleanup of the text.

Effect of Mirror Shape on the Caustic

  The rectangular mirror in the previous blog used a rectangular grid of x,y values for the reflection points on the mirror to obtain curves for the caustic. For a circular mirror one has to filter the points of this grid so that for the radius we have r≤1 and the resulting caustic is somewhat more compact.

The Caustic in the Focal Plane

  One can get an idea of what the caustic looks like in the focal plane, the x,y-plane with z=1, by plotting curves of constant x for the reflection point on the mirror. The x values were stepped by 0.2 from -1.0 to 1.0 and on each curve the y values step by 0.1 from -1.0 to 1.0. The small red marker indicates the point on a curve where y=-1.0.

The curves for -x and +x appear to be mirror images of each other. The bounds of the caustic would be the outer edge or envelope of these curves so one gets a symmetrical shape for the caustic.

Closeup of the Off-Axis Focus of a Parabolic Mirror

  One can get more detail on the off-axis focus of the parabolic mirror found in the last blog. We find that the curves of equal action are caustics with a cusp on the left. Here the step in path length has been reduced to Δℓ=0.002 and the path length for the curve with cusp at z=1 is ℓ=2. The curve moves upwards as the path length increases.

Notice that the density of points increase near the cusp. The corresponding points on the curves are on the rays of the "system." A peculiarity of Excel is that points in a plot are not centered on the data point but displaced somewhat. The upper left corners of the square point markers are closest to the data points. The same is true for the focal point. These curves are the parts of the surfaces of constant action in the plane of the plot. To get the caustic for the focal plane one has to project the constant action surface onto this plane. The "density of points" determines the relative brightness of the caustic.

Focusing Properties of a Parabolic Mirror

  Hamilton as an astronomer would have been interested in optics for its use in studying the properties of the mirrors used in telescopes. A parabolic mirror can be used to bring a source on its optical axis to a sharp focus. The mirror is the blue curve at the bottom. The red lines are incoming plane wave surfaces from a very distant source and the relative change in action is constant. The green curves are the outgoing surfaces of constant action. The reference surface is the plane z=1 where ℓ=0 and the steps in action were set equal to 1/4 unit.

The focus appears to be quite good. When the incoming rays are slightly off axis the focus is less precise. Below the incoming rays move downward and to the right. The outgoing rays move upward and also slightly to the right coming to an approximate focus to the right of the previous focal point.

So we have an effective focal plane at z=1. The equation for the mirror is z=A(x²+y²) with A=0.25. The focal length of a parabolic mirror is f=1/(4A).

Spherical Mirror Surface of Constant Path Length

  As mentioned previously for a spherical mirror the positions of a source image varies with changes in the relative position of the observer. As the rays to and from a reflection point vary with changes in the reflection point's position on the mirror the point on the final ray at constant total path length sweeps out a surface that is everywhere perpendicular to the rays.

In the plot above the three bent rays have a total path length equal to 5 units. The ends of the dashed lines indicate the positions of the apparent source images determined by the parallax from very small changes in position. The purple dashed line are normals to the final rays at Σℓ=5 units. Thus the surface of constant action will depend on the both the sum of path lengths for the rays and the shape of the surface of reflection. As the reflection point moves further away from the source the radius of curvature for the surface decreases.

Images In Spherical Mirrors

  One can also use ray tracing and parallax to find the images within spherical mirrors. One uses the conditions for the shortest path to determine the reflection points on the sphere. The intersection of two rays can then be used to determine the parallax position of the source image.

One can use path lengths to determine the surfaces of constant action. The directions of the final rays are all perpendicular to these surfaces. The image point within the sphere varies with the position of the observer.

Parallax Position From Two Rays

  One can use parallax directly to determine the apparent position of the source of the final rays. All one needs is a position on two rays and the direction of the rays at those points. Using this data one can compute the position, x̄, where the rays will cross.

Using a minus sign with μ makes the calculation a little easier. Here's the plot from the last blog for review.

The calculation shows that the two final rays radiate from the image of the luminous source.

Parallax and the Focal Point of the Reflected Rays

  Hamilton's description of a surface of constant action is contained in III [13] and uses the sum of the separate pathlengths. The system of final reflected rays all focus on an image of the luminous source. This is why the method of images used in the last blog works. One can illustrate this with a simple problem. The first row of data below contains the position of the source, the nearest point on the mirror, the normal to the mirror and the reflected image of the source. The second row contains two points, f̄ and ḡ, used to select the rays, their reflected images and the points of reflection on the mirror. The next row gives the bent rays and the final row gives the ray extensions to the source image.

The plot shows the shift in the direction of the apparent source due to parallax associated with changes in the position of observation.

The green line represents the arc whose action or path length is the same as that of f̄ and the direction of the fnal rays to the source image are all perpendicular to this surface. For reflection the surfaces of constant action are spheres due to the spherical propagation of light.

Multiple Reflections With Two Mirrors

  As mentioned previously the images of the target focal point can be used to find the reflection points on a mirror. Expressed alternatively this is where the line from the source to the a focal image intersects the plane of the mirror. We compute images and successive images using the reflection matrix operators. Upper indices in brackets can be used to designated the alternating sequence of reflections.

The intersection x̄ can be found by projecting the ray between point source x̄₀ and the focal image f̄ onto the mirror. If the the order of a double reflection of a ray is mirror 1 followed by mirror 2 one needs to use image f⁽²¹⁾ for the first intersection and f⁽²⁾ for the second reflection. The sequence of reflection for f⁽²¹⁾ is a reflection of the focal point f through mirror 2 followed by a second through mirror 1.

Using this method we can obtain a sequence of multiple reflections.

Plotting the data gives a better picture of what's happening.

An observer at the focal point on the right will see multiple images of the luminous source on the left with the directions indicated by the directions of the converging rays and a distance equal to the total path length the reflection segments.

Please note that the path length or "action" is a local minimum, i.e., when compared with infinitesimally neighboring rays.

Reflection From Two Mirrors - A 2nd Solution

  If we change the order of reflection by exchanging the values for the proximal points, r̄₁ and r̄₂, and recomputing we get a second least action solution.

A check shows the reflection points are on the mirrors and the direction cosine requirements for reflection are met. The two solutions are nearly identical except for the relative positions of the reflection points. The total of the  path lengths is the same for both problems.

*projections on left should read ρ₁·B₁

Here's the plot in the y,z-plane.

*edit (Jul 2): typo

Reflection From Two Mirrors

  In Part I, section III of Hamilton's Theory of Systems of Rays he states "...if it be possible to find a mirror, which shall reflect to a given focus the rays of a given system, those rays must be perpendicular to a series of surfaces..." but it is unclear what surfaces he is talking about. Are they surfaces of constant action? Perhaps a simple double reflection problem will help clarify matters.

Let's place a luminous body at the origin and find the positions of reflection from two plane mirrors needed to arrive at some chosen point, r̄₃. The planes of reflection can be specified by the position in each plane that is closest to the origin, say, r̄₁ and r̄₂. The lines connecting these points to the origin will be perpendicular to their respective mirror.

We need the sum of the path lengths to be a minimum subject to the constraints that the reflection points be on the mirror or equivalently that the difference between the reflection points and proximal points is perpendicular to the mirror normals.

Solving for the minimum total path length gives the conditions on the directions of the path segments required for reflection from each mirror.

From the relations connecting the reflected rays with the incident rays we can define two matrix operators, P and Q, associated with the mirrors.

We can verify that the inverse of these operators is the transpose and that their determinant is unity. This tells us the magnitudes of the vectors operated on by these matrices are unchanged. When we look at the action of the operators on vectors perpendicular to the mirror and those parallel to it we see that vectors are reflected through the plane of the mirror.

These operators can be used to find the apparent source of illumination for each mirror, r̄'₀ and  r̄''₀.

Using a trick at least as old as Hero of Alexandria we can use the apparent sources of light to determine the direction of a line segment and its length in reverse order.

Choosing arbitrary values for the positions of r̄₁, r̄₂ and r̄₃ we can now solve for the unknowns using the method shown above.

Using the basis vectors for the polar coordinates for the position of each mirror we can show that the reflection points are on the mirrors and that the requirements for reflection are satisfied. The fact that the determinant of the matrix containing the direction vectors is not zero tells us that the reflected rays do not all lie in the same plane.

*projections on left should read ρ₁·B₁

Here's a plot of the reflected rays in the y,z-plane.

If the positions of the mirrors, r̄₁ and r̄₂, are chosen arbitrarily along with the mirror normals one can use ê_i(ê_i·r̄_i) for the proximal points.

Supplemental (July 1): Instead of finding images for the light source one could determine the mirror images of the target point and then solve for the reflection points in order. Being able to straighten out the bent line segments enables one to solve for the unknowns. The surface of constant action through the target point would have an apparent radius of curvature about the final image of the source.

Edit (July 1): Corrected a minor error. The reflection operator is its own inverse. It is also equal to its transpose.

*edit (Jul 2): typo