From light’s speed to Snell’s law
This page is dedicated to the simplified study of the propagation of light within the eye and main propagation media of interest in ophthalmology such as those which constitute ocular tissues. Snell’s law is a fundamental property in geometrical optics and at the origin of the formula of vergence. We will see how to establish this law from the elementary properties which govern the propagation of light.
The important properties of light
On the scale of the Universe, light moves rather slowly; it takes about 100,000 years to travel through our galaxy, which is just one structure among many within an isolated cluster of galaxies. We will call here light the small portion of the light spectrum visible to our eyes. Whatever the frequency of the light wave, the speed is identical for a given propagation medium.
A star in a night sky corresponds to a basic image: the rays emitted diverge from the star, in all directions of space. Some of these rays can be picked up by an observer who directs his eyes towards the star. Due to the star’s enormous distance from the eye (the closest star, Proxima Centauri, is located 40,208,000,000,000 km away and it nearly takes 4.3 years for its light to reach the Earth), the rays picked up are considered parallel. The cornea and the lens must deflect (we speak of « refraction ») these rays towards the retina.
It would certainly be closer to a certain physical reality to consider the light emission as that of a flux of photons in all directions of space. However, in the context of paraxial calculus, the particulate aspect of light is of little interest. The same goes for the wave aspect of light, but we will see later that this point is questionable with the notion of optical path, which connects with the wave front theory these considerations on the light propagation.
Finally, placing oneself in paraxial conditions implies that only the rays which take a path close to the optical axis will be considered: these rays are called paraxial. We can, however, consider that they correspond to the path assumed to be that of the photons picked up by the eye near the center of the pupil.
On our scale, light travels at a tremendous speed; nearly 300,000,000 meters per second. It only takes about 0.0000000001 seconds (100 000 femtoseconds) for light to pass through the eye and travel the distance between the cornea and the retina! It would take even less time if light traveled at the same speed through the eye as it did through a vacuum of space. However, before being absorbed by photoreceptors in the retina, light from the surrounding world passes through transparent ocular media where it slows down (due to interactions with the atomic structures that make up matter).
Certain properties allow simple laws to be established which are useful in predicting the path of light through ocular structures:
The speed of light is reduced in dense materials (ex: cornea, aqueous, lens, vitreous) but not its “pulse” (frequency)+++
Therefore, the period of time (T) elapsing between each crest is constant.
The value of the refractive index (n) of a medium is defined as the ratio between the (length of a) wavelength in a vacuum and the wavelength (of same frequency) in the medium considered.
The refractive index is also equivalently defined as the ratio between the speed of light in the vacuum and in the material: this definition hides, however, the dependence of the wavelength on the refractive index.
A fundamental property concerns the time taken by light to connect two points. By virtue of the principle of least action, light takes the shortest path in duration (time). This route is not always the shortest from a geomtric point of view (distance).
If the travel time of the light information between two points is minimal, the number of unit periods of oscillations is necessarily minimal. The optical path is the length corresponding to the number of oscillations accomplished by the luminous electric field between two points.
Snell’s Law, shown in the previous illustration, concerns the path of a light ray that encounters a medium with a different refractive index (where light slows down if the RI is greater and vice versa). It stipulates that the angle of penetration (we speak of refracted angle, labeled « r ») of this light ray in a given medium is smaller than the angle of incidence (labeled « i ») if the index of this refracting medium (nr) is greater than the index of the medium incident (ni) – it happens when light leaves the surrounding air to enter corneal tissue. Conversely, the refracted angle is greater than the angle of incidence at the interface between the corneal stroma and aqueous humor in the anterior chamber.
The angle is defined by the direction normal (locally perpendicular) to the surface at the point of penetration of the ray. It is interesting to take a step back because if we think carefully, the behavior of an isolated ray cannot be « understood » without a wave conception of light propagation! We cannot in fact understand how an isolated ray « would know » which direction it must take when it changes its propagation medium. Snell’s law is therefore not intuitive, on the contrary, but it is relatively easy to demonstrate with simple geometric reasoning.
Demonstration of Snell’s law
Snell’s law states that we can calculate the direction of the refracted ray as a function of that of the incident ray if we know the refractive index of each medium (ni for the incident medium, nr for the refracting medium). The angles are related to the line perpendicular to the point of incidence O, called the line “normal to the surface”.
Snell’s law can be written simply: ni x sin (i) = nr x sin (r).
Snell’s law is fundamental because it allows the formulation of the vergence formula, so let’s demonstrate it.
In the medium on the left, the wavefront moves at speed vi. On the medium on the right, its speed (vr) decreases. The wavefront forms an angle θi with the normal to the surface and is deflected by an angle θr after refraction. By virtue of Huygens’ principle, the time taken to go from C to B is the same as that to go from A to D. This time is equal to the distance divided by the speed, and we can thus write:
CB / vi = AD / vr
From the geometry of the figure, and by observing that θi is equal to the angle (CAB) and that θi is equal to the angle (ABD) we can express that the distance CB = AB sin (θi) and the distance AD = AB sin (θr)
The preceeding equation becomes: AB sin (θi) / vi = AB sin (θr) / vr.
We can simplify by AB and multiply both sides of the equation by a constant c (speed of light in vacuum) to obtain: sin (θi) c / vi = sin (θr) c / vr.
The ratio between the speed of light in vacuum and light in the medium considered is equal to the refractive index of this medium: ni = c / vi and nr = c / vr
We finally obtain the Snell’s law equation: ni sin (θi) = nr sin (θr)
Snell’s law and optical path
Between the two points, the travel time is minimal. It is therefore not possible to have fewer unit periods (which correspond to unit paths of one wavelength) between these two points. The minimal number of wavelengths corresponds to the optical path.
if a point receives rays along which the optical path is identical from the source, it is an image point because the light waves will be in phase; they will have accomplished the same number of oscillations from the source and interfere constructively there.
The careful reader will probably have deduced from this that if one adds a whole number of wavelengths to a given path, the effect will be unchanged. This principle based on the similarity of the optical path regardless of the whole number of wavelengths added to it is used for the design of diffractive optics. For this, it is necessary that these optics include patterns that make it possible to modulate the optical path by an order of magnitude equal to that of the wavelength. These patterns are however much smaller than the orders of magnitude used in paraxial optics; the paraxial radius of the anterior face typical of an artificial lens implant measures a few millimeters, while the diffractive steps of bi or trifocal implants have a maximum thickness of a few microns.