Principal planes: explanations
The principal planes of an optical system are useful for studying in a simplified way its behavior under paraxial conditions.
They are used to establish a simplified model which makes it possible to predict the path of light rays through an optical system: this system can be simple (a thick lens), or more complex: an assembly of lenses. Biometric computational models that consider the cornea and the IOL as thick lenses use the notion of the principal plane. The principal planes have no physical existence: they are a theoretical construction intended to simplify the problem of the path of light rays within an optical system, under paraxial conditions.
If we know the position of the principal planes of the optical system, we can predict their behavior with respect to rays close to the optical axis, the direction these rays will take after refraction by the system – under paraxial conditions.
There are two principal planes: one is called the principal object plane, the other is called the principal image plane. This page aims to demystify the concept of “principal planes” and “cardinal points”, which often appear to be quite “esoteric” to non-specialists!
Justification of principal planes
The underlying problem which justifies the use of this model is the « thickness » of the optical systems.
There are two special situations, for which the problem is relatively simple:
Dioptre vs Thin lens vs Thick lens
Referring to the above diagram:
1) The system has an infinite thickness: the case of the spherical diopter. In this case, the ray is refracted, according to Snell’s law, and continues to travel indefinitely within the diopter.
2) The system has zero thickness (or is very thin, with a thickness that can be neglected in practice). In this case, the behavior of the ray is predicted by the theory of thin lenses (it is enough to know the position of the focal points of the lens). If we know the location of the object focus of the thin lens, we can determine the path of the light ray.
3) However, when the thickness of the system is intermediate (and it consists of one thick lens, or more), the path taken by the light ray is more difficult to predict. The thickness crossed by the light ray must be taken into account: the ray emerges from the system at a different height than the entry point of the incident point. This is because the incident ray parallel to the optical axis undergoes two successive refractions. We will focus on this next.
Refraction by a thick lens
In the case of a thin lens, one can act as if the incident parallel rays were « instantaneously » deflected towards the focal point. The height (distance to the optical axis) of the entrance ray is equal to the height of the exit ray.
In the case of a more complex system such as a thick lens, there is a path « inside » the system that cannot be overlooked: it is responsible for a variation in the height at which the ray emerges. The only certainty is that the emerging ray from an incident ray close and parallel to the optical axis passes through the image focus.
We place ourselves here in ideal paraxial conditions, even if the diagrams, for didactic reasons, include representations apparently outside the paraxial situation.
The incident light rays (left) come from a distant source (at infinity) and hit the left surface of the thick lens, where they undergo the first refraction. The refracted rays then propagate in a straight line and then meet the straight surface of the lens where they undergo the second refraction and are focused on the image focal point.
To perform the exact geometric trace and the calculation of the path of light through the lens (e.g. ray tracing using Snell’s law), each incident ray generates two consecutive refraction calculations (corresponding to the first and the second refraction).
Light is likely to take the “shortest” optical path (the route that takes the least amount of time between two points) between the entrance and exit of a thick lens; it would certainly be possible to calculate this path if we know the geometry of the system and its refractive indices, but it is simpler to define the existence of two planes, called the principal object plane and the principal image plane, which materialize the intersection between the incident rays (respectively emerging) parallel to the optical axis and the emerging rays (respectively incident) which pass through the image focus (respectively object).
The layout of the principal planes
Everything happens « as if » an incident light ray close and parallel to the optical axis was not deflected … until it meets the principal image plane: it is then refracted towards the focus.
Likewise, by using the principle of reversibility of the path of light, we can establish that a ray passing through the object focus propagates without deviation until it meets the principal object plane, and then be instantly deflected in a direction parallel to the object. optical axis. This principle is used to « trace » the location of the principal planes.
Drawing of the principal image plane
Principal image plane
To determine the location of the principal image plane, it is sufficient to draw an incident ray parallel to the optical axis and to know the path of the emerging ray (which passes to the image focal point). The path of these rays is extended, and their intersection is in the principal image plane, which is (in strict paraxial conditions) perpendicular to the optical axis.
If we repeat this for several incident rays parallel to the optical axis, we obtain the following plot:
In this figure, the incident rays (in purple) and the emerging rays (in red) have been extended in dotted lines. Their intersection (cross-sectional dotted plots – yellow circles) defines the principal image plane. It is interesting to note that the envelope of these points is a straight line (a plane) near the optical axis, that is to say in paraxial conditions: at a distance, this surface is slightly curved. The concept of the principal plane only applies under paraxial conditions.
It is possible to define a principal object plane by reasoning symmetrically.
Principal object plane
The location of the principal object plane is obtained by the same method:
A similar plot can be made for the incident ray passing through the object focal point and refracted parallel to the optical axis, in order to find the principal object plane. The intersection between the principal planes and the optical axis defines the principal points denoted P and P’. The focal length should be measured from one of these points to the corresponding focal point, and not from the top (vertex) of the lens or optical system (PF: object focal length and PF’: image focal length).
When using a « thick lens eye model » for IOL power-related calculations, the position of the IOL used for paraxial computations refers to the position of its principal object plane. In such a thick lens model, the effective lens position refers to the distance between the principal image plane of the cornea and the principal object plane of the IOL.
Rather than calculating the path of each incident ray (ex: ray-tracing), it is preferable to replace the complex system (thick lens) with a simple system, made up of the principal planes.
The distances used for paraxial calculations with thick lenses are defined with respect to the principal planes. It is therefore necessary to determine the position of the principal planes of a system composed of thick lenses (eg cornea + IOL) to accurately calculate its refractive properties.
If the thick lens is in the air, the principal image plane is the same distance from the center of the lens as the principal object plane. When we study a system where the incident medium has a refractive index different from that of the medium where the image is formed (this is the case of the eye), then this symmetry disappears.
If the refractive index difference between the lens and the medium on its right, the focal distance will be modified, as well as the position of the principal image plane:
At the end of such construction, we can consider that we have reduced a « complex » optical system to a simplified system, which would be equivalent to a thin lens if the space between the principal planes would collapse.
In reality, the principal planes are only planar in the vicinity of the optical axis, in paraxial conditions: if we studied more at a distance the real shape of the surfaces grouping the points whose image by the optical system does not undergo magnification one would observe slightly curved surfaces.
The magnification of an object located in the principal object plane (virtual object in the case where this principal object plane is in the lens or the optical system) is equal to 1: the corresponding image (virtual and located in the principal image plane ) is the same size and orientation as the object.
The two principal planes have the property that a ray emerging from the lens appears to have crossed the rear principal plane at the same distance from the axis that the ray appeared to cross the front principal plane, as viewed from the front of the lens.
The lens can be treated as if all of the refraction happened at the principal planes. The magnification from one principal plane to the other is +1. The principal points P and P’ are the points where the principal planes cross the optical axis.
Position of principal planes
The position of the principal planes is equal to the ratio between the object or image focal length of the entire system, respectively, and:
– the object focal length of the posterior face for the principal object plane.
-the image focal length of the anterior face for the principal image plane.
This is shown schematically in the following figures:
We can interpret them like this; the smaller f is, the greater the total power of the lens is compared to the power of the opposite face. The principal plane is therefore close to the vertex considered (anterior for the main object plane, posterior for the main image plane). This is important for understanding the effect of the Coddington form factor on the position of the principal planes of intraocular lenses. These planes move towards the more powerful surface and are otherwise centered when the implant has symmetrical surfaces.
The next figure recapitulates the formulas, which gives the distance between the principal plane of the thick lens of radii Ra & Rp and its vertices Sa & Sp (intersections with the optical axis of lens surfaces):
D is calculated according to Gullstrand’s formula, which takes into account the power of the front surface of the lens (of radius Ra), the power of the posterior surface of the lens (of radius Rp), and the thickness (d) of the lens as well as its index.
f and f’ are the object and image focal length of the thick centered system, n1 the refractive index of the medium of the incident ray, d the thickness of the lens, n2 the refractive index of the lens, and n3 the refractive index of the medium of the refracted ray. f and f’ are referenced to the principal object and image planes.
The position of the principal planes can be calculated using the formula mentioned above from the geometry of the implant: (power, curvature of the surfaces, refractive index, thickness of the implant), and by replacing n1 with the air refractive index (1) and n3 by the value of the aqueous humor’s refractive index, ex: 1.337).
The curvatures (power) of the front and rear faces of the thick lens, the variations in the refractive index of the media logically influence the position of each principal plane.
The position of the principal planes logically moves towards the most optically powerful surface. When the two surfaces of the thick lens are of equivalent power, the principal planes occupy a middle position. If the lens has an asymmetric geometry (like most intraocular implants used in cataract surgery), we cannot, therefore, identify the position of the principal planes with that of the center of the lens, or suppose that these principal planes will be located at equal distance from the center of the implant.
Principal planes and intraocular lenses (IOLs)
Intraocular lens implants are thick lenses: in order to improve refraction control after implantation, and to establish reliable comparisons in terms of positioning, it is important that the position of a given IOL is measured from its principal object plane. The position of the principal object plane with respect to the vertex of the IOL does not depend on its effective position in the eye, but the refractive effect of the implant depends on its effective position (i.e. the anatomical position of the ‘implant in the eye after cataract surgery AND the effective position of the principal object plane).
We could identify the position of the implant as the distance between the anterior vertex of the cornea and the anterior surface of the implant, but given the different geometries, refractive indices, etc. of implants as « thick lenses », this choice would expose to uncertainties.
The position of the principal object plane of the IOL defines the effective position of the IOL.
The focal length of the implant is defined from the principal image point (this length is equal to the distance between the principal image point and the image focus).
The differences in the position of the principal planes with respect to the geometry of the implant partly explain the need of adjusting constant values such as “constant A” (SRK T formula) or the triplet « a0, a1 and a2 » (Haigis formula).