Vergence in the pseudophakic eye
Definition of vergence
Vergence is a useful parameter in paraxial optics: it is proportional to the inverse length and is expressed in diopters (D or δ).
Vergence corresponds to the ratio between the medium’s refractive index and the distance between the interface considered and the focus.
Let n be the refractive index of the medium considered, and f the distance to the focus:
V = n / f
The vergence of an optical system is positive for a convergent system (light propagates to the right) and negative for a divergent system (light propagates to the left): it takes the same sign as the image focal length.
In the case of an optical system immersed in air or vacuum, the vergence can be defined simply as the inverse of the image focal length.
Vergence is an advantageous quantity for predicting the focal point (s) of an optical system. If we know the vergence, we know the focus’s position, which is equal to the inverse of the vergence divided by the refractive index of the propagation medium.
Light propagation to and in the eye
Consider a point source that emits light in all directions. We only represent a part of the light emitted here, in the form of wave trains, rays, and wavefronts, even if our calculations will involve vergence. The diagrams are not drawn to scale, of course.
A wavefront is a virtual construction: it corresponds to the envelope of points in space in the same phase state whose variations are represented by the wavetrains (sinusoids). A ray is another virtual construct that corresponds to the local direction of movement of the wavefront.
We have seen that vergence is a magnitude that is equal to the inverse of a distance. It is equal to the propagation medium’s refractive index divided by the distance from (negative vergence) or towards (positive vergence) the focus. Thus, the further one is located from the source, the more the vergence decreases (in absolute value). Far away from the source, the vergence is null.
An image of the source point can be formed from part of the rays/wave trains captured by an optical system designed to converge the light towards a point focus.
Suppose there is a spectacle lens in the path of light far away from the source. Light propagates in the air (n=1). This spectacle glass can modify the path of the light rays. The question asked is: what becomes of the vergence after the passage through the glass?
To answer this question, it suffices to use the additive properties of vergence. The vergence of a distant source (to infinity) is zero near the spectacle lens.
In reality, things are often more complex because there is usually an eye behind a spectacle lens. And this spectacle lens serves to modify the incident vergence of light so that the image from a distant source is perceived as sharp by that eye, which assumes that this image is formed in the retina plane. If this is true, then we know that the vergence of a wavefront which leaves the posterior face of the lens (or of the implant if the eye is pseudophakic) is necessarily equal to the ratio between the refractive index of the vitreous and the distance separating the posterior surface of the lens from the fovea. (more here)
To compute the wavefront’s vergence leaving the crystalline lens (or implant), we must calculate the effect on the vergence of the cornea, the lens, and the distance between them and the retina. The method to be applied is always the same: we calculate the vergence at the entry then at the exit of a refractive element (surface or thin lens).
After passing through the glass, the wavefront propagates towards the cornea, which it reaches after having crossed the distance between the spectacle glass and the eye. The vergence just before the corneal plane is therefore easy to calculate.
The cornea is considered here as a thin lens. Its vergence (VK expressed in diopter, for example, 43 D) adds up to VBCP, and at the exit of the cornea, the vergence VACP is easy to calculate. It is important to realize that the distance at which the wavefront would focus is equal to the refractive index of the aqueous humor (na) to VACP.
We can continue to study the propagation of the vergence: the next step is to subtract the distance separating the cornea’s posterior face from the anterior face of the lens. The reciprocal of this distance, multiplied by the aqueous humor’s refractive index, provides the value of the vergence in contact with the implant (or lens).
This type of calculation is used to produce an IOL power calculation formula known as a « thin lens IOL power formula »: the vergence values corresponding to the cornea and to the implant are used, which are considered as infinitely thin lenses.
To gain precision, it may be interesting to consider the cornea and the artificial lens implant as « thick lenses« . We must then apply the same method as above, but detailing the effect of each of the surfaces of the cornea and the intraocular implant. This procedure is detailed in the next paragraphs, which contain an interactive calculation form.
Vergence through a pseudophakic eye model
The cornea and intraocular lens (IOL), considered thick lenses, can be analyzed in vergence. The vergence can be traced by noting its change at each surface, using the vergence formula. S1 denotes the plane of the anterior corneal surface, S2 denotes the plane of the posterior surface, and S3 denotes the plane of the anterior surface of the IOL. S1S3 is the anterior chamber depth from the anterior corneal surface.
Once the local vergence value is known, the distance to the focus can be determined (assuming that the light does not meet any other surface and propagates in a medium of the constant index). We subtract from this distance the distance with the next refractive surface to calculate the vergence when reaching this refractive surface.
Each computational step is shown here (not to scale).
We start with a point source located at a finite distance from the cornea: its vergence is negative and equal to the inverse of this distance. After refraction by the four surfaces of the model eye (front face then back of the cornea, front face then back of the artificial lens implant), the vergence is modified. Its final value makes it possible to calculate the distance to the image focus from the back surface of the implant. If this distance is equal to that which separates the retina from the rear surface of the implant, the retina is said to be conjugated with the source point and will form a clear image of it.
Vergence in the pseudophakic eye: online computation
Numerical computations can be performed here (scroll down for results):