The vergence formula is very useful in ophthalmic optics. It is used to calculate the position where an object’s image is formed after refraction of light by an optical system: this system can be simple (a thin lens) or more complex (cornea + lens).
Using the vergence formula allows various calculations to be made without calculating the path of many rays through the optical system. It allows, among other things, to establish a simple formula for calculating the implant’s power during cataract surgery. It also allows formulas to be established to convert a spectacle lens’s power from a contact lens power (and vice versa) as a function of the distance between the spectacle plane and the eye.
The vergence formula involves the source object’s position, its distance in the incident medium with the optical system, and the vergence or optical power of this system.
What is vergence?
Vergence – analogous to the paraxial optical power – is expressed in diopters (D or δ). For an optical surface, it is equal to the inverse (reciprocal) of the distance to the focus – where the clear image is formed (more precisely from the principal image plane to the focal point in paraxial optics) multiplied by the refractive index of the medium considered.
Let n be the refractive index of the medium considered, and f the distance to the focus:
V = n / f
Vergence corresponds to the ratio between the medium’s refractive index and the distance between the interface considered and the focus.
In a phakic eye, the vergence depends on the vergence of the cornea, the crystalline lens, and the distance that separates them. During accommodation, the lens’s vergence increases (the lens deforms and takes on a greater curvature).
In the pseudophakic eye, the vergence depends on the vergence of the cornea, of the IOL, and the distance that separates them.
At this point, it is important to realize that vergence and distance are related by an inverse relationship. If we use a formula for calculating the vergence at a point in space, we can easily deduce the distance that separates this point from the focus – in the case, of course, where the light continues its path in the same propagation medium.
This page is devoted to the formula of vergence: its origin is reported, and few applications are mentioned.
Origin of the vergence formula
The vergence formula can be established from the law establishing the path (refraction) of light rays, Snell’s (or Snell-Descartes) law, applied in paraxial conditions. It derives from approximations that can be made under these conditions if we consider the small region of the optical system close to the optical axis (paraxial conditions) and where the rays form a small angle with this axis. The vergence formula allows you to calculate the position of a source’s image without needing to use Snell Descartes’ formula for all rays (ray tracing), which is tedious and requires the use of dedicated computer software.
When the light source is placed “at infinity,” the distance between the diopter’s apex and the formed image is the focal length (F).
Consider a general case, where the source is located not at infinity but a distance O and is surrounded by a medium of refractive index n. The optical system is a spherical diopter with radius R and index n’. We know n, n’, O (the distance from the source to the diopter), and R, the radius of curvature of the diopter’s apex.
When the light source is closer than infinity, the image of the source shifts beyond F. To compute this distance; we need a simple formula, named the vergence formula, which can be obtained at the cost of a few approximations (inset box), linked to the small magnitude of h (height of the incident ray to the optical axis). The vergence formula is therefore only valid for points located close to the axis (paraxial conditions).
R represents the radius of curvature of the diopter (if the diopter is perfectly spherical) or its apical radius (if it is not perfectly spherical). Knowing R is necessary if our formula is to be applied to points near the optical axis. In the cornea, R is close to 8 mm, and paraxial conditions apply for points within the central 1mm. Still, they can be extended with reasonable loss of accuracy to the central 3 mm zone, at least in normal situations.
Establishing the vergence formula
This paragraph may be skipped by the reader with little geometry enthusiasm.
The vergence formula derives from Snell’s law, which allows us to write: n sin θi = n ’sin θr
Under paraxial conditions, we can replace sin θi with the value θi in radians. The equation therefore becomes:
n θi = n’ θr
We must now express θi and sin θt as a function of variables that are easier to measure: (remember that we know the position of O, the value of R, n, and n ’).
By observing the figure above, we can put: θi = α + γ
The following figure makes it possible to express the angle of incidence θi as the sum of the ray angle with the optical axis α and the angle between the segment connecting the point of entry and the center of curvature and the optical axis γ.
Likewise, it is possible to verify that θr = γ – β
We can then replace θi = α + γ and θr = γ – β in Snell’s law to obtain:
n (α + γ) = n’(γ – β)
n α + n γ = n’γ – n’ β
n α + n ’β = γ (n’ – n)
We can express the angles α γ and β as a function of h, O (distance from the source point to the optical vertex), i, and R (remember that we know the values O and R).
Under paraxial conditions (for small angles):
tan α ≈ α = h / -O = – h / O (the distance O is negative by convention; the distance is measured to the left)
tan γ ≈ γ = h / R
tan β ≈ β = h / I
At this stage, it would suffice to know h to find the value of I. Unfortunately, we do not know this value. However, if we replace the values of the angles with these ratios in the expression:
n α + n’β = γ (n’ – n)
We obtain :
-n h / O + n’ h / I = h (n’ – n) / R
We can simplify the formula by h, and we finally get:
-n / O + n’/ I = (n’ – n) / R
Rearranging terms, we obtain the vergence formula:
n’/ I = n / O + (n’ – n) / R
We can then calculate I since we have « got rid » of the unknowns: the angles and the height of incidence h! The formula of vergence is aptly named since all its terms correspond to a vergence:
n’/ I is the vergence of the image Vi
n / O is the vergence of the object Vo
(n’ – n) / R is the optical power P of the diopter.
It is important to note that the vergence of the diopter depends both on the difference in refractive index, and on the radius of curvature of the apex of the diopter.
We can thus write:
Vo + P = Vi
Using the vergence formula
We appreciate this formula’s power, which only requires knowing the distance O of the source object with the diopter (of which we know the curvature R and the index n’) to calculate the image’s distance.
To calculate these vergences, it is necessary to consider the sign of the distances: negative to the left (diverging light), positive to the right (converging light). Hence, O<0.
The inverse of a distance (in meters) is expressed in diopters (remember that n and n ‘are dimensionless because they correspond to the ratio between the wavelength in the media, which is equal to the ratio between the speed of light in a vacuum and the speed of light in the media).
For example, if the diopter has an index of 1.37 and a radius of curvature R of 8 mm, the diopter’s vergence in the air is equal to (1.33 – 1) / 0.008 or 41.25 diopters. These are values close to those used to model the paraxial properties of the anterior surface of the cornea.
The vergence formula can be applied to study the vergence in the eye, i.e. the propagation of an incident light beam through ocular media.
Using the vergence formula applied to a double « cornea + crystalline » system, we can establish a paraxial formula for calculating IOL power.
Outside paraxial conditions
The vergence formula is based on a paraxial approximation; it cannot be applied at a distance from the optical axis to predict the focal length of an incident beam.
We are going to show here that spherical optics (circular in section) are not stigmatic apart from paraxial conditions. Even if this apparently takes us away from our main topic, it is useful to forge links to the concepts of asphericity and to introduce some fundamental notions that can be useful in cataract surgery and refractive surgery.
To begin with, we will recall some useful equations, concerning the profile of a circular section. It suffices to remember the Pythagorean theorem to establish the equation of a circle centered on the origin of a Cartesian coordinate system. However, when a circle describes the curvature of an optical surface, it is shifted so that the origin of the Cartesian coordinate system coincides with the vertex of the circle. We can easily establish the equation of the shifted circle.
If we consider this profile as that of a circular diopter of index n, we can determine its vergence and its focal length in paraxial conditions.
But what happens for a more peripheral ray?
To answer this question, we can establish the conditions for a diopter to be perfectly stigmatic for a source located at infinity; it suffices for the optical path, a notion that we have discussed previously, to be identical for any incident ray coming from a distant source (it is assumed here that the rays propagate in the same medium after having encountered the spherical diopter). For a given wavelength, the optical path is a concept specific to wave optics; it is equal to the physical distance multiplied by the refractive index of the propagation medium (the refractive index is the ratio between the wavelength considered in a vacuum and the wavelength considered in the medium; the index is an eigenvalue at a specific wavelength). The frequency of light waves is unchanged regardless of the medium, but the speed of propagation decreases with its density, which causes a shortening of the wavelength)
If the optical path length from source to the image is the same for all refracted rays, perfect stigmatism will occur. Intuitively, it is conceivable that the rays located near the vertex represent an optical path where the path in the air is shorter than that of the more peripheral rays, which meet the surface of the diopter a little further. These rays take a certain lead, which will have to be compensated for at the level of the path in the medium of the diopter.
We can thus establish the equation which establishes this condition of stigmatism and the equality between the optical paths for a ray coincident with the optical axis and a peripheral (non paraxial) ray coming from a distant source. Using the geometry of the figure and the Pythagorean theorem, we can establish an equation that includes our useful parameters: the radius of curvature of the diopter R, its refractive index n, and the coordinates of the Cartesian system (y,z).
By identifying the equation describing the profile of a perfectly stigmatic diopter with that of a circle, we realize that there appears a difference with the equation of the circle. The factor weighting the term in z^2 contains a constant (usually named Q) which characterizes the asphericity (literally the « defect or absence of sphericity »). The equation thus rearranged corresponds to the « Baker’s equation » and describes, when Q = -1 / n^2, the profile of a perfectly stigmatic optic, often referred to as « Cartesian oval« .
The corneal profile is slightly aspherical and often modeled by a conical section using two parameters: the apical or « osculating » radius (R) and the asphericity parameter Q.
The following diagram gathers the equations and terms useful for the study of the conical sections applied to the corneal profile. The prolate vertex (or tip) of the ellipse has a curvature that decreases from the center 0 towards the periphery (up to the oblate tip).
For paraxial calculations intended to estimate the optical power of the cornea, the radius of curvature of the apical radius must be used. In practice, it is provided by the keratometric measurement (in mm, or in diopters after a conversion which can use a reduced refractive index, or a paraxial calculation that takes into account the posterior face of the cornea).
The refractive index of the corneal stroma is equal to 1.376. The theoretical value of the corneal asphericity which would make the corneal diopter a perfectly stigmatic surface is equal to Q = -0.528. The average asphericity of the profile of the human cornea is measured close to Q = -0.25. It is slightly less « prolate ».
Even if the radius of curvature decreases at the periphery, this reduction is not large enough to counter the increase in the angle of incidence of the peripheral rays coming from distant sources. The optical power of the anterior surface of the human cornea increases slightly physiologically towards the periphery. Even if the radius of curvature decreases at the periphery, this reduction is not large enough to counter the increase in the angle of incidence of the peripheral rays coming from distant sources. The optical power of the anterior surface of the human cornea increases slightly physiologically towards the periphery. But it is not the latter that is used for calculations involving the vergence formula in paraxial conditions:
Near the apex, the profile of the osculating circle and that of the perfectly stigmatic aspherical corneal surface can be confounded in practice.