Impact of internal curvature gradient on the power and accommodation of the crystalline lens

Experimental studies on the GRIN distribution within the crystalline lens, especially those based on MRI, suggest that young lenses may have a strong gradient of curvature of the iso-indicial surfaces. This curvature gradient appears to decrease with age and eventually could even become negative in old lenses. Our purpose was to study the effect of this curvature gradient by comparing the paraxial power and its increment due to accommodation of the GRIN lens with that of a lens with a homogenous refractive index. To this aim we developed an approximated theoretical method, which was first validated thorough direct comparison with paraxial computations using ZEMAX.


Introduction
Experimental studies on the GRIN distribution within the crystalline lens, especially those based on MRI, 1 suggest that young lenses may have a strong gradient of curvature of the iso-indicial surfaces.This curvature gradient appears to decrease with age and eventually could even become negative in old lenses. 2Our purpose was to study the effect of this curvature gradient by comparing the paraxial power and its increment due to accommodation of the GRIN lens with that of a lens with a homogenous refractive index.To this aim we developed an approximated theoretical method, which was first validated thorough direct comparison with paraxial computations using ZEMAX.

Methods
Let us assume a constant negative gradient -G for the curvature radius of the iso-indicial surfaces R(z) = R ant -Gz (and a similar expression for the posterior part), which means a positive gradient for the curvature: dC/dz = G/R²(z).The case G = 1 corresponds to concentric iso-indicial surfaces, which was the value assumed in most GRIN lens models. 3Then the paraxial power, for both the anterior and posterior halves of the lens can be estimated using the thin lens approximation: The first terms in both expressions represent the contribution of the surface; n 0 is the refractive index of the surrounding medium (1.336) and n s is the refractive index of the lens surface; R ant (> 0) and R post (< 0) are the curvature radii of the anterior and posterior surfaces.The second terms correspond to the internal GRIN structure, where t is the axial lens thickness and t a is the thickness of the anterior part.The integral is the sum of all the instantaneous contributions to the power along Now we compute the total lens power using the thick lens equation: Here we use the approximation: where n n is the maximum refractive index of the nucleus.This method was validated by comparing results with those obtained with ZEMAX © in different cases (see next section).

Results
From Equation (1) it is straightforward to demonstrate that P = P homogeneous + GP GRIN , with P GRIN positive (for both anterior and posterior parts of the lens).This means that when G = 0 (no curvature gradient) the power of the lens is equal to that of a homogeneous lens with n = n n (nucleus).When G > 0, then the power is higher than that of the homogenous lens.This was confirmed in Figure 2, Gradient of curvature radius (G)

Lens accommodation
Paraxial power increase (D) which shows the lens power (left) and lens accommodation (right) versus G, for different types of lens models, with the same parameters (surface radii and refractive index).The parameters are taken from a 20-year-old adaptive GRIN lens model, 3 both at 0 D and at 8 D of accommodation.The model was also implemented in ZEMAX for cross validation of our theoretical approach.As we can see there is close agreement between the theoretical approximation and the ZEMAX implementations of the models.
We can observe a nearly quadratic (accelerated) increase of both lens power and accommodation (power addition) with the curvature gradient G.As expected from theory, at G = 0 the curve coincides with the case of a homogeneous lens with n = n n .For G = 1, both curves (left and right) show a close agreement with the ZEMAX implementation of the GRIN model with concentric isoindicial surfaces.We can observe (the right panel) that the effect of the gradient G is an even stronger enhancement (higher acceleration) of accommodation.

Conclusion
The curvature gradient of the iso-indicial surfaces of the GRIN structure of the lens appears to play an essential role in substantially enhancing both its power and its accommodation response.This parameter seems important to understand the optics of the human lens as well as their changes with age and accommodation.

Figure 2 :
Figure 2: Lens power (left) and accommodation (right) versus gradient of curvature radius.