A freeform progressive addition lens (PAL) is an optical lens composed of different optical powers, with a curvature that is not constant. It can achieve smooth focusing within a range of focal distances, from distant to near, providing a more natural adjustment for users. This type of lens fully meets both physiological and psychological needs, making it increasingly favored by the middle-aged and presbyopic population. The design of the meridional power distribution plays a crucial role in the astigmatic distribution, the lens' distance and near vision area, the width of the corridor, and the astigmatic gradient, all of which are essential for the wearer's comfort. The channel width of the lens is closely related to the design of meridional power, contour line distribution, and sagittal height surface profile. Lenses with a wide channel design exhibit lower image distortion, chromatic aberration, and spherical aberration. Moreover, they also feature smoother transition zones, reducing the adaptation period, and offering a more accurate and natural visual experience while minimizing eye fatigue and dizziness. The current design of meridional power does not adequately consider its impact on the overall channel width, and there is a lack of efforts to broaden the channel width. This results in lenses having a relatively high level of astigmatism within a single pupil size. Therefore, we propose a new meridional power distribution based on cumulative distribution functions and analyze the curvature of the function's impact on the width of the image dispersion center. Additionally, the overall sagittal surface shape overlay and the reduction of meridional power distribution curvature are employed to increase the channel width of progressive addition lenses.

To widen the progressive channel of progressive addition lenses, we propose a novel approach. Firstly, a method utilizing the cumulative distribution function is introduced for designing the meridional power distribution, and a comparative analysis is conducted with the commonly used octave polynomial function and trigonometric function. Subsequently, we superimpose sagittal height surface profiles calculated from different channel lengths of meridional power and contour lines to achieve the optimization of channel width, facilitating a smooth transition in gradient changes. Then, we superimpose two functions to derive a new function, thereby altering the curvature of the meridional power function to optimize the channel width. Finally, we fabricate and evaluate three sets of lenses using a freeform machining tool and measure instruments to analyze the impact of this optimization method on the meridional power, astigmatic distribution, astigmatic gradient, and other optical performance aspects of progressive addition lenses.

The proposed meridional power distribution based on the cumulative distribution function is feasible. Compared to the octave polynomial function, the curvature values of the cumulative distribution function decrease, leading to an increased width of astigmatism. However, the channel width is smaller than that of the trigonometric function (Fig. 3). Sagittal height surface profiles are calculated by cumulative distribution functions with different channel lengths and two types of contour line distributions, weighted and superimposed, resulting in a new surface profile (Fig. 4). This significantly widens the progressive channel width, achieves a smooth transition in astigmatic gradient changes, and results in maximum astigmatism distributed on both sides of the nasal area of the lens (Fig. 11). From the perspective of meridional power, a linear combination of two functions forms a new function (Fig. 5), achieving a smooth transition in the central meridional power, widening the progressive channel width, reducing the rate of focal power change, and minimizing peripheral maximum astigmatism (Fig. 12, Table 4). The machining results align closely with simulation results, demonstrating that this optimization method effectively achieves the optical performance enhancement of freeform progressive addition lenses.

We propose a new meridional power distribution based on the cumulative distribution function. We conduct a comparative analysis to assess the impact of this function, as opposed to an octave polynomial function and a trigonometric function, on the absolute curvature values affecting the width of intermediate astigmatism. To address the issue of narrow channel width, we employ different channel lengths of the cumulative distribution function and two types of contour line distributions for sagittal height surface profile calculations. Weight values are assigned for weighted superimposition, leading to the creation of a new surface profile and a significant widening of the channel width. Additionally, from the perspective of meridional power, a linear combination of two functions is employed to form a new function, facilitating a smooth transition in the central meridional power and reducing the rate of focal power change. Finally, we conduct optical simulations to analyze lens focal power and manufacture and quality-test three designed lenses using a freeform machining tool to validate the accuracy of experimental results. Building on our findings, future research can further focus on optimizing the design methods for meridional power, aiming to discover more effective mathematical functions or technical approaches to enhance overall lens width, thus achieving superior optical performance in the design of progressive addition lenses.