# Aspherical Lens Design Using Genetic Algorithm for Reducing Aberrations in Multifocal Artificial Intraocular Lens

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology of Optical IOL Design

## 3. Theory and Formula Analysis

#### 3.1. Aspherical Lens

Initial Conditions of Design | |
---|---|

Anterior radius of the crystalline lens | 0 mm |

Posterior radius of the crystalline lens | 0.0806 mm |

Anterior thickness of the crystalline lens | 1.59 mm |

Posterior thickness of the crystalline lens | 2.43 mm |

**Figure 1.**Two-dimensional (2D) layout diagram of the initial crystalline lens [17]. (

**a**) 5 mm; (

**b**) 6 mm.

**Table 2.**Third-order aberrations in the initial human eye model. SA: spherical aberration; TCO: tangential coma; RMS: root mean square.

Aberration | SA | TCO | RMS | |
---|---|---|---|---|

Pupil Size | ||||

5 mm | −0.016148 | −0.033068 | 0.005315 | |

6 mm | −0.027904 | −0.047618 | 0.01264 |

**Figure 4.**The 2D layout illustrating 550 degrees myopia and 175 degrees astigmatism. (

**a**) 5 mm; (

**b**) 6 mm.

Aberration | SA | TCO | RMS | |
---|---|---|---|---|

Pupil Size | ||||

5 mm | −0.073302 | −0.052464 | 0.537989 | |

6 mm | −0.126666 | −0.075548 | 0.698584 |

**Figure 6.**Modulation transfer function (MTF) plot for 550 degrees myopia and 175 degrees astigmatism. (

**a**) 5 mm; (

**b**) 6 mm.

Surface# | Surface Type | Y Radius | Thickness | Glass | Refract Mode | Y Semi-Aperture |
---|---|---|---|---|---|---|

#Object | Sphere | 0 | Infinity | - | Refract | - |

#1 | Conic | 0.1287 | 0.6 | “cornea” | Refract | 4.6712 |

#2 | Conic | 0.1563 | 5.195 | “humor” | Refract | 4.3066 |

#Stop | Sphere | 0 | 0 | “humor” | Refract | 3 |

#4 | Conic | 0.0707 | 0.0915 | ACRYLIC_SPECIAL | Refract | 2.0413 |

#5 | Sphere | 0 | 1.2623 | ACRYLIC_SPECIAL | Refract | 2.2640 |

#6 | Conic | −0.0853 | 16.27 | “humor” | Refract | 2.6105 |

#Image | Sphere | −0.0909 | 0 | “humor” | Refract | 7.7009 |

Surface# | Surface Type | Y Radius | Thickness | Glass | Refract Mode | Y Semi-Aperture |
---|---|---|---|---|---|---|

#Object | Sphere | 0 | Infinity | - | Refract | - |

#1 | Conic | 0.1287 | 0.6 | “cornea” | Refract | 4.6712 |

#2 | Conic | 0.1563 | 5.195 | “humor” | Refract | 4.3066 |

#Stop | Sphere | 0 | 0 | “humor” | Refract | 3 |

#4 | Conic | 0.0755 | 0.9283 | ACRYLIC_SPECIAL | Refract | 2.0408 |

#5 | Sphere | 0 | 1.0016 | ACRYLIC_SPECIAL | Refract | 2.2688 |

#6 | Conic | −0.0849 | 16.27 | “humor” | Refract | 2.5282 |

#Image | Sphere | −0.0909 | 0 | “humor” | Refract | 7.6499 |

Surface# | Surface Type | Y Radius | Thickness | Glass | Refract Mode | Y Semi-Aperture |
---|---|---|---|---|---|---|

#Object | Sphere | 0 | Infinity | - | Refract | - |

#1 | Conic | 0.1287 | 0.6 | “cornea” | Refract | 4.6712 |

#2 | Conic | 0.1563 | 5.195 | “humor” | Refract | 4.3066 |

#Stop | Sphere | 0 | 0 | “humor” | Refract | 3 |

#4 | Conic | 0.0819 | 0.9763 | ACRYLIC_SPECIAL | Refract | 2.0413 |

#5 | Sphere | 0 | 1.2627 | ACRYLIC_SPECIAL | Refract | 2.2640 |

#6 | Conic | −0.0702 | 16.27 | “humor” | Refract | 2.6105 |

#Image | Sphere | −0.0909 | 0 | “humor” | Refract | 7.7009 |

Surface# | Surface Type | Y Radius | Thickness | Glass | Refract Mode | Y Semi-Aperture |
---|---|---|---|---|---|---|

#Object | Sphere | 0 | Infinity | - | Refract | - |

#1 | Conic | 0.1287 | 0.6 | “cornea” | Refract | 4.6712 |

#2 | Conic | 0.1563 | 5.195 | “humor” | Refract | 4.3066 |

#Stop | Sphere | 0 | 0 | “humor” | Refract | 3 |

#4 | Conic | 0.0791 | 1.1739 | ACRYLIC_SPECIAL | Refract | 2.0408 |

#5 | Sphere | 0 | 1.1021 | ACRYLIC_SPECIAL | Refract | 2.2688 |

#6 | Conic | −0.0723 | 16.27 | “humor” | Refract | 2.5282 |

#Image | Sphere | −0.0909 | 0 | “humor” | Refract | 7.6499 |

Aberration | SA | TCO | RMS | |
---|---|---|---|---|

Pupil Size | ||||

5 mm | 0.01 | −0.0162 | 0.032718 | |

6 mm | 0.01 | −0.01781 | 0.038357 |

**Figure 7.**The 2D layout after applying the built-in optimization method in CODE V. (

**a**) 5 mm; (

**b**) 6 mm.

**Figure 8.**Spot size diagram after applying the built-in optimization method in CODE V. (

**a**) 5 mm; (

**b**) 6 mm.

#### 3.2. Principle of Genetic Algorithm

_{roulette}(i) = (fit

_{max}+ fit

_{min}) – fit(i) for i = 1,2,..., pop_size

_{max}and fit

_{min}are the maximum and minimum values of fin(i).

_{1}, x

_{2},…, x

_{2}) and Y = (y

_{1}, y

_{2},…, y

_{2}). The offspring of these two parents is the sequence Z = (z

_{1}, z

_{2},…, z

_{2}) and its crossover genes are defined by:

_{m}is defined and a random number between 0 and 1 is generated in the GA process. The mutation operation is implemented when α > p

_{m}. The mutation operation used in the proposal is defined as:

_{1}to w

_{2}are assigned the values all of 1. The population size, generation, crossover rate, and mutation rate are set to 100, 70, 0.8 and 0.2 respectively in running the optimization.

Aberration | SA | TCO | RMS | |
---|---|---|---|---|

Pupil Size | ||||

5 mm | 0.032994 | −0.011763 | 0.020633 | |

6 mm | 0.014618 | −0.010201 | 0.031816 |

**Table 10.**Improvement rate of third-order aberrations after applying the proposed GA optimization method for pupil size 5 mm.

Quality Characteristics | SA | TCO |
---|---|---|

550 degrees myopia and 175 degrees astigmatism condition | −0.073302 | −0.052464 |

GA optimization | 0.032994 | −0.011763 |

Difference values | 0.040308 | 0.040701 |

Improvement rate (%) | 53.98% | 77.57% |

**Table 11.**Improvement rate of third-order aberrations after applying the proposed GA optimization method for pupil size 6 mm.

Quality Characteristics | SA | TCO |
---|---|---|

550 degrees myopia and 175 degrees astigmatism condition | −0.126666 | −0.075548 |

GA optimization | 0.014618 | −0.010201 |

Difference values | 0.112048 | 0.065347 |

Improvement rate (%) | 88.45% | 86.49% |

**Table 12.**Comparisons of third-order aberrations of the built-in and proposed optimization methods for 550 degrees myopia and 175 degrees astigmatism for pupil size 5 mm.

Quality Characteristics | SA | TCO | RMS |
---|---|---|---|

CODE V optimization | 0.01 | −0.016200 | 0.032718 |

GA optimization | 0.032994 | −0.011763 | 0.020633 |

Difference values | 0.022 | 0.004437 | 0.012 |

Improvement rate (%) | −229% | 27.38% | 36.93% |

**Table 13.**Comparisons of third-order aberrations of the CODE V built-in and proposed optimization methods for 550 degrees myopia and 175 degrees astigmatism for pupil size 6 mm.

Quality Characteristics | SA | TCO | RMS |
---|---|---|---|

CODE V optimization | 0.01 | −0.01781 | 0.038357 |

GA optimization | 0.014618 | −0.010201 | 0.031816 |

Difference values | 0.004618 | 0.007609 | 0.006541 |

Improvement rate (%) | −46.18% | 42.72% | 17.05% |

Optimal State | 10 lp/mm | 20 lp/mm | 30 lp/mm |
---|---|---|---|

Myopic and astigmatism | 0.003 | 0.001 | 0 |

Internal CODE V | 0.768 | 0.356 | 0.145 |

GA | 0.901 | 0.657 | 0.389 |

Improvement rate(%) | 14.76% | 45.81% | 62.72% |

Optimal State | 10 lp/mm | 20 lp/mm | 30 lp/mm |
---|---|---|---|

Myopic and astigmatism | 0.005 | 0 | 0.007 |

Internal CODE V | 0.695 | 0.253 | 0.115 |

GA | 0.783 | 0.391 | 0.177 |

Improvement rate(%) | 11.23% | 35.29% | 35.02% |

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Yen, C.-T.; Jin, S.-C.
Aspherical Lens Design Using Genetic Algorithm for Reducing Aberrations in Multifocal Artificial Intraocular Lens. *Materials* **2015**, *8*, 6309-6325.
https://doi.org/10.3390/ma8095305

**AMA Style**

Yen C-T, Jin S-C.
Aspherical Lens Design Using Genetic Algorithm for Reducing Aberrations in Multifocal Artificial Intraocular Lens. *Materials*. 2015; 8(9):6309-6325.
https://doi.org/10.3390/ma8095305

**Chicago/Turabian Style**

Yen, Chih-Ta, and Shih-Cyuan Jin.
2015. "Aspherical Lens Design Using Genetic Algorithm for Reducing Aberrations in Multifocal Artificial Intraocular Lens" *Materials* 8, no. 9: 6309-6325.
https://doi.org/10.3390/ma8095305