This paper presents ray tracing algorithms to evaluate the geometrical modulation transfer function (GMTF) of optical lens system. There are two kinds of ray tracings methods that can be applied to help simulate the point spread function (PSF) in the image plane, for example, paraxial optics and real ray tracings. The paraxial optics ray tracing is used to calculate the first-order properties such as the effective focal length (EFL) and the entrance pupil position through less cost of computation. However, the PSF could have a large tolerance by only using paraxial optics ray tracing for simulation. Some formulas for real ray tracing are applied in the sagittal and tangential line spread function (LSF). The algorithms are developed to demonstrate the simulation of LSF. Finally, the GMTF is evaluated after the fast Fourier transform (FFT) of the LSF.

Optical products are very popular for people today. The camera function has also been widespread applied for various circumstances [

The MTF is the amplitude term of optical transfer function (OTF) that is similar to the transfer function of linear system. The transfer function is regarded as a major characteristic in the linear system. We cannot derive or evaluate the output signal without being given the transfer function. There are some proposals that present their methods to approach the transfer function [

The MTF can be evaluated from either the geometrical optics or diffraction calculation [

Although the geometrical PSF can be used to determine the system performance, the MTF is always investigated to look into the image resolution of lens system. By the Fourier transform of the geometrical PSF, the MTF can be calculated to analyze the resolution of image in the lens systems. The organization of this paper is as follows. The relation between the lens system and linear system is explained in Section

The transfer function of a linear system is to represent the relation between input and output signals, as shown in Figure

The relation between input and output for a linear system.

From the linear system concept, we can obtain the transfer function by replacing the input

The relation between the object points and their image responses.

A camera lens is used to catch an object with multiple points into its image plane. Each object point is like a point source to diverge its power toward the outside. In geometrical optics, the direction of wavefront propagation is regarded as the rays. The ray tracing is to investigate how the rays travel from an object point to image plane. There are two processes used in ray tracing, that is, refraction process and transfer process.

An optical lens system is usually assembled by multiple lens elements. In order to collect the rays from an object point at image plane, the materials in both sides of a surface must have different refractive indices, which results in the rays deviating at the conjunctive surface. Figure

A ray traces from one material to the other material.

From the triangles PCA and

After refraction process, a ray travels from one surface to another surface. There is only one material between two surfaces. Figure

A ray travels between two surfaces.

The common lens system is to employ lens elements with spherical surfaces. Let us discuss a real ray transfer process between spherical surfaces [

A ray propagates between two spherical surfaces [

Figure

A point

The refraction of real ray tracing follows Snell’s law. Figure

The presentation of Snell’s law on vector form.

A lens system with three lens elements shown in Figure

The simulation lens parameters.

Radius |
Thickness |
Refractive index | |
---|---|---|---|

Object | Infinity | Infinity | 1 |

1 | 22.014 | 3.259 | 1.62 |

2 | −435.76 | 6.008 | 1 |

3 | −22.213 | 1 | 1.62 |

4 | 20.292 | 4.75 | 1 |

5 | 79.684 | 2.952 | 1.62 |

6 | −18.390 | 42.208 | 1 |

typeXYZ TProcess(typeXYZ p_1,typeLMN LMN,double d,double C,double

double x0,y0,z0;

double delta,F,G;

double tmp;

typeXYZ p;

x0=p_1.x+LMN.L/LMN.N

y0=p_1.y+LMN.M/LMN.N

z0=0;

F=C

G=LMN.N-C

tmp=G

delta=F/(G+

p.x=x0+LMN.L

p.y=y0+LMN.M

p.z=LMN.N

return p;

typeLMN RProcess(typeLMN LMN,typeXYZ p,double C,double n,double np,double cosI){

double cosIp;

double tmp;

double K;

typeLMN LMNp;

tmp=n/np;

tmp=1-tmp

cosIp=sqrt(tmp);

K=(np

tmp=n

LMNp.L=tmp/np;

tmp=n

LMNp.M=tmp/np; //

tmp=n

LMNp.N=tmp/np;

return LMNp;

The structure of the simulation lens system.

We collect these tracing rays on image plane to investigate the LSF. Sampling the rays on the image plane can count the number of the rays to imitate the power distribution. The LSP is sampled based on taking

The simulated LSF at field 0°.

The simulated LSF at field 14°: (a)

The simulated LSF at field 20°: (a)

Modulation transfer function by proposed simulation.

Modulation transfer function by using Zemax.

This paper presents that the transfer function of a lens system is similar to that of a linear system. A transfer function of the linear system can be evaluated through inputting an impulse signal, making the fact that the output would be the transfer function. The same concept of linear system is used to respect a point source at object space as the impulse signal to simulate its response at image space. The paraxial ray tracing is applied to calculate the essential lens properties like the EFL and pupil position. However, the image of a point source would reveal a large tolerance when only using paraxial optics ray tracing for simulation. The real ray tracing method is applied to evaluate the LPF. The algorithms for ray tracing were also developed to simulate the LPF of a lens system. After the FFT process, we can evaluate the MTF to determine the resolution of a lens system.

There is no conflict of interests regarding the publication of this manuscript.