## 1.

## INTRODUTION

Lens design (LD) is a familiar topic for us. Nowadays, lens design become more and more important due to the increasing demand for advanced optical system, such as high FOV Helmet-Mounted Display (HMD) system and hyper numerical aperture lithography system. In our opinion, lens design could be divided into two parts. One can be called as lens optimization, another one is known as first order structure generating. For a long time many researchers focused on the former work, and a series of achievements had been published. As we know, lens optimization is not a simple work which can not be solved in the form of analytic method. In terms of lens optimization, people usually optimized the curvature of lens and the distance between each lens. A large number of algorithms had been proposed on this issue. However, few people investigate the method that could generate first order optical structure.

Delano first put forward the concept of Delano diagram ^{[1]}. He described the first order quantity of optical system in a Delano diagram. It is an excellent and very simple method to represent the properties of optical system. According to his theory, if the Lagrange invariant is known, all the first order quantity can be calculated easily. Later, F. J. López-López developed the Delano diagram theory ^{[3]} and R. J. Pegis published a paper for some design example with use of Delano diagram ^{[2]}. In 1988, Mechae E. Harrigan design a grin rod by virtue of Delano diagram ^{[4]}. David Kessler applied this theory into laser system design ^{[5]} and S. L. Zhuang applied it into the design of prototype lenses ^{[6]}. But it is a pity that there not existed a general method for calculating the initial optical structure.

In this paper, we will give a general method that can generate a first order structure of an infinite conjugated distance system here. For further study, we have developed a Graphic User Interface (GUI) based on MFC. Then a practical system is designed with the method we proposed.

## 2.

## OVERIEW OF DELANO DIAGRAM

## 2.1

### The optical system’s first order quantity in Delano diagram

The basic Delano diagram is shown in Fig.1. From Fig.1, we can see that it is an optical system which contains three optical components. The *y* and *y _{bar}* represent the height of marginal ray and chief ray, respectively. The point

*J*and

*M*are object point and image point, respectively. Here,

*A*,

*B*and

*C*represent three optical components, respectively. Point P in this figure denotes the system’s principle plane which is an intersect point of the line

*JA*and line

*CM*. The line

*OF’*is parallel to the line

*JA*, which represent the image focal length of the schematic system in Fig.1. The expression for focal length has been given in Delano’s paper.

Here, the parameter *L* is the Lagrange invariant which can be expressed as

*L* can be known according to the design requirement. In formula (2), *u*_{bar} and *u* represent angle of field of view (FOV) and aperture angle, respectively. The *y*_{bar} stands for the height of chief ray, and *y* is the height of the first paraxial ray accordingly. Further, point *S’* and point *S”* represent entrance pupil and excite pupil respectively. The distance between two adjacent components can be expressed as in Delano’s theory

For show how can we calculate the image focus length, we set an example here. In Fig.1, We also can know that line *OF ^{′}_{c}* parallel to line BC. On the basis of equation (1), the image focal length of component

*C*can be obtained as

With a similar method, the focal length of component *B* and component *C* can all be calculated.

## 2.2

### The control point quantity

Thus far, the relational expressions of focal length and distance between each optical component have been shown. But, how can we get optimal parameters? As we know, when we design an optical system, the focal length *f* and focal ration *F#* is the basic data of the optical system. To obtain the optimal parameters, we should find some control points. Unfortunately, Delano did not tell us how we can constrain the optical system which satisfies the design requirement in his original paper. To clarify our method, a schematic diagram is shown in Fig.2.

Considering a very simple structure, the optical system is composed of only one optical component, as Fig.2 shows. Here, we define the object height as *HJ* and the image height as *HM*, which is respectively represented by *OJ* and *OM* in Fig.2. Assume that *D* is the entrance pupil diameter. The chief ray angle then can be expressed as

According to the geometrical relationship in Fig.2, the coordinate of point P can be denoted by

For further study, we can solve the point *F′* which is shown as below according to analytic geometry method.

When we design an optical system, the focal length, object height and image height are all known to us as basic parameters. So, combing Eq. (1) to Eq. (7), the point P can be expressed as below by given parameter.

According to Eq. (7), (8) and (9), we can see that the parameters are represented by known quantity. If the optical system is composed of multiple components, the calculation method is same. During primary lens design, the object height, image height and the parameter of each optical component should be constrained to satisfy the design requirements.

## 3.

## THE WAY TO OBTAIN THE BEST SOLUTION

## 3.1

### objective function

Now, we consider that the issue how to optimize these parameters. Assume that the number of the optical component is N. Fig.3 is the schematic Delano diagram for multiple components.

The first component and the last component locate in line *JP _{1}* and line

*MP*

_{N}, respectively. Considering that the L is positive, so we need the line from original point to last point is clockwise. We can see that infinite possible first-order solutions can be existed from Fig.3. A new problem has arisen, which is the best design result? As far as we know, the aberration is induced by deflection of ray transmission. If the total deflection angle of all components got its minimum, we think it will be the best first-order optical structure. Then the objective function can be expressed as

The marginal and chief ray angle can be obtained by the paraxial ray tracing. The angle also can be expressed by *y* and *y*_{bar}. The rest problem is that how we make *Obj* be the minimum. We will discuss the algorithm which can minimize the objective function bellow.

## 3.2

### Particle swarm algorithm

Many algorithms about optical design have been proposed recent years. As we know, the most popular local search method is Damped Least Squares. It is widely used as an optimized method in commercial software such as ZEMAX and CODE V. Here, we do not adapt this algorithm because it may easily fall into local minimum. To overcome this problem, global searching algorithm has been adopted to get the best solve. Particle swarm algorithm is a stochastic optimization algorithm based on swarm intelligence. It has been widely used in many fields due to its fast convergence speed and good robustness. Hua Qin firstly applied this algorithm in optical design, and a good design results had been obtained. Every particle represent a potential solve during particle swarm optimization. We write the velocity of the *i*^{th} particle in the *N* dimensional solve space as

Similarly, the position of *i*^{th} particle can be written as

Eq. (10) can serve as fitness value. The position and velocity for each particle should be updated in the way as followed where *P*_{i} is the local optimal solution and *P*_{ibest} is the global optimal solution.

## 4.

## AN EXAMPLE OF OPTICAL DESIGN WITH THE PROPOSED METHOD

Combine the physical model and particle swarm algorithm, we make a program in MATLAB at the beginning, and then a GUI has been built based on Microsoft Foundation Classes (MFC). Fig.4 is a glimpse of the GUI.

We show an example of Head Mounted Display (HMD) optical system here. This is an infinity conjugated system. The image height is 32.5mm. The FOV is 90 degree to increase the size of eyebox and the bigger eyebox will benefit the users’ illusion of immersion. The entrance pupil diameter is 8mm. So the Lagrange invariant is 4. In order to decrease the total axial length and leave enough space for the other optical element, we set the number of optical component is 2 to get an initial structure. Table.1 shows the first order structure based on modified Delano diagram and particle swarm algorithm. Fig.5(a) is a simple ray trace diagram drawn by the proposed GUI and (b) shows the 2D layout diagram in ZEMAX with the surface type treated as paraxial.

## Table 1.

The first order structure of the HMD system

Element | Distance/mm | Focal/mm | ybar/mm | y/mm |
---|---|---|---|---|

Entrance pupil | 16.000 | - | 0 | 4.000 |

2 | 18.000 | 49.535 | 16.000 | 4.000 |

3 | 26.343 | 160.028 | 24.323 | 2.547 |

image | 32.500 | 0 |

Then, for obtaining practice lens data, we firstly set the lens as equal curvature, and the glass can be made BK7 glass, then put the parameters into commercial optical software such as ZEMAX to balance the aberrations. In ZEMAX software, we set the glasses type substituted. After lens optimizations, the image quality of the proposed optical system can easily meet the requirement. Fig.6 represented the ultimate structure.

The FOV here is very large and this can result in the large distortion. Fig.7 is the ultimate distortion diagram. We can see the maximum distortion is about 10% and this is acceptable here. The spot diagram is shown in Fig.8. And the MTF is shown in Fig.9.

For further study, we give the Delano diagram of the initial structure and final structure in Fig.10. In Fig.10, we can see the final diagram is similar to the initial Delano diagram.

In this paper, we proposed a useful method for generating first-order structure combined Delano diagram and particle swarm algorithm. For designer to compute an infinity conjugated system, we just need to know the FOV, image height, Lagrange invariant and component numbers. And then the best optimal first order structure can be calculated. This initial first order structure can be loaded into ZEMAX, and it can be easily converted a practical structure. The proposed method can provide a technical reference for optical system design.

This work is supported by National Science and Technology Major Project of China (Gran NO.2011 ZX02402), International Science & Technology Cooperation Program of China (Gran NO.2011DFR10010), Science and Technology Commission of Shanghai Municipality under Grant 14YF1406300 and Chinese Academy of Sciences Visiting Professorship for Senior International Scientist (Gran NO.2013T1G0041).