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.

Fig.1

Delano diagram

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.

Fig.2

Schematic diagram for constrain points

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.

3.1

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.

Fig.3

Schematic diagram for multiple components in Delano diagram

The first component and the last component locate in line JP₁ 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.