The image quality of a lens design can be good but the cost of the glass from which it is made can be quite high. Kidger [1] considered variation in cost between glasses used for this purpose. In the large size lenses, the cost of expensive glass could be prohibitive; but in small size lenses, the materials make up only a small percentage of the total cost of the lens, so expensive materials might well be acceptable in this case. Chromatic aberration occurs because of the different refractive indices of lenses for different wavelengths of light [2,3]. If two different types of glasses are combined into a thin two-element system, a paraxial chromatic aberration will develop [4]. Robb [5] developed a method using two different types of glass for the correction of axial color for at least three wavelengths. Certain combinations might be found for correction at four and five wavelengths. In 1983, Sharma and Gopal [6] used the double-graph technique to produce doublet designs. Then, in 2001, Rayces and Rosete-Aguilar [7,8] described a method to select pairs of glasses for both thin cemented achromatic doublets and thin aplanatic achromatic doublets with a reduced secondary spectrum. In another study, Banerjee and Hazra [9] used a genetic algorithm for the structural design of cemented doublets. The aim of this work is to minimize chromatic aberration by using a low-cost optimal double-prism method. An efficient approach for finding an optimum design is also proposed.

An illustration of ray tracing in and out of a prism is shown in Fig. 1. The angle of a ray in the normal direction from the prism surface is positive in the anticlockwise

direction, and negative in the clockwise direction. In this figure, I₁ and I₂ are the incident angles of the first and the second surfaces, respectively; I₁' and I₂' are the refractive angles of the first and the second surfaces, respectively; A is the apex angle of the prism. The sign of the apex angle is positive in the vertical, and negative in the inverse direction. The ray thus deviates through an angle of (I₁'？ I₁) at the first surface. At the second surface, the ray deviates by (I₂'－I₂), so the angle of deviation D of the ray is given by

If we consider real ray tracing, the deviation angle [10] presented by

If all the angles of the design are small, the equations for the paraxial ray tracing can be obtained, and the paraxial deviation angle [11] given by

The paraxial primary color ε_F,C is the difference in deviation angles between F line (0.48613 μm) and C line (0.65627 μm) as shown. Thus

where D_d is the paraxial deviation angle of d line (0.58756 μm); V_d=(n_d - 1)/(n_F - n_C) is the Abbe number; and n_C, n_d, n_F are the refractive indices of C, d, and F lines, respectively. Consequently, the paraxial second spectrum ε_d,C is the difference in the deviation angles between d line and C line. This can show as

where P_d,c=(n_d-n_C)/(n_F-n_C) is the relative partial dispersion.

In the paraxial ray tracing of doublet prisms, the angle of deviation of d-line light is defined as 3°, so the primary color ε_F,C is zero [4,5]. These equations can be written as

where the paraxial deviation angles of the doublet prisms are D_d1 and D_d2, respectively. The primary chromatic aberration is ε_F,C. Solving for D_d1 and D_d2 from the above equations, we obtain

and

The apex angles of the doublet prisms are expressed as

and

Accordingly the paraxial primary color is zero, but there are still some paraxial secondary spectra. Thus the paraxial secondary spectrum can be defined as

where the paraxial primary chromatic aberrations of the doublet prisms are ε_F,C1 and ε_F,C2, respectively.

We choose the Schott glass [12] for the design because of the large number of types of glasses that are available. The Abbe numbers of the different glasses have been ranked. There are 119 different optical glasses all with different prices. The price of N-BK7 is the lowest. The relative price (RP) is found by comparison and the results are indexed. To avoid using the most expensive types of glass in the design, the twenty-nine types with a relative price RP ≥ 17 as well as those with no marked prices are eliminated. The costs of glasses such as N-KZFS11, N-PK51, and N-LASF31A, are much higher than the others. Those with no marked prices are molding glasses or new types of glasses. The internal transmittance is the transmittance of light excluding reflection loss. The N-SF6HT and N-SF57HT glasses offer improved transmittance in the visible spectral range especially in the blue-violet area. Moreover, since the V_d, n_d, and P_d,C of N-SF6HT and N-SF57HT are all the same as those of N-SF6 and N-SF57, the corrected chromatic aberrations will be almost the same. Thus we can neglect the N-SF6HT and N-SF57HT glasses. A total of seventy types of glasses were chosen for doublet prisms design to correct chromatic aberration.

In Eqs. (2) and (3), it can be seen that the there is a difference in the deviation angles between the real ray and the paraxial ray. The deviation angle of the paraxial ray is unconcerned with the incident angle I₁, but the deviation angle of the real ray is related to the incident angle I₁. When the incident angle increases, the real and paraxial chromatic aberrations will be very different. The real chromatic aberration is corrected for optimization. The damped least-squares method [13-15] is applied for an optimization design of the chromatic aberration. A merit function is defined as the summation of the squared values of the weighting differences between the aberrations and their target values. The formula can be written as

where m is the total summation number; the w_i is the weighting factor; e_i is the aberration and t_i is the target value. We define the function f_i(x₁, x₂, ‥‥, x_n) as

Before optimization, the n variables are denoted as x₁₀, x₂₀, ···., x_n0; the m aberrations before the optimization are f₁₀, f₂₀, ···., f_m0. After the optimization process, the variables are denoted as x₁, x₂, ···, x_n, and the aberrations as f₁, f₂, ···., f_m. Here, we define a matrix A, in which the elements are

We then get the equation

where A^T is the transpose matrix of A; I is a unit matrix, p is a damping factor; and f₀ is the matrix containing the elements f₁₀, f₂₀,···. f_m0. If x and x₀ are the matrices containing the elements x₁, x₂, ···, x_n, and x₁₀, x₂₀, ···, x_n0, respectively, we can obtain

The doublet prisms have two apex angles. The angle of deviation of d line is 3°, and the primary color is eliminated. The steps are repeated to reduce the secondary spectra of the doublet prisms. Using Eq. (12), the correct doublet prisms combination can be found by choosing the smaller (P_dC1-P_dC2) and the larger (V_d1-V_d2). When the (P_dC1-P_dC2)/(V_d1-V_d2) is close to zero, the chromatic aberration is smaller. Figure 2 shows the relative partial dispersion with respect to the V_d number. We chose six groups from A to F for minimizing the paraxial chromatic aberration. The design results are listed in Table 1, where the CA is the area of the chromatic aberration curve. Figure 3 shows the chromatic aberration curves. The

horizontal ideal line, which denotes the angle of deviation of d line is 3°, has been set to zero for the chromatic aberration. The other lines are described as the chromatic aberration of doublet prisms groups from A to F.

We choose group A from Table 1 as an example as the initial value. The doublet prisms are made of N-PSK52A and N-SK5 glasses. We set I₁ and I₃ as the incident angles for the first surface of the first and the second prisms, respectively; A₁ and A₂ are the apex angles of the first and the second prisms, respectively. An illustration of the ray tracing of the doublet prisms is shown in Fig. 4. At the initial values, we set I₁ = 0, I₃ = 0, A₁ = 24.215°, and A₂ = -15.336°, as listed in Table 2. We can calculate the D_d = 4.148°, ε_d,C = 6.332×10^-3, ε_F,C = 2.078×10^-2, and CA= 9.083×10^-4 for the real ray tracing. The D_d and chromatic aberrations between the real and the paraxial rays (ε_F,C, ε_d,C, CA) are very different. The real chromatic aberration is corrected by an optimization program.

The merit function consists of three terms. The first term is the deviation angle D_d of the real ray for the doublet prisms, the second is the real primary color aberration ε_F,C of the doublet prisms, and the last is the real secondary spectrum ε_d,C of the doublet prisms. If the target values are t_Dd=3, t_εd,C=0, and t_εF,C=0, then the merit function is given by

where the weighting factors are w₁= 1, w₂= 20, w₃= 20, respectively. During the doublet prisms optimization, we

will consider some sort of aberration balance, a sensible choice of weighting factors is essential if we are to achieve the best possible performance. In the optimization process, we think that two aberrations of ε_d,C and ε_F,C are more rigorous than that of D_d, and then the target values of both |ε_d,C| and |ε_F,C| are twenty times smaller than those of |D_d-3|. It is therefore sensible, often but not always, for weighting factors to be smaller for larger target values.

We use four variables as x₁, x₂, x₃, and x₄, to represent the incident angle I₁ of the first prism, the incident angle I₃ of the second prism, the apex angle A₁ of the first prism, the apex angle A₂ of the second prism, respectively. Before optimization, the variables are denoted as x₁₀, x₂₀, x₃₀, and x₄₀, which correspond to I₁= 0, I₃= 0, A₁= 24.215, and A₂= -15.336, respectively. The optimization results are listed in Table 3, and the real chromatic aberration curve is shown in Fig. 5.

Except for fixing the deviation angle of real ray D_d = 3°, we can optimize the area CA of the real chromatic aberration curve to obtain an optimization design. The merit function is defined as

where the target values are t_Dd=3, and t_CA=0. The optimized results are listed in Table 4, and the real chromatic aberration curve is shown in Fig. 6.

As mentioned before, figure 4 shows the ray tracing of the doublet prisms, where I₁, I₁′, A₂, I₄, and I₄′ are negative and A₁, I₂, I₂′, I₃, and I₃′ are positive. When the incident angle of the first prism is I₂ > θ_C, the total internal reflection of the ray appears. The critical angle θ_C is given by

where n is the refractive index of the prism.

We choose group B from Table 5 as an example. The two types of glass used in the doublet prisms are N-BAK2 and N-LAK34, their refractive indices are n_d1=1.53996 and n_d2=1.72916, respectively, and the critical angles are θ_C1=40.494° and θ_C2=-35.332°, respectively. In order to avoid total reflection, the incident angle I₁ of the first surface of

the first prism must be

and the incident angle I₃ of the first surface of the second prism is required to be

A flow-chart of the optimization program for the doublet prisms design is shown in Fig. 7. First, the program selects (P_dC1-P_dC2)/(V_d1-V_d2), the minimal value of the doublet glasses. Second, the program uses the paraxial ray equations from Eq. (6) to Eq. (11) to fix the deviation angle D_d =3°, and eliminate the primary color. Then, the two apex angles are obtained. Third, the program sets t_Dd=3, t_？dC=0, and t_？F,C=0 to optimize the real primary color. Finally, t_Dd=3 and t_CA=0 are used to optimize the area of the real chromatic aberration curve, until the real chromatic aberration is a minimum. The

results for the optimized designs A to F are listed in Table 5. The chromatic aberration curves A to E corresponding to the optimal designs are shown in Fig. 8. This indicates that the proposed design method is effective in minimizing the chromatic aberration.

A low-cost optimal double-prism method combined with the developed MATLAB program to correct chromatic aberration has been presented. In comparison of the doublet-prism designs shown in Tables 1 and 5, shows that the areas between the paraxial and real chromatic aberration curves are similar. We can quickly find the best combination of doublet prisms by choosing the materials with small differences in relative partial dispersion and large differences in V_d number, and minimizing the real

chromatic aberration of doublet prisms by an optimization program.