A Low-cost Optimization Design for Minimizing Chromatic Aberration by Doublet Prisms
- Author: Sun Wen-Shing, Tien Chuen-Lin, Sun Ching-Cherng, Lee Ching-Chun
- Organization: Sun Wen-Shing; Tien Chuen-Lin; Sun Ching-Cherng; Lee Ching-Chun
- Publish: Journal of the Optical Society of Korea Volume 16, Issue4, p336~342, 25 Dec 2012
A low-cost optimal double-prism method is proposed by using the developed MATLAB program to correct chromatic aberration. We present an efficient approach to choose a couple of low-cost glasses to obtain a low aberration double prism. The doublet prisms were made of two lead-free glasses. The relative partial dispersion of the two lead-free glasses is identical and their Abbe numbers are different greatly. The proposed design aims to minimize chromatic aberration, such as in apochromats, for paraxial ray tracing. Finally, an optimization design for real ray tracing can be evaluated by the chromatic aberration curve with a minimal area.
Doublet prisms , Optimization , Chromatic aberration , Ray tracing
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  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 . Robb  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  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  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,
I1and I2are 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; Ais 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 Dof the ray is given by
If we consider real ray tracing, the deviation angle  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  given by
The paraxial primary color
εF,Cis the difference in deviation angles between F line (0.48613 μm) and C line (0.65627 μm) as shown. Thus
Ddis the paraxial deviation angle of d line (0.58756 μm); Vd= (nd - 1)/(nF - nC)is the Abbe number; and nC, nd, nFare the refractive indices of C, d, and F lines, respectively. Consequently, the paraxial second spectrum εd,Cis the difference in the deviation angles between d line and C line. This can show as
Pd,c=( nd- nC)/( nF- nC) is the relative partial dispersion.
where the paraxial deviation angles of the doublet prisms are
Dd1and Dd2, respectively. The primary chromatic aberration is εF,C. Solving for Dd1and Dd2from the above equations, we obtain
The apex angles of the doublet prisms are expressed as
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
εF,C1and εF,C2, respectively.
We choose the Schott glass  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
Vd, nd, and Pd,Cof 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
mis the total summation number; the wiis the weighting factor; eiis the aberration and tiis the target value. We define the function fi( x1, x2, ‥‥, xn) as
Before optimization, the
nvariables are denoted as x10, x20, ···., xn0; the maberrations before the optimization are f10, f20, ···., fm0. After the optimization process, the variables are denoted as x1, x2, ···, xn, and the aberrations as f1, f2, ···., fm. Here, we define a matrix A, in which the elements are
We then get the equation
ATis the transpose matrix of A; Iis a unit matrix, pis a damping factor; and f0is the matrix containing the elements f10, f20,···. fm0. If xand x0are the matrices containing the elements x1, x2, ···, xn, and x10, x20, ···, xn0, 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 (
PdC1- PdC2) and the larger ( Vd1- Vd2). When the ( PdC1- PdC2)/( Vd1- Vd2) is close to zero, the chromatic aberration is smaller. Figure 2 shows the relative partial dispersion with respect to the Vdnumber. 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 Dd= 4.148°, εd,C= 6.332×10-3, εF,C= 2.078×10-2, and CA= 9.083×10-4 for the real ray tracing. The Ddand chromatic aberrations between the real and the paraxial rays ( εF,C, εd,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
Ddof the real ray for the doublet prisms, the second is the real primary color aberration εF,Cof the doublet prisms, and the last is the real secondary spectrum εd,Cof the doublet prisms. If the target values are tDd=3, tεd,C=0, and tεF,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
εd,Cand εF,Care more rigorous than that of Dd, and then the target values of both | εd,C| and | εF,C| are twenty times smaller than those of | Dd-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
Dd= 3°, we can optimize the area CA of the real chromatic aberration curve to obtain an optimization design. The merit function is defined as
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> θC, the total internal reflection of the ray appears. The critical angle θCis given by
nis 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
nd1=1.53996 and nd2=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 (
PdC1- PdC2)/( Vd1- Vd2), 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 Dd=3°, and eliminate the primary color. Then, the two apex angles are obtained. Third, the program sets tDd=3, t？dC=0, and t？F,C=0 to optimize the real primary color. Finally, tDd=3 and tCA=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 Vd number, and minimizing the real
chromatic aberration of doublet prisms by an optimization program.
[FIG. 1.] Angle of deviation of a prism.
[FIG. 2.] Pd,C - Vd mapping for the selected doublet prisms.
[FIG. 3.] Minimizing paraxial chromatic aberration curves for doublet prisms groups A to F.
[TABLE 1.] Design data and area of paraxial chromatic aberration curves for doublet prisms A To F
[TABLE 2.] Initial values for optimization of the doublet prism design (group A)
[FIG. 4.] Ray tracing in the doublet prisms.
[TABLE 3.] Design results (group A) for target values: tDd=3, t？d,D=0, and t？F,C=0
[TABLE 4.] Design results (group A) for target values: tDd=3, tCA=0
[FIG. 5.] Real chromatic aberration curve for group A for optimized target values: tDd=3, t？d,C=0, and t？F,C=0
[FIG. 6.] Real chromatic aberration curve for group A for optimized target values: tDd=3,tCA=0.
[TABLE 5.] Optimization designs for doublet prisms from group A to group F
[FIG. 7.] Flow chart of optimization program for doublet prisms.
[FIG. 8.] Chromatic aberration curves A to E for the optimal designs.