A Lowcost Optimization Design for Minimizing Chromatic Aberration by Doublet Prisms
 Author: Sun WenShing, Tien ChuenLin, Sun ChingCherng, Lee ChingChun
 Organization: Sun WenShing; Tien ChuenLin; Sun ChingCherng; Lee ChingChun
 Publish: Journal of the Optical Society of Korea Volume 16, Issue4, p336~342, 25 Dec 2012

ABSTRACT
A lowcost optimal doubleprism method is proposed by using the developed MATLAB program to correct chromatic aberration. We present an efficient approach to choose a couple of lowcost glasses to obtain a low aberration double prism. The doublet prisms were made of two leadfree glasses. The relative partial dispersion of the two leadfree 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.

KEYWORD
Doublet prisms , Optimization , Chromatic aberration , Ray tracing

I. INTRODUCTION
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 twoelement 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 doublegraph technique to produce doublet designs. Then, in 2001, Rayces and RoseteAguilar [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 lowcost optimal doubleprism method. An efficient approach for finding an optimum design is also proposed.
II. METHODS
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,
I_{1} andI_{2} are the incident angles of the first and the second surfaces, respectively;I_{1}' andI_{2}' 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 (I_{1}' ？I_{1} ) at the first surface. At the second surface, the ray deviates by (I_{2}'－I_{2} ), so the angle of deviationD of the ray is given byIf we consider real ray tracing, the deviation angle [10] presented by
2.1. Single Prism Paraxial Chromatic Aberration
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. Thuswhere
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; andn_{C} ,n_{d} ,n_{F} are the refractive indices of C, d, and F lines, respectively. Consequently, the paraxial second spectrumε_{d,C} is the difference in the deviation angles between d line and C line. This can show aswhere
P_{d,c} =(n_{d} n_{C} )/(n_{F} n_{C} ) is the relative partial dispersion.2.2. Doublet Prisms Paraxial Chromatic Aberration
In the paraxial ray tracing of doublet prisms, the angle of deviation of dline light is defined as 3°, so the primary color
ε_{F,C} is zero [4,5]. These equations can be written aswhere the paraxial deviation angles of the doublet prisms are
D_{d1} andD_{d2} , respectively. The primary chromatic aberration isε_{F,C} . Solving forD_{d1} andD_{d2} from the above equations, we obtainand
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} andε_{F,C2} , respectively.2.3. Schott Glasses Selection
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 NBK7 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 twentynine types with a relative price RP ≥ 17 as well as those with no marked prices are eliminated. The costs of glasses such as NKZFS11, NPK51, and NLASF31A, 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 NSF6HT and NSF57HT glasses offer improved transmittance in the visible spectral range especially in the blueviolet area. Moreover, since the
V_{d} ,n_{d} , andP_{d,C} of NSF6HT and NSF57HT are all the same as those of NSF6 and NSF57, the corrected chromatic aberrations will be almost the same. Thus we can neglect the NSF6HT and NSF57HT glasses. A total of seventy types of glasses were chosen for doublet prisms design to correct chromatic aberration.2.4. Merit Function
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
I _{1}, but the deviation angle of the real ray is related to the incident angleI _{1}. 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 leastsquares method [1315] 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 aswhere
m is the total summation number; thew_{i} is the weighting factor;e_{i} is the aberration andt_{i} is the target value. We define the functionf_{i} (x_{1} ,x_{2} , ‥‥,x_{n} ) asBefore optimization, the
n variables are denoted asx_{10} ,x_{20} , ···.,x_{n0} ; them aberrations before the optimization aref_{10} ,f_{20} , ···.,f_{m0} . After the optimization process, the variables are denoted asx_{1} ,x_{2} , ···,x_{n} , and the aberrations asf_{1} ,f_{2} , ···.,f_{m} . Here, we define a matrixA , in which the elements areWe then get the equation
where
A^{T} is the transpose matrix ofA; I is a unit matrix,p is a damping factor; andf_{0} is the matrix containing the elementsf_{10} ,f_{20} ,···.f_{m0} . Ifx andx_{0} are the matrices containing the elementsx_{1} ,x_{2} , ···,x_{n} , andx_{10} ,x_{20} , ···,x_{n0} , respectively, we can obtainIII. RESULTS
3.1. Minimizing the Paraxial Chromatic Aberration
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 theV_{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. Thehorizontal 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.
3.2. OPtimization Design For The Real Chromatic Aberration
We choose group A from Table 1 as an example as the initial value. The doublet prisms are made of NPSK52A and NSK5 glasses. We set
I _{1} andI _{3} as the incident angles for the first surface of the first and the second prisms, respectively;A _{1} andA _{2} 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 setI _{1} = 0,I _{3} = 0,A _{1} = 24.215°, andA _{2} = 15.336°, as listed in Table 2. We can calculate theD_{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. TheD_{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 aberrationε_{F,C} of the doublet prisms, and the last is the real secondary spectrumε_{d,C} of the doublet prisms. If the target values aret_{Dd} =3,t_{εd,C} =0, andt_{εF,C=0} , then the merit function is given bywhere the weighting factors are
w_{1} = 1,w_{2} = 20,w_{3} = 20, respectively. During the doublet prisms optimization, wewill 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} andε_{F,C} are more rigorous than that ofD_{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
x_{1} ,x_{2} ,x_{3} , andx_{4} , to represent the incident angleI _{1} of the first prism, the incident angleI _{3} of the second prism, the apex angleA _{1} of the first prism, the apex angleA _{2} of the second prism, respectively. Before optimization, the variables are denoted asx_{10} ,x_{20} ,x_{30} , andx_{40} , which correspond toI _{1}= 0,I _{3}= 0,A _{1}= 24.215, andA _{2}= 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 aswhere the target values are
t_{Dd} =3, andt_{CA} =0. The optimized results are listed in Table 4, and the real chromatic aberration curve is shown in Fig. 6.3.3. Total Internal Reflection
As mentioned before, figure 4 shows the ray tracing of the doublet prisms, where
I _{1},I _{1}′ ,A _{2},I _{4}, andI _{4}′ are negative andA _{1},I _{2},I _{2}′ ,I _{3}, andI _{3}′ are positive. When the incident angle of the first prism isI_{2} >θ_{C} , the total internal reflection of the ray appears. The critical angleθ_{C} is given bywhere
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 NBAK2 and NLAK34, their refractive indices are
n_{d} _{1}=1.53996 andn_{d} _{2}=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 I_{1} of the first surface ofthe first prism must be
and the incident angle
I _{3} of the first surface of the second prism is required to be3.4. Design of The Optimization Program
A flowchart of the optimization program for the doublet prisms design is shown in Fig. 7. First, the program selects (
P_{dC} _{1}P_{dC} _{2})/(V_{d} _{1}V_{d} _{2}), 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 angleD_{d} =3°, and eliminate the primary color. Then, the two apex angles are obtained. Third, the program setst_{Dd} =3,t_{？dC} =0, andt_{？F,C} =0 to optimize the real primary color. Finally,t_{Dd} =3 andt_{CA} =0 are used to optimize the area of the real chromatic aberration curve, until the real chromatic aberration is a minimum. Theresults 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.
IV. CONCLUSION
A lowcost optimal doubleprism method combined with the developed MATLAB program to correct chromatic aberration has been presented. In comparison of the doubletprism 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.

[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.