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.
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
Angle of deviation of a prism.
direction, and negative in the clockwise direction. In this figure,
I_{1}
and
I_{2}
are the incident angles of the first and the second surfaces, respectively;
I_{1}'
and
I_{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 deviation
D
of the ray is given by
If 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. 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.
 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 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.
 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}
, and
P_{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 angle
I
_{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
[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}
(
x_{1}
,
x_{2}
, ‥‥,
x_{n}
) as
Before optimization, the
n
variables are denoted as
x_{10}
,
x_{20}
, ···.,
x_{n0}
; the
m
aberrations before the optimization are
f_{10}
,
f_{20}
, ···.,
f_{m0}
. After the optimization process, the variables are denoted as
x_{1}
,
x_{2}
, ···,
x_{n}
, and the aberrations as
f_{1}
,
f_{2}
, ···.,
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
f_{0}
is the matrix containing the elements
f_{10}
,
f_{20}
,···.
f_{m0}
. If
x
and
x_{0}
are the matrices containing the elements
x_{1}
,
x_{2}
, ···,
x_{n}
, and
x_{10}
,
x_{20}
, ···,
x_{n0}
, respectively, we can obtain
III. 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 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
P_{d,C}  V_{d} mapping for the selected doublet prisms.
Minimizing paraxial chromatic aberration curves for doublet prisms groups A to F.
Design data and area of paraxial chromatic aberration curves for doublet prisms A To F
Design data and area of paraxial chromatic aberration curves for doublet prisms A To F
Initial values for optimization of the doublet prism design (group A)
Initial values for optimization of the doublet prism design (group A)
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.
 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}
and
I
_{3}
as the incident angles for the first surface of the first and the second prisms, respectively;
A
_{1}
and
A
_{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 set
I
_{1}
= 0,
I
_{3}
= 0,
A
_{1}
= 24.215°, and
A
_{2}
= 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
w_{1}
= 1,
w_{2}
= 20,
w_{3}
= 20, respectively. During the doublet prisms optimization, we
Ray tracing in the doublet prisms.
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
x_{1}
,
x_{2}
,
x_{3}
, and
x_{4}
, to represent the incident angle
I
_{1}
of the first prism, the incident angle
I
_{3}
of the second prism, the apex angle
A
_{1}
of the first prism, the apex angle
A
_{2}
of the second prism, respectively. Before optimization, the variables are denoted as
x_{10}
,
x_{20}
,
x_{30}
, and
x_{40}
, which correspond to
I
_{1}
= 0,
I
_{3}
= 0,
A
_{1}
= 24.215, and
A
_{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 as
Design results (group A) for target values: tDd=3, tєd,D=0, and tєF,C=0
Design results (group A) for target values: t_{Dd}=3, t_{єd,D}=0, and t_{єF,C}=0
Design results (group A) for target values: tDd=3, tCA=0
Design results (group A) for target values: t_{Dd}=3, t_{CA}=0
Real chromatic aberration curve for group A for optimized target values: t_{Dd}=3, t_{єd,C}=0, and t_{єF,C}=0
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
.
 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}
, and
I
_{4}
′
are negative and
A
_{1}
,
I
_{2}
,
I
_{2}
′
,
I
_{3}
, and
I
_{3}
′
are positive. When the incident angle of the first prism is
I_{2}
>
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 NBAK2 and NLAK34, their refractive indices are
n_{d}
_{1}
=1.53996 and
n_{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 of
Real chromatic aberration curve for group A for optimized target values: t_{Dd}=3,t_{CA}=0.
the first prism must be
and the incident angle
I
_{3}
of the first surface of the second prism is required to be
 3.4. 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 angle
D_{d}
=3°, and eliminate the primary color. Then, the two apex angles are obtained. Third, the program sets
t_{Dd}
=3,
t_{єdC}
=0, and
t_{єF,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
Optimization designs for doublet prisms from group A to group F
Optimization designs for doublet prisms from group A to group F
Flow chart of optimization program for doublet prisms.
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.
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 curves A to E for the optimal designs.
chromatic aberration of doublet prisms by an optimization program.
Acknowledgements
The authors gratefully appreciate the support of the National Science Council of Taiwan, the Republic of China, under project numbers NSC 982221E008021MY3, NSC 1002623E008002ET and NSC 1012221E035055.
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