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A novel approach for structural synthesis of zoom systems
L. N. Hazra* and S. Pal
Department of Applied Optics & Photonics, University of Calcutta
92, A. P. C. Road, Kolkata, India
ABSTRACT
A new approach for ‘ab initio’ synthesis of thin lens structure of zoom lenses is reported. This is accomplished by an
implementation of evolutionary programming, based on Genetic Algorithm, which explores the available configuration
space formed by powers of individual components and inter-component separations. Normalization of the variables is
carried out to get an insight on the optimum structures. The method has been successfully used to get thin lens structures
of mechanically compensated, optically compensated, and linearly compensated zoom lens systems by suitable
formulation of merit function of optimization. Investigations have been carried out on three component and four
component zoom lens structures. Illustrative numerical results are presented.
Keywords: Optically compensated zoom lenses; mechanically compensated zoom lenses; evolutionary programming;
genetic algorithm; thin lens design;
1. INTRODUCTION
Advantage of getting images of different magnifications on an image plane has made zoom lenses1-4 immensely popular
in recent times. Mainly two classes of zoom lens structures, viz. mechanically compensated zoom lenses5-9 and optically
compensated10-12 or in more general case linearly compensated13, 14 zoom lenses are in use. In both cases, the
magnification of the zoom system is changed by moving one or more components with respect to the other. Then, the
obvious change in the position of image is balanced by shifting another set of components along the axis. Traditionally
the group of component that is more responsible for the change of overall focal length is called the ‘variator’, and the
other set of component that is used to maintain the image over a fixed image plane is known as the ‘compensator’. In a
mechanically compensated zoom lens system the final image plane is held firmly over a fixed image plane by moving
the variator and the compensator independently. The movements of the components hold a nonlinear relationship with
the change in overall power of the zoom lens [Fig. 1(a)]. On the other hand, in an optically/ linearly compensated zoom
lens system, small axial shift of the image plane is allowed to get a linear motion of the moving components. In optically
compensated zoom lenses, which are a special class of linearly compensated zoom lenses, the moving components are
coupled together to maintain a linear relationship between the movements of individual components[Fig. 1(b)]. Recently
attempts are being made to change the over all focal length of the zoom system by using variable focal length lens
components, keeping inter-component separation fixed15, 16
This paper describes a method for searching globally or quasi-globally optimal thin lens structures of zoom lenses
within the multivariate space formed by design variables, i.e. powers of individual components and inter-component
separations. This is accomplished by an implementation of evolutionary programming based on genetic algorithm (GA).
17, 18 We present some results of our investigations on the use of evolutionary programming in structural design of three
types of zoom lenses, namely, mechanically, optically and linearly compensated zoom lenses. The next section describes
our implementation of evolutionary programming for the purpose. Section 3 presents some illustrative results for zoom
lenses for different magnification ratios. Concluding remarks are put forward in the last section.
*lnhaphy@caluniv.ac.in; Phone +913323587302; Fax +913323519755
Current Developments in Lens Design and Optical Engineering XI; and Advances in Thin Film Coatings VI,
edited by R. Barry Johnson, Virendra N. Mahajan, Simon Thibault, Proc. of SPIE Vol. 7786
778607 · © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.858881
Proc. of SPIE Vol. 7786 778607-1
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WA
Tel e
Z
WA
Tel e
WA
Tel e
r
1
Zr
2
Z
(a) (b) (c)
Fig. 1 Thee component (a) mechanically (b) optically (c) linearly compensated zoom system
2. EVOLUTIONARY PROGRAMMING
The evolution of species in nature by survival of the fittest is the basis of evolutionary programming.19-23 The latter
searches a globally or quasi-globally optimum result in a given hyperspace formed by degrees of freedom of a problem.
2.1. Design variables
Design of the thin lens structure of an N-component zoom lens involves determination of a set of N powers of the
components, and (N-1) inter-component separations. These (2N-1) parameters decide the overall power of the system,
and they constitute the design variables for this optimization problem. All these design variables are normalized in terms
of required wide angle power ( WA
k ) of the zoom system. Normalized power ( i
k
~) of ith
component and normalized
separation ( i
d
~
) between ith and (i+1)th components are given below,
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
WA
i
ik
k
m
k1
~ (1a)
iWAi dkd =
~
(1b)
where, i
k is actual power of ith
component, and i
d is the actual separation between ith and (i+1)th components. The
factor
()
m1 in the right hand side of (1a) is used to facilitate the search of zoom structure with larger zoom ratio.
Introduction of this factor increases the value of component power, deduced from a normalized power, for large zoom
ratios. This is necessary to maintain the image plane on a fixed image plane or to keep image plane oscillation within the
limit in spite of a large zoom ratio. Generally the range of normalized power for a positive component is (0, +1) and that
for a negative component is (-1, 0). The range of variation in normalized values for inter-component separations is (0,
+1). These ranges for normalized powers and inter-component separations are found to be quite effective in conducting
searches for the optimum thin lens structure of zoom lens with the commonly used values for zoom ratio and wide angle
power. However, the ranges of these variations may have to be extended, if suitable solutions are not obtained in specific
situations.
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2.2. Optimization process
The total process of optimization consists of five major stages that are discussed below.
I. Initial population: In the evolutionary programming based on GA, each design variables is represented by a fixed
string of 1’s and 0’s. The string is called gene. Each gene is represented by a Gray code to ensure that adjacent
phenotypes always correspond to adjacent genotypes. The length of a gene gene
i
l, corresponds to th
ivariable is
i
ii
gene
i
xx
l
ε
minmax
2
log −
= (2)
where, max
i
x and min
i
xare maximum and minimum value of the ith variable and i
ε
is the required accuracy. All
these genes, representing all variables of the optimization problem, collectively form a chromosome. The length
of a chromosome ( chr
l) is therefore given by,
∑
=
=p
i
gene
i
chr ll
1
(3)
where, p is the number of variables in the problem. Initially a population of such chromosome is created by
randomly choosing 1 or 0 for each bit of the chromosome.
II. Fitness Function: Each member of the population is first decoded in two steps to get the value for the design
variables in decimal. In the first step Gray coded chromosome is converted into its binary form, and then decimal
value for each variable is obtained by decoding that binary string. The decimal values obtained are normalized
values of powers and inter-component separations and actual values are calculated using (1). With these values,
the Gaussian properties of the structure are determined, and then compared with the desired specifications for the
zoom system, to calculate the fitness of a member. The methods of evaluation Gaussian parameters of the thin
lens structures of three different zoom lenses and also formation of fitness function differ slightly from each other
and are described in section 2.3 and 2.4, respectively.
III. Generation of offspring: It starts with the selection of parents by using tournament selection based on the fitness
value of chromosomes calculated in previous stage. Single point crossover and single point mutation is then
carried out between two parent chromosomes to get two child chromosomes. Fitness function of each offspring is
evaluated following the process described above.
IV. Elitism: The fittest member of the population in a generation is stored as the elite member and used in the next
generation.
V. Genetic diversity check: The genetic diversity (g)of a population is defined as
∑×
=PopSize
ichr
i
lPopSize
d
g (4)
where i
d is the Hamming distance, and PopSize is the size of the population.26 If the genetic diversity between the
members of the population becomes small, the possibility of getting improved solution becomes low. Therefore, if
g falls below a prespecified threshold genetic diversity, say g
~
, only the elite member is retained and the remaining
member of the population is reinitialized by randomly generating the chromosomes again.
These five steps are repeated for a prespecified number, known as Generations, before termination.
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2.3. Determination of zoom range
A paraxial ray trace is carried out, assuming the object at infinity, at every stage of the evaluation of Gaussian parameters
of the thin lens structure. The following ray tracing formulae are used.
11
1
++
+
−=
=−
iiii
iiii
udhh
khuu (5)
where, hi is the height of the paraxial marginal ray (PMR) at ith
component, and the convergence angles of the PMR
before and after the ith
component are ui and ui+1 respectively. Separation between ith and (i+1)th component is di. Note
that in paraxial calculations we are using the notation and sign convention of Born and Wolf24 and Hopkins.25 The
equivalent focal length of a N-component zoom system is obtained using following equation
1
1
+
=
N
eq u
h
f (6)
Positions of the variator and the compensator at the two ends of zoom range are then determined for three types of zoom
structures by using slightly different approaches.
2.3.1. Mechanically compensated zoom lenses
Initial focal length ( eq
f) of the system decides direction of axial shift of the variator. First, the variator, a prespecified
component in the zoom system, is given sequentially an axial shift by a small amount
ε
in both directions, and the
resulting focal lengths of the zoom system are calculated. The direction providing a change of the focal length towards
the desired wide angle focal length is chosen as the direction of axial movement of the variator. The change in focal
length corresponding to the new position of the variator is accompanied by a change in position of the image plane. The
compensator, another prespecified component in the zoom system, is then shifted along the axis by small amount
ε
in
both directions. The direction providing a reduction in shift of the image plane is chosen as the preferred direction. The
correct position of the compensator for exact compensation of the image plane is then determined by using bi-section
search technique. The latter essentially involves repeated halving of the interval within which the compensator needs to
be located axially for providing exact compensation. These two processes, the variation of focal length and then the
compensation of image plane is repeated till the movement of any component is restricted by the mechanical constraints
or when the overall power becomes equal to the desired wide angle power. At the end of this process, the focal length of
the system gets its minimum value min
f. Next, a search for axial locations of the variator and the compensator is carried
out to achieve the long focal length ( max
f) by following a technique similar to the one used for searching the wide angle
position of the system. The value of
ε
used in our computation is typically taken as (1/50) of the minimum value of
inter-component separations of the system.
2.3.2. Optically/ linearly compensated zoom lenses
In an optically/ linearly compensated zoom lens systems, movement of the components maintain linear relationship
between each other. Therefore, after a shift of the components, by an amount Z, in the positive direction, the changed
value of inter-component separations of an N-component zoom lens are,
()
Zrdd
Zrdd
Zrdd
N
N
NN 1
1
1
*
1
22
*
2
11
*
1
1
...
−
−
−− −+=
+=
−=
(7)
Here it assumed that the first component is moved together with the other components [Fig. 1(b) & 1(c)]. For the sake of
simplicity we have assumed 1
rto be +1. The other factors
(
)
132 N-, ....., , , iri
=
can have any positive or negative value.
In the special case of optically compensated zoom lenses 121 ... −
=
=
=
N
rrr =+1. The available limits of axial shift of the
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moving component in the left side, ( L
Z) and right side, ( R
Z) of the current position can be easily calculated using
(7). R
Z is the maximum positive value of Z for which at least one of the values of separations, *
N-
** , .... , d, dd 121 , becomes
zero and others acquire zero or positive value. Similarly, L
Z is the minimum negative value of Z for which at least one
of the values of separations, *
N-
** , .... , d, dd 121 , becomes zero and others acquire zero or positive value. The moving
components are first set at extreme left position by providing a shift of an amount L
Z from their current position, and
the equivalent focal length of the overall system L
eq
f, is determined. Similarly, equivalent focal length of the overall
system, R
eq
f, when the coupled component is set at extreme right position, by providing a shift of an amount RL ZZ
+
from their current position, is also determined. Keeping in view the requirements for focal lengths in the telephoto and
wide angle position, WA
f and Tel
f respectively, four cases may arise.
Case I. Both TelWA ff , are outside the range ( L
eq
f, R
eq
f)
min
f =
R
eq
f, if R
eqWA ff −<L
eqWA ff −
=
L
eq
f, if L
eqWA ff −<R
eqWA ff −.
max
f =
R
eq
f, if L
eq
ff =
min
=
L
eq
f, if R
eq
ff =
min .
Case II.
WA
f is within ( L
eq
f, R
eq
f), but Tel
f lies outside
max
f =
R
eq
f, if R
eqTel ff −<L
eqTel ff −
=
L
eq
f, if L
eqTel ff −<R
eqTel ff −.
Case III.
Tel
f is within ( L
eq
f, R
eq
f), but WA
f lies outside
min
f =
R
eq
f, if R
eqWA ff −<L
eqWA ff −
=
L
eq
f, if L
eqWA ff −<R
eqWA ff −.
Case IV. Both TelWA ff , inside ( L
eq
f, R
eq
f)
In Case IV, bisection search techniques are utilized to determine the axial positions of the moving components yielding
equivalent focal lengths WA
f and Tel
f. This technique essentially involves bracketing of the exact location of the moving
components that yield the desired focal length, within two axial locations of the moving components, and gradually
halving this interval to achieve desired accuracy in this axial location. For practical purposes, min
f=WA
f and max
f=Tel
f
at these axial locations of the coupled components. For Case II and Case III, axial location of the moving components
yielding equivalent focal lengths WA
f and Tel
f, respectively are determined by using similar bisection search techniques.
min
f and max
f in the corresponding cases are assigned as min
f=WA
f and max
f=Tel
f respectively.
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2.4. Fitness function
Before calculating fitness function of the system, a merit function (
Φ
) is calculated for the system. For mechanically
compensated system, it consists of the percentage of errors in the short and long focal length when compared with the
desired focal length for the system at its wide and telephoto positions respectively. The percentages for both the errors
are calculated with respect to the desired short focal length of the system. Along with these two errors, in case of
optically/ linearly compensated zoom lens system the merit function have an additional part indicating error in the
position of the image plane. This error is also expressed in percentage of the desired short focal length of the system.
Incidentally, it may be noted that an inadvertent error occurred in the expression for merit function in our earlier report.23
In expression (11) of that report, min
f and max
f need to be interchanged to obtain the correct expression.
2.4.1. Mechanically compensated zoom lenses
Merit function of a mechanically compensated zoom system is calculated using the formula,
2
max
2
2
2
min
111 ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−×+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−×=Φ
TelWA f
f
m
f
f
ωω
(9)
where, 21 ,
ω
ω
and 3
ω
are the weighting factors.
2.4.2. Optically/ linearly compensated zoom lenses
For optically/ linearly compensated zoom systems, the shift of the final image plane, Δ, is determined with respect to a
fixed component in the system at fifty equispaced intermediate positions of the moving component within the total range
of axial movement. The merit function for an optically/ linearly compensated zoom lens is,
2
3
2
max
2
2
2
min
111 ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ
×+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−×+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−×=Φ
WA
m
TelWA ff
f
m
f
f
ωωω
(10)
where, m
Δ is maximum axial shift of the image plane during travel of the coupled components from one end to other
[Fig. 2]. 21 ,
ω
ω
and 3
ω
are the weighting factors.
The fitness function, Ψ, is defined as
Φ+
=Ψ 1
1 (11)
Expression (11) is used to calculate fitness value from the merit function for both mechanically and optically/ linearly
compensated zoom lenses. Target of the optimization process is to increase
Ψ
so that min
kand max
k gets closer to the
required focal lengths WA
kand Tel
k respectively.
Δ
k
Δ
m
k
k
WA
Tel
Fig. 2. m
Δ: maximum axial shift of the image plane over the range of power ( TelWA kk , )
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3. ILLUSTRATIVE RESULTS
Some preliminary results on the use of evolutionary programming in the design of thee types of zoom lenses are shown
in Fig. 3 – 11. For optically compensated zoom lenses shown in figures 3 and 4, the first and the third components are
coupled moving components. For the linearly compensated zoom lenses shown in figures 5 and 6, 1
1=r and 2
ris given
in the corresponding figure captions. Inter-component separations at wide angle position for all zoom structures are also
given. Fig. 7 presents a 3X mechanically compensated zoom lens. Figs. 8 – 10 present three different designs for 4X
mechanically compensated zoom lens. Fig. 11 presents the design for a 15X mechanically compensated zoom lens. For
each lens with zoom ratio m:1 the design data is presented for mff TelWA
=
=
and 1 . Axial shifts of the image plane over
the zoom range are presented for each of the optically and linearly compensated zoom lenses. The nonlinear movement
of the moving components is also demonstrated for each of the mechanically compensated zoom lenses.
0.0 1.0
k
i
0.40.2 0.6 0.8-0.2
Tele
WA
0.004-0.004
Δ
k
0.5
1.0
(a) (b)
Fig. 3. (a) Component movement (b) axial shift of image plane for a 2X optically compensated zoom lens system. Maximum
movement of the coupled components, Z= -0.307. Inter-component separations: d1=0.228 and d2=0.685
0.0 1.5
k
i
0.5 1.0-0.5
Tele
WA
0.032-0.032
Δ
k
0.25
1.0
(a) (b)
Fig. 4. (a) Component movement (b) axial shift of image plane for a 4X optically compensated zoom lens system. Maximum
movement of the coupled components, Z= -0.556. Inter-component separations: d1=0.796 and d2=0.569
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0.0 1.0
k
i
0.5-0.25
Tele
WA
0.003-0.003
Δ
k
0.5
1.0
(a) (b)
Fig. 5. (a) Component movement (b) axial shift of image plane for a 2X linearly compensated zoom lens system.
Maximum movement of the first component Z= -0.452 and 2
r=0.5. Inter-component separations: d1=0.565 and d2=0.424
0.0 1.0
k
i
0.5 1.5-0.25
Tele
WA
0.024-0.024
Δ
k
0.25
1.0
(a) (b)
Fig. 6. (a) Component movement (b) axial shift of image plane for a 4X linearly compensated zoom lens system.
Maximum movement of the first component Z= -0.214 and 2
r=2.0. Inter-component separations: d1=1.091 and d2=0.593
0.0 0.2 0.4-0.2-0.4-0.6
Tele
WA
k
i
Fig. 7. Component movement for a 3X mechanically compensated zoom lens system.
Inter-component separations: d1=0.114 and d2=0.295
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0.0 0.5 1.0 1.5
Tel e
WA
k
i
Fig. 8. Component movement for a 4X mechanically compensated zoom lens system.
Inter-component separations: d1=0.508, d2=0.556 and d3=0.667
0.0 0.5 1.0
Tel e
WA
k
i
Fig. 9. Component movement for a 4X mechanically compensated zoom lens system.
Inter-component separations: d1=0.127, d2=0.905 and d3=0.270
0.0 0.66 1.32
Tel e
WA
k
i
Fig. 10. Component movement for a 4X mechanically compensated zoom lens system.
Inter-component separations: d1=0.048, d2=0.778 and d3=0.841
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0.0 0.5 1.0 1.5
Tel e
WA
k
i
Fig. 11. Component movement for a 15X mechanically compensated zoom lens system.
Inter-component separations: d1=0.619, d2=0.619 and d3=0. 524
4. CONCLUDING REMARKS
Preliminary results presented in this report shows the feasibility of using an evolutionary programming based on GA in
optimal synthesis of thin lens structure of zoom lenses. It may be noted that choice of the ranges for the design variables
and the weighting factors in merit function significantly affect output of the optimization algorithm. Also choice of the
factors i
r plays a critical role in the synthesis of linearly compensated zoom lenses. Further investigations on these aspects of the
problem are currently in progress.
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