Paraxial design of an optical element with variable focal length and fixed position of principal planes

Abstract

In this article, we analyze the problem of the paraxial design of an active optical element with variable focal length, which maintains the positions of its principal planes fixed during the change of its optical power. Such optical elements are important in the process of design of complex optical systems (e.g., zoom systems), where the fixed position of principal planes during the change of optical power is essential for the design process. The proposed solution is based on the generalized membrane tunable-focus fluidic lens with several membrane surfaces.

Full text

Paraxial design of an optical element with
variable focal length and fixed position of
principal planes
ANTONÍN MIKŠAND PAVEL NOVÁK*
Czech Technical University in Prague, Faculty of Civil Engineering, Department of Physics, Thakurova 7, 16629 Prague, Czech Republic
*Corresponding author: pavel.novak.3@fsv.cvut.cz
Received 12 February 2018; revised 6 April 2018; accepted 6 April 2018; posted 10 April 2018 (Doc. ID 323074); published 3 May 2018
In this article, we analyze the problem of the paraxial design of an active optical element with variable focal
length, which maintains the positions of its principal planes fixed during the change of its optical power.
Such optical elements are important in the process of design of complex optical systems (e.g., zoom systems),
where the fixed position of principal planes during the change of optical power is essential for the design process.
The proposed solution is based on the generalized membrane tunable-focus fluidic lens with several membrane
surfaces. © 2018 Optical Society of America
OCIS codes: (080.2468) First-order optics; (080.2740) Geometric optical design; (080.3630) Lenses; (110.1080) Active or adaptive
optics; (080.3620) Lens system design; (220.3630) Lenses.
https://doi.org/10.1364/AO.57.003714
1. INTRODUCTION
Tunable-focus fluidic lenses [1–18] are fascinating optical com-
ponents that enable a continuous change of focal length, and
their properties differ from the properties of classical glass lenses
with a fixed focus. During the design process of optical systems
composed both of classical optical elements and tunable-focus
fluidic lenses, it is necessary to know positions of the principal
planes of such elements in both cases [19–22].
Generally, the position and mutual distance of principal
planes of tunable optical elements with variable focal length
vary with the change of optical power of such an element.
Therefore, when using these elements in complex optical sys-
tems, such as zoom systems for example [23–36], one has to
consider that due to this behavior, the mutual axial distances
between individual elements of the zoom systems change dur-
ing the zoom operation. To design a zoom with such elements,
one would have to know the dependence of position of prin-
cipal planes on optical power of each individual element and
either compensate those changes by the mutual movement
of individual elements, or use this knowledge during the design
process to calculate the necessary values of optical powers of
individual elements that compose the zoom system. The first
case would result in the loss of the main benefit of application
of optical elements with variable focal length in complex optical
systems, which is a much simpler mechanical construction of
the system due to the fact that there are no moving elements
(change of magnification is achieved just by the variation of
optical powers of individual elements). The second approach
would lead to severe difficulties in calculation of the optical
powers of individual elements.
To avoid this situation, it is possible to use optical elements
with variable focal length, designed in such a way that the po-
sition and mutual distance of principal planes of these elements
remains fixed during the change of optical power of such an
element. The aim of this work is to perform a paraxial analysis
of the problem and show a possible solution as to how to realize
a paraxial design of a simple tunable optical element with
fixed positions of principal planes during the change of
optical power. The proposed simple solution is based on the
generalized membrane tunable-focus fluidic lens with several
membrane surfaces.
2. AN OPTICAL SYSTEM WITH VARIABLE
FOCAL LENGTH AND CONSTANT POSITION OF
PRINCIPAL PLANES
In the paraxial optical design, one can derive the following
formulas for the fundamental paraxial imaging parameters of
the optical system [22,28]:
φ−γ,sFδ∕γ,s0
F0−α∕γ,(1)
sδ−1∕m
γ,s0−
β−αs
δ−γs,ms0γα,(2)
where sFand s0
F0are distances of the object and image focal
points from the first and last surface of the optical system, s
and s0are distances of the object and image planes from the
3714 Vol. 57, No. 14 / 10 May 2018 / Applied Optics Research Article
1559-128X/18/143714-06 Journal © 2018 Optical Society of America

first and last surface of the optical system, φis the optical
power, mis the transverse magnification of the optical system,
and paraxial parameters α,β,γ, and δ, which depend on the
inner structure of the optical system, can be generally expressed
by employing Gaussian brackets [22,28]. In the case of the op-
tical system composed of thin lenses, one can replace the term
“surface”with the term “lens”in the previous text.
Consider now the optical system composed of three thin
elements, for example, lenses as given schematically in Fig. 1.
We used the following denotation: d1and d2are distances be-
tween individual elements of the optical system along the op-
tical axis, ξand ξ0are the object side (front) and the image side
(rear) principal planes, Hand H0are the object side (front) and
the image side (rear) principal points, and ηand η0denote the
object and image planes of the optical system. The meaning of
other symbols is clear from Fig. 1.
Fundamental paraxial parameters (functions) α,β,γ, and δ
of the optical system are then given by the following expressions
(in the case of three thin elements in air) [22,28]:
α1−d2φ1φ2−φ1φ2d1−φ1d1,(3)
βd1d2−φ2d1d2,(4)
γ−φ−φ1φ2φ3φ1φ2d1φ2φ3d2
φ1φ3d1d2−φ1φ2φ3d1d2,(5)
δ1−d1φ2φ3−d2φ3d1d2φ2φ3,(6)
where φ1,φ2, and φ3are values of the optical power of
individual elements of the optical system, and d1and d2are
distances between the elements of the optical system. For three
thin lenses in air, one can therefore obtain the following
equations:
φ−γφ1φ2φ3−φ1φ2d1−φ2φ3d2
−φ1φ3d1d2−φ2d1d2,(7)
sFφ2d1φ3d1φ3d2−φ2φ3d1d2−1∕φ,(8)
s0
F0−φ1d1φ1d2φ2d2−φ1φ2d1d2−1∕φ,(9)
s0d1d2−φ2d1d2−ss0
F0φ
sF−sφ,(10)
m1∕φs−φsF,(11)
where φiis the optical power of the ith lens, diis the mutual
distance of the i1th and ith lens, sis the axial object distance
measured from the first lens, s0is the axial image distance mea-
sured from the last lens, sFand s0
F0are the axial distances of the
object and image focal points from the first and last lens of the
thin lens optical system, and mis the transverse magnification
of the optical system. In the case of three refractive surfaces, the
previous relations are still valid when we perform the following
substitution:
di→di∕ni1,φini1−ni∕ri,(12)
where niand ni1are the refractive index in front of and
behind the ith optical surface, respectively, and riis the radius
of curvature of the ith optical surface (i1,2,3).
By using Eqs. (7)–(10), one can obtain the following rela-
tions for optical powers of individual thin elements of the three-
element optical system with variable focal length and fixed
position of principal planes:
φ1d1d2−sHs0
H0φs0
H0d1−sH
d1s0
H0−sH−φsHs0
H0,(13)
φ2d1d2−sHs0
H0−φsHs0
H0
d1d2
,(14)
φ3d1d2−sHs0
H0−φsHd2s0
H0
d2s0
H0−sH−φsHs0
H0,(15)
where sHsF−fsF1∕φis the axial position of the
front principal plane ξwith respect to the first thin lens
element, and s0
H0s0
F0−f0s0
F0−1∕φis the axial position
of the rear principal plane ξ0with respect to the last (third)
thin lens element (see Fig. 1).
Let us further deal with the case of a cemented doublet,
which is schematically pictured in Fig. 2, where r1,r2, and
r3are the radii of curvature of individual optical surfaces of
the doublet, d1and d2are the axial thicknesses of individual
lenses of the doublet, nk(k1,2,3,4) are refractive indices of
individual optical media, Fand F0are object and image focal
points of the doublet, respectively, Hand H0are principal
points of the doublet, and ξand ξ0are principal planes of
the doublet. The meaning of other symbols is evident
from Fig. 2.
Fig. 1. Three-element thin lens optical system. Fig. 2. Scheme of a doublet.
Research Article Vol. 57, No. 14 / 10 May 2018 / Applied Optics 3715

Let us now further assume the case of classical cemented
doublet in air (n1n21). According to Eq. (12) and
Eqs. (7)–(9), it holds that
φφ1φ2φ3−φ1φ2d1∕n2−φ2φ3d2∕n3
−φ1φ3d1∕n2d2∕n3−φ2d1d2∕n2n3,(16)
sFd1n3φ2φ3φ3d2n2−φ2φ3d1d2−n2n3
n2n3φ,(17)
s0
F0
−d2n2φ1φ2−φ1d1n3φ1φ2d1d2n2n3
n2n3φ:
(18)
For given axial thicknesses d1and d2, the optical powers of
individual refractive surfaces can then be expressed as
φ1d1n2d2∕n3−sHs0
H0φs0
H0d1−n2sH
d1s0
H0−sH−φsHs0
H0,(19)
φ2d1n3d2n2n2n3s0
H0−sH−φsHs0
H0
d1d2
,(20)
φ3d2n3d1∕n2−sHs0
H0−φsHd2n3s0
H0
d2s0
H0−sH−φsHs0
H0:(21)
The radii of curvature of individual surfaces can be
obtained as
r1n2−1
φ1
,r2n3−n2
φ2
,r31−n3
φ3
,(22)
where the optical powers φiare calculated from Eqs. (19)–(21).
Let us now assume the case of a tunable-focus liquid mem-
brane doublet, whose optical surfaces are made of elastic mem-
branes, and the volume between the surfaces is filled with
liquid. By changing of the pressure of the liquid, the radii of
curvature of membrane surfaces r1,r2, and r3change, and also
the axial distances d1and d2between the optical surfaces
change. The simplified scheme of such a doublet is the same
as in the case of the classical cemented doublet given in Fig. 2.
The practical realization of such an optical element would
however require the pressure acting on individual optical mem-
brane surfaces to be different to enable independent change of
individual membrane curvatures (arbitrary values of r1,r2, and
r3). This could be achieved, for example, by dividing the cham-
bers filled by the liquid using a separating transparent optical
element. The separating optical element can be constructed as a
simple thin plane parallel plate made of a solid optical material
with optical properties that are very close to optical properties
of the liquid in the chamber, that is, having practically the same
refractive index, dispersion, etc.
A simplified scheme of a possible design of such an element
is given in Fig. 3. The first lens of the doublet depicted in Fig. 2
is divided into two chambers, Chambers I and II, by the plane
parallel plate PPP1, and both chambers are filled with the same
liquid having refractive index n2. The plane parallel plate PPP1
is made of material with optical properties similar to the optical
liquid used in Chambers I and II.
Similarly, the second lens of the doublet depicted in Fig. 2is
divided into two chambers, Chambers III and IV, by the plane
parallel plate PPP2, and both chambers are filled with the same
liquid having the refractive index n3. The plane parallel plate
PPP2is made of material with optical properties similar to the
optical liquid used in Chambers III and IV. Now by changing
the pressures pI,pII,pIII , and pIV of optically transparent liquids
in Chambers I–IV, one can control independently the radii r1,
r2, and r3of the membrane surfaces. In such a case, the effect of
the inserted plane parallel plates PPP1and PPP2would be neg-
ligible, and one can use a simplified model of three refracting
surfaces given in Fig. 2. Considering this fact in further analysis,
we will assume the simplified optical scheme of three refracting
surfaces given in Fig. 2.
By changing the pressure of the liquids in the chambers
between the membranes, the radii of curvature of membrane
surfaces r1,r2, and r3will change. In addition, the axial dis-
tances d1and d2between the optical surfaces will also change.
The situation is therefore different than in the case of a classical
doublet design, where one assumes given fixed values of thick-
nesses d1and d2as input parameters, and points M1,M2, and
M3at the edge then change their positions with the change of
the required focal length of the fixed-focus classical doublet. In
the case of a tunable-focus liquid membrane doublet, the posi-
tions of points M1,M2, and M3are fixed during the change of
the focal length (i.e., the edge thicknesses of individual liquid
membrane lenses, which are given by the construction of a tun-
able-focus liquid membrane doublet, are constant during the
focal length change). If we want the positions of principal
planes ξand ξ0with respect to fixed points M1and M3to re-
main constant during the change of the focal length of the
doublet, we have to use the following procedure to calculate
its paraxial parameters:
1. We choose an initial value of focal length f0f0
0and
positions of principal planes sHV1Hand s0
H0V3H0, axial
separations d1and d2, and refractive indices of liquids n2and
n3. From Eqs. (13)–(15), we calculate the optical powers of
individual refracting surfaces, and further by using Eq. (22),
we obtain the radii of curvature of surfaces r1,r2, and r3.
2. We calculate sags of individual refractive surfaces l1,l2,
and l3using the following equation:
lih2∕ri
1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−h∕ri2
p,i1,2,3,(23)
Fig. 3. Simplified scheme of practical realization of the tunable-fo-
cus optical element having three membrane optical surfaces with inde-
pendently changeable radii of curvature.
3716 Vol. 57, No. 14 / 10 May 2018 / Applied Optics Research Article

where his the perpendicular distance of point M1from the
optical axis of the doublet (Fig. 2), and riis the radius of cur-
vature of the i-th surface. Using the calculated sag values, one
can determine the position of points M1,M2, and M3.
3. For the initial value of focal length f0f0
0, we then
calculate the distance Δ1of the front principal plane ξfrom the
plane perpendicular to the optical axis that goes through point
M1and distance Δ3of the rear principal plane ξ0from the
plane perpendicular to the optical axis that goes through point
M3. These distances can be calculated simply as
Δ1sH0−l10,Δ3s0
H00
−l30,(24)
where the lower index 0 denotes that these values are calculated
for the focal length f0
0. The calculated distances Δ1and Δ3are
then required to be fixed during the change of the focal length
of the doublet.
4. The axial position sHof the object side principal plane ξ
(i.e., distance of the object side principal plane from the vertex
of the first optical surface) is then given as sHl1Δ1, and
similarly, the axial position s0
H0of the image side principal plane
ξ0(i.e., distance of the image side principal plane from the ver-
tex of the last optical surface) is then given as s0
H0l3Δ3.
5. Axial thicknesses d1and d2of the first and the second
lens of the doublet are then given by
d1d10 l1−l2,d2d20 l2−l3,(25)
where d10 M1M2and d20 M2M3, and l1,l2, and l3are
the sag values of refractive surfaces in points V1,V2, and V3.
6. By changing the focal length of the doublet, the value of
distances sHand s0
H0changes. The system must satisfy the
following conditions for all values of the focal length f0:
sH−l1Δ1,s0
H0−l3Δ3,(26)
which ensures that the requirement of the stability of the prin-
cipal planes positions during the focal length change is satisfied.
7. To calculate the optical powers of individual refractive
surfaces of the doublet, we substitute Eq. (26) into the formulas
for the calculation of position of principal planes:
sHf0sF,s0
H0s0
F0−f0,(27)
and thus obtain:
f0sF−l1Δ1,s0
F0−f0−l3Δ3:(28)
The values of surface sags l1,l2, and l3used in the previous
equations are calculated using Eq. (23).
By substitution of Eqs. (16)–(18) and Eqs. (23)–(26) into
Eq. (28), one obtains a set of nonlinear equations. By solving
these equations for different values of the focal length (optical
power) of the system from the selected range, one obtains the
radii of curvatures r1,r2, and r3of individual optical surfaces of
the doublet. The given problem is thus solved. The rigorous
analytic solution of the resulting equations cannot be found
in a reasonably simple form. For practical reasons, it is therefore
better to formulate the solving of the given equations set as an
optimization problem, and then by using the well-known iter-
ative methods of mathematical optimization, one can find the
values of radii of curvature numerically [37,38].
There are various possibilities how to formulate the corre-
sponding optimization problem and how to solve it numeri-
cally. For the purpose of the verification of the proposed
method, we formulated the optimization problem in the fol-
lowing form:
min Fr1,r2,r3,(29)
where the merit function Fis given as
Fr1,r2,r3F12F22F32,(30)
whereas
F1r1,r2,r3f0
req −1∕φ∕f0
req,(31)
F2r1,r2,r31∕φsF−l1−Δ1∕Δ1,(32)
F3r1,r2,r3s0
F0−1∕φ−l3−Δ3∕Δ3,(33)
where the values of sF,s0
F0, and φare given by substitution of
Eq. (22) into Eqs. (16)–(18), sags liare given by Eq. (23), f0
req
is the required focal length of the whole system, Δ1is the re-
quired fixed distance of the front principal plane ξfrom the
plane perpendicular to the optical axis that goes through point
M1, and Δ3is the required fixed distance of the rear principal
plane ξ0from the plane perpendicular to the optical axis that
goes through point M3. For given positions of points M1,M2,
and M3, the values of individual radii of curvature r1,r2, and r3
of the three elastic membranes can be obtained by solving the
optimization problem given by Eq. (29) for required values of
f0
req,Δ1, and Δ3. The proposed method was implemented in
MATLAB [39], and the solution was searched for numerically,
using the MultiStart method (with fmincon local solver
using the default interior-point method) available in the Global
Optimization Toolbox [40].
3. EXAMPLES
Let us now show three examples of application of the derived
equations for the calculation of parameters of optical systems
with constant position of principal planes for different focal
length values.
A. Example 1
As the first example, we assume an optical system composed of
three thin lenses in air with variable focal len gth and fixed mutual
positions, where the change of the focal length of the system is
performed by the change of optical powers of individual thin
lens elements. The positions of the principal planes are fixed dur-
ing the change of the focal length. Table 1shows the paraxial
parameters of such an optical system calculated using
Eqs. (13)–(15). All values in Table 1are given in millimeters.
Table 1. Paraxial Parameters of the Thin Lens System
Composed of Three Tunable-Focus Lenses with Variable
Focal Length and Fixed Position of Principal Planes
d14,d24,sH3.985,s0
H0−3.985
f050 100 150 200
f0
11062.6667 1062.6667 1062.6667 1062.6667
f0
246.0293 84.7448 117.7612 146.2508
f0
31062.6667 1062.6667 1062.6667 1062.6667
sF−46.0150 −96.0150 −146.0150 −196.0150
s0
F046.0150 96.0150 146.0150 196.0150
Research Article Vol. 57, No. 14 / 10 May 2018 / Applied Optics 3717

It is obvious from Eqs. (13)–(15) that, generally, the optical
powers φ1,φ2, and φ3of individual thin lens elements will
change with the change of the focal length of the system.
However, in our case of a symmetrical optical system with
equal distances between the lenses (d1d24mm) and po-
sitions of principal planes located symmetrically (sH−s0
H0
3.895 mm), the optical powers φ1and φ3of the first and the
last element are the same, and they remain fixed during the
change of focal length of the system. The only parameter that
changes with the change of focal length of the system is the
optical power of the second element φ2, as can be seen from
Table 1.
B. Example 2
As the second example, let us show the calculation of the
parameters of the classical cemented doublet for several differ-
ent focal lengths with the same initial parameters d1and d2
(i.e., constant values of axial thicknesses of the doublet). We
require that all the doublets with different focal lengths (but
same axial thicknesses d1and d2) have the same position of
principal planes. Table 2shows the parameters of such doublets
calculated using Eqs. (19)–(22). All linear dimension values in
Table 1are given in millimeters.
Axial thicknesses of individual lenses of the doublet
d16mmand d26.6mmfor selected values of refractive
indices n21.5and n31.65 were chosen in such a way that
the positions of object and image focal points sFand s0
F0are the
same for given value of the focal length f0, as in Example 1.
As it is obvious from Eqs. (19)–(21), all the edge points M1,
M2, and M3would move with the change of the focal length of
the doublet in general case. However, for our special case, the
positions of points M1and M3would remain fixed, and only
point M2will move with the focal length variation, as one can
see from Table 2.
C. Example 3
As the third example, we will assume an active tunable-focus
liquid membrane doublet, whose optical surfaces consist of thin
elastic optically transparent membranes, and the volume in-
between the bounding surfaces is filled with transparent liquid.
Such an active optical element can change the value of its focal
length by changing the pressures of the liquid in different
chambers, thus changing the individual curvature of membrane
surfaces (see Figs. 2and 3). The calculations were performed
for the same refractive indices of media n21.5and
n31.65 between the membranes, as in the case of the
previous example of classical doublet for comparison purposes.
The calculation was performed using Eqs. (16)–(18) and
Eqs. (23)–(28), as described in the previous section. The search
for a solution was formulated as the optimization problem
given by Eqs. (29)–(33), which was then solved with the help
of the Global Optimization Toolbox of MATLAB [39,40], as
described in the last paragraph of Section 2.
The first doublet in Table 2(f050 mm) was chosen as
the initial state for calculation of parameters Δ14.0085 mm
and Δ3−4.0031 mm (initial axial separations were therefore
d16mmand d26.6mm). Further, the value of the half
diameter of the adaptive doublet h5mmwas chosen. The
calculated paraxial parameters of the tunable-focus doublet are
given in Table 3(all linear dimensions are given in millimeters).
The designed doublet has continuously variable focal length
and constant positions of principal planes (in paraxial space).
As it can be seen from Table 3, the positions of the edge
points M1,M2, and M3stay fixed during the change of focal
length of the system, and the positions Δ14.0085 mm and
Δ3−4.0031 mm of principal planes with respect to fixed
points M1and M3are fixed during the change of focal length
f0of the system.
Since we used the first column of Table 2from Example 2 as
the initial state of the tunable doublet in Example 3, the first
columns in Tables 2and 3show the same values of paraxial
parameters. However, for different focal lengths, one can see
that for the case of tunable-focus liquid membrane doublet,
the axial thicknesses d1and d2change their value as well as the
positions sHand s0
H0of principal planes with respect to the first
and last surface vertices V1and V3.
4. CONCLUSION
Our work was focused on the paraxial analysis of the problem
of design of an optical element with variable focal length, which
maintains positions of principal planes fixed during the change
of optical power. Such optical elements are important in design
of more complex optical systems (e.g., zoom systems), where
the fixed position of principal planes during the change of
optical power of such an optical element is essential during
the design process. Equations for calculation of such a system
in terms of thin lenses were provided, and a simplified
version of a real-world active optical element based on elastic
Table 2. Paraxial Parameters of Cemented Doublets
with Fixed Position of Principal Planes for Different Values
of Focal Length
n1n41,n21.5,n31.65,d16,d26.6,
sH3.985,s0
H0−3.985
f050 100 150 200
r1−531.3333 −531.3333 −531.3333 −531.3333
r26.9044 12.7117 17.6642 21.9376
r3690.7333 690.7333 690.7333 690.7333
sF−46.0150 −96.0150 −146.0150 −196.0150
s0
F046.0150 96.0150 146.0150 196.0150
Table 3. Paraxial Parameters of Tunable-Focus Liquid
Membrane Doublet with Fixed Position of Principal
Planes
n1n41,n21.5,n31.65,Δ14.0085,Δ2−4.0031
f050 100 150 200
d16.0000 6.5757 6.7360 6.8122
d26.6000 5.9139 5.7175 5.6235
r1−531.3333 −185.4538 −147.4632 −133.1933
r26.9044 8.9667 9.9613 10.5470
r3690.7333 147.6370 121.0455 111.6101
sF−46.0150 −96.0589 −146.0763 −196.0854
s0
F046.0150 96.0816 146.1002 196.1090
sH3.9850 3.9411 3.9237 3.9146
s0
H0−3.9850 −3.9184 −3.8998 −3.8910
3718 Vol. 57, No. 14 / 10 May 2018 / Applied Optics Research Article

membrane surfaces actuated by the pressure of optically trans-
parent liquid was presented and analyzed. The equations that
enable us to solve the problem of tunable-focus liquid mem-
brane doublet were derived, and the solution to these equations
using the optimization methods was proposed. The derived for-
mulas were verified by several examples.
Funding.
ˇ
Ceské Vysoké Učení Technické v Praze (
ˇ
CVUT)
(SGS18/105/OHK1/2T/11).
REFERENCES
1. G. Li, “Adaptive lens,”Prog. Opt. 55, 199–284 (2010).
2. H. Ren and S. T. Wu, Introduction to Adaptive Lenses (Wiley, 2012).
3. http://www.varioptic.com/.
4. http://www.optotune.com/.
5. S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for minia-
ture cameras,”Appl. Phys. Lett. 85, 1128–1130 (2004).
6. H. Ren, D. Fox, P. A. Anderson, B. Wu, and S.-T. Wu, “Tunable-
focus liquid lens controlled using a servo motor,”Opt. Express 14,
8031–8036 (2006).
7. H. Ren and S.-T. Wu, “Variable-focus liquid lens,”Opt. Express 15,
5931–5936 (2007).
8. R. Peng, J. Chen, and S. Zhuang, “Electrowetting-actuated zoom lens
with spherical-interface liquid lenses,”J. Opt. Soc. Am. A 25, 2644–
2650 (2008).
9. S. Xu, Y. Liu, H. Ren, and S.-T. Wu, “A novel adaptive mechanical-
wetting lens for visible and near infrared imaging,”Opt. Express 18,
12430–12435 (2010).
10. A. Miks, J. Novak, and P. Novak, “Generalized refractive tunable-
focus lens and its imaging characteristics,”Opt. Express 18, 9034–
9047 (2010).
11. A. Miks and J. Novak, “Third-order aberrations of the thin refractive
tunable-focus lens,”Opt. Lett. 35, 1031–1033 (2010).
12. H. Choi, D. S. Han, and Y. H. Won, “Adaptive double-sided fluidic lens
of polydimethylsiloxane membranes of matching thickness,”Opt. Lett.
36, 4701–4703 (2011).
13. A. M. Watson, K. Dease, S. Terrab, C. Roath, J. T. Gopinath, and V.
M. Bright, “Focus-tunable low-power electrowetting lenses with thin
parylene films,”Appl. Opt. 54, 6224–6229 (2015).
14. P. Zhao, C. Ataman, and H. Zappe, “Gravity-immune liquid-filled
tunable lens with reduced spherical aberration,”Appl. Opt. 55,
7816–7823 (2016).
15. K. Wei, H. Huang, Q. Wang, and Y. Zhao, “Focus-tunable liquid lens
with an aspherical membrane for improved central and peripheral res-
olutions at high diopters,”Opt. Express 24, 3929–3939 (2016).
16. N. Hasan, H. Kim, and C. H. Mastrangelo, “Large aperture tunable-
focus liquid lens using shape memory alloy spring,”Opt. Express
24, 13334–13342 (2016).
17. B. Jin, H. Ren, and W.-K. Choi, “Dielectric liquid lens with chevron-
patterned electrode,”Opt. Express 25, 32411–32419 (2017).
18. W. Kim, H. C. Yang, and D. S. Kim, “Wide and fast focus-tunable
dielectro-optofluidic lens via pinning of the interface of aqueous
and dielectric liquids,”Opt. Express 25, 14697–14705 (2017).
19. W. T. Welford, Aberrations of the Symmetrical Optical Systems
(Academic, 1974).
20. M. Born and E. Wolf, Principles of Optics (Oxford University, 1964).
21. A. Miks, Applied Optics (Czech Technical University, 2009).
22. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
23. A. D. Clark, Zoom Lenses (Adam Hilger, 1973).
24. A. Miks, J. Novak, and P. Novak, “Method of zoom lens design,”Appl.
Opt. 47, 6088–6098 (2008).
25. A. Mikšand J. Novák, “Method of first-order analysis of a three-
element two-conjugate zoom lens,”Appl. Opt. 56, 5301–5306
(2017).
26. R. Peng, J. Chen, C. Zhu, and S. Zhuang, “Design of a zoom lens
without motorized optical elements,”Opt. Express 15, 6664–6669
(2007).
27. N. Savidis, G. Peyman, N. Peyghambarian, and J. Schwiegerling,
“Nonmechanical zoom system through pressure-controlled tunable
fluidic lenses,”Appl. Opt. 52, 2858–2865 (2013)
28. A. Miks and J. Novak, “Three-component double conjugate zoom lens
system from tunable focus lenses,”Appl. Opt. 52, 862–865 (2013).
29. A. Miks and J. Novak, “Paraxial imaging properties of double conju-
gate zoom lens system composed of three tunable-focus lenses,”Opt.
Lasers Eng. 53,86–89 (2014).
30. G. Lan, T. F. Mauger, and G. Li, “Design of high-performance adaptive
objective lens with large optical depth scanning range for ultrabroad
near infrared microscopic imaging,”Biomed. Opt. Express 6,
3362–3377 (2015).
31. D. Lee and S.-C. Park, “Design of an 8x four-group inner-focus zoom
system using a focus tunable lens,”J. Opt. Soc. Korea 20, 283–290
(2016).
32. L. Li, D. Wang, C. Liu, and Q.-H. Wang, “Zoom microscope objective
using electrowetting lenses,”Opt. Express 24, 2931–2940 (2016).
33. L. Li, D. Wang, C. Liu, and Q.-H. Wang, “Ultrathin zoom telescopic
objective,”Opt. Express 24, 18674–18684 (2016).
34. L. Li, R.-Y. Yuan, J.-H. Wang, and Q.-H. Wang, “Electrically optofluidic
zoom system with a large zoom range and high-resolution image,”
Opt. Express 25, 22280–22291 (2017).
35. D. Kopp, T. Brender, and H. Zappe, “All-liquid dual-lens optofluidic
zoom system,”Appl. Opt. 56, 3758–3763 (2017).
36. A. Mikšand P. Novák, “Double-sided telecentric zoom lens consisting
of four tunable lenses with fixed distance between object and image
plane,”Appl. Opt. 56, 7020–7023 (2017).
37. L. E. Scales, Introduction to Non-linear Optimization (Springer, 1985).
38. M. Aoki, Introduction to Optimization Techniques: Fundamentals and
Applications of Nonlinear Programming (Macmillan, 1971).
39. https://www.mathworks.com/products/matlab.html.
40. https://www.mathworks.com/products/global-optimization.html.
Research Article Vol. 57, No. 14 / 10 May 2018 / Applied Optics 3719