Cylindrical adaptive lenses
O.A.Zayakin1, M.Yu.Loktev2
P.N.Lebedev Physical Institute, Russian Academy of Sciences,
Novo-Sadovaya st., 221, Samara, 443011 Russia.
G.D.Love3, A.F.Naumov4
Department of Physics & School of Engineering, University of Durham,
South Road, Durham, DH1 3LE, United Kingdom
The correction of low-order aberrations is important in many adaptive optics applications. Modal
cylindrical adaptive lenses can be used to correct several low order aberrations. Furthermore, the same
technology can be used for creating arrays of controllable lenses. The most significant feature of these cylindrical
lenses is a modal control system based on nematic liquid crystals. Modal control allows the precise control of the
spatial phase distribution in order to achieve an aberration-free lens. This has been investigated both by computer
simulation and experiment. We found that the introduction of a 180-degree phase shift between the second or
higher order harmonics and no phase shift between the first harmonic components of the control voltages
improves the optical performance of the device. These extra harmonics eliminate the strong dependence of the
liquid crystal orientation on the impedance of the device. This is especially important for devices with small
apertures. It also was found that modal cylindrical lens controlled by two-harmonic voltages can produce a slitlike beam whose transverse structure has the shape of a pulse which remains unchanged over a long range in the
direction of propagation.
We investigated a device with two-crossed one-dimensional control electrodes and produced a lens with
controllable focus and astigmatism.
1. Construction and operation principle of cylindrical modal liquid crystal lens
The correction of a low-order aberrations, such as defocus and astigmatism, is important in optical system design for
many applications. Liquid crystal (LC) phase modulators are attractive devices for aberration correction due to their high
transparency, small power consumption and low control
voltages. Modal LC lenses are described in ref [1], and they
are very simple to produce.
Let us consider the structure of a one-dimensional
LC corrector (fig. 1). A thin LC layer is formed by the two
glass substrates which have transparent electrodes deposited
on them. One of these electrodes (ground electrode) has a low
50–200 / resistance, and the other (control electrode) has
a high 3–8 M/ resistance. The control electrode has two
linear equidistant contacts on the periphery of the cell
aperture. Dielectric spacers with a fixed thickness set the
Fig.1. Construction of cylindrical modal LC lens.
thickness of the LC layer. The initially planar alignment of the
LC on the substrates surfaces is determined by the alignment
coating . The control voltages, U1 and U2, are applied to each contact. U1 and U2 are given by,
1
E-mail: olegz@fian.samara.ru
E-mail: loktev@fian.samara.ru
3
E-mail: g.d.love@durham.ac.uk
4
E-mail: a.f.naumov@durham.ac.uk
2
(1)
where U10 and U20 are the amplitudes, ck are the harmonic
coefficients,  is the fundamental frequency, and k is the phase
of each harmonic. The electrical analogue of this corrector is the
circuit with distributed parameters shown in fig 2. The resistance
RLC corresponds to a leakage current across the LC layer.
In the case of a sinusoidal control signal with the
frequency , the voltage amplitude distribution along the LC
corrector aperture is described by the following equation
 2U
x 2
 S c
U
  S gU ,
t
(2)
Fig.2. Electrical circuit equivalent to a
cylindrical modal LC lens.
10
wavelengths
n

U1  U10  ck sin kt

k 1
,

n

U 2  U 20  ck sin(kt   k )
k 1

8
6
4
2
where s is the sheet resistance of ground and control electrodes,
c and g are specific capacitance and conductance of the LC layer,
respectively. Based on equation (2), a numerical model of the LC
0
lens was developed. A theoretical analysis of the LC lens for a
0
2
4
6
8
sinusoidal control voltage (n = 1) was performed in ref [2]. Let
Rms voltage, V
us consider the LC lens behavior for symmetrical boundary
Fig.3.
Electro-optic
response of 25 m layer of
conditions (U10 = U20, 1 = 0). Due to the reactive character of
the
LC
E49.
the distributed RC-circuit and a leakage current through the LC,
the AC voltage across the LC falls from the periphery to the
center. In [2] it is shown that in this case a quasi-parabolic voltage distribution is produced. The dependence of the LC
extraordinary refractive index on the field strength results in a variation of the phase of a light beam propagating through the
device. The dependence of phase in a layer of LC E49 with a thickness of 25 m versus rms voltage for a wavelength of
0.633 m is shown in fig. 3. If the rms voltage does not extend from the quasi-linear region of the LC electro-optic response then
the phase distribution along the aperture will be quasiparabolic too. For linearly polarized light with the polarization axis parallel
to the direction of the LC initial planar orientation this modulator operates as a cylindrical lens with the following focal distance
F
l 2
,
4(  c   e )
(3)
where l is an aperture width,  is a wave length, c and e are values of phase delay in a center and at an edge of aperture
respectively. Hence, the focal length of LC lens is determined by the value of a voltage difference between the periphery and
center. This value is determined mainly by the frequency of the applied voltage. The range of working frequencies for certain LC
lenses depends on the reactive parameters of the LC cell - the sheet resistance of the control electrode, the specific capacitance
and the conductance of the LC layer.
The cylindrical LC lens control technique described above was investigated in previous works [3,4]. This work is
devoted to the theoretical and experimental investigation of cylindrical LC lenses operating in various modes. It is shown that the
focal length may be controlled not only by means of frequency but by introducing a phase shift between the control voltage
components as well. In this case the role of the reactive parameters of the cell is reduced. We have found that practically, the
most convenient way to control a LC lens is with a two-frequency signal with a phase shift 1 = 0 between the first harmonic
components and 2 =  between the second harmonic components. The possibilities of this control mode will be demonstrated in
the following section.
Fig.4. Optical set up for the investigation of a cylindrical modal LC lens control modes.
2. Investigation of cylindrical modal LC lens control modes
We have investigated the optical performance of cylindrical modal
LC lenses using a Michelson interferometer. The optical setup is shown in
fig. 4. We placed the adaptive lens in one arm of the interferometer so that
the axis of the LC's extraordinary axis was parallel to the polarization
direction. The interference fringe patterns were projected on to the CCD
camera. A computer with a sound card and an amplifier was used to control
the LC lens. The interferometer was adjusted to give fringes normal to the
linear contacts, so that the lens's operation causes a bending of these
fringes. The interferograms are shown on the left hand side of fig. 5, and
are accompanied by the rms voltage distributions calculated using the
numerical model of the cylindrical lens (right).
Fig. 5a shows an example of lens operation by frequency control.
The required phase variation is caused by the reactive character of the
device, and its value increases with frequency for a fixed control voltage. If
we set the phase shift 1= between components U1 and U2 of the
sinusoidal control voltage then the value of phase variation does not
depend on frequency. It can be explained by the fact that when oscillations
of U1 and U2 occur in anti-phase then the instantaneous voltage distribution
in the lens is antisymmetric, and the voltage in the aperture center is always
zero. However, because the rms voltage in central region falls below the
threshold value of the LC, the phase delay value is constant in this region
and has a discontinuity at its edges. It results in undesirable diffraction at
the edges of the central region shown in fig.5b.
This effect can be eliminated by the use of a two-harmonic control
voltage (fig. 5c). The phase shift between the components of the first
Fig.5. Interference fringe patterns (left)
and voltage distributions (right) of a
cylindrical modal LC lens in different
control modes:
(a) U10=U20=5 V, f=4 kHz, c1=1, c2=0,
1=2=0;
(b) U10=U20=5 V, f=100 Hz, c1=1, c2=0,
1=, 2=0;
(c) U10=U20=5 V, f=100 Hz, c1=0.44721,
c2=0.89442, 1=0, 2=;
(d) U10=U20=5 V, f=1 kHz, c1=c2=0.7071,
1=0, 2=.
10
10
I=I(z), a.u.
I=I(x), a.u.
on
6
4
off
2
I=I(z), a.u
8
8
6
4
0
-4
-2
0
2
x, mm
40
20
2
0
a
60
0
20
4
b
40
60
z, cm
80
100
20
c
40
60
80
100
z, cm
Fig.6. Experimental intensity distributions (a) 0.5 m away from cylindrical modal LC lens and (b) along
the optical axis of a lens in “axicon” mode: U10=U20=5.21 V, f=100 Hz, c1=0.568, c2=0.823, 1=0,
2=; (c) calculated intensity distribution along the optical axis of an ideal parabolic lens with 0.5 m
focal length.
harmonic is 1 = 0 and between the components of the second harmonic is 2 = . The frequency of the first harmonic must be
sufficiently small so that it does not vary across the lens and its voltage must exceed the LC threshold value over all of the
aperture. It is equivalent to making a pretilt of the LC. We can control the pretilt angle by changing the amplitude of the first
harmonic. The required phase variation is caused mainly by the second harmonic. The discontinuities in the fringes is absent
because the voltage does not fall below the threshold value. In this case the optical performance of the adaptive lens is similar to
a bi-prism with a small angle. Like an axicon, such a bi-prism allows the focusing of the light beam into a slit-like beam whose
transverse structure has the shape of a pulse with a slowly varying transverse shape. Such beams are called “pseudonondiffracting” beams [5]. Control of the second harmonic amplitude in this mode results in a change of phase variation that is
equivalent to a change of prism angle. It results in the shift of the beam waist along the optical axis. Thus, this control mode
allows the realisation of an “adaptive axicon”.
The experimental transverse structure of the light beam in the "axicon” mode, at a distance of 50 cm behind the lens
compared with the intensity distribution of a non-focused beam is shown on Fig.6a. It was obtained using the following control
voltage parameters: U10 = U20 = 5.21 V, f = 100 Hz, c1 = 0.568, c2 = 0.823, 1 = 0, 2 = . Experimental intensity distribution
along the optical axis behind the cylindrical modal LC lens in “axicon” mode is shown on fig. 6b, and calculated intensity
distribution behind ideal parabolic lens with 0.5 m focal lens is shown on fig.6c. It is seen that in “axicon” mode the beam waist
height changes slower than for ordinary focused beam.
Two-harmonics control allow the significant reduction of the range of working frequencies. This is especially important
for microlenses fabrication. At the consideration of frequency lens control in [2] it was shown that lens demonstrates the best
optical performance maintaining the following criterion
l 2 s g 2  2 c 2 ~ 1 ,
(4)
where g is proportional to frequency. Hence, for small lenses (small l) it is necessary to increase the frequency and (or) sheet
resistance of the control electrode s. The technology of the evaporation of high resistance transparent electrodes allows the
production of coatings with sheet resistances up to 16 M/ [6]. For instance, let us consider a cylindrical modal LC lens with a
width l=1 mm, thickness of LC layer d = 25m, and LC E49 with birefringence n = 0.26. For these parameters the minimum
focal length is 1.2cm. Computer simulation of this lens for a focal length of 3 cm results in the following optimal control voltage
parameters: amplitude U10 = U20 = 5.19 V and frequency f = 89 kHz. However at high frequencies ~ 90 - 100 kHz undesirable
anomalous growth of conductivity of the LC become apparent [7]. The formation of the required phase variation through the
phase shifts between components of the harmonics at low frequencies allows this problem to be solved. Fig. 5d shows that the
combined influence of the frequency and the phase shifts between the components of the control voltages on the lens profile. It
c12
 c22
1.
(5)
The phase shifts between components of control voltages were
fixed: 1 = 0 and 2 = . At given parameters we carried out a gradient
optimization of the phase profile by frequency. The value of the rms
deviation of the phase profile from a paraboloid obtained at each step was
compared with the best result obtained at previous steps of algorithm. For
each focal length we calculated ~200 steps. The results are shown in
fig. 7. It can be seen that the use of a two-harmonic control voltage allows
the reduction of phase aberrations. For instance, at a focal length 1.5 m
the rms deviation of phase profile from an ideal parabola was 0.071  for
one-frequency, and 0.012  for two-frequency control. This value is
comparable with the phase calculation error caused by the fact that the
voltage distribution was calculated with an accuracy of 0.001 V.
However, the aberrations sharply increase when approaching the minimal
theoretical limit of focal lengths (48 cm here), and the value of the rms
phase deviation for F  0.7 m is almost the same for one and two
harmonics. In this case the aberrations are caused mainly by nonlinearity
of the LC electro-optical response
0.6
RMS, wavelengths
can be seen that a quasi-parabolic phase profile is produced. In practice we
use simultaneous control by frequency and phase shifts between harmonic
components. This possibility is especially important because it gives us an
additional degree of freedom for the minimization of phase aberrations. If
we vary the frequency of the control signal and the proportions of
harmonics coefficients we can significantly diminish the phase aberrations
in comparison with one-frequency control.
We verified this assumption by numerical simulation. The
optimization-based search of optimal two-harmonics voltages was carried
out by a Monte-Carlo method combined with gradient descent technique.
On each step of the algorithm we generated the amplitude and harmonic
coefficients by means of a random number generator with the single
limitation
One harmonic
Two harmonics
0.4
0.2
0.0
0
1
2
3
4
F, m
Fig.7. Dependencies of the minimal rms
deviation of the phase profile from a parabolic
versus focal length for one (a) and two (b)
harmonics control voltages (numerical
simulation).
.3. Astigmatic phase modulator
En route to achieving more complicated two-dimensional phase
distributions, we have fabricated an astigmatic phase modulator. It
consists of two crossed control electrodes separated by the LC layer. The
control voltage addressing scheme is shown in fig 8.
We studied the operation of the modulator by placing it between
two crossed polarizers. The angles between the direction of initial planar
orientation of the LC and axes of the polarizers was 45. It can be seen
from fig. 9a that when the amplitudes and frequencies of the control
voltages U1 and U2 are equal then the modulator operates as a spherical
lens. The phase variation along the aperture increases with frequency (Fig.
9a,b). It results in an increase of lens optical power. Furthermore, it is
possible to achieve astigmatism with orthogonal axes, as it is shown in Fig
9c. We may control the amplitude of the astigmatism by changing the
phase shift between U1 and U2. The possibility of simultaneous focus and
astigmatism control is a very important in many adaptive optics
Fig.8. Scheme of control voltage addressing for
astigmatic LC phase modulators.
Fig.9. Interference patterns of astigmatic LC
phase modulator obtained by positioning the
astigmatic LC phase modulator between crossed
polarizers: (a) U10 = U20 = 10 V, f =1 kHz;
(b) U10 = U20 = 10 V, f = 2 kHz; (c) U10 = U20 =
10 V, f = 2 kHz;  = 1.57 rad.
application.
Conclusion
In this work we have demonstrated experimentally the possibility of cylindrical modal LC lenses controlled by the
introduction of phase shifts between the control voltages. In this case the reactive parameters of the LC lens – specific
capacitance and conductance of the LC and the sheet resistances of the control electrode – do not have a strong influence on the
value of phase variation along the lens aperture. Therefore, it will be possible to produce controlled microlenses with small phase
aberrations.
It was also demonstrated that the use of cylindrical modal LC lens enables the focussing of a light beam into a slit-like
beam with a narrow beam waist stretched along the optical axis. In a computer simulation it was shown that combined influence
of the frequency of control voltages and phase shifts between its harmonics may improve the optical performance of the device.
We have manufactured a two-dimensional LC phase corrector based on two crossed control electrodes. Its optical
performance is similar to a lens with controlled focus and astigmatism.
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