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Adaptive interferometer based on wave mixing in a photorefractive crystal
under alternating electric field
Alexei A. Kamshilina)
University of Joensuu, Department of Physics, P.O. Box 111, FIN-80101 Joensuu, Finland
Alexander I. Grachev
A. F. Ioffe Physics-Technical Institute, Polytekhnicheskaya 26, 194021 St. Petersburg, Russia
共Received 26 March 2002; accepted 23 August 2002兲
Coupling of two waves with different polarization states in a photorefractive crystal under an ac
electric field allows us to achieve the linear transformation of a small transient phase shift into the
intensity modulation without any output polarization filtering. This can lead to design of an adaptive
interferometer with sensitivity that reaches the classical homodyne detection limit. © 2002
American Institute of Physics. 关DOI: 10.1063/1.1515131兴
Optical detection of extremely small surface displace-
ment 共such as that caused by ultrasound兲of real objects is an
important task for nondestructive remote evaluation of mate-
rial quality.1Two-wave coupling via dynamic hologram re-
corded in a photorefractive 共PR兲crystal is known as the sim-
plest and most efficient technique for demodulation of a
transient phase shift encrypted in a speckled wave, such as
that reflected from a rough surface.2Different configurations
of the optical setup allowing for linear phase demodulation
in PR crystals have been proposed.3–6 Two of them possess
sensitivity that approaches the classical homodyne detection
limit.5These are direct intensity detection of the beam trans-
mitted through the crystal under an external dc field 共the
hologram recording in the drift mode兲and differential detec-
tion of two parts split from the transmitted beam by a prop-
erly oriented polarization beam-splitter when the hologram is
recorded in the diffusion or mixed mode. In any PR phase
demodulator, the signal 共intensity modulation兲is defined pri-
marily by the modulus of the space-charge field induced by
the intensity pattern. It is known that in PR crystals with high
mobility–lifetime product of charge carriers, the space-
charge field created under an ac-field is Qtimes higher than
under a dc field.7,8 Here Qis the quality factor of the space-
charge waves, which depends on the external field and the
grating spacing and in some crystals it can be much higher
than the unity.7Therefore, the possibility of linear phase de-
modulation using PR crystal under ac field would be advan-
tageous, considering in addition that there is no screening
effect for an ac field. On one hand, however, direct phase
demodulation requires formation of at least a partly non-
shifted grating, which is not the case for recording under an
ac field.2,5 On the other hand, installation of a polarization
beam-splitter5or a polarization filter6results in spurious in-
tensity modulation caused by an external ac field applied to
the crystal. In this letter we show that direct phase demodu-
lation can be achieved in PR crystals under an ac field by
mixing two waves with different polarization states. It was
found that the sensitivity of the proposed configuration ap-
proaches that of a classical interferometer, and the system
operates well even applying ac electric field of the sinusoidal
form.
Since the fastest PR crystals belong to the cubic symme-
try (4
¯
3mand 23 groups兲, we concentrate on these symmetry
groups. In these crystals, anisotropic nature of light diffrac-
tion is especially important, providing that the energy and
polarization exchange between interacting light waves can-
not generally be held apart.8It is usually assumed that the
polarization states of the object and reference beams are the
same 共more frequently, linear兲to provide the highest contrast
of an interference pattern and consequently, the largest sig-
nal. In our configuration we exploit the polarization degree
of freedom. In the so-called transverse geometry8共the grat-
ing vector is parallel to the axis 关1
¯
10兴and the light beams
propagate along the axis 关110兴兲, light diffraction is accompa-
nied by the rotation of the plane of polarization for the lin-
early polarized light. In this geometry 共and not only兲mixing
of waves with linear and elliptical polarization allows for
linear phase-to-intensity transformation. It occurs because
the phase difference ⌬
between orthogonal components (x
and y) of the elliptically polarized light beam is transferred
into a non-zero phase difference in the interference term be-
tween the transmitted part of the object beam and diffracted
part of the reference beam with the optimal ⌬
being
/2.
Considering the important role of vectorial light interaction
in the proposed technique, we call it vectorial wave mixing
共VWM兲.
Evolution of light wave amplitudes in the crystal at the
steady state is governed by the vectorial system of coupled
wave equations:8
dAS
dz ⫽i
ˆ共EA,
兲•AS⫹iESC共EA,AS,AR兲•
␥
ˆ•AR,
共1兲
dAR
dz ⫽i
ˆ共EA,
兲•AR⫹iESC
*共EA,AS,AR兲•
␥
ˆ•AS.
Here ASand ARare the light amplitude 共Jones vectors兲of the
signal and reference waves, respectively. The 2⫻2 matrix
ˆ(EA,
) includes the effects of optical activity
and bire-
fringence induced by the external field EA, while the 2⫻2
matrix
␥
ˆgoverns waves coupling via the space-charge-field
a兲Electronic mail: alexei.kamshilin@hut.fi
APPLIED PHYSICS LETTERS VOLUME 81, NUMBER 16 14 OCTOBER 2002
29230003-6951/2002/81(16)/2923/3/$19.00 © 2002 American Institute of Physics
Downloaded 11 Oct 2002 to 130.233.188.144. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp
grating ESC . In the undepleted-pump approximation (m
Ⰶ1) and in the nonsaturated regime 共we refer here to the
tarps saturation兲, the space-charge field ESC is given by
ESC⫽iQmEA共nonlocal grating recorded under an ac field兲7
or by ESC⫽mEA共local grating under a dc field兲with m
⫽(AR•AS
*)/(
兩
AR
兩
2⫹
兩
AS
兩
2). Currently, we are not able to
present an analytical solution of Eq. 共1兲for arbitrary crystal
orientation and arbitrary polarization states of the interacting
beams. Nevertheless, we show in Fig. 1 two typical ex-
amples of the numerical solution of Eq. 1 calculating the
signal-to-noise ratio 共SNR兲as a function of the crystal length
zto compare performance of direct phase demodulation un-
der dc and ac fields. For the sake of simplicity the graphs of
Fig. 1 were calculated for nonabsorbing crystal of the group
4
¯
3m(
⫽0) in the transverse geometry. The external field
EA⫽10 kV/cm and the electro-optic coefficient r41
⫽5 pm/V 共typical for CdTe and Bi12TiO20) were used in
these calculations. All data in Fig. 1 are normalized to the
SNR of a classical homodyne interferometer. To achieve di-
rect phase demodulation in this geometry under a dc field
共curve a), input polarizations were set to be linear, making
an angle of 45° to the axis 关1
¯
10兴. As known, the direct phase
demodulation under a dc field possesses sensitivity ap-
proaching the classical homodyne detection limit.5Curve b
was calculated for the nonlocal grating 共under an ac field兲
when the reference beam was circularly polarized but the
object beam is linearly polarized, with the plane of polariza-
tion being orthogonal to 关1
¯
10兴. As seen from curve c, fur-
ther adjustment of the input polarization states allows us to
reach the SNR of a classical interferometer. In the last case
the reference beam is elliptically polarized with an ellipticity
of 1:10, and the major axis of the ellipsis making an angle of
91° with 关1
¯
10兴, while the object beam is linearly polarized,
making an angle of 84° to 关1
¯
10兴.
Experiments were carried out with a Bi12TiO20 crystal
共symmetry group 23兲in the transverse geometry. The sample
dimensions are 2.02⫻3.37⫻3.86 mm3, where the first mea-
surement is the interelectrode distance and the second is the
light interaction length. The sample was antireflection coated
and it possesses an absorption coefficient of 0.75 cm⫺1re-
sulting in the static light transmission of 75% at the wave-
length of 632.8 nm. Acalibrated electro-optic modulator was
inserted in the object beam to modulate its phase. We have
measured the SNR of the interferometer when the light
power of the object beam was 0.2 mW. At this power level,
the photodetector 共New Focus model # 1801兲operates in the
shot-noise-limited regime. The reference-to-object beam in-
tensity ratio was R⫽120 and the amplitude of the phase
modulation was 0.1 radians at 3 kHz. The polarization states
of the object and reference beams were linear at 88° with
关1
¯
10兴and elliptical with an ellipticity of 1:3.5 and the major
axis at the angle of 124° with 关1
¯
10兴, respectively.
The measured intensity modulation was recalculated to
the equivalent surface displacement and normalized to a ho-
modyne detection limit
␦
lim , expressed in nm
冑
W/Hz.9
␦
lim
corresponds to the minimal detectable displacement of the
object surface (SNR⫽1) in a classical interferometer using
the object beam power of P0共the same as incident on the
crystal兲and the detection bandwidth of ⌬f:9
␦
lim⫽
4
冑
h
•⌬f
2
•P0,共2兲
where is the wavelength, h
is the photon energy, and
is
the quantum efficiency of the detector. Dependence of the
relative sensitivity is plotted as a function of the external ac
field E0in Fig. 2 by circles. The solid line in Fig. 2 repre-
sents the theoretical dependence calculated using Eq. 共1兲
with the parameters of our particular sample and configura-
tion, including input polarization states, crystal absorption,
and optical activity. One can see that the theory fits the ex-
periment quite well. Sensitivity deterioration observed at
E0⬎10 kV/cm is associated with increased light-induced
scattering 共known as the fanning effect兲, which was not in-
cluded in the theoretical consideration. The minimal relative
sensitivity achieved in our experiment is 1.5, which is
slightly better than the previously reported best sensitivity
共1.6兲for direct phase-demodulation configuration using a
CdZnTe crystal.9
In Fig. 3 we plot the signal measured by the photodetec-
tor when a sinusoidal external electric field is applied to the
crystal. As one can see, the light power is modulated at the
FIG. 1. SNR as a function of the light interaction length calculated for
different configurations of direct phase demodulation via wave mixing in the
transverse geometry of a PR crystal. 共a兲mixing of linearly polarized waves
under a dc field; 共b兲mixing of linearly and circularly polarized waves under
an ac field; and 共c兲mixing of linearly and elliptically polarized wave under
an ac field. The SNR is normalized to that obtained with a classical inter-
ferometer.
FIG. 2. The normalized sensitivity 共circles兲of the adaptive VWM interfer-
ometer with the Bi12TiO20 crystal as a function of the external electric field.
The solid line is the theoretical dependence calculated using Eq. 共1兲. The
measurements were carried out using a He-Ne laser at ⫽632.8 nm.
2924 Appl. Phys. Lett., Vol. 81, No. 16, 14 October 2002 A. A. Kamshilin and A. I. Grachev
Downloaded 11 Oct 2002 to 130.233.188.144. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp
frequency of phase modulation which proves the linear
phase-to-intensity transformation by nonlocal space-charge-
field grating recorded under an ac field. Note that the ampli-
tude of the signal is modulated by only about 10% 共and there
are no transient artifacts兲when a sine wave form is used for
the applied field. Such a behavior is opposite to an interfer-
ometer with polarization filtration,6and it is more practical
since it does not require any electronic triggering system. In
the proposed configuration without output polarization filter-
ing, the modulation of the average level of the transmitted
intensity may arise from the modulation of diffraction effi-
ciency of the dynamic hologram due to dependence of the
polarization state of the reference beam inside the crystal on
the external electric field.10 However, such a dependence can
be softened by a proper adjustment of the input polarization
state that we demonstrate experimentally in Fig. 3. Note that
this adjustment decreases the maximal achievable SNR of
the system by less than 10%.
We have also found that in our configuration, coupling
of the circularly polarized reference beam with the object
beam polarized linearly and parallel to the axis 关1
¯
10兴leads
to the same rate of the phase-to-intensity transformation as
coupling of the same reference beam with the object beam
having orthogonal polarization, but the sign of the intensity
change is opposite. This feature allows designing of an adap-
tive interferometer capable to operate with a nonpolarized
object beam. It can be achieved by installation of a polariza-
tion analyzer before the crystal, mixing the reference beam
and two orthogonally polarized parts of the object beam in-
side the crystal, and differential intensity detection of both
transmitted object beams by two photodiodes. Due to differ-
ential detection, the amplitude noise of the laser can be can-
celled in this configuration. In this interferometer, the polar-
ization beam-splitter serves to create two orthogonally
polarized beams from a nonpolarized object beam that are
then mixed in the crystal with the reference beam. In some
sense the last configuration looks like that proposed in Ref.
11, but such distinctive features as space-charge enhance-
ment by an ac field and installation of the polarization beam-
splitter before the crystal lead to the advantages discussed
earlier.
In conclusion, we have demonstrated that coupling of
two waves with different polarization states in a PR crystal
of cubic symmetry under an ac electric field allows linear
transformation of a small transient phase shift into the inten-
sity modulation without any output polarization filtering. An
adaptive interferometer based on the proposed configuration
can possess the sensitivity approaching to the classical ho-
modyne detection limit. It was shown that the relative sensi-
tivity of an interferometer with a Bi12TiO20 crystal under an
ac field is higher than the sensitivity of the previously re-
ported adaptive interferometers with PR crystals. It is worth
noting that the recording of a dynamic hologram under an ac
field allows focusing of interacting beams inside the crystal
共in contrast to recording under a dc field兲, thus resulting in
higher intensity and faster response time, which is very im-
portant from the practical point of view.
The authors acknowledge Dr. M. B. Klein for useful
discussions and Dr. V. V. Prokofiev for growth and prepara-
tion of Bi12TiO20 crystals.
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FIG. 3. Signal wave form measured with the VWM interferometer when a
sinusoidal external field of 15.6 kV/cm at 1.0 kHz is applied to the Bi12TiO20
crystal. The phase of the object beam is modulated at the frequency of 50
kHz. Trace 共a兲is a signal from the photo-receiver; trace 共b兲is a voltage
applied to the crystal.
2925Appl. Phys. Lett., Vol. 81, No. 16, 14 October 2002 A. A. Kamshilin and A. I. Grachev
Downloaded 11 Oct 2002 to 130.233.188.144. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp
