J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Two-dimensional Numerical Simulation of Microwave
Imaging Reﬂectometry
Zhongbing Shi1) , Yoshio Nagayama2) , Daisuke Kuwahara3) , Tomokazu Yoshinaga2) , Masaharu
Sugito2) , Soichiro Yamaguchi4)
The Graduate University for Advanced Studies, Toki 509-5292, Japan
2)
National Institute for Fusion Science, Toki 509-5292, Japan3)
Department of Energy Science, Tokyo Institute of Technology, Tokyo 152-8550, Japan
4)
Kansai University, Suita 564-8680, Japan
1)(Received: 3 September 2008 / Accepted: 7 November 2008)

This work describes a numerical study of the two dimensional turbulent ﬂuctuations in microwave
imaging reﬂectometry (MIR) system. The numerical results show that the ﬂuctuations of the cutoﬀ
surface can be obtained from the phase of the reﬂected waves when these are collected with a wide
aperture optical lens system, which is used to form an image of the cutoﬀ onto the detector surface.
A laboratory arrangement of the MIR system is made. The preliminary tests show that the MIR
system works well at the in-focus condition when the ﬂuctuation amplitude is much smaller than the
wavelength of the launching wave. However, the reﬂected signals are distorted at strong ﬂuctuations.
The relationships between the ﬂuctuation amplitude and the wavenumber of the ﬂuctuation which may
cause the distortions of the reﬂected signals, are discussed.
Keywords: Microwave imaging Reﬂectometry (MIR), density ﬂuctuation
development in several fusion devices, such as LHD,
DIII-D and ASDEX-U [4, 7]. The further understanding of the received signals is very important to develop
the next generation of MIR system. Although the encouraging initial results have been obtained in TEXTOR and LHD, there remain many issues [4, 7, 11],such as the fringe jump, the antenna array and receiver array. Among them, the optical aberration is
one of the biggest issues. Therefore, the optical lensmust be considered in the simulation.
Simulations have been performed at the standard
reﬂectometry without imaging optics [6, 8, 12]. In thiswork, the numerical simulation model based on the
Huygens-Fresnel equation is used to simulate the wave
propagation in a microwave imaging system. The optical lens is considered in this model. A laboratory
arrangement of the MIR system is made. The experimental results show that the MIR system works well at
the in-focus condition when the ﬂuctuation amplitude
is much smaller than the wavelength of the launching wave. However, the reﬂected signals are distorted
at strong ﬂuctuations. Both the simulation and theexperiments show that the distortions of the reﬂectedsignals depend on not only the ﬂuctuation amplitude
but also the wavenumber of the ﬂuctuation. Theirrelationships are discussed by the geometrical approximation.
Section 2 contains a description of the numerical simulation based on the Huygens-Fresnel equation,
which is used for the propagation of the reﬂected signals. The simulation results in the case of diﬀerent
amplitude and the phase ﬂuctuations are shown. The

1. Introduction
Microwave reﬂectometry is a powerful tool tomeasure the plasma density proﬁle and density ﬂuctuation [1–3]. In a reﬂectometry system, the microwave
illuminates the plasma, and the wave is reﬂected by
the cutoﬀ layer in the plasma. The phase diﬀerence
between the incident wave and the reﬂected wave is detected. The reﬂected wave is mainly modiﬁed by thedensity ﬂuctuations close to the cutoﬀ region, and thephase ﬂuctuation is proportional to the density ﬂuctuation as φ ∝ Ln ñ/n, where Ln = n/(dn/dr)r=rc is thereverse density gradient length at the cutoﬀ surface.
The wave propagation is well understood in a onedimensional (1D) reﬂectometer system [1]. However,in the two-dimensional (2D) and three-dimensional(3D) conﬁgurations the complicated interference pattern appears on the detection plane, because the reﬂected wave propagates in diﬀerent directions. Therefore, it is very diﬃcult to extract any useful information from the ﬂuctuations [1, 4–6].
To correct the disturbed wave front, the opticalimaging technique is used in the reﬂectometry, named
as microwave imaging refelctometry (MIR) [1,4]. Thisdiagnostic system uses the wide aperture optics toform an image of the reﬂected surface onto a 2D detector array located at the image plane. The time evolutions of 2D pictures of density ﬂuctuation at the cutoﬀ
surface are captured, just like a movie. The feasibilities of MIR for turbulence measurement have been
investigated in theories and experiments [1, 4, 7–10]
intensively. So far, the MIR diagnostics are underauthor’s e-mail: shi.zhongbing@nifs.ac.jp

109©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Z. Shi et al., Two-Dimensional Numerical Simulation of Microwave Imaging Reflectometry

arrangement of the MIR is described in section 3. The
weak and strong ﬂuctuations obtained from MIR system are compared and explained. The summary anddiscussions are given in section 4.

poloidal plasma density ﬂuctuation causes a phasemodulation with poloidal wavenumber kp . Assume
the ﬂuctuation is localized near the cutoﬀ surface andthe ﬂuctuation radius is much smaller than the cutoﬀ
radius, the cutoﬀ surface is given as:

2. Numerical simulation
2.1 ModelR(x, z) = x2 /R0 + (z − z0 )2 /R0�+
aj sin(kpj (σj )z + θj )The electric ﬁeld in all space can be calculated
from the Maxwell equation. It is used in the reﬂectometer model for the propagation of reﬂected signals.
In this simulation, we assume the wave is only reﬂected
at the cutoﬀ surface in vacuum. The full wave equation can be solved by means of Fourier method, given
as follows [12]:
�
√ 2 2
1 ∞
E(x, z) =Ez (kz )ei(kz z+x k0 −kz ) dkz (1)
2π −∞
� ∞
E0 (z)e−ikz z dz
(2)
Ez (kz ) =where a is the ﬂuctuation amplitude, the wavenumber
kpj has the standard deviation σj , θj represents theinitial phase, z0 is the vertical shift of the cutoﬀ surface, and R0 is the cutoﬀ radius without ﬂuctuation.
In general, the concave mirror is used in the MIR
optics system. The role of the concave mirror is similar
to the convex lens, which can make a convergent beam.
Therefore, the mirror system in MIR can be reduced.
Figure 1 shows the schematic diagrams of MIR. The
image of the cutoﬀ surface at the detector screen ismade by the optical lens M. The wave propagation
direction rotates angle α by the optical lens M. The
electric ﬁeld changes as:−∞

where k0 is the wavenumber of the incident microwave,
E0 (z) is the distribution of the incident microwave at
position x0 , given as:
22

2

E0 (z) = Ex0 (z)e−z /w eik0 z /ρ eiφE(x, z) = EM (x, z)e−iα2= EM (x, z)e−i0.5k0 z /

(3)

2.2

where � is plasma permittivity which is a functionof plasma density and magnetic ﬁeld (X-mode). A

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Numerical results

The amplitude modulation of the reﬂected wave ismainly caused by the perpendicular (azimuthal) ﬂuctuation. To obtain a large amplitude modiﬁcation,
we assume that the ﬂuctuation amplitude is a = λ0 ,where λ0 is the wavelength of the incident wave. Figure 2(a) shows the contour plot of the amplitude ofelectric ﬁeld in the imaging system. Here, we assume
the cutoﬀ surface is at x = 40 cm. The launching
beam with the frequency of 20 GHz and the beamwidth of 20 cm is plane wave. The beam is reﬂected
and modiﬁed by the cutoﬀ surface. An aperture witha diameter of 30 cm is set at x = 67 cm. The optical lens with a focal length of 50 cm is arranged at
x = 140 cm. Therefore, the inverse image is madeat x = 240 cm according to the principle of geometrical optics. We assume the poloidal wavenumber is

x0

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√ 2

where EM (x, z) is the electric ﬁeld at the optical lens
M and f1 is the focal length. If the focal length isinﬁnite, the mirror is reduced to a plane mirror.
The wide aperture lens is necessary to obtain adeﬁnitional image. Small optical lens may lose someof the reﬂected beam and reduce the imaging depth.
Practically, the lens size should be at least 2 times
larger than the image size. On the other hand, thesolution of the electric ﬁeld in this model is based on
the WKB approximation which requires the ﬂuctuations should be in the condition k < k0 (k0 L� )1/3 ,where L� is the scale length of the plasma permittivity: L� = 1/(d�/dx)|x=xcut [1].

here, Ex0 (z) is the amplitude of the electric ﬁeld at the
cutoﬀ surface. It is a constant in a 1D model while
it is modiﬁed by the reﬂected surface in a 2D model.
Ex0 (z) = cos(2 arctan(dφ/dz)) is used in this simula2
2tion. The second term e−z /w denotes the Gaussian
incident beam, and w is the 1/e fold of the Gaussian2
beam intensity. The third term eik0 z /ρ comes fromthe curvature eﬀects of the cutoﬀ surface and the incident wave front. ρ is the eﬀective curvature radius.
The curvature eﬀect becomes prominent at the inner
reﬂection. The forth term eiφ represents the phasemodulation by density ﬂuctuation.
The phase of the incident wave is calculated in
the 1D geometrical optics as:� xcutφ = 2k0
�1/2 dx
(4)

� ��

(5)

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Fig. 1 The model of the reﬂectometer imaging system.

110

Z. Shi et al., Two-Dimensional Numerical Simulation of Microwave Imaging Reflectometry

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The phase modulation of the reﬂected wave ismainly produced by the radial ﬂuctuations of the cutoﬀ layer. Figure 3 plots the complex amplitude of thereﬂected waves at (a) in-focus, (b) weak-focus and (c)out-of-focus conditions of the ﬂuctuation amplitude
λ0 /4 , and at the (d) in-focus of ﬂuctuation amplitude 3λ0 /2. Therefore, the phase ﬂuctuations in theradial direction are φ� = 1.0π and φ� = 6.0π due to thebeam illuminating back and forth. The other parameters are the same as those in ﬁgure 2. In the caseof weak ﬂuctuation, the Lissajous’ plot is a half circle
at the in-focus condition. When the focus becomes
weak, the Lissajous’ plot changes to moon shape. At
the out-of-focus condition, the Lissajous’ plot exhibits
large and random ﬂuctuations. In the case of strong
ﬂuctuation, the Lissajous’ plot shows deformed circles
at the in-focus position. Therefore, the phase is distorted at strong ﬂuctuation.

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Fig. 4 Schematic view of the laboratory arrangement

Fig. 2 (a) The contour plot of the electric ﬁeld intensity
in the imaging system. (b) The electric ﬁeld amplitude proﬁles at x = 40 cm (solid line), x = 66 cm(dashed line) and x = 240 cm (long dashed line).

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3. Laboratory test of MIR
3.1 Arrangement of MIR system���Figure 4 shows the Schematic view of the test MIR
equipment. It consists of optical elements (i.e., mirrors, lenses, beam splitters and antennas) and a 2D
receiver system. The radiofrequency (RF) wave illuminating from the horn antenna is reﬂected by ﬁrst
beam splitter (BS1 ), and comes to the main mirror(M). The main mirror, which is an elliptic concave
mirror with the size of 50 cm in diameter, makes aparallel illumination beam to the reﬂector. The planeand rotation reﬂectors are used. The plane reﬂector
moves back and forth in experiments. The rotation
reﬂector is made by a thick wood disk with a sinusoid corrugation with wavelength kθ and depth 2a .
The disk covers with the aluminum foil. The disk isdriven by an electric motor in experiments. The reﬂected wave is collected by the collection mirror (M)and is separated from the illumination beam by theﬁrst beam splitter (BS1 ). The local oscillation (LO)wave and the reﬂected wave are mixed at the second
beam splitter (BS2 ). The beam is focused at the an-�=Fig. 3 The complex plots of the reﬂected waves with φ
1.0π at (a) in-focus, (b) weak-focus, (c) out-of-focus
� = 6.0π.
conditions, and (d) in-focus of φ

kp = 0.2 cm−1 with the deviation of 0.1 cm−1 . Theinterference pattern is observed outside of the cutoﬀ
surface. The beam becomes convergent when it passes
through the optical lens, and the image of the cutoﬀ surface is made at the image plane. Figure 2(b)shows the amplitude distributions of the electric ﬁeld
at x = 40 cm (solid line), x = 66 cm (dashed line)
and x = 240 cm (long dashed line). The peak of theelectric ﬁeld amplitude in cutoﬀ plane is at z = 2 cm
while it is at z = −2 cm in the image plane, whichmatch the principle of optics imaging. The proﬁle ofthe electric ﬁeld amplitude at x = 66 cm shows large
ﬂuctuation. Therefore, the reﬂected wave is deformedaway from the image plane.

111

Z. Shi et al., Two-Dimensional Numerical Simulation of Microwave Imaging Reflectometry

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tenna plane. Therefore, an image of the plasma ﬂuctuation is made. A new pyramidal 2D antenna witha wide band frequency response array is developed.
The detail discussions about the antenna array are
discussed by Kuwahara [13].
The launching microwave frequency is 48 GHz.
The LO wave with the frequency of 48.11 GHz is madeby mixing the RF wave (48 GHz) and the lower frequency wave (110 MHz) at an up-converter. By mixing the reﬂected wave and the LO wave, the 2-D mixer
array makes intermediate frequency (IF) signal of 110
MHz. This IF signal contains the amplitude A and thephase φ of the density ﬂuctuation in plasma. Here,the phase φ indicates the vibrational motion of thereﬂection layer, and the amplitude A represents thereﬂected power. The amplitude is obtained by rectifying the IF signal with a diode detector. The phase
is obtained by comparing the IF frequency and themixed signal by the IQ demodulator. I and Q signals correspond to in-phase signal (I = A cos(φ)) and
quadrature signal (Q = A sin(φ)), respectively.

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Fig. 6 Waveforms obtained by the plane reﬂector with
� = 0.7π and the simulation withphase ﬂuctuation φ� = 1.0π. (a) The phase, I, Q andphase ﬂuctuation φamplitude from top to bottom, (b) the Lissajous’
curves of test and simulation.

3.3
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Plane reﬂector

Measurements are taken with a series of ﬂuctuations when the reﬂector is arranged at the in-focus
region. Figure 6 shows the waveforms in the case of
weak ﬂuctuation by using the plane reﬂector. The
phase is obtained from the I and Q signals. The phaseﬂuctuation is about φ� = 0.7π when the plane reﬂector
moves back and forth with the ﬂuctuation amplitudeof 1.1 mm along the x direction. Simulation with thephase ﬂuctuation of φ� = 1.0π is also given in thispicture. The signals look quite similar although they
have diﬀerent initial phases. The phase ﬂuctuation isquasi-sinusoid which reﬂects the cutoﬀ surface moves
back and forth. The reﬂected amplitude is modiﬁed
while it is constant in the simulation. According to the
1D/2D reﬂectometer model, the ﬂuctuation in the x(radial) direction can only cause the phase ﬂuctuation,
not the amplitude ﬂuctuation. In this test, the amplitude ﬂuctuation is small, and it might be caused by
the misalignment of the plane reﬂector. The plane reﬂector didn’t move along the x direction exactly. The
Lissajous’ curve shows a circular arc with the angle of
0.7π. Therefore, the movement of the cutoﬀ surface
in radial direction is reproduced by MIR system.

Calibration

To obtain an in-focus image of the reﬂector, thedetector array should be arranged at the image plane.
The position of the detector array can be calculated by
the principle of geometrical optics. The in-focus image is obtained when the reﬂector is set at about 2 maway from the collection mirror (M) at present setup.
Figure 5 shows the cross-correlation of the reﬂectedsignals as a function of the distance between the reﬂector and the collection mirror. The cross-correlation
is calculated by the two signals with the distance of
1 cm in the horizontal direction. Here, the plane reﬂector is used. The cross-correlation is high when thereﬂector is located at the region between 1.7 m to 2.15
m. This is the in-focus region which agrees with thegeometrical optical estimation. Note that the crosscorrelation is not exactly one at the in-focus region.
It is caused by either the misalignment of the reﬂector
or the optical abbreviation of the MIR system. This
problem can be solved by carefully adjusting the optical system. The low correlation at x > 2.2 m and x< 1.6 m means out-of-focus condition in MIR system
which may cause the phase and amplitude distortions.

112

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Fig. 8 Waveforms in the strong ﬂuctuations (φk = 0.55 cm−1 ). (a) the phase, I, Q and reﬂected
amplitude from top to bottom, (b) the lissajous’
curve of the reﬂected signal.

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Z. Shi et al., Two-Dimensional Numerical Simulation of Microwave Imaging Reflectometry

though the Lissajous’ curve is deformed, it seems thatthe phase ﬂuctuation is not so serious distorted by thestrong ﬂuctuation. The phase modulation shows a sinusoidal trace which is similar as the reﬂector surface.
It agrees with the longer correlation length measured
by IQ detectors in reﬂectometer [2, 14]. This result
suggests that the phase measurement is necessary toobtain the high correlation signals, especially for thestrong and high k ﬂuctuations.
The phase distortion depends on not only the ﬂuctuation amplitude but also the wave number of the
ﬂuctuation. A rotation reﬂector with 12 sinusoidal
corrugations is made. The modulated amplitude of
the corrugation is about 1.0 cm. Therefore, the phaseﬂuctuation is about φ� = 6.4π, and the wavenumber isabout k = 0.55 cm−1 . The waveforms of phase, IQ
and amplitude signals have strong and complicatedﬂuctuations shown in ﬁgure 8. The Lissajous’ plotshows circles with deformed circles. The phase rapidly
increases when the reﬂected amplitude becomes large.
The phase stops at small reﬂected amplitude, leading
to the runaway-phase phenomenon which is known instandard reﬂectometry [11]. One possible reason of
the runaway-phase phenomenon is that the cutoﬀ surface reﬂects the launching beam out of the optical lens
when the deviation angle of the cutoﬀ surface is too
big, leading to small reﬂected power at the detector
surface. The auto-gain ampliﬁer in the phase detector

Rotation reﬂector

Figure 7 shows the waveforms obtained by the rotation reﬂector whose surface is corrugated with three
amplitude modulations. The waves are indicated by
S, M and L in ﬁgure 7(a). Here, the S, M and L
represent the small (φ� = 2/3π), medial (φ� = 4/3π)
and large (φ� = 2π) phase ﬂuctuations, respectively.
The ﬂuctuation amplitudes of S, M and L are 0.1 cm,
0.2 cm and 0.3 cm on the reﬂector, respectively. The
wavenumbers of the waves are about k = 0.2 cm−1 .
The ﬂuctuation signals appear periodically, and theangular velocity of the reﬂector is estimated about
14.0 rads−1 in this test. In the case of small phaseﬂuctuation, the Lissajous’ curve shows a standard circular arc. The reﬂected amplitude is also modulated
which is in consistent with the phase ﬂuctuation. The
Lissajous’ curve becomes a deformed circular arc in
the case of medial phase ﬂuctuation, and the reﬂected
amplitude shows two large sharp peaks with higher
frequency ﬂuctuations. The amplitude becomes spurious peaks and the Lissajous’ curve shows a deformed
circle at the strong phase ﬂuctuation. The amplitude
distortion may be caused by the strong interference
eﬀect near the reﬂector surface due to the strong ﬂuctuation or the optical abberation of MIR system. Al-

113

Z. Shi et al., Two-Dimensional Numerical Simulation of Microwave Imaging Reflectometry

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ameter of the optical lens. L is the distance between
the optical lens and the reﬂector surface. The parallel
launching beam is deﬂected an angle 2β by the reﬂector surface. The phase lost will happen when the lens
can’t collect the main reﬂected beam. If we assume
the modulated wave is sinusoid and L >> D, the relationship between the ﬂuctuation amplitude a and thewavenumber k is given as������������
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ak <

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for the in-focus imaging. The phase lost happens
when the ﬂuctuation amplitude is much larger thanD/(4kL). As shown in ﬁgure 7 (b), the wavenumber k of the modulated wave is estimated about 0.2
cm−1 . Therefore, the ﬂuctuation amplitude should besmaller than 0.16 cm for the phase ﬂuctuation without distortion. In the case of small ﬂuctuation, theﬂuctuation amplitudes is about 0.1 cm, so the circular IQ signals are obtained. In the case of medial andlarge ﬂuctuations, the ﬂuctuation amplitudes are 0.2
cm and 0.3 cm, respectively. So the IQ signals are
distorted. Therfore, MIR can’t obtain the clear imageof the large amplitude ﬂuctuations with a � D/(4kL)
due to the diﬀraction eﬀect of the reﬂected wave.

doesn’t work well with small signals. Another reason
might be the strong ﬂuctuations may cause a complicated interference pattern near the reﬂected surface,
MIR can’t restore the strong ﬂuctuations, and the interference signals are obtained. Therefore, MIR fails
to make the clear image in the strong ﬂuctuations.

4. Summary and discussions
In summary, the numerical study of the 2D ﬂuctuations in MIR system based on the Huygens-Fresnel
equation is carried out to simulate the wave propagation. The optical lens is considered in this model.
The ﬂuctuation of the cutoﬀ surface is obtained fromthe phase of the reﬂected waves when the MIR system arranges at the in-focus condition. A laboratory
arrangement of the MIR system is made. The preliminary tests show that the focus of optics in MIR
system is one of the key issues to obtain a deﬁnitional reﬂected image. The MIR system works well at
the in-focus condition when the ﬂuctuation amplitude
is much smaller than the wavelength of the launching wave. The Lissajous’ curves from both simulation and experiments have a circular arc shape when
MIR works at the in-focus condition. However, thereﬂected signals are distorted and the phase is lost at
strong ﬂuctuations.
This simulation considers the optical lens in the
wave equation at the laboratory scenario. It can beused to calibrate the optics and explain the experimental signals. In the present imaging system, the incident beam is perpendicular to the cutoﬀ surface if the
optical system is well designed and arranged. Therefore, the refractive eﬀect is not so serious as expectedin the plasma. The approximation in this work might
be used for the O-mode plasma. However, it is nottrue under the most conditions. The full wave equation simulation should be used for the actual plasma
test because of the strong diﬀractive eﬀect. Several
similar works have been carried out [8–10]. Even so,
the phase distortion is still a crucial problem for MIR.
The phase distortion can be estimated from theoptical arrangement in MIR system. Figure 9 shows
the schematic illustration of the beam diﬀraction in
MIR system. Where, n is the surface normal and β
is the cutting angle of the cutoﬀ surface. D is the di-

Acknowledgements

This work is carried out as one of the NINS Imaging Science Project (Grant No. NIFS08KEIN0021).
This work is also supported by NIFS (Grant
No. NIFS08ULPP525), and SOKENDAI (Grant
No. NIFS08GLPP003).

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