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the threshold are shown to decelerate to velocities comparable to 5.0 keV particles. The interaction of the retarding potential grid system happens on the nanosecond time scale and does not affect the timing of the measurements. The detection efficiency of the system is changed because of the use of a stripping foil, instead of the stripping cell and focusing lens. The detecting efficient is a combination of the efficiency of neutral conversion to ion in the stripping foil and the scattering of the ions in the foil.12 With the use of large grid mounts and the focusing nature of the lens system, the scattering and the asymmetry effects of the potential well are minimized. Therefore, since the same thickness and density foil used in Refs. 3 and 4 will be used in this system, the detection efficiency for that system would be a good approximation for this system.
V. CONCLUSION
(axes in inches)
m m/s
FIG. 5. The electron particle trajectory due to the retarding potential grid system potential well in Fig. 4.
potential decreases to zero within the custom four-way vacuum tube, showing that the particles should only slow down to their theorized values. The electron and ion particle trajectories with initial velocities that correspond to 35 keV neutrals are shown in Figs. 5 and 6, respectively. The electrons trajectories are shown to get bent perpendicular to the beam of ions or accelerated back toward MST. The ions with velocities over
The deuterium ion energy distribution in MST plasmas is measured from charge exchange neutrals in the energy range 0.34–45 keV by the CNPA and ANPA. The CNPA energy range covers the bulk ions to the beginning of the fast ion tail (0.34–5.2 keV) with high-energy resolution (25 channels) while the ANPA spans the vast majority of the fast ion tail distribution (∼10–45 keV) with low energy resolution (10 channels). To gain more knowledge of the energy distribution for ion energization studies, a retarding potential grid system has been designed to be put in front of the CNPA, decelerating the fast ion tail and enabling it to be detected by the low energy, high energy resolution channels.
ACKNOWLEDGMENTS
We would like to thank Matthew Leeds for the Solid Works drawings. We would also like to thank the Department of Energy for supplying funding for this work. 1 E.
D. Mezonlin et al., Rev. Sci. Instrum. 78, 053504 (2007).
2 J. B. Titus, E. D. Mezonlin, and J. A. Johnson III, Phys. Plasmas 21, 062511
(2014). Eilerman, J. K. Anderson, J. A. Reusch, D. Liu, G. Fiksel, S. Polosatkin, and V. Belykh, Rev. Sci. Instrum. 83, 10D302 (2012). 4 J. A. Reusch, J. K. Anderson, V. Belykh, S. Eilerman, D. Liu, G. Fiksel, and S. Polosatkin, Rev. Sci. Instrum. 83, 10D704 (2012). 5 R. M. Magee, D. J. Den Hartog, S. T. A. Kumar, A. F. Almagri, B. E. Chapman, G. Fiksel, V. V. Mirnov, E. D. Mezonlin, and J. B. Titus, Phys. Rev. Lett. 107, 065005 (2011). 6 J. Waksman, J. K. Anderson, M. D. Nornberg, E. Parke, J. A. Reusch, D. Liu, G. Fiksel, V. I. Davydenko, A. A. Ivanov, N. Stupishin, P. P. Deichuli, and H. Sakakita, Phys. Plasmas 19, 122505 (2012). 7 V. I. Krassovsky, Proc. IRE 47, 289, 1959. 8 W. B. Hanson, D. R. Zuccaro, C. R. Lippincott, and S. Sanatani, Radio Sci. 8, 333, doi:10.1029/RS008i004p00333 (1973). 9 I. H. Hutchinson, Principles of Plasma Diagnostics (Cambridge University Press, Cambridge, 1987). 10 R. E. Fox and J. A. Hipple, Rev. Sci. Instrum. 19, 462 (1948). 11 J. A. Hipple, J. Phys. Colloid Chem. 52, 456 (1948). 12 S. Polosatkin, Fusion Sci. Technol. 59, 259 (2011). 3 S.
(axes in inches)
m//s
FIG. 6. The ion particle trajectory due to the retarding potential grid system potential well in Fig. 4.
REVIEW OF SCIENTIFIC INSTRUMENTS 85, 11D403 (2014)
2D image of local density and magnetic fluctuations from line-integrated interferometry-polarimetry measurementsa) L. Lin,b) W. X. Ding, and D. L. Brower Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095, USA
(Presented 2 June 2014; received 30 May 2014; accepted 11 June 2014; published online 2 July 2014) Combined polarimetry-interferometry capability permits simultaneous measurement of lineintegrated density and Faraday effect with fast time response (∼1 μs) and high sensitivity. Faraday effect fluctuations with phase shift of order 0.05◦ associated with global tearing modes are resolved with an uncertainty ∼0.01◦ . For physics investigations, local density fluctuations are obtained by inverting the line-integrated interferometry data. The local magnetic and current density fluctuations are then reconstructed using a parameterized fit of the polarimetry data. Reconstructed 2D images of density and magnetic field fluctuations in a poloidal cross section exhibit significantly different spatial structure. Combined with their relative phase, the magnetic-fluctuationinduced particle transport flux and its spatial distribution are resolved. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884788] I. INTRODUCTION
On the Madison Symmetric Torus (MST) reversed field pinch (RFP), a multi-chord far-infrared (wavelength λ = 432 μm) laser-based Faraday-effect polarimetry-interferometry system has been used to measure equilibrium and fluctuating density and magnetic fields.1 The diagnostic consists of 11 vertically viewing single-pass chords with separation ∼ 0.08 m, covering nearly the entire plasma cross section (minor radius a = 0.52 m and major radius R0 = 1.5 m). The polarimetry technique employed measures the phase difference between left-hand and right-hand circularly polarized waves passing through the plasma, thereby obtaining Faraday effect directly related to the magnetic field parallel to the beam propagation direction. Combined interferometry capability, by adding an external local oscillator, permits simultaneous measurement of the internal spatial structure of density, current density, and magnetic field with fast time response (up to 1 MHz) and high sensitivity (capable of measuring 0) measure˜ m,n and ments are necessary to unambiguously determine |n| δ m, n .4 To improve the accuracy and spatial resolution, the inboard and outboard data are used simultaneously, which over˜ m,n and δ m, n . Consequently, a minimization apconstrains |n| proach is preferred over direct numerical calculation.3 The reconstruction of density fluctuation associated with the m/n = 1/6 tearing mode at ∼ 20 kHz (in the laboratory frame of reference) is shown in Fig. 1. The reconstructed rmsamplitude [Fig. 1(a)] and phase [Fig. 1(b)] profiles of lineintegrated density fluctuations match well with experimental measurements. The rms-amplitude of line-integrated density ˜ fluctuation (∫ ndz) reaches zero near the magnetic axis, primarily due to phase cancellation along this chord arising ˜ phase profile from the m = 1 nature of the mode. The ∫ ndz exhibits a π shift across the magnetic axis, consistent with an m = 1 perturbation. The structure of local n˜ [Fig. 1(c)] differs ˜ from that of ∫ ndz. The local n˜ peaks near r ∼ 0.3 m where the equilibrium density gradient is large. An additional smallscale structure inside r < 0.2 m, corresponding to the same location as the phase profile jump [Fig. 1(d)], is also noted.
where n0 is the equilibrium density, and Bθ0 is the equilibrium poloidal magnetic field, cF = e3 λ2 /(8π 2 c3 ε0 m2e ). From the standard picture of the resistive tearing mode, a parameterized function is assumed for the toroidal current density fluctuation, j˜φ = j0 exp[−(r − rs )2 /w 2 ], which contains three free parameters: amplitude j0 , location rs , and width w. Once j˜φ is identified, b˜r and b˜θ are obtained from solving Ampere’s Law ∇ × b˜ = u0 ˜j and Gauss’ Law ∇ · b˜ = 0 under the assumption of ∇ · ˜j = 0 and the radial toroidal current density j˜r = 0. For simplicity, these equations are solved in a cylindrical geometry with boundary condition b˜r (a) = 0. By com˜ the bining the solutions for b˜r and b˜θ with measured n0 , n, ˜ f it ) can be numerically expected polarimetry fluctuations ( calculated, which are a function of j0 , rs , and w. Free parameters (j0 , rs , and w) are then determined by minimizing ˜ f it ) and measured ( ˜ exp ) the difference between the fitted ( polarimetry fluctuations, f it 2 f it exp 2 ˜i − ˜ iexp b˜θ,a − b˜θ,a 2 + , (4) χ = σ ,i σb˜θ,a ˜ i where σ ˜ is the uncertainty of measured polarimetry fluctuation and the subscript “i” corresponds to the channel index. f it As shown in Eq. (4), the difference between fitted (b˜θ,a ) and exp measured (b˜θ,a ) edge poloidal magnetic fluctuation normalized by its experimental uncertainty (σb˜θ,a ) is also included in the minimization. The reconstruction of polarimetry fluctuation associated with the m/n = 1/6 tearing mode is shown in Fig. 2, where ˜ exp ) with maximum phase the experimental Faraday effect ( ◦ shift 0.08 is resolved with an uncertainty below 0.01◦ . The fitted polarimetry amplitude [Fig. 2(a)] and phase [Fig. 2(b)] profiles remain close to measurements. The line-integrated Faraday effect profile is very different from the density fluctuation profile [Fig. 1(a)] due to the embedded magnetic field fluctuation information. The radial magnetic fluctuation [Fig. 2(c)] peaks at ∼ 28 G for r = 0.2 m (near the m/n
III. POLARIMETRY MEASUREMENTS
˜ from poAnalysis of Faraday-effect fluctuations ( ) larimetry is more complicated, since it contains contributions ˜ radial magnetic (b˜r ), and poloidal magnetic from density (n), (b˜θ ) fluctuations,7 +√a 2 −x 2 ˜ ˜ θ0 cos θ + n0 b˜r sin θ + n0 b˜θ cos θ )dz, (nB = cF √ − a 2 −x 2
(3)
FIG. 2. (a) rms-amplitude and (b) phase profiles of Faraday-effect polarimetry fluctuations, where symbols are from measurement, solid line (red) is from reconstruction, and vertical dashed line marks the magnetic axis. (c) rms-amplitude and (d) phase profiles of reconstructed local radial (solid line) and poloidal (red dashed line) magnetic fluctuations. The data point near the plasma edge in (c) is from external coil measurements.
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= 1/6 resonant surface) and maintains a constant phase across the minor radius [Fig. 2(d)]. The poloidal magnetic fluctuation changes the sign across the resonant surface [Fig. 2(d)] and has maximum amplitude ∼ 24 G [Fig. 2(c)]. These features are roughly consistent with the standard tearing mode picture. IV. 2D IMAGE OF LOCAL DENSITY, CURRENT DENSITY, AND MANGETIC FLUCTUATIONS FIG. 4. Magnetic fluctuation-induced electron particle flux induced by the m/n = 1/6 tearing mode.
After resolving the local amplitude and phase of density, toroidal current, and magnetic fluctuations, a detailed 2D perturbation structure in the poloidal cross section can be visualized, as shown in Fig. 3. All fluctuating quantities clearly exhibit an m = 1 pattern. The density fluctuation [Fig. 3(a)] contains multiple structures with the maximum peaking near r = 0.3 m with width ∼ 0.2 m. The two beanshaped structures near the core (∼ 0.15 m from the magnetic axis) show a shift in the poloidal direction relative to the structures near the edge. The toroidal current fluctuations [Fig. 3(b)] peak ∼ 0.2 m from the magnetic axis with the width ∼ 0.08 m. The magnetic fluctuation vectors [Fig. 3(b)] form two closed loops with opposite direction around the toroidal current sheets, consistent with Ampere’s law. In addition to a different spatial structure, local density fluctuations are misaligned with the magnetic and current fluctuations.
librium reconstruction. Flux dependence on multi-field fluctuating quantities is more complicated, requiring fluctuation rms-amplitude (n˜ rms and b˜r,rms ), their coherence (γn˜ b˜r ) and relative phase (δn˜ b˜r ). The phase information is of particular importance for calculating fluctuation-induced flux, as a relative phase near π /2 leads to negligible e even for largeamplitude fluctuations. Combining local density (Fig. 1) and radial magnetic (Fig. 2) fluctuations, e induced by the m/n = 1/6 tearing mode across the minor radius is determined and shown in Fig. 4. The structure of e peaks at r ∼ 0.15 m and is quite different spatially from radial magnetic or density fluctuations, highlighting the importance of multi-field measurements.
V. PARTICLE FLUX
VI. SUMMARY
One major goal of measuring fluctuations is to evaluate their impact on confinement. Fluctuation-induced particle flux ˜ ( e ) can arise from coherent interaction between density (n) and radial magnetic (b˜r ) fluctuations,8
This paper presents reconstruction of local density and magnetic fluctuations associated with a global tearing mode from line-integrated polarimetry-interferometry measurements. Along with providing the 2D spatial pattern of the local density and magnetic perturbations in the poloidal cross section, their relative phase is also resolved. This permits the evaluation of magnetic fluctuation-induced particle flux, thereby allowing direct assessment of anomalous transport in a high-temperature toroidal plasma device.
e =
V//,e n˜ rms b˜r,rms γn˜ b˜r cos δn˜ b˜r , B0
(5)
where the electron parallel velocity (V//,e ) and the equilibrium magnetic field strength (B0 ) can be obtained from equi-
ACKNOWLEDGMENTS
This work is supported by the U.S. Department of Energy. 1 W.
FIG. 3. (a) Density and (b) toroidal current density fluctuations in the poloidal cross section, where the vector arrows in (b) represents magnetic fluctuations projected onto the poloidal plane, including both radial and poloidal components.
X. Ding, D. L. Brower, W. F. Bergerson, and L. Lin, Rev. Sci. Instrum. 81, 10D508 (2010). 2 T. F. Yates, W. X. Ding, T. A. Carter, and D. L. Brower, Rev. Sci. Instrum. 79, 10E714 (2008). 3 H. K. Park, Plasma Phys. Controlled Fusion 31, 2035 (1989). 4 N. E. Lanier, D. Craig, J. K. Anderson, T. M. Biewer, B. E. Chapman, D. J. Den Hartog, C. B. Forest, S. C. Prager, D. L. Brower, and Y. Jiang, Rev. Sci. Instrum. 72, 1039 (2001). 5 P. J. Green and B. W. Silverman, Nonparametric Regression and Generalized Linear Models (Chapman and Hall, London, 1994). 6 L. Lin, W. X. Ding, D. L. Brower, W. F. Bergerson, and T. F. Yates, Rev. Sci. Instrum. 81, 10D509 (2010). 7 W. X. Ding, D. L. Brower, B. H. Deng, D. Craig, S. C. Prager, and V. Svidzinski, Rev. Sci. Instrum. 75, 3387 (2004). 8 W. X. Ding, D. L. Brower, G. Fiksel, D. J. Den Hartog, S. C. Prager, and J. S. Sarff, Phys. Rev. Lett. 103, 025001 (2009).
REVIEW OF SCIENTIFIC INSTRUMENTS 85, 11D404 (2014)
Design of vibration compensation interferometer for Experimental Advanced Superconducting Tokamaka) Y. Yang,1 G. S. Li,1,2 H. Q. Liu,1 Y. X. Jie,1,b) W. X. Ding,3 D. L. Brower,3 X. Zhu,1 Z. X. Wang,1 L. Zeng,1 Z. Y. Zou,1 X. C. Wei,1 and T. Lan2 1
Institute of Plasma Physics, Chinese Academy of Science, Hefei, Anhui 230031, People’s Republic of China University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China 3 Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, California 90095, USA 2
(Presented 2 June 2014; received 31 May 2014; accepted 18 June 2014; published online 11 July 2014) A vibration compensation interferometer (wavelength at 0.532 μm) has been designed and tested for Experimental Advanced Superconducting Tokamak (EAST). It is designed as a sub-system for EAST far-infrared (wavelength at 432.5 μm) poloarimeter/interferometer system. Two Acoustic Optical Modulators have been applied to produce the 1 MHz intermediate frequency. The path length drift of the system is lower than 2 wavelengths within 10 min test, showing the system stability. The system sensitivity has been tested by applying a periodic vibration source on one mirror in the system. The vibration is measured and the result matches the source period. The system is expected to be installed on EAST by the end of 2014. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4886455] I. INTRODUCTION
A far-infrared laser polarimeter/interferometer system has been designed and installed for Experimental Advanced Superconducting Tokamak (EAST).1 This diagnostic will measure the density and Faraday angle profile information and contribute to the plasma current profile control. In the polarimeter/interferometer system, the plasma density measurement is carried out by the interferometer, where the phase shift between the reference channel and detecting channel gives the line integrated electron density. The vibration (either from the machine or the system) will introduce an additional change in the path length of the optics system, which is an error for the phase of the density measurement.2–6 The error is higher, especially, during the plasma start-up phase since the density term is low and the vibration term is large or during the change of the toroidal magnetic field. The most severe potential vibration of EAST is expected from the retro reflectors locating at the high field side wall in the chamber. In order to compensate the vibration, we designed a second laser (wavelength at 0.532 μm) interferometer as a sub-system for EAST far-infrared polarimeter/interferometer. The system has been tested in the lab and planned to be installed on EAST by the end of 2014. II. SYSTEM DESCRIPTION
The principle of interferometer for plasma density measurement is known as ϕ = ϕp = c ne dl, (1) a) Contributed paper, published as part of the Proceedings of the 20th
Topical Conference on High-Temperature Plasma Diagnostics, Atlanta, Georgia, USA, June 2014. b) Author to whom correspondence should be addressed. Electronic mail: [email protected]. 0034-6748/2014/85(11)/11D404/3/$30.00
where ϕ p is the phase shift by plasma, c is a constant proportional to the wave length, and ne is the local plasma density. By considering the vibration in the optical path, the phase shift between the reference channel and the detecting channel is 2π ϕ = ϕp + ϕv = c ne dl + L, (2) λ where L is the path length change by vibration, ϕ v is the vibration term and is proportional to L, and λ is the wave length of the working laser. For CO2 pumped HCOOH laser (wavelength at 432.5 μm) and EAST low density (∼1 × 1019 m−3 ) case, the ϕ v term is ∼8% (80%) in the total phase shift for a 0.1 mm (1 mm) vibration. So the vibration compensation is needed. A visible laser (wavelength at 0.532 μm) has been selected as the second laser to compensate the vibration. For this wavelength, the dominant term in Eq. (2) is the vibration term and the plasma term can be neglected. The modulation frequency is set at 1 MHz by two Acoustic Optical Modulators (AOMs). One AOM produces a first order beam which has an 80 MHz frequency shift from the original/zero order beam. The other AOM produces an 81 MHz frequency shifted beam. These two first order beams then interfere and an IF signal at 1 MHz is created carrying the vibration information. The layout of the vibration compensation system is presented in Figure 1. The system will settle on a vertical optical board in front of the “O” window on EAST. The main part of the board is for the HCOOH laser polarimeter/interferometer. The upper part of the board is currently reserved for the vibration compensation system. The vibration compensation system shares part of the optical path with one HCOOH laser interferometer channel (5 in all and possibly upgraded to 11). Such arrangement implies that the assumed vibration is dominant from the retro reflectors and is identical for all channels. The outputs from one AOM are composed of the zero order
85, 11D404-1
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FIG. 1. System layout of vibration compensation interferometer. The black/red/blue dashed box shows the optical path for 1 channel HCOOH laser/visible laser/both lasers.
beam, the first order beam and the higher order beams. An aperture is applied after the AOM, to allow only the first order beam to pass. A heterodyne interferometer scheme is designed to measure the phase shift. Avalanche Photodetectors (APD120 Series, Thorlabs Company) have been selected as the detectors for the 1 MHz IF signal. Other main components in the system are beam splitters (for visible laser and FIR laser, respectively), mirrors, quarter wave plate (for FIR laser), retro reflector, Polymethylpentene (TPX) lens, and TPX window. Bench test of these components has been finished. For each component, we tested the transmission efficiency for the visible laser. The power dependence (decreasing) on the optical path length has been measured and fit. Based on these data, the relative power (to the source power) at the two detectors are 0.0935% and 0.046%, respectively. For a 20 mW laser source, such optical arrangement can produce 18.7 μW and 9.2 μW signal in front of the detector. According to the detector’s bench performance, these signals could be detected. III. BENCH TEST OF THE SYSTEM A. System stability test
As described in Sec. II, the IF of the system is 1 MHz. These IF data are shown in Figure 2 (upper part). To test the system stability, this experiment was done when the system is mechanically stationary and without plasma. There is a phase difference between the two channels signal, which means the initial path length difference between the reference and detecting channels. In an ideal case (without spatial perturbation or any drift), such phase difference value should be constant in time. The phase difference, δϕ, has been measured for 400s (bottom part of Figure 2). Such time scale is currently the longest EAST pulse length. The result shows that during this time, the largest phase drift is within 3π . This means the drift in the system’s path length is within 1.5λ (∼0.8 μm). The
FIG. 2. The intermediate frequency signal and the stability of optical path.
drift could from any spatial perturbation in the lab. Comparing with the possible vibration from the machine which needs to be compensated (∼0.1 mm), this drift is in the order of 1%. On the other hand, if the machine vibration is ∼0.1 mm, there will be ∼180 fringe jumps for such a vibration. This may bring some difficulty in the fast fringe counting during the plasma discharges. B. Measurement of applied vibration
In order to simulate the machine vibration in the experiment, an external vibration has been applied to the system. This artificial vibration is introduced by setting an electric motor on one mirror (by screws). The frequency of the motor can be controlled and modified (
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Rev. Sci. Instrum. 85, 11D402 (2014)
the threshold are shown to decelerate to velocities comparable to 5.0 keV particles. The interaction of the retarding potential grid system happens on the nanosecond time scale and does not affect the timing of the measurements. The detection efficiency of the system is changed because of the use of a stripping foil, instead of the stripping cell and focusing lens. The detecting efficient is a combination of the efficiency of neutral conversion to ion in the stripping foil and the scattering of the ions in the foil.12 With the use of large grid mounts and the focusing nature of the lens system, the scattering and the asymmetry effects of the potential well are minimized. Therefore, since the same thickness and density foil used in Refs. 3 and 4 will be used in this system, the detection efficiency for that system would be a good approximation for this system.
V. CONCLUSION
(axes in inches)
m m/s
FIG. 5. The electron particle trajectory due to the retarding potential grid system potential well in Fig. 4.
potential decreases to zero within the custom four-way vacuum tube, showing that the particles should only slow down to their theorized values. The electron and ion particle trajectories with initial velocities that correspond to 35 keV neutrals are shown in Figs. 5 and 6, respectively. The electrons trajectories are shown to get bent perpendicular to the beam of ions or accelerated back toward MST. The ions with velocities over
The deuterium ion energy distribution in MST plasmas is measured from charge exchange neutrals in the energy range 0.34–45 keV by the CNPA and ANPA. The CNPA energy range covers the bulk ions to the beginning of the fast ion tail (0.34–5.2 keV) with high-energy resolution (25 channels) while the ANPA spans the vast majority of the fast ion tail distribution (∼10–45 keV) with low energy resolution (10 channels). To gain more knowledge of the energy distribution for ion energization studies, a retarding potential grid system has been designed to be put in front of the CNPA, decelerating the fast ion tail and enabling it to be detected by the low energy, high energy resolution channels.
ACKNOWLEDGMENTS
We would like to thank Matthew Leeds for the Solid Works drawings. We would also like to thank the Department of Energy for supplying funding for this work. 1 E.
D. Mezonlin et al., Rev. Sci. Instrum. 78, 053504 (2007).
2 J. B. Titus, E. D. Mezonlin, and J. A. Johnson III, Phys. Plasmas 21, 062511
(2014). Eilerman, J. K. Anderson, J. A. Reusch, D. Liu, G. Fiksel, S. Polosatkin, and V. Belykh, Rev. Sci. Instrum. 83, 10D302 (2012). 4 J. A. Reusch, J. K. Anderson, V. Belykh, S. Eilerman, D. Liu, G. Fiksel, and S. Polosatkin, Rev. Sci. Instrum. 83, 10D704 (2012). 5 R. M. Magee, D. J. Den Hartog, S. T. A. Kumar, A. F. Almagri, B. E. Chapman, G. Fiksel, V. V. Mirnov, E. D. Mezonlin, and J. B. Titus, Phys. Rev. Lett. 107, 065005 (2011). 6 J. Waksman, J. K. Anderson, M. D. Nornberg, E. Parke, J. A. Reusch, D. Liu, G. Fiksel, V. I. Davydenko, A. A. Ivanov, N. Stupishin, P. P. Deichuli, and H. Sakakita, Phys. Plasmas 19, 122505 (2012). 7 V. I. Krassovsky, Proc. IRE 47, 289, 1959. 8 W. B. Hanson, D. R. Zuccaro, C. R. Lippincott, and S. Sanatani, Radio Sci. 8, 333, doi:10.1029/RS008i004p00333 (1973). 9 I. H. Hutchinson, Principles of Plasma Diagnostics (Cambridge University Press, Cambridge, 1987). 10 R. E. Fox and J. A. Hipple, Rev. Sci. Instrum. 19, 462 (1948). 11 J. A. Hipple, J. Phys. Colloid Chem. 52, 456 (1948). 12 S. Polosatkin, Fusion Sci. Technol. 59, 259 (2011). 3 S.
(axes in inches)
m//s
FIG. 6. The ion particle trajectory due to the retarding potential grid system potential well in Fig. 4.
REVIEW OF SCIENTIFIC INSTRUMENTS 85, 11D403 (2014)
2D image of local density and magnetic fluctuations from line-integrated interferometry-polarimetry measurementsa) L. Lin,b) W. X. Ding, and D. L. Brower Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095, USA
(Presented 2 June 2014; received 30 May 2014; accepted 11 June 2014; published online 2 July 2014) Combined polarimetry-interferometry capability permits simultaneous measurement of lineintegrated density and Faraday effect with fast time response (∼1 μs) and high sensitivity. Faraday effect fluctuations with phase shift of order 0.05◦ associated with global tearing modes are resolved with an uncertainty ∼0.01◦ . For physics investigations, local density fluctuations are obtained by inverting the line-integrated interferometry data. The local magnetic and current density fluctuations are then reconstructed using a parameterized fit of the polarimetry data. Reconstructed 2D images of density and magnetic field fluctuations in a poloidal cross section exhibit significantly different spatial structure. Combined with their relative phase, the magnetic-fluctuationinduced particle transport flux and its spatial distribution are resolved. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884788] I. INTRODUCTION
On the Madison Symmetric Torus (MST) reversed field pinch (RFP), a multi-chord far-infrared (wavelength λ = 432 μm) laser-based Faraday-effect polarimetry-interferometry system has been used to measure equilibrium and fluctuating density and magnetic fields.1 The diagnostic consists of 11 vertically viewing single-pass chords with separation ∼ 0.08 m, covering nearly the entire plasma cross section (minor radius a = 0.52 m and major radius R0 = 1.5 m). The polarimetry technique employed measures the phase difference between left-hand and right-hand circularly polarized waves passing through the plasma, thereby obtaining Faraday effect directly related to the magnetic field parallel to the beam propagation direction. Combined interferometry capability, by adding an external local oscillator, permits simultaneous measurement of the internal spatial structure of density, current density, and magnetic field with fast time response (up to 1 MHz) and high sensitivity (capable of measuring 0) measure˜ m,n and ments are necessary to unambiguously determine |n| δ m, n .4 To improve the accuracy and spatial resolution, the inboard and outboard data are used simultaneously, which over˜ m,n and δ m, n . Consequently, a minimization apconstrains |n| proach is preferred over direct numerical calculation.3 The reconstruction of density fluctuation associated with the m/n = 1/6 tearing mode at ∼ 20 kHz (in the laboratory frame of reference) is shown in Fig. 1. The reconstructed rmsamplitude [Fig. 1(a)] and phase [Fig. 1(b)] profiles of lineintegrated density fluctuations match well with experimental measurements. The rms-amplitude of line-integrated density ˜ fluctuation (∫ ndz) reaches zero near the magnetic axis, primarily due to phase cancellation along this chord arising ˜ phase profile from the m = 1 nature of the mode. The ∫ ndz exhibits a π shift across the magnetic axis, consistent with an m = 1 perturbation. The structure of local n˜ [Fig. 1(c)] differs ˜ from that of ∫ ndz. The local n˜ peaks near r ∼ 0.3 m where the equilibrium density gradient is large. An additional smallscale structure inside r < 0.2 m, corresponding to the same location as the phase profile jump [Fig. 1(d)], is also noted.
where n0 is the equilibrium density, and Bθ0 is the equilibrium poloidal magnetic field, cF = e3 λ2 /(8π 2 c3 ε0 m2e ). From the standard picture of the resistive tearing mode, a parameterized function is assumed for the toroidal current density fluctuation, j˜φ = j0 exp[−(r − rs )2 /w 2 ], which contains three free parameters: amplitude j0 , location rs , and width w. Once j˜φ is identified, b˜r and b˜θ are obtained from solving Ampere’s Law ∇ × b˜ = u0 ˜j and Gauss’ Law ∇ · b˜ = 0 under the assumption of ∇ · ˜j = 0 and the radial toroidal current density j˜r = 0. For simplicity, these equations are solved in a cylindrical geometry with boundary condition b˜r (a) = 0. By com˜ the bining the solutions for b˜r and b˜θ with measured n0 , n, ˜ f it ) can be numerically expected polarimetry fluctuations ( calculated, which are a function of j0 , rs , and w. Free parameters (j0 , rs , and w) are then determined by minimizing ˜ f it ) and measured ( ˜ exp ) the difference between the fitted ( polarimetry fluctuations, f it 2 f it exp 2 ˜i − ˜ iexp b˜θ,a − b˜θ,a 2 + , (4) χ = σ ,i σb˜θ,a ˜ i where σ ˜ is the uncertainty of measured polarimetry fluctuation and the subscript “i” corresponds to the channel index. f it As shown in Eq. (4), the difference between fitted (b˜θ,a ) and exp measured (b˜θ,a ) edge poloidal magnetic fluctuation normalized by its experimental uncertainty (σb˜θ,a ) is also included in the minimization. The reconstruction of polarimetry fluctuation associated with the m/n = 1/6 tearing mode is shown in Fig. 2, where ˜ exp ) with maximum phase the experimental Faraday effect ( ◦ shift 0.08 is resolved with an uncertainty below 0.01◦ . The fitted polarimetry amplitude [Fig. 2(a)] and phase [Fig. 2(b)] profiles remain close to measurements. The line-integrated Faraday effect profile is very different from the density fluctuation profile [Fig. 1(a)] due to the embedded magnetic field fluctuation information. The radial magnetic fluctuation [Fig. 2(c)] peaks at ∼ 28 G for r = 0.2 m (near the m/n
III. POLARIMETRY MEASUREMENTS
˜ from poAnalysis of Faraday-effect fluctuations ( ) larimetry is more complicated, since it contains contributions ˜ radial magnetic (b˜r ), and poloidal magnetic from density (n), (b˜θ ) fluctuations,7 +√a 2 −x 2 ˜ ˜ θ0 cos θ + n0 b˜r sin θ + n0 b˜θ cos θ )dz, (nB = cF √ − a 2 −x 2
(3)
FIG. 2. (a) rms-amplitude and (b) phase profiles of Faraday-effect polarimetry fluctuations, where symbols are from measurement, solid line (red) is from reconstruction, and vertical dashed line marks the magnetic axis. (c) rms-amplitude and (d) phase profiles of reconstructed local radial (solid line) and poloidal (red dashed line) magnetic fluctuations. The data point near the plasma edge in (c) is from external coil measurements.
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= 1/6 resonant surface) and maintains a constant phase across the minor radius [Fig. 2(d)]. The poloidal magnetic fluctuation changes the sign across the resonant surface [Fig. 2(d)] and has maximum amplitude ∼ 24 G [Fig. 2(c)]. These features are roughly consistent with the standard tearing mode picture. IV. 2D IMAGE OF LOCAL DENSITY, CURRENT DENSITY, AND MANGETIC FLUCTUATIONS FIG. 4. Magnetic fluctuation-induced electron particle flux induced by the m/n = 1/6 tearing mode.
After resolving the local amplitude and phase of density, toroidal current, and magnetic fluctuations, a detailed 2D perturbation structure in the poloidal cross section can be visualized, as shown in Fig. 3. All fluctuating quantities clearly exhibit an m = 1 pattern. The density fluctuation [Fig. 3(a)] contains multiple structures with the maximum peaking near r = 0.3 m with width ∼ 0.2 m. The two beanshaped structures near the core (∼ 0.15 m from the magnetic axis) show a shift in the poloidal direction relative to the structures near the edge. The toroidal current fluctuations [Fig. 3(b)] peak ∼ 0.2 m from the magnetic axis with the width ∼ 0.08 m. The magnetic fluctuation vectors [Fig. 3(b)] form two closed loops with opposite direction around the toroidal current sheets, consistent with Ampere’s law. In addition to a different spatial structure, local density fluctuations are misaligned with the magnetic and current fluctuations.
librium reconstruction. Flux dependence on multi-field fluctuating quantities is more complicated, requiring fluctuation rms-amplitude (n˜ rms and b˜r,rms ), their coherence (γn˜ b˜r ) and relative phase (δn˜ b˜r ). The phase information is of particular importance for calculating fluctuation-induced flux, as a relative phase near π /2 leads to negligible e even for largeamplitude fluctuations. Combining local density (Fig. 1) and radial magnetic (Fig. 2) fluctuations, e induced by the m/n = 1/6 tearing mode across the minor radius is determined and shown in Fig. 4. The structure of e peaks at r ∼ 0.15 m and is quite different spatially from radial magnetic or density fluctuations, highlighting the importance of multi-field measurements.
V. PARTICLE FLUX
VI. SUMMARY
One major goal of measuring fluctuations is to evaluate their impact on confinement. Fluctuation-induced particle flux ˜ ( e ) can arise from coherent interaction between density (n) and radial magnetic (b˜r ) fluctuations,8
This paper presents reconstruction of local density and magnetic fluctuations associated with a global tearing mode from line-integrated polarimetry-interferometry measurements. Along with providing the 2D spatial pattern of the local density and magnetic perturbations in the poloidal cross section, their relative phase is also resolved. This permits the evaluation of magnetic fluctuation-induced particle flux, thereby allowing direct assessment of anomalous transport in a high-temperature toroidal plasma device.
e =
V//,e n˜ rms b˜r,rms γn˜ b˜r cos δn˜ b˜r , B0
(5)
where the electron parallel velocity (V//,e ) and the equilibrium magnetic field strength (B0 ) can be obtained from equi-
ACKNOWLEDGMENTS
This work is supported by the U.S. Department of Energy. 1 W.
FIG. 3. (a) Density and (b) toroidal current density fluctuations in the poloidal cross section, where the vector arrows in (b) represents magnetic fluctuations projected onto the poloidal plane, including both radial and poloidal components.
X. Ding, D. L. Brower, W. F. Bergerson, and L. Lin, Rev. Sci. Instrum. 81, 10D508 (2010). 2 T. F. Yates, W. X. Ding, T. A. Carter, and D. L. Brower, Rev. Sci. Instrum. 79, 10E714 (2008). 3 H. K. Park, Plasma Phys. Controlled Fusion 31, 2035 (1989). 4 N. E. Lanier, D. Craig, J. K. Anderson, T. M. Biewer, B. E. Chapman, D. J. Den Hartog, C. B. Forest, S. C. Prager, D. L. Brower, and Y. Jiang, Rev. Sci. Instrum. 72, 1039 (2001). 5 P. J. Green and B. W. Silverman, Nonparametric Regression and Generalized Linear Models (Chapman and Hall, London, 1994). 6 L. Lin, W. X. Ding, D. L. Brower, W. F. Bergerson, and T. F. Yates, Rev. Sci. Instrum. 81, 10D509 (2010). 7 W. X. Ding, D. L. Brower, B. H. Deng, D. Craig, S. C. Prager, and V. Svidzinski, Rev. Sci. Instrum. 75, 3387 (2004). 8 W. X. Ding, D. L. Brower, G. Fiksel, D. J. Den Hartog, S. C. Prager, and J. S. Sarff, Phys. Rev. Lett. 103, 025001 (2009).
REVIEW OF SCIENTIFIC INSTRUMENTS 85, 11D404 (2014)
Design of vibration compensation interferometer for Experimental Advanced Superconducting Tokamaka) Y. Yang,1 G. S. Li,1,2 H. Q. Liu,1 Y. X. Jie,1,b) W. X. Ding,3 D. L. Brower,3 X. Zhu,1 Z. X. Wang,1 L. Zeng,1 Z. Y. Zou,1 X. C. Wei,1 and T. Lan2 1
Institute of Plasma Physics, Chinese Academy of Science, Hefei, Anhui 230031, People’s Republic of China University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China 3 Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, California 90095, USA 2
(Presented 2 June 2014; received 31 May 2014; accepted 18 June 2014; published online 11 July 2014) A vibration compensation interferometer (wavelength at 0.532 μm) has been designed and tested for Experimental Advanced Superconducting Tokamak (EAST). It is designed as a sub-system for EAST far-infrared (wavelength at 432.5 μm) poloarimeter/interferometer system. Two Acoustic Optical Modulators have been applied to produce the 1 MHz intermediate frequency. The path length drift of the system is lower than 2 wavelengths within 10 min test, showing the system stability. The system sensitivity has been tested by applying a periodic vibration source on one mirror in the system. The vibration is measured and the result matches the source period. The system is expected to be installed on EAST by the end of 2014. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4886455] I. INTRODUCTION
A far-infrared laser polarimeter/interferometer system has been designed and installed for Experimental Advanced Superconducting Tokamak (EAST).1 This diagnostic will measure the density and Faraday angle profile information and contribute to the plasma current profile control. In the polarimeter/interferometer system, the plasma density measurement is carried out by the interferometer, where the phase shift between the reference channel and detecting channel gives the line integrated electron density. The vibration (either from the machine or the system) will introduce an additional change in the path length of the optics system, which is an error for the phase of the density measurement.2–6 The error is higher, especially, during the plasma start-up phase since the density term is low and the vibration term is large or during the change of the toroidal magnetic field. The most severe potential vibration of EAST is expected from the retro reflectors locating at the high field side wall in the chamber. In order to compensate the vibration, we designed a second laser (wavelength at 0.532 μm) interferometer as a sub-system for EAST far-infrared polarimeter/interferometer. The system has been tested in the lab and planned to be installed on EAST by the end of 2014. II. SYSTEM DESCRIPTION
The principle of interferometer for plasma density measurement is known as ϕ = ϕp = c ne dl, (1) a) Contributed paper, published as part of the Proceedings of the 20th
Topical Conference on High-Temperature Plasma Diagnostics, Atlanta, Georgia, USA, June 2014. b) Author to whom correspondence should be addressed. Electronic mail: [email protected]. 0034-6748/2014/85(11)/11D404/3/$30.00
where ϕ p is the phase shift by plasma, c is a constant proportional to the wave length, and ne is the local plasma density. By considering the vibration in the optical path, the phase shift between the reference channel and the detecting channel is 2π ϕ = ϕp + ϕv = c ne dl + L, (2) λ where L is the path length change by vibration, ϕ v is the vibration term and is proportional to L, and λ is the wave length of the working laser. For CO2 pumped HCOOH laser (wavelength at 432.5 μm) and EAST low density (∼1 × 1019 m−3 ) case, the ϕ v term is ∼8% (80%) in the total phase shift for a 0.1 mm (1 mm) vibration. So the vibration compensation is needed. A visible laser (wavelength at 0.532 μm) has been selected as the second laser to compensate the vibration. For this wavelength, the dominant term in Eq. (2) is the vibration term and the plasma term can be neglected. The modulation frequency is set at 1 MHz by two Acoustic Optical Modulators (AOMs). One AOM produces a first order beam which has an 80 MHz frequency shift from the original/zero order beam. The other AOM produces an 81 MHz frequency shifted beam. These two first order beams then interfere and an IF signal at 1 MHz is created carrying the vibration information. The layout of the vibration compensation system is presented in Figure 1. The system will settle on a vertical optical board in front of the “O” window on EAST. The main part of the board is for the HCOOH laser polarimeter/interferometer. The upper part of the board is currently reserved for the vibration compensation system. The vibration compensation system shares part of the optical path with one HCOOH laser interferometer channel (5 in all and possibly upgraded to 11). Such arrangement implies that the assumed vibration is dominant from the retro reflectors and is identical for all channels. The outputs from one AOM are composed of the zero order
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FIG. 1. System layout of vibration compensation interferometer. The black/red/blue dashed box shows the optical path for 1 channel HCOOH laser/visible laser/both lasers.
beam, the first order beam and the higher order beams. An aperture is applied after the AOM, to allow only the first order beam to pass. A heterodyne interferometer scheme is designed to measure the phase shift. Avalanche Photodetectors (APD120 Series, Thorlabs Company) have been selected as the detectors for the 1 MHz IF signal. Other main components in the system are beam splitters (for visible laser and FIR laser, respectively), mirrors, quarter wave plate (for FIR laser), retro reflector, Polymethylpentene (TPX) lens, and TPX window. Bench test of these components has been finished. For each component, we tested the transmission efficiency for the visible laser. The power dependence (decreasing) on the optical path length has been measured and fit. Based on these data, the relative power (to the source power) at the two detectors are 0.0935% and 0.046%, respectively. For a 20 mW laser source, such optical arrangement can produce 18.7 μW and 9.2 μW signal in front of the detector. According to the detector’s bench performance, these signals could be detected. III. BENCH TEST OF THE SYSTEM A. System stability test
As described in Sec. II, the IF of the system is 1 MHz. These IF data are shown in Figure 2 (upper part). To test the system stability, this experiment was done when the system is mechanically stationary and without plasma. There is a phase difference between the two channels signal, which means the initial path length difference between the reference and detecting channels. In an ideal case (without spatial perturbation or any drift), such phase difference value should be constant in time. The phase difference, δϕ, has been measured for 400s (bottom part of Figure 2). Such time scale is currently the longest EAST pulse length. The result shows that during this time, the largest phase drift is within 3π . This means the drift in the system’s path length is within 1.5λ (∼0.8 μm). The
FIG. 2. The intermediate frequency signal and the stability of optical path.
drift could from any spatial perturbation in the lab. Comparing with the possible vibration from the machine which needs to be compensated (∼0.1 mm), this drift is in the order of 1%. On the other hand, if the machine vibration is ∼0.1 mm, there will be ∼180 fringe jumps for such a vibration. This may bring some difficulty in the fast fringe counting during the plasma discharges. B. Measurement of applied vibration
In order to simulate the machine vibration in the experiment, an external vibration has been applied to the system. This artificial vibration is introduced by setting an electric motor on one mirror (by screws). The frequency of the motor can be controlled and modified (
