L. Lei,1 B. Tobias,1 C. W. Domier,1 N. C. Luhmann, Jr.,1 G. J. Kramer,2 E. J. Valeo,2. W. Lee,3 G. S. Yun,3 and H. K. Park3. 1University of California at Davis, ...
REVIEW OF SCIENTIFIC INSTRUMENTS 81, 10D904 共2010兲
A synthetic diagnostic for the evaluation of new microwave imaging reflectometry diagnostics for DIII-D and KSTARa… L. Lei,1 B. Tobias,1 C. W. Domier,1 N. C. Luhmann, Jr.,1 G. J. Kramer,2 E. J. Valeo,2 W. Lee,3 G. S. Yun,3 and H. K. Park3 1
University of California at Davis, Davis, California 95616, USA Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA 3 Pohang University of Science and Technology, Pohang, Gyeongbuk 790-784, South Korea 2
共Presented 18 May 2010; received 17 May 2010; accepted 4 June 2010; published online 4 October 2010兲 The first microwave imaging reflectometry 共MIR兲 system for characterization of fluctuating plasma density has been implemented for the TEXTOR tokamak 关H. Park et al., Rev. Sci. Instrum. 75, 3787 共2004兲兴; an improved MIR system will be installed on DIII-D and KSTAR. The central issue remains in preserving phase information by addressing antenna coupling between the reflection layer and the detector array in the presence of plasma turbulence. A synthetic diagnostic making use of coupled full-wave diffractive codes has been developed in geometries and applied to a variety of optical arrangements. The effectiveness of each scheme is quantitatively compared with respect to the fluctuation levels accessible in the simulation. © 2010 American Institute of Physics. 关doi:10.1063/1.3464461兴 I. INTRODUCTION
A prototype microwave imaging reflectometry 共MIR兲 system for the acquisition of two-dimensional 共2D兲 turbulent fluctuation measurements was demonstrated on the TEXTOR tokamak in 2004.1 However, it was found that in some cases the experimental measurements were inconsistent with the laboratory performance of the MIR system.2 Continued studies to improve the performances of this MIR system and to explain the deficiencies observed in plasma experiments will be needed before such system may be implemented on DIII-D and KSTAR. Full-wave simulations are needed for complete characterization of the plasma region including diffraction and refraction effects in order to evaluate the present and future optical coupling approaches. To this end, the fullwave reflectometer code FWR2D 共Ref. 3兲 has been used in conjunction with independent full-wave simulations of MIR system optics and antenna sources. Interfacing FWR2D and MIR optical design simulations will aid in understanding the effects of imperfections and nonidealities in fabricated lens systems, while simultaneously testing a variety of optical configurations with different boundary conditions. In this paper, we use FWR2D in a numerical survey of candidate MIR configurations. A primary and key consideration that must be appropriately addressed in the design of an MIR system is the establishment of appropriate boundary conditions for the optical receiver and illumination systems such that the optical systems preserve the coupling of transmitter and receiver phase information within the well defined standards, allowing for the detection of turbulent fluctuations. In Sec. II, the concepts for numerical simulations in both FWR2D and optical design software are discussed. The results are shown in Sec. III, and a comparison of different a兲
Contributed paper, published as part of the Proceedings of the 18th Topical Conference on High-Temperature Plasma Diagnostics, Wildwood, New Jersey, May 2010.
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arrangements is presented in Sec. IV. A conclusion is given in Sec. V. II. SIMULATIONS
A two-dimensional full-wave simulation 共FWR2D兲 code developed at PPPL and validated successfully against laboratory experiments3,4 is used in conjunction with beam propagation descriptions of MIR optical systems 共optical design simulation兲. By coupling the output of FWR2D complex field data to the imaging properties generated in simulation of the optical system, a useful synthetic diagnostic may be realized. This method greatly reduces simulation time by allowing many optical configurations to be interchangeably mated to a single set of simulated plasma effects, resulting in an efficient design tool. Figure 1 illustrates the simulation concept, where FWR2D simulation includes an illumination with a beam coming from a horn antenna and focused at the cutoff, and the optical design simulation includes an arbitrary receiver. In the case shown, both the illumination and the receiver are configured to image the cutoff layer, making use of the concept of a virtual cutoff5 in order to derive the geometric properties of both beams. In the study presented here, three different microwave beam patterns are simulated in FWR2D: a standard short gain horn with a rapidly diverging illumination pattern 共wide beam case兲, a collimated incident beam incident beam 共collimated beam case兲, and an incident beam focused with phase front radius of curvature matching at the virtual cutoff 共focused case兲 such that the incoming and outgoing beam patterns are identical in FWR2D 共the case seen in Fig. 1兲. In optical simulation there are two different arrangements: a collimated horn receiver, similar to the collimated illumination case 共collimate beam兲, and a receiver system focused at the virtual cutoff, as described in the focused illumination 共focused case兲.
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FIG. 1. 共Color online兲 A schematic representation of the simulation approach used to evaluate an approach for imaging in MIR systems illustrates the major components and their integration. FWR2D simulation provides an illumination source and simulates the plasma region in three parts: vacuum region, paraxial region, and full-wave region for incident and reflected beams. Optical simulation acts as a receiver by providing a complex amplitude distribution for the E-field of a reciprocal beam generated at the detector with which the output of FWR2D may be convolved. In the above setup, an incident beam is focused at the virtual cutoff, verified by the exact correspondence of incoming and outgoing beams in FWR2D.
For the case shown in Fig. 1, the incident beam from is focused at the cutoff and reflected to the antenna plane where convolution with the electric field generated in the optical simulation takes place. The electric field in the optical simulation is collected using a wide aperture lens and imaged at the detector by evaluating a reciprocal Gaussian beam originating there. Similar processes are performed for other simulation arrangements and all combinations of illumination and receiver configurations. There are a total of six different cases simulated in this study. In FWR2D, plasma parameters for a midsize tokamak equilibrium similar to those available on TEXTOR, ASDEX-U, DIII-D, or KSTAR are used. A microwave beam frequency of 77.5 GHz with X-mode polarization is assumed. A turbulent radial correlation length of x = 2 cm is chosen, and the poloidal correlation length is set to y = 2 cm; the mean value of the fluctuation wave number is chosen such that kx = 0 and kx = 0; the mean distribution of fluctuation wave numbers are chosen to be ⌬kx = 1 cm−1 and ⌬ky = 1 cm−1. Density fluctuation levels ranging from 0.2% to 2% are simulated. For each fluctuation level, a statistically significant number of random density fluctuation profiles are generated. FWR2D
III. RESULTS
The quantitative interpretation of the measured signals in terms of density fluctuations at the reflecting layer is straightforward, provided that these fluctuations primarily affect the phase of the reflected waves.6 The relationship between power spectra of the phase ⌫共kx兲 and power spectra of permittivity ⌫共kx兲 for a simple plane-stratified medium is given by5 ⌫共kx兲 = 2
k20Ln 2 关C 共w兲 + S2共w兲兴⌫共kx兲, 兩kx兩
共1兲
FIG. 2. 共Color online兲 Histogram information for multiple random density fluctuation simulations at differing mean fluctuation amplitude. As fluctuation amplitude is increased, the received signal is strongly distorted by signal amplitude modulation, making interpretation of receiver phase challenging.
where Ln is the density scale length, C共w兲 and S共w兲 are Fresnel integrals with w = 共2兩kx兩L / 兲1/2, and kx represents the fluctuation wave number. From this, one may obtain an ideal statistical relationship between the amplitude distribution of the density fluctuations and the measured phase. Further analysis reveals a fundamental limit on the amplitude of fluctuations which may be measured as5
2n ⬍ 1/共3/2ML2⌬kx兲,
共2兲
where 2n is the variance of density fluctuation, L is the scale length of plasma permittivity, and M = 1 for O-mode or M = 2 for X-mode. In this study, L = 52.1 cm, this gives n ⬍ 1.9%. I/Q plots are generated in this study as histograms of the real and imaginary parts of the convolved fields at the simulation interface, where the output field from FWR2D is mated to an external simulation of the receiver optics by simple convolution of the electric fields at the boundary, resulting on a statistical sampling of simulations in a time series output for this synthetic diagnostic. Figure 2 shows the I/Q plots for different density fluctuation levels where an incident beam with 3 m focal length propagates to the cutoff and the reflected beam is received with an imaging lens. The phase and amplitude distribution of the histograms increase as the density fluctuation levels increase. While the correlation coefficient between the density fluctuation and the receiver phase is not presented in this paper, a comparison of the amplitude and phase statistics serves as a useful measure of the system characteristics. With a small turbulence, the virtual cutoff is in-focus and the image of the cutoff is reconstructed. This is evident in the limited amplitude distortion of the receive signal. In the latter case, only noise is present as the reflected waves reveal large and random amplitude modulation causing hundreds for “phase jumps” across the origin. A histogram of signal amplitude fitted to Rice probability distribution for these cases is shown in Fig. 3 for statistical analysis. The Rice distribution is shown to converge to the Rayleigh distribution as indirect signal paths from transmitter to the receiver become stronger than the direct path signal. Figure 3 shows that the mean amplitude of the re-
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FIG. 3. 共Color online兲 Amplitude distributions for different density fluctuation levels. In this case, at a density fluctuation level of approximately 1%, the distribution begins to converge to a Rayleigh distribution and density fluctuation information is no longer preserved in the received data.
ceived signals deviates significantly from unity as the density fluctuation levels increase beyond 1%, which is well below the limitation predicted by Eq. 共2兲. IV. COMPARISON BETWEEN DIFFERENT SIMULATION ARRANGEMENTS
Statistical comparisons of the amplitude spread A and phase spread of the received signals versus density fluctuation amplitude are shown in Figs. 4共a兲 and 4共b兲, respectively. In Fig. 4共a兲, if we compare the wide-collimate case 共diverging beam in FWR2D—collimated horn antenna in optical simulation兲 with the wide-focal case 共imaging receiver in optical simulation兲, it is apparent that amplitude distortion
in the wide-collimate case presents itself much faster than for the wide-focal case as fluctuation levels are increased. More data in this fluctuation amplitude regime are required, but these limited results suggest that receiver imaging is more important than focusing of the illumination source. Correlation of density fluctuation and receiver phase, as has been previously evaluated for the purpose of contrasting optical and synthetic imaging,6 is required for a more complete picture. However, the statistical data in Fig. 4共b兲 remain meaningful. For the cases tested, loss of information through excess phase perturbation relative to density fluctuation, not amplitude distortion, is the limiting factor in diagnostic utility. This is evidence by the mean deviation in phase approaching that of an ergotic I/Q histogram at low fluctuation levels. Further, the optical properties of the receiver system are found to have a complicated relationship to the detected phase distribution and may result in a positive or a negative deviation from the ideal case proposed by Eq. 共1兲. This is suggested by the data for a 0.5% fluctuation, for which the highly focused system is the central data point. In all cases studied here, a ␦n / n ⬍ 1% limit for detection is observed, which remains below theoretical limits.7,8 V. CONCLUSION
An extensive numerical survey of the candidate MIR configurations is being performed by addressing antenna coupling between the reflection layer and the detector array in the presence of plasma turbulence. Understanding the coupling effectiveness provides new insights for evaluation of the previously obtained experimental data and direct contribution to continuing MIR system development. Enhancements of this simulation technique will integrate the recently developed three-dimensional 共3D兲 platform, FWR3D 共Ref. 9兲 and make use of the experimentally obtained plasma profiles and optical system characterizations from the TEXTOR simultaneous ECEI/MIR system.1 ACKNOWLEDGMENTS
This work was supported by U.S. DOE Grant Nos. DEFG02-99ER54531 and DE-AC02-09CH11466 and by POSTECH. Thanks are due to Eliot Feibush for helping with the FWR2D simulations. 1
FIG. 4. 共Color online兲 共a兲 Amplitude spread vs density fluctuation level and 共b兲 phase spread vs density fluctuation level for six different simulated couplings. In all cases, phase distribution is random when fluctuation levels exceed 1%. 共Notations for each case referred to Sec. II, for example, wide focal: wide beam case from FWR2D and focused beam from optical simulation, etc.兲
H. Park, E. Mazzucato, T. Munsat, C. W. Domier, M. Johnason, N. C. Luhmann, Jr., J. Wang, Z. Xia, I. G. J. Classen, A. J. H. Donne, and M. J. van de Pol, Rev. Sci. Instrum. 75, 3787 共2004兲. 2 T. Munsat, E. Mazzucato, H. Partk, C. W. Domier, N. C. Luhmann, Jr., A. J. H. Donne, and M. van de Pol, Plasma Phys. Controlled Fusion 45, 469 共2003兲. 3 E. J. Valeo, G. J. Kramer, and R. Nazikian, Plasma Phys. Controlled Fusion 44, L1 共2002兲. 4 G. J. Kramer, R. Nazikian, and E. Valeo, Plasma Phys. Controlled Fusion 44, L11 共2002兲. 5 E. Mazzucato, Rev. Sci. Instrum. 69, 6 共1998兲. 6 G. J. Kramer, R. Nazikian, and E. Valeo, Plasma Phys. Controlled Fusion 46, 695 共2004兲. 7 G. J. Kramer, R. Nazikian, and E. Valeo, Rev. Sci. Instrum. 74, 3 共2003兲. 8 E. Mazzucato, T. Munsat, H. Park, B. H. Deng, C. W. Domier, N. C. Luhmann, Jr., A. J. H. Donne, and M. J. van de Pol, Phys. Plasmas 9, 5 共2002兲. 9 E. J. Valeo, G. J. Kramer, and R. Nazikian, Proceedings of the 18th Topical Conference, 2009, Vol. 1187, p. 649.
