Digital interferometry applied to transient dense plasmas

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(1)3384. IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 12, DECEMBER 2012. Digital Interferometry Applied to Transient Dense Plasmas Cristian Pavez, José Pedreros, Carlos Curín, Gonzalo Muñoz C., and Leopoldo Soto. Abstract—The work presented in this paper proposes a simple interferometric technique of recording and digital processing, which allows the obtaining of three possible interferometric records of variations at the refraction index, both in large scale and in small scale. The experimental setup basically consists of the Mach–Zehnder interferometer and a digital camera (with CMOS technology) coupled to a PC. The interferometric arrangement originally forms a pattern of very thin parallel fringes (15– 20 lines/mm on the CMOS device), which could eventually be used to obtain information of the refractivity of the phase object to submillimeter scales, looking directly to the original interferogram. The digital interferometric technique shown here is for triple exposure: the first exposure contains the plasma information and the next two are only referential records (reference interferometric patterns), similar to optical holographic interferometry, but instead of each exposure occurring on the same holographic plate, the record is done in consecutive captures of a digital acquisition system. One of the referential records is captured in the same condition as the one with plasma and the other one slightly changing the incidence angle of the reference beam. Thus, with this digital interferometry technique, it is possible to obtain (for the same plasma) a microinterferogram, which is an interferogram in fringes of infinite width and an interferogram in fringes of finite width. The technique is depicted in three different cases, namely, synthetic interferograms of a Gaussian plasma profile (simulated), laser-produced plasma, and a Z-pinch discharge at the SPEED2 generator. The first case describes the actual scope of the technique. For the other two pulsed plasmas, an Nd–YAG power laser (at the second harmonic, 532 nm) is used to produce the fringe pattern. Index Terms—Digital interferometry, plasma density measurement, refractive optical diagnostics.. Manuscript received May 1, 2012; revised August 23, 2012 and September 27, 2012; accepted September 30, 2012. Date of publication November 16, 2012; date of current version December 7, 2012. This work was supported by Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) under Grant 11090377. C. Pavez is with the Comisión Chilena de Energía Nuclear, Santiago 650-0687, Chile, with the Center for Research and Applications in Plasma Physics and Pulsed Power (P4 ), Santiago, Chile, and also with the Universidad Nacional Andrés Bello, Santiago 7600713, Chile (e-mail: [email protected]). J. Pedreros is with the Departamento de Ingeniería Eléctrica, Universidad de Santiago de Chile, Santiago 9170019, Chile. C. Curín is with the Universidad Nacional Andrés Bello, Santiago, Chile, and also with the Universidad de Santiago de Chile, Santiago 9170019, Chile. G. Muñoz C. is with the Pontificia Universidad Católica de Chile, Santiago 7820436, Chile. L. Soto is with the Comisión Chilena de Energía Nuclear, Santiago 650-0687, Chile, with the Center for Research and Applications in Plasma Physics and Pulsed Power (P4 ), Santiago, Chile, with the Universidad Nacional Andrés Bello, Santiago 7600713, Chile, with the University of Concepción, Concepción 4070386, Chile, and also with the University of Talca, Talca 3340000, Chile. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPS.2012.2223237. I. I NTRODUCTION. D. IGITAL interferometry started when a digital acquisition system was incorporated into the interferometric diagnosis. Thus, this technique can be plainly defined as the electronic record of an irradiance pattern of some interferometric scheme. Some of the interferometric diagnostics that fall into this definition are those that record the interference pattern directly on a charge-coupled device (CCD), without an additional process. This is the most used in interferometry of Z-pinch plasmas. A hybrid technique, which combines the optical holography with the digital image acquisition, was developed in [1] and [2]. In this technique, the optical reconstruction of the hologram is handled and recorded on a CCD. The saved information is processed to determine the interferometric phase directly from a set of image irradiance measurements. On the other hand, from the digital interferometry techniques, in which digital processing is necessary for the formation of the interference pattern, the most important is the digital speckle pattern interferometry (DSPI), which is sometimes called as “digital image plane holography” [3], [4], or simply “digital holography.” In this methodology, the interference pattern corresponds to a lowfrequency modulation of the high-frequency speckle noise [5]. An interferometric technique based in a method of digital processing similar to the one developed with DSPI [5] is described in [6]. With this method, the fringe pattern appears as the result of a digital operation, without using the speckle pattern as an information carrier. The work presented in this paper proposes a simple pulsed interferometric technique of recording and digital processing. With only three exposures (one with a phase object and two reference records), this technique allows the obtaining of simultaneous interferometric records of variations of the refractive index at large and small scale, similar to the optical method proposed in [7]. Each record is made with a parallel fringe interference pattern of high frequency (microinterferogram), of which spacing is between 15 and 20 lines/mm, or higher, depending on the CCD resolution. With digital processing (as in [6]) of the three saved images, we generate interference patterns with fringes of finite and infinite width, similar to optical holographic interferometry, but instead of each exposure occurring on the same holographic plate, the record is done in consecutive captures of a digital acquisition system. It should be noted that a submillimetric inspection on the microinterferometric record (on an enlarged image) would allow one to measure changes in the refractive index to small scale inside a macroscopic phase object. On the other hand, the direct visualization of the plasma microinterferogram possesses the. 0093-3813/$31.00 © 2012 IEEE.

(2) PAVEZ et al.: DIGITAL INTERFEROMETRY APPLIED TO TRANSIENT DENSE PLASMAS. 3385. attributes of an image plane shadowgram, with which it is possible to have more detailed information of the plasma edges. Thus, the technique implemented here integrates the benefits of digital interferometry, as developed in [6], with the potential of the interferometric technique developed in [7]. In addition, the same information can be processed as a digital hologram (image plane hologram) [8], [9]. Finally, the resulting interferograms can be used by some reconstruction algorithms of the interference phase, such as the Fourier transform method or the fringe tracking method within the disturbed area. Algorithms such as these have been used at density measurements of pinch phase in plasma focus discharges [10]. II. O PERATING P RINCIPLE The principle of the digital interferometry technique presented here consists of the digital capture of the microinterferometric pattern produced by two coherent laser beams. This procedure is done both before and after the optical path length is altered in one of the beams with the presence of plasma (phase objects). One of the two images is then subtracted pointwise from the other, and the result is displayed on a video monitor. The resulting image contains interference fringes of high frequency modulated by a fringe pattern, which are related to the optical path-length variation, as shown in [6]. The mathematical expression for the subtraction of the irradiance between both microinterferometric records is given by If = |I1 − I2 | = 2(Io Ir )1/2 [cos(Δψ1 ) − cos(Δψ1 + Δψ)] Δψ Δψ 1/2 If = 4(Io Ir ) sin Δψ1 + (1) sin 2 2 where Δψ = Δψ2 − Δψ1 = 2π(δ2 − δ1 )/λ is the phase difference between the object beams, δ1 and δ2 are the optical path lengths of these beams, and λ is the wavelength of the laser. The first sine function [in (1)] corresponds to high-frequency carrier fringes, which can be deleted using a low-pass filter. Since the second sine describes the fringes that appear when the argument of that function is a multiple of π, we can write λf = Δδ, where f is an integer that represents the order of dark fringes. The technique is exemplified with a group of three synthetic interferograms, for which amplified sections are shown in Fig. 1, in order to make the fringe pattern readable. The interferogram with the plasma information (plasma microinterferogram) is generated with a density Gaussian plasma profile [see Fig. 1(a)]. The reference synthetic interferograms (without plasma) correspond to parallel fringe patterns: one with the same frequency as the fringe pattern of the plasma microinterferogram [see Fig. 1(b)], i.e., 13 pixels between each black fringe, or equivalent to 15 lines/mm for the CMOS device used in our experimental setup (pixel size 5.2 μm × 5.2 μm) and the other one with a frequency (lines density) slightly higher in order to generate a suitable finite-width interferogram after the digital processing, namely, an interferogram with an appropriate number of fringes for the later interferometric analysis. In Fig. 1(c), the line density is increased to 17 lines/mm. At the. Fig. 1. Enlarged synthetic microinterferograms. (a) Plasma interferogram. (b) Reference interferogram with the same frequency as the fringe pattern of the plasma interferogram. (c) Reference interferogram with a frequency slightly higher.. interferograms with laser, the best results were obtained for microinterferometric patterns of very high frequency (more than 10 lines/mm), fringe patterns of high contrast, and an increase in the frequency of the reference interferogram between 5% and 10%. For higher increases in the frequency, the interferogram in fringes of finite width shows a low number of fringes on the plasma region, which could limit the spatial resolution along the plasma axis. Experimentally, the increase in the line density is obtained by changing the incidence angle of the reference beam of the interferometric assembly. Digital processing, as aforementioned, is carried out with the interferograms in Fig. 1. In Fig. 2, it is possible to observe the interferograms in fringes of infinite width [see Fig. 2(a)] as finite width [see Fig. 2(b)], which are results of the processing between the interferograms in Fig. 1(a) and (b) and the interferograms in Fig. 1(a) and (c), respectively. In both cases, in addition to the absolute value of the difference between the intensity fringe patterns, the low-pass filter and equalizing are applied on the resulting interferograms. Fig. 2(c) shows an enlarged zone of the digital interferogram in fringes of finite width, corresponding to Fig. 2(b) without the digital filtering process. Here, it is possible to observe the high-frequency pattern modulated by the fringe patterns, which are related to the optical pathlength variation. It should be noted that the technique used to obtain the interference phase from the interferograms defines the level of digital processing that the interferograms require, namely, the low-pass filter characteristics and the improving contrast at the interferograms by means of an equalization process. On the other hand, by the adjustment of the spatial cutoff frequencies, it is possible to filter out low-frequency background variations, high-frequency speckle noise, and other.

(3) 3386. IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 12, DECEMBER 2012. Fig. 3.. Gray level profile for each interferogram.. not automated, and they are applied according to the need of improving the image quality for later interferometric analyses. In order to verify that the total number of fringes inside the plasma region is the same for the three interferograms [see Fig. 1(a) and Fig. 2(a) and (b)], a gray level profile is read along a perpendicular line to the symmetry axis of the plasma for each interferogram and at the same position (over the black line of the fringe pattern). The graph in Fig. 3 shows the gray level profile for each interferogram (microinterferogram, infinite width, and finite width). With the information obtained from the graph, it is possible to observe an agreement among the maxima and minima of each gray level profile. This reports that each interferogram contains the same information about the phase shift. III. E XPERIMENTAL S ETUP. Fig. 2. (a) Interferograms in fringes of infinite width. (b) Interferograms in fringes of finite width. (c) Enlarged section of the digital interferogram in fringes of finite width without the filtering process.. disturbances that lead to spatial–frequency components not expected in the interferogram. The low-pass filter used here corresponds to a linear filter, i.e., low-pass type, which is applied on the propagation vector domain (Fourier space). In this space, the spatial cutoff frequency is chosen slightly higher to the frequency that modulates the high-frequency pattern. In the equalization process, the histogram of the image is modified by means of linear functions, in order to improve the image contrast and to reduce the contribution of zero order (continuous component). These two methods of image processing are. The experimental tests of the technique presented here were carried out on an experimental layout, as shown in Fig. 4. Basically, it consists of a Mach–Zehnder interferometer, which forms the original interferometric pattern (microinterferometric pattern); a digital camera with CMOS technology; a PC for storage and data processing (JPEG image format of 8 bits, using only the green matrix); and, finally, a pulsed test plasma generated in two different ways: the first formed by focusing a laser beam of high power in air and the second by forming a hollow gas-embedded Z-pinch guided by the conical geometry of the electrodes in the SPEED2 generator (3.98-μF equivalent Marx generator capacity, 150 kV, 2.1 MA in short circuit, 60 kJ, 400-ns rise time, and dI/dt ∼ 1013 A/s). For the case of the laser-produced plasma, a Nd–YAG power laser of 8-ns pulsewidth (1064 nm) is used both to produce the plasma and to diagnose it (532 nm), as shown in Fig. 4. It is important to note that any internal change from RAW to JPEG format could generate a distortion of the intensity pattern due to nonlinearity of the transformation. In order to avoid ambiguities in the information processed, we test the intensity profile of the interference pattern recorded by the digital device, ensuring that this has a sinusoidal profile..

(4) PAVEZ et al.: DIGITAL INTERFEROMETRY APPLIED TO TRANSIENT DENSE PLASMAS. Fig. 4.. 3387. Schematic of the experimental setup and the two mechanisms to produce the pulsed dense plasma.. Fig. 5. (a) Microinterferogram of laser-produced plasma spark. (b) Reference interferogram with a frequency slightly higher than the microinterferogram. (c) Interferogram in fringes of finite width reconstructed.. IV. E XPERIMENTAL R ESULTS In the experiment with laser-produced plasma, the image formation optical system of imaging formation is composed of a biconvex lens of focal length f = 200 mm and diameter 50 mm (magnification m = 8.9), an aperture of 5 mm placed at the focus position in order to remove the plasma light, and a Canon digital camera, model Rebel Xsi, with a CMOS size of 14.8 mm × 22.2 mm (5.2-μm pixel size). In this experiment, only the interferogram in fringes of finite width was considered. Since the maximum phase shift measured inside the plasma is about 2π, the line shift effect is not appreciable in the interferogram in fringes of infinite width. Fig. 5(a). shows the interferogram of the plasma (microinterferogram) and an enlargement of the region enclosed by the white square, enhancing the original fringe pattern. In Fig. 5(b), the reference interferogram is shown next to their respective enlargement of the region inside the white box, as shown in Fig. 5(a). The interferogram in fringes of finite width resulting from the digital processing between the microinterferograms in Fig. 5(a) and (b) is displayed in grayscale in Fig. 5(c). It is clear, from Fig. 5(a), that the direct observation of the interferometric pattern (microinterferogram record) is illegible, particularly when the density of interferometric fringes is very high. However, this characteristic allows us to record gradients (second derivative) at the plasma refraction index as in the image plane shadowgram, which give us information about the plasma structure instead (inhomogeneities, edges, and shock waves). In the application of the technique to a plasma pinch, a hollow gas-embedded discharge between two conical electrodes driven by SPEED2 generator at 60 kJ was used. The discharges were performed in air as filling gas at 33 mbar with an electrode gap of 20 mm. Both the interferogram of the plasma and the interferogram in infinite width are shown in Fig. 6(a) and (b), respectively. From the interferogram, the formation of a hollow plasma channel with filamentous structures that are most notorious near the electrodes is clearly observed. From the interferogram in infinite width, the regions with greater symmetry are more easily observed; similarly, the technique allows easily identifying the surfaces with equal phase change. The shaded region on the anode surface corresponds to leakage currents on the insulator that separate the anode–cathode region due to the bad coupling between the generator and the load. However, this defect does not represent a problem for the purpose to be shown here. From the information delivered by the microinterferogram in Fig. 6(a), a mapping of the phase shift was made. The mapping is shown in Fig. 7. With the information of the phase shift, and assuming cylindrical symmetry for the plasma, a density profile is computed and depicted in Fig. 8. The electron density profile shows a hollow channel with a maximum density value of about 1 × 1024 m−3 (with an estimated error of about 5%)..

(5) 3388. IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 12, DECEMBER 2012. Fig. 8. Hollow electron density profile of the interferogram shown in Fig. 6(a).. Fig. 6. (a) Interferogram of the hollow pinch. (b) Interferogram in fringes of infinite width of interferogram (a).. the advantage to have three interferograms for the plasma’s analysis with the complementary information, which allow us to obtain quantitative information from the inhomogeneous or dark zones that would be impossible to obtain if we use only photographic plates as recording media. It is also important to observe that the performance of the technique described here depends on both the mechanical stability of the experimental setup and the stability of the irradiance level between each exposure (laser stability). The last situation can be digitally solved with some equalized processing, previous to the main digital processing. The mechanical instabilities can generate small relative displacements between successive interferometric records, which produce, at the interferograms in fringes of infinite width, a small displacement of the maximum and minimum values in the fringe pattern. For the case of the experimental results in Fig. 6, the mechanical instability was generated by the electromechanical shutter of the digital reflex camera, and it was manually solved by a digital process, in which the electronic image (digital matrix) is displaced to the correct position. This problem can be easily solved by replacing the digital reflex camera with a CCD camera with electronic shutter. In this paper, all digital subtraction processes were manually developed. For the time being, an automation process of the digital processing is beyond the scope of this research stage. V. S UMMARY AND C ONCLUSION. Fig. 7. Mapping of phase shift between both exposures.. It is important to note that this technique does not improve the sensitivity in the interferometric recording; thus, for a typical Z-pinch (1 mm in diameter), the minimum detectable value is of about 1 × 1023 m−3 . However, the interferograms thus obtained can be later analyzed as any other interferogram, with. A modified interferometric technique of multianalysis has been presented. The potentiality of the technique is exemplified by three kinds of plasmas, namely, a simulated Gaussian profile of density and two pulsed plasmas, which reveal the versatility and advantages of the technique, which are summarized in: 1) The possibility to obtain large-scale features of the object from the interferograms digitally constructed and then to assess the small-scale structure directly by magnifying the plasma microinterferogram. A clear example of this is shown with the microinterferogram and interferogram in Fig. 5(a) and (b), respectively. For inhomogeneity of 50-μm size within the plasma,.

(6) PAVEZ et al.: DIGITAL INTERFEROMETRY APPLIED TO TRANSIENT DENSE PLASMAS. only the microinterferogram would be able to record the phase change produced by the refractivity of the inhomogeneity. 2) The optical setup for the microinterferometric record produces an equivalent image to an image plane shadowgram, which give us information about the plasma structure instead (inhomogeneities, edges, and shock waves). 3) The simultaneous availability of the interferograms, in fringes of finite and infinite width, gives complementary information about the shift-phase distribution. On one hand, the interferogram in fringes of infinite width allows a direct mapping of the isophase surfaces, particularly in phase objects with axial symmetry as the Z-pinch. On the other hand, the interferogram in fringes of finite width provides the sign of the interference phase without the ambiguity that appears with the interferogram in fringes of infinite width. 4) Finally, the recorded information of the plasma microinterferogram can be processed as a digital hologram (image plane hologram) [8], [9]. R EFERENCES [1] M. D. Watt and C. Vest, “Digital interferometry for flow visualization,” Exp. Fluids, vol. 5, no. 6, pp. 401–406, Nov. 1987. [2] S. M. Tieng, W. Z. Lai, and T. Fijiwara, “Holographic temperature measurement on axisymmetric propane-air, fuel-lean flame,” Meas. Sci. Technol., vol. 3, no. 12, pp. 1179–1187, Dec. 1992. [3] E. Marquardt and J. Richter, “Digital image holography,” Opt. Eng., vol. 37, no. 5, pp. 1514–1519, May 1998. [4] S. Schedin, G. Pedrini, and H. J. Tiziani, “Pulsed digital holography for deformation measurements on biological tissues,” Appl. Opt., vol. 39, no. 16, pp. 2853–2857, Jun. 2000. [5] T. Kreis, Handbook of Holographic Interferometry. Hoboken, NJ: Wiley, 2005, ch. 7. [6] I. Lira and L. E. Moreno, “Digital interferometry of phase objects,” Meas. Sci. Technol., vol. 8, no. 5, pp. 493–500, May 1997. [7] L. Soto, H. Chuaqui, and R. Saavedra, “Interferometry of phase micro inhomogeneities within macroscopic objects,” Meas. Sci. Technol., vol. 8, no. 8, pp. 875–879, Aug. 1997. [8] U. Schnars and W. Juepner, Digital Holography. New York: SpringerVerlag, 2005, pp. 71–85. [9] C. Pavez, J. Pedreros, and L. Soto, “Pulsed digital interferometry applied to plasma as phase objects,” in Proc. 8th Int. Conf. Dense Z-Pinches, Biarritz, France, 2011, (unpublished). [10] P. Kubes, M. Paduch, T. Pisarczyk, M. Scholz, T. Chodukowski, D. Klir, J. Kravarik, K. Rezac, I. Ivanova-Stanik, L. Karpinski, K. Tomaszewski, and E. Zieliñska, “Interferometric study of pinch phase in plasma-focus discharge at the time of neutron production,” IEEE Trans. Plasma Sci., vol. 37, no. 11, pp. 2191–2196, Nov. 2009.. Cristian Pavez was born in Santiago, Chile, in 1972. He received the B.S. and M.S. degrees in physics from the Pontificia Universidad Católica de Chile, Santiago, in 1998 and 2005, respectively, and the Ph.D. degree in 2007 from the Universidad de Concepción, Concepción, Chile, developing his Ph.D. thesis in experimental plasma physics at the Plasma Laboratory of the Comisión Chilena de Energía Nuclear (CCHEN). Since December 2008, he has been a Researcher with the Thermonuclear Plasma Department, CCHEN, and since 2011, he has been an Adjoint Assistant Professor with the Universidad Nacional Andrés Bello, Santiago. His main research interests are in dense transient plasmas, including Z-pinch, plasma focus and capillary discharges, transient plasma diagnostics, optical and digital holography, optical and digital interferometry, and optical refractive diagnostics.. 3389. José Pedreros was born in Santiago, Chile, in 1984. He received the B.S. degrees in applied physics and physics engineering in 2007 and 2010, respectively, from the Universidad de Santiago de Chile, Santiago, where he is currently working toward the M.S. degree in electrical engineering. Since January 2012, he has been a Digital Image Processing Consultant with the Thermonuclear Plasma Department of the Comisión Chilena de Energía Nuclear. His research interest includes soft switching converts and digital image processing.. Carlos Curín was born in Santiago, Chile, in 1964. He received the B.S. degrees in electrical engineering sciences and education in physical sciences and mathematics from the Universidad de Santiago de Chile, Santiago, in 1992 and 2005, respectively. He is currently working toward the Magister’s degree in science physics at the Universidad Nacional Andrés Bello, Santiago, working on his thesis in experimental plasma physics, which is being developed at the laboratory of the Thermonuclear Plasma Department of the Chilean Nuclear Energy Commission (CCHEN). He is currently a Professor of Physical Sciences with the Universidad Nacional Andrés Bello and with the Universidad de Santiago de Chile. His main interest is the teaching of experimental physics.. Gonzalo Muñoz C. was born in Santiago, Chile, in 1987. He received the B.S. degree in physics from the Universidad de Chile, Santiago, in 2010. He is currently working toward the Ph.D. degree at the Pontificia Universidad Católica de Chile, Santiago. In 2011, he was with the Plasma Laboratory of the Comisión Chilena de Energía Nuclear, mainly focused in interferogram analysis.. Leopoldo Soto was born in Santiago, Chile, in 1964. He received the B.S., M.S., and Ph.D. degrees from the Pontificia Universidad Católica de Chile, Santiago, in 1989, 1990, and 1993, respectively, all in physics. He is currently the Head of the Thermonuclear Plasma Department of the Comisión Chilena de Energía Nuclear (CCHEN) and the Director of the Center for Research and Applications in Plasma Physics and Pulsed Power (P4 ), CCHEN, University of Talca, Chile. He is an Associate Full Professor with the Universidad Nacional Andrés Bello; the Ph.D. program in physics of the University of Concepción, Chile; and the Ph.D. program in applied science of the University of Talca. His main research interests are related to dense transient plasmas, pulsed power, and applied optics, including Z-pinch, plasma focus, capillary discharges, pulsed-power miniature devices, transient plasma diagnostics, holography, interferometry, and optical refractive diagnostics. In 1999, he was awarded with a Presidential Chair in Science by the President of Chile. In 2007, he was elected as a Fellow of the Institute of Physics, U.K. He was the President of the Chilean Physical Society for two periods, from April 2003 to April 2008..

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