| Citation: |
Minyong SHEN, Jibo ZHANG, Yao ZHANG, Yinxian JIE, Haiqing LIU, Jinlin XIE, Weixing DING. Forward modelling of the Cotton-Mouton effect polarimetry on EAST tokamak[J]. Plasma Science and Technology, 2024, 26(3): 034015. DOI: 10.1088/2058-6272/ad15df
|
Measurement of plasma electron density by far-infrared laser polarimetry has become a routine and indispensable tool in magnetic confinement fusion research. This article presents the design of a Cotton-Mouton polarimeter interferometer, which provides a reliable density measurement without fringe jumps. Cotton-Mouton effect on Experimental Advanced Superconducting Tokamak (EAST) is studied by Stokes equation with three parameters (s1,s2,s3). It demonstrates that under the condition of a small Cotton-Mouton effect, parameter s2 contains information about Cotton-Mouton effect which is proportional to the line-integrated density. For a typical EAST plasma, the magnitude of Cotton-Mouton effects is less than 2π for laser wavelength of 432 μm. Refractive effect due to density gradient is calculated to be negligible. Time modulation of Stokes parameters (s2, s3) provides heterodyne measurement. Due to the instabilities arising from laser oscillation and beam refraction in plasmas, it is necessary for the system to be insensitive to variations in the amplitude of the detection signal. Furthermore, it is shown that non-equal amplitude of X-mode and O-mode within a certain range only affects the DC offset of Stokes parameters (s2,s3) but does not greatly influence the phase measurements of Cotton-Mouton effects.
The purpose of this paper is to propose a method for continuous and reliable high-precision measurements of plasma electron density [1] using a polarimeter interferometer [2] based on the Cotton-Mouton effect (hereafter abbreviated as CM) on EAST tokamak or on future fusion devices. This measurement can provide key information for density control of magnetic fusion reactors [1]. Unlike conventional interferometers that require time accumulation for phase wrap-up, the CM effect induces a phase shift of approximately 15 degrees on the central chord of EAST when a wavelength of 432 μm is chosen, effectively compensating for the disadvantage of fringe jumps when the phase difference exceeds 2π after the interruption and reconnection of the interferometer signal [3]. To guarantee the density accurate measurement, CM effect-based polarimetry can be operated simultaneously with an interferometer that employs the same probing beam [4]. Furthermore, due to the instabilities arising from laser oscillation and beam refraction in plasmas, the system is designed to be insensitive to variations in the amplitude of the detection signal [5].
There are two types of polarimeters: those based on the Faraday rotation (hereafter referred to as FR) effect and those based on the CM effect [6]. These magneto-optical effects depend on the magnetic field aligned parallel or perpendicular to the propagation direction of electromagnetic waves in the plasma [4], respectively. Therefore, the choice of one of these effects depends on the optical beam path. For density measurement, the Faraday rotation effect is not suitable due to variation of magnetic field parallel to the beam path in poloidal section. Hence, the CM polarimeter becomes a good option for high density measurement without fringe jump [7]. This paper limits the configuration to within a poloidal section.
In the context of superimposed elliptical modulation, achieving exacting power equality between two orthogonal laser beams can be nearly impossible [8]. This study aims to demonstrate that, even under this condition, accurate measurements of the relatively small CM effects can still be obtained through a phase comparator, while remaining insensitive to changes in the detection signal amplitude. Furthermore, we elucidate the rationale behind selecting radiation with wavelengths falling within the far-infrared (FIR) range. Although a Cotton-Mouton polarimeter has recently been developed on the Compact Helical System (CHS) [4] and is proposed for the International Thermonuclear Experiment Reactor (ITER) [9], its applications are fewer than those of Faraday polarimeters. We will develop a CM effect polarimeter system that utilizes a 432.6 μm formic acid laser on EAST tokamak. In section 2 we introduce typical parameters of EAST plasma and present some optical quantities related to polarimetry. In section 3, we utilize Stokes equation and Mueller matrix to investigate evolution of polarization of lasers in EAST plasma. By using time modulation of Stokes parameters, one can determine elements of Mueller matrix which relates to plasma parameters. In section 4, we study the coupling between FR effect and CM effect. An explicit nonlinear relationship is proposed between CM phase shift and χ for phase difference not much less than one. Section 5 provides a brief summary.
EAST is a magnetic confinement tokamak fusion device, with a plasma major radius of approximately 1.9 m, an average minor radius of approximately 0.45 m, and a magnetic field strength of about 2.4 T along the magnetic axis. Currently, there are five vertical ports in EAST device, evenly spaced at intervals of 90 mm in the R-direction, located at R = 1640 mm, 1730 mm, 1820 mm, 1910 mm, and 2000 mm, respectively. The vertical port at R = 1910 mm approximately passes through the magnetic axis. The laser beam is injected from the upper port and detects the line-averaged electron density at the lower window of the device.
The magnetic configuration of EAST shot #98958 at 21 s is shown in figure 1, where the red dashed line represents the last closed magnetic surface. A right-handed orthogonal coordinate system is established, with the vertical direction defined as the Z-axis, the direction from the high-field side to the low-field side in the major radial direction as the X-axis, and the toroidal direction as the Y-axis. In the simulations performed in this study, the incident position is set within the range of R0∈(1640mm,2000mm).
x0 is defined as
| x0=(R0−1910)mm. | (1) |
The vertical channel passing through the magnetic axis is defined as x0=0, which corresponds to the range of x0∈(−270mm,90mm) in the minor radial direction. Electron density (ne) and toroidal magnetic field distribution (Bt) for EAST shots #71320 at 3.3 s and #98958 at 21 s are shown in figures 2 and 3, respectively.
In order to study the evolution of the polarization state of electromagnetic waves in magnetized plasmas, the following conventional quantities are introduced to describe the elliptical polarization state of a laser beam.
A top view of elliptical polarization of laser beam is shown in figure 4, where beam propagates along the negative direction of the Z-axis defined in figure 1.
In the electric field coordinate system X-Y, α is the arctangent of the ratio of electric field amplitude in the Y direction to that in the X direction.
| tanα=EyEx. | (2) |
Generally, along the propagation direction, the projection trajectory of the polarization electric field vector’s endpoint onto the X-Y plane is an ellipse. Polarized ellipse in the elliptical coordinate system X′-Y′, χ is the ellipticity, referred to as the arctangent of the ratio of the short axis of the ellipse to the long axis
| tanχ=ba. | (3) |
ψ is the azimuthal angle of the polarization plane, referred to as the angle between the long axis of the ellipse X′ and the X-axis in the electric field coordinate system. ψFR is the change in the angle of the polarization plane due to Faraday effect
| ψFR=CFRλ2∫B‖ | (4) |
where the constant C_{\rm{FR}}=\dfrac{e^{3}}{8 {\text{π}}^{2} \varepsilon_{0} m_{{\rm{e}}}^{2} c^{3}} , \lambda is the wavelength of the laser, B_{\|} is the magnetic field parallel to the direction of light propagation, n_{{\rm{e}}} is the electron density. When B_{\|} , n_{{\rm{e}}} , \lambda and l are expressed in the International System of Units (SI), C_{\rm{FR}} has a numerical value of 2.6312 \times 10^{-13} , the unit of \psi_{\rm{FR}} is rad.
Plasma can be viewed as an anisotropic birefringent medium. The mode of the electromagnetic wave in which the electric field is parallel to the magnetic field direction is the O-mode, while the X-mode is perpendicular to the magnetic field. The passage of linearly polarized light through birefringent medium results in ellipsoidalization, known as the Cotton-Mouton effect.
Upon entering a plasma, incident polarization states will all split into fast and slow modes, each with distinct optical path lengths upon exiting. The term “fast” indicates a smaller optical path length. The slow state is delayed relative to the fast state, and this delay is due to the difference in optical path lengths, which describes the relative phase change between the two states [10].
As shown in figure 5, a retarder splits light into two linearly orthogonal polarized components. Half-wave linear retarder delays one of the linearly polarized components by half a wavelength and is useful for changing the direction of linearly polarized light. With a vertical fast axis (lower refractive index), the component perpendicular to the magnetic field direction is referred to as the ‘X-mode’. The X-mode enters the half-wave plate along the fast axis and undergoes 2 and 1/4 cycles. The O-mode, on the other hand, traverses 2 and 3/4 cycles along the slow axis, resulting in a phase delay of {\text{π}} . Their combined light transforms from right-circularly polarized light to left-circularly polarized light.
As shown in figure 4, an inscribed circle can be constructed using the length of the minor axis of the ellipse as its diameter. Constructing a line parallel to line X' and passing through point A, let AC be parallel to X' . Similarly, constructing another line parallel to line X' and passing through point B, let BD be parallel to X' . These lines intersect the inner circle at points C and D, respectively. The angle \angle COD is geometrically referred to as the eccentric anomaly and is denoted as \delta . It is also known as the orthogonal electric vector phase. \delta\mathrm{_{CM}} is the change in the orthogonal electric vector phase angle due to the CM effect [11].
| \begin{array}{l} \delta_{\rm{CM}}=C_{\rm{CM}} \lambda^{3} \int B_{\perp}^{2} n_{{\rm{e}}} {\rm{d}} l, \end{array} | (5) |
where the constant C_{\rm{CM}}=\dfrac{e^{4}}{16 {\text{π}}^{3} \varepsilon_{0} m_{{\rm{e}}}^{3} c^{4}} , B_{\perp} is perpendicular to the direction of light propagation of the magnetic field. When B_{\perp} , n_{{\rm{e}}} , \lambda and l are expressed in SI, C_{\rm{CM}} has a numerical value of 2.4568 \times 10^{-11} , the unit of \delta_{\mathrm{CM}} is rad.
Figure 6 illustrates the phase shift of the CM effect in the X -direction for each chord. The maximum phase shift of the CM effect does not exceed 20°, thus avoiding the occurrence of fringe jumps that would result from exceeding 2 {\text{π}} .
To achieve a phase shift magnitude of approximately 20° the wavelength of the far-infrared laser was chosen as 432.6 \mu m. According to equation (5), figure 7 illustrates that the phase shift is proportional to the cube of the wavelength. Due to the refractive effect, the plasma region traversed by the same chord slightly varies at different wavelengths, so the curve in this figure is not strictly a cubic function.
Plasma can exhibit refractive effects due to density gradients. The propagation of light, as described by Born in [12], is governed by Snell’s Law, which relates the incident angle ( \theta_{1} ) and the refracted angle ( \theta_{2} ) to the absolute refractive indices of the two media, denoted as {n}_{1} and {n}_{2} :
| \begin{array}{l}n_1\sin\theta_1=n_2\sin\theta_2.\end{array} | (6) |
This law describes the relationship between the angles of incidence and refraction when light passes from one medium to another. By calculating the refractive index vector at each spatial step and determining the displacement after passing through the plasma, the quasi-optical deviation caused by the presence of plasma is inferred. This displacement must be kept as small as possible so that probing beam can exit from vacuum windows for signal detection.
119 \mu {\rm{m}} represents the CH3OH laser, 195 \mu {\rm{m}} represents the DCN laser, 337 \mu {\rm{m}} represents the HCN laser, 432.6 \mu {\rm{m}} represents the CH _3 COOH laser, and 462 \mu {\rm{m}} represents the solid source. figures 8 and 9 use data from shot \# 71320, which can be observed that the wavelength does not change the trend of lateral offset and phase shift. However, due to the proportional relationship between phase shift and \lambda^3 , the wavelength significantly affects the magnitude of the measured values. When the wavelength is smaller than the chosen wavelength of the formic acid laser in this paper, the peak phase shift is less than 10°, which will affect the resolution of the polarimeter interferometer [13].
Figure 9 illustrates the variation in the magnitude of transverse displacement caused by the refractive effects in each channel along the X -direction, which is on the order of millimeters. After consulting the engineering drawings, it was determined that the diagnostic window has a diameter of 70 mm. During the discharge, the laser generates a refraction of approximately 2 mm, and the far-infrared laser spot diameter is around 30 mm. As long as the optical alignment is accurate on the tabletop during experiments, it can be observed that the currently selected wavelength meets the requirements of the diagnostic window size in EAST, enabling the reception of laser signals emitted in the opposite direction.
When the probe beam is incident vertically, B_{\perp} is equivalent to B_{{\rm{t}}} , which is approximately constant along the light path, as shown in figure 3. The density chord integral \int n_{{\rm{e}}} {\rm{d}} l is measured using the interferometer at EAST. The CM effect along this chord is calculated by transforming it through equation (5), as \delta_{\mathrm{CM}}=C_{\rm{CM}}B_{\perp}^2\lambda^3\int_{ }^{ }n_{\rm{e}}{\rm{d}}l , so the measurement of electron density along the central chord of the poloidal section in the plasma is most appropriately performed using a vertical port CM effect polarimeter interferometer. It can be seen that the phase shift of the CM effect is expected to be around 15° throughout the EAST discharge, a vertical-view Cotton-Mouton polarimeter interferometer is capable of providing reliable density measurement. In figure 10, there is a sharp and brief density jump at t=73\enspace{\rm{s}} , which is caused by the interferometer’s fringe jump, and the CM effect polarimeter interferometer will effectively circumvent this anomaly.
| \begin{array}{l} \sin 2 \chi=\dfrac{2 E_{x} E_{y}}{E_{x}^{2}+E_{y}^{2}} \sin {\delta}=\sin 2 \alpha \sin {\delta}. \end{array} | (7) |
The CM effect acts directly on the phase shift change {\rm{d}}{\delta}_{\rm{CM}} , which leads to the ellipticity \chi change through equation (7) [8].
Figure 11 illustrates the noticeable variations in ellipticity along the central chord, demonstrating the theoretical feasibility of measuring the CM effect through the ellipticity information carried by the polarized ellipse.
Two concepts should be clarified here. The term “orthogonal” in the context of orthogonal electric vector phase refers to the perpendicular relationship in the X-Y coordinate system of the electric field. The polarization ellipse is the trajectory traced by the endpoints of the electric field vector after one cycle, i.e., a 360° change in the phase angle. From the extremal point B of E_x to the extremal point A of E_y through a certain phase angle, which is the mapped \angle COD, not necessarily 90°, so-called “quadrature”. CM effect {\delta}_{\rm{CM}} acts directly on the change of phase angle of the orthogonal electric vector, which is a concept of polarization optics. The “phase angle” here is not equivalent to the phase change of the light intensity.
As discussed above the measurement of polarized ellipse is non-trivial, it requires two measurements of E_x and E_y . Accurately measuring the absolute electric field amplitude can be very challenging. Instead the signal is modulated to incorporate the information of the CM effect within the phase difference of the Stokes parameters. One can modulate the polarized ellipse in time so that phase difference between E_x and E_y can be determined. In this section we utilize Stokes equation, Jones vector and Mueller matrix to describe evolution of polarization state in EAST. By detecting Stokes parameters one can determine plasma density, magnetic field.
The purpose of introducing Stokes parameters is to provide a mathematical representation of the polarization state of electromagnetic waves. Stokes parameters encompass polarization characteristics such as intensity, degree of polarization, and polarization ellipse orientation. They enable a quantitative description and analysis of the polarization state of light [14].
Introducing the Poincaré sphere with a unit radius of 1 in the (s_1, s_2, s_3) space facilitates the study of polarization, as depicted in figure 12. Here, all possible polarization states ( \psi , \chi ) are uniquely mapped by points P on the unit sphere surface. Establishing an orthogonal Cartesian coordinate system with the center of the sphere as the origin, the three directional components can be represented using a one-row-three-column matrix of Stokes parameters, corresponding to a latitude of 2 \chi and a longitude of 2 \psi .
The normalized Stokes parameters can be defined as [15]
| \begin{split} {\boldsymbol{s}}(z=0) &=\left(\begin{array}{c} \cos 2 \chi \cos 2 \psi \\ \cos 2 \chi \sin 2 \psi \\ \sin 2 \chi \end{array}\right) =\left(\begin{array}{c} \cos 2 \alpha \\ \sin 2 \alpha \cos {\delta} \\ \sin 2 \alpha \sin {\delta} \end{array}\right) \\& =\left(\begin{aligned} \frac{\langle E_{x}^{2}\rangle-\langle E_{y}^{2}\rangle}{\langle E_{x}^{2}\rangle+\langle E_{y}^{2}\rangle} \\ \frac{2\langle E_{x} E_{y} \cos {\delta}\rangle}{\langle E_{x}^{2}\rangle+\langle E_{y}^{2}\rangle} \\ \frac{2\langle E_{x} E_{y} \sin {\delta}\rangle}{\langle E_{x}^{2}\rangle+\langle E_{y}^{2}\rangle} \end{aligned}\right), \end{split} | (8) |
where,
| \begin{array}{l} s_{1}^{2}+s_{2}^{2}+s_{3}^{2}=s_{0}^{2}=1. \end{array} | (9) |
Two orthogonal polarizations are represented by antipodal points on the equator of the Poincaré sphere. Taking the X and O beams under investigation in this paper as examples, the horizontal polarization is characterized by the Stokes parameters {\boldsymbol{s}}_{\rm{H}}=(1,0,0) , and the vertical polarization is characterized by {\boldsymbol{s}}_{\rm{V}}=(-1,0,0) . When polarizations propagate through a homogeneous non-absorbing medium, they experience different characteristic refractive indices, denoted as \mu_1 and \mu_2 , respectively, satisfying \mu_1>\mu_2 .
\Omega represents the angular velocity of the rotation of the Stokes parameters on the Poincaré sphere [16]. The fast axis aligns with the direction of the \Omega vector, passing through the two points on the Poincaré sphere that represent the characteristic orthogonal polarizations, s_{\rm{H}} and s_{\rm{V}} . An introduction to the fast axis can be found in figure 5. Therefore, the evolution of polarization can be expressed by a vector equation [17].
| \begin{array}{l} \dfrac{{\rm{d}} {\boldsymbol{s}}(z)}{{\rm{d}} z}={\boldsymbol{\Omega}} \times {\boldsymbol{s}}(z), \end{array} | (10) |
| \begin{array}{l} {\boldsymbol{\Omega}}={\rm{d}} {\text{Δ}} \varphi / {\rm{d}} z=\left(\mu_{1}-\mu_{2}\right) \omega / c=\left(\begin{array}{c} C_{\rm{CM}} \lambda^{3} n_{{\rm{e}}}\left(B_{y}^{2}-B_{x}^{2}\right) \\ -2 C_{\rm{CM}} \lambda^{3} n_{{\rm{e}}} B_{y} B_{x} \\ C_{\rm{FR}} \lambda^{2} n_{{\rm{e}}} B_{z} \end{array}\right). \end{array} | (11) |
Here, the magnitudes of C_{\rm{FR}} and C_{\rm{CM}} are given by equation (4) and equation (5), respectively; \Omega_{1,2,3} represent the angular velocities of the Stokes parameters rotating around the s_{1,2,3} axes, and each component is contributed by the CM effect {\rm{d}}{\delta}_{\rm{CM}} , cross-terms, and the FR effect {\rm{d}}\psi_{\rm{FR}} . The quantities \Omega_{1,2,3} reflect the optical properties of the plasma medium. By setting an appropriate incident laser polarization direction and measuring the changes in polarization optical quantities, \Omega_{1,2,3} can be determined, providing information about plasma density, magnetic field, and other parameters.
Figure 13 illustrates when the central chord of the light beam varies along the negative direction of the Z -axis, the impact of the CM effect on three components of the Stokes parameters. The initial Stokes parameters are given by the superposition of horizontally and vertically linearly polarized light, i.e., {\boldsymbol{s}}(z=0)=(0,1,0) . The variation in the amplitude of s_3 is −0.2444 rad, which can be determined through the vector cross product in equation (10). Since, s_3\sim\int{\rm{d}}{\delta}_{\rm{CM}}\cdot s_2(0){\rm{d}} z , the CM effect on EAST can be approximated as s_3=0.2508 rad, relating plasma parameter.
For application of polarized ellipse time modulation technique, we present the time-dependent Stokes parameter using Jones vectors. From equation (8), it can be observed that the Stokes parameters represent the temporal ensemble average of the orthogonal components of the electric field and cannot reflect the frequency modulation of a single beam of light. Therefore, the Jones vector representation is introduced, which employs a 2 \times 1 matrix to represent the x and y components of the electric field [17],
| \begin{array}{l} \left[\begin{array}{l} E_{x} \\ E_{y} \end{array}\right]=\left[\begin{array}{c} E_{0 x} {\rm{e}}^{\rm{i} {\delta}_{1}} \\ E_{0y} {\rm{e}}^{\rm{i} {\delta}_{2}} \end{array}\right]. \end{array} | (12) |
The electric field resulting from the superposition of the X-mode and O-mode is
| \begin{array}{l} {\boldsymbol{E}} = {\boldsymbol{E}}_{\rm{O}} +{\boldsymbol{E}}_{\rm{X}}\\ \;\;\;\; =E_{\rm{aO}}\exp \left[{\rm{i}}\left(\omega_{1} t\right)\right]\left(\begin{array}{l} 1 \\ 0 \end{array}\right)+E_{\rm{aX}}\exp \left[{\rm{i}}\left(\omega_{2}t\right) \right]\left(\begin{array}{l} 0 \\ 1 \end{array}\right) \\ \;\;\;\;= \exp \left[{\rm{i}}\left(\omega_{1} t\right)\right]\left(\begin{array}{l} E_{\rm{aO}} \cdot \exp [{\rm{i}}({\text{Δ}} \omega t)] \\ E_{\rm{aX}} \cdot\exp [{\rm{i}}(0)] \end{array}\right), \end{array} | (13) |
where, {\text{Δ}} \omega=\omega_{2}-\omega_{1} represents the intermediate frequency of the X-mode and O-mode, and the subscript ‘a’ denotes the amplitude, and {\delta}_1 and {\delta}_2 are initial phases for each component. From equation (2), the amplitude ratio \alpha can be obtained as
| \tan \alpha=\frac{{\rm{Re}}(E_{y})}{{\rm{Re}}(E_{x})}=\frac{E_{\rm{aX}}}{E_{\rm{aO}}}. | (14) |
The parameter \alpha is a constant that is solely related to the ratio of the transmitted power. Furthermore, since s_{1}=\cos (2 \alpha) , the trajectory of elliptical modulation is reflected on the Poincaré sphere as a constant value of s_{1} . Phase difference
| \begin{array}{l} {\delta}={\delta}_{2}-{\delta}_{1}={\text{Δ}} \omega t, \end{array} | (15) |
the trajectory of the modulation process on the Poincaré sphere is represented by a circle drawn from the periodic variation of the phase angle.
In figure 14, the end of the electric field vector traces a more complex shape, which can be interpreted as a time-varying polarization state resembling an evolving ellipse. This distinctive shape is known as a Lissajous figure, which is formed by the parametric equation. To a Stokes polarimeter, this beam appears as unpolarized light and cannot be distinguished.
From equation (8), the Stokes parameters of the synthesized light resulting from the superposition of horizontal and vertical polarization modulation can be obtained as
| \begin{array}{l} {\boldsymbol{s}}(z=0) =\left(\begin{array}{c} \cos 2 \alpha \\ \sin 2 \alpha \cos {\text{Δ}} \omega t \\ \sin 2 \alpha \sin {\text{Δ}} \omega t \end{array}\right), \end{array} | (16) |
where, \alpha=\arctan\dfrac{E_{\rm{aX}}}{E_{\rm{aO}}} [17].
Non-uniform power emission will introduce deviations in the intensity amplitude of the measurement by affecting the magnitude of s_1 . The elliptical modulation trajectory under different transmission ratio conditions is shown in figure 15, represented by a series of circles with their centers falling on the s_1 axis. The standard form in an orthogonal Cartesian coordinate system between the variables s_2 and s_3 can be written as:
| \begin{array}{l} s_2^2+s_3^2=\sin^2 (2 \alpha), \end{array} | (17) |
which is represented by the yellow dashed line. In general, there exists a nonlinear relationship between {\text{Δ}}\chi and {\delta}_{\rm{CM}} .
Specifically, when two orthogonally polarized beams are equally transmitted with equal power, i.e., E_{\rm{aX}}=E_{\rm{aO}} or \tan \alpha=1 , from equation (7) it can be deduced that {\delta} and ellipticity \chi have a linear relationship. The relationship between the variables s_2 and s_3 is defined by the equation s_2^2+s_3^2=\sin^2 (2 \times 45 {\text{°}} ) , forming a meridian loop by \psi=90\text{°} and \psi=-90 {\text{°}} . The circle can be analogously compared to geographical east longitude 90\text{°} and west longitude 90\text{°} , as indicated by the purple dashed line in figure 15.
The output polarization state can be characterized using either Stokes parameters or, equivalently, \psi and \chi . The former is often more convenient since the detector signals are directly related to the three components of the Stokes parameters. For the design and analysis of polarization measurements, the Mueller matrix M is required to comprehensively describe the effects of optical components or plasma media on the polarization state [18]. This matrix is the characteristic of plasma (density and magnetic field etc). The output Stokes vector relates to Mueller matrix (plasma parameters) as the following expression [17]:
| \begin{array}{l} {\boldsymbol{s}}(z)={\boldsymbol{M}}(z) \cdot {\boldsymbol{s}}(0). \end{array} | (18) |
Due to the non-commutativity of the Mueller matrices for the two effects, it is not possible to simply integrate each effect separately to calculate the total effect. Therefore, when both effects are strong, the coupling cannot be separated [19]. When at least one of the two effects is small, approximate analytic solutions of the evolution equation can be obtained. When the FR effect is small, we consider two cases for discussion: the CM effect where {\delta}_{\rm{CM}} \ll 1 or {\delta}_{\rm{CM}} \sim 1 . The dimensionless parameters W_{1,2,3} are used for analysis [17].
| \begin{split} & W_{j} =\int_{z_{0}}^{z_{1}} \;{\rm{d}} z \Omega_{j}(z) \\ & W_{j k} =\int_{z_{0}}^{z_{1}} \;{\rm{d}} z \Omega_{j}(z) \int_{z_{0}}^{z} \;{\rm{d}} z^{\prime} \Omega_{k}\left(z^{\prime}\right)\\ &W_{j k l} =\int_{z_{0}}^{z_{1}} \;{\rm{d}} z \Omega_{j}(z) \cdot \int_{z_{0}}^{z} \;{\rm{d}} z^{\prime} \Omega_{k}\left(z^{\prime}\right) \cdot \int_{z_{0}}^{z^{\prime}} {\rm{d}} z^{\prime \prime} \Omega_{l}\left(z^{\prime \prime}\right). \end{split} | (19) |
The parameters W_{1,2,3} correspond to the CM effect {\delta}_{\rm{CM}} , the cross term, and the FR effect \psi_{\rm{FR}} respectively.
(1) W_{1,2,3} \ll 1
| \begin{split} {\boldsymbol{M}}(z_1)=& {\boldsymbol{M}}^{(0)}+{\boldsymbol{M}}^{(1)}+{\boldsymbol{M}}^{(2)}+{\boldsymbol{M}}^{(3)}+\cdots\\ =&{{1}}+ \int_{z_{0}}^{z_{1}} \;{\rm{d}} z {\boldsymbol{A}}(z)+\int_{z_{0}}^{z_{1}} \;{\rm{d}} z {\boldsymbol{A}}(z) \cdot \int_{z_{0}}^{z} \;{\rm{d}} z^{\prime} {\boldsymbol{A}}(z) \\&+\int_{z_{0}}^{z_{1}} \;{\rm{d}} z {\boldsymbol{A}}(z) \cdot \int_{z_{0}}^{z} \;{\rm{d}} z^{\prime} {\boldsymbol{A}}\left(z^{\prime}\right) \cdot \int_{z_{0}}^{z^{\prime}} {\rm{d}} z^{\prime \prime} {\boldsymbol{A}}\left(z^{\prime \prime}\right)+\cdots. \end{split} | (20) |
According to equation (18), {\boldsymbol{A}} satisfies the following equation:
| \begin{array}{l}\dfrac{{\rm d}\boldsymbol{M}}{{\rm d}z}=\boldsymbol{A}(z)\cdot\boldsymbol{M}(z),\end{array} | (21) |
where now
| \begin{array}{l} {\boldsymbol{A}}(z)=\left(\begin{array}{ccc} 0 & -\Omega_{3} & \Omega_{2} \\ \Omega_{3} & 0 & -\Omega_{1} \\ -\Omega_{2} & \Omega_{1} & 0 \end{array}\right), \end{array} | (22) |
where the components of the matrices {{M}}^{(1)}, {{M}}^{(2)}, {{M}}^{(3)}\cdots are given by
| \begin{split} & {{M}}^{(1)}_{j k}= \int_{Z_{0}}^{Z_{1}} \;{\rm{d}} Z A_{j k}(Z) \\ &{{M}}^{(2)}_{j k}= \int_{Z_{0}}^{Z_{1}} \;{\rm{d}} Z A_{j m}(Z) \int_{Z_{0}}^{Z} \;{\rm{d}} Z^{\prime} A_{m k}\left(Z^{\prime}\right) \\ & {{M}}^{(3)}_{j k}= \int_{Z_{0}}^{Z_{1}} \;{\rm{d}} Z A_{j m}(Z) \int_{Z_{0}}^{Z} \;{\rm{d}} Z^{\prime} A_{m n}\left(Z^{\prime}\right) \int_{Z_{0}}^{Z^{\prime}} {\rm{d}} Z^{\prime \prime} A_{n k}\left(Z^{\prime \prime}\right) \cdots .\quad\end{split} | (23) |
where again repeated indices imply summation [20].
When the plasma effect is small, the Mueller matrix can be expanded to second order,
| \begin{split} &{{\boldsymbol{M}}\left(z_{1}\right) \approx }\\ &\left(\begin{array}{ccc} {1-\left(W_{2}^{2}+W_{3}^{2}\right) / 2} & {-\left(W_{3}-W_{21}\right) }& {\left(W_{2}+W_{31}\right) }\\ {\left(W_{3}+W_{12}\right) }& {1-\left(W_{1}^{2}+W_{3}^{2}\right) / 2 }& {-\left(W_{1}-W_{32}\right) }\\ {-\left(W_{2}-W_{13}\right) }& {\left(W_{1}+W_{23}\right) }& {1-\left(W_{1}^{2}+W_{2}^{2}\right) / 2} \end{array}\right). \end{split} | (24) |
For a tokamak having up/down symmetry with respect to the equatorial plane, the following relations hold: W_{2}=0 , W_{13}=W_{31}=\dfrac{W_{1} W_{3}}{2} , W_{12}=-W_{21} , and W_{23}=-W_{32} . These relations are investigated under the assumption that W_{1}^{2} \ll 1 and W_{3}^{2} \ll 1 , therefore [17],
| \begin{array}{l} {\boldsymbol{M}}\left(z_{1}\right) \approx\left[\begin{array}{ccc} 1 & -\left(W_{3}-W_{21}\right) & W_{1} W_{3}/2 \\ W_{3}-W_{21} & 1 & -\left(W_{1}-W_{32}\right) \\ W_{1} W_{3}/2 & W_{1}-W_{32} & 1 \end{array}\right]. \end{array} | (25) |
Based on equations (16) and (18), and using the the auxiliary angle formula, the second component s_2 of the Stokes parameters can be expressed as follows:
| \begin{split} {s}_{2}(z)= & M_{21} \cos 2 \alpha+ M_{22} \sin 2 \alpha \cos \left({\text{Δ}}\omega t\right)\\&+M_{23} \sin 2 \alpha\sin \left({\text{Δ}}\omega t\right) . \end{split} | (26) |
So, multiplying M_{21} by the constant \cos 2 \alpha only affects the DC offset of s_2 , while multiplying M_{22} and M_{23} by terms involving \left({\text{Δ}}\omega t\right) affects both the amplitude and phase of s_2 .
| \begin{split} {s}_{2}(z)=& \left(W_{3}-W_{21}\right) \cos 2 \alpha +\sin 2 \alpha \cos \left({\text{Δ}}\omega t\right)\\& -\left(W_{1}-W_{32}\right) \sin 2 \alpha\sin \left({\text{Δ}}\omega t\right) \\ =& \left(W_{3}-W_{21}\right) \cos 2 \alpha +\sin 2 \alpha \cdot \sqrt{1+\left(W_{1}-W_{32}\right)^2} \\& \cdot \cos \left[\left({\text{Δ}}\omega t\right)+\tan ^{-1}\left(W_{1}-W_{32}\right)\right]. \end{split} | (27) |
The Stokes parameter s_2 of laser polarization state after plasma is modulated at frequency ( {\text{Δ}}\omega ). It contains information about the chord integral W_1 , The phase of s_2 does not include quantities related to the power ratio \alpha . One can determine W_1 by phase measurement instead of amplitude measurement ( \sin 2\alpha ). In the process of orthogonal superposition signal radiation, the ({\text{Δ}}\omega t) changes while \alpha remains constant. The simplest case is E_{{\rm a}x}=E_{{\rm a}y},\alpha=\text{π}/4 , where two laser radiations have equal power.
The errors resulting from non-equal power radiation is discussed here. Furthermore, parameters in Mueller matrix may not be very small. These approximations can introduce finite corrections. A comparison is made between the definitional formula (equation (10)) and the computational formula (equation (27)) for s_2 , and the results are plotted in figure 16. Without loss of generality, we set the modulation phase ({\text{Δ}}\omega t) to be 90°, the phase variation of the AC component is reflected in a larger amplitude variation.
In the case of non-uniform power emission, without loss of generality, the power ratio is assumed as \alpha = \dfrac{{\text{π}}}{6} . Under equal power emission, the result from the definitional formula is s_{\rm{2d}} = 0.252 , while the result from the computational formula is s_{\rm{2e}} = 0.248 . Under non-uniform power emission, the result from the definitional formula is s_{\rm{2d}} = 0.263 , while the result from the computational formula is s_{\rm{2e}} = 0.260 . The errors introduced by neglecting higher-order terms in the computational formula are only −1.59% and −1.14% respectively. Now let us perform an order of magnitude analysis on the sources of error. As shown by equation (26), {s}_{2}(z) consists of three terms, each containing M_{21}, M_{22} , and M_{23} . The errors arise from neglecting terms of order three and higher, beyond {\boldsymbol{M}}^{(3)} , in the derivation from equation (20) to equation (24). In the previous section, we assumed {\text{Δ}}\omega t=90\text{°} , making the second term in equation (26) equal to 0. From equation (22) and equation (23), we can derive the expressions for {{M}}^{(3)}_{21} and {{M}}^{(3)}_{23} with respect to W
| \begin{array}{l}{M}_{21}^{(3)}=-W_{333}-W_{322}-W_{311},\end{array} | (28) |
| \begin{array}{l}{M}_{23}^{(3)}=W_{111}+W_{122}+W_{133}.\end{array} | (29) |
In equation (26), since W is always positive, when \alpha increases, the weight of M_{23} becomes greater, leading to an overestimation of {s}_{2} by neglecting {{M}}^{(3)}_{21} . Conversely, when \alpha decreases, the weight of M_{21} increases, resulting in an underestimation of {s}_{2} by neglecting {{M}}^{(3)}_{23} .
As previously mentioned, in the case of vertically incident CM effects on the poloidal section, the subscripts indicate that index 3 is attributed to CM effects and index 1 is attributed to FR effects. Therefore, \left|-W_{333}\right| > \left|W_{111}\right|, \left|-W_{322}\right| > \left|W_{122}\right|,\ \left|-W_{311}\right| > \left|W_{133}\right| . In other words,
| \begin{array}{l}|M_{21}^{(3)}| > |M_{23}^{(3)}|.\end{array} | (30) |
This indicates that the error introduced when \alpha takes a smaller value must not be greater than that introduced when \alpha={\text{π}}/4 .
This indicates the feasibility of the proposed heterodyne measurement scheme with small errors. It can be understood that the non-equal power component only generates a DC offset of \left(W_{3}-W_{21}\right) \cos 2 \alpha and a signal attenuation of \sin 2 \alpha times at the mixer stage. It is important to emphasize that the derivation in this subsection cannot be extended to cases where W_{1}\ll 1 is not satisfied.
(2) W \approx W_{1} ; W_{2,3} \ll 1
For example, in the case of vertical propagation in a future tokamak, it may not always satisfy the condition \left|W_{1}\right| \ll 1 in practical scenarios. Now let us write the analytical expression for the first-order expansion of {s}_{2}(z) when W \sim {\cal{O}}\left(1\right) .
Similarly to equation (26), the second component s_2 of the Stokes parameters is given by [17]
| \begin{split} {s}_{2}(z) =&Q_{1}^{(0)} \cos 2 \alpha+\cos W_{1}\sin 2 \alpha \cos \left({\text{Δ}}\omega t\right)\\&-\sin W_{1} \sin 2 \alpha\sin \left({\text{Δ}}\omega t\right) \\ = &Q_{1}^{(0)} \cos 2 \alpha+\sin 2 \alpha \cdot \cos \left[\left({\text{Δ}}\omega t\right)+W_{1}\right], \end{split} | (31) |
where,
| Q_{1}^{(0)} =\int_{0}^{z} \;{\rm{d}} z^{\prime}\left(\Omega_{2}^{\prime} \sin W_{1}^{\prime}+\Omega_{3}^{\prime} \cos W_{1}^{\prime}\right) . | (32) |
The derivation process of expanding s_2(z) to the first order when W \sim {\cal{O}}\left(1\right) is provided in Appendix A.
To simulate a larger CM effect, the typical toroidal magnetic field magnitude B_{\rm{t}} is multiplied by a factor of \sqrt{3} , effectively amplifying the CM effect by a factor of 3. figure 17 is plotted to illustrate the measurement under this scenario. Without loss of generality, we set the modulation phase \left({\text{Δ}}\omega t\right) to be 90{\text{°}}, the phase variation of the AC component is reflected in a larger amplitude variation. The power ratio \alpha in the case of unequal power emission is \dfrac{{\text{π}}}{6} , the same as in figure 16.
In the case of equal power emission, the result from the definition formula is s_{\rm{2d}}=0.694 , while the result from the calculation formula is s_{\rm{2e}}=0.746 . Similarly, in the case of unequal power emission, the definition formula yields s_{\rm{2d}}=0.646 , and the calculation formula yields s_{\rm{2e}}=0.692 . The errors introduced by neglecting higher-order terms in the calculation formula are +7.49 \% and +7.12 \% respectively.
If a longer wavelength laser is chosen to achieve a larger CM effect, this heterodyne measurement scheme will introduce larger errors. Moreover, the longer the wavelength, the more pronounced the refractive effect becomes, so the detector no longer receives the laser signal precisely at the center, leading to a decrease in signal-to-noise ratio. Therefore, in practical experiments, the CM effect is typically not allowed to become excessively large (around {\delta}_{\rm{CM}}\sim {\cal{O}}\left(1\right) ) during electron density measurements.
In summary, under the condition of non-equal power emission of orthogonal light, the superimposed elliptical modulation can still accurately measure relatively small CM effects through a phase comparator, while being insensitive to variations in the detected signal’s amplitude. In order to achieve an appropriate magnitude of the CM effect, the wavelength of the radiation used should fall within the far infrared (FIR) range. The error arises from neglecting W_{1}^{2} and W_{3}^{2} , so non-uniform power emission only affects the signal-to-noise ratio, with minimal impact on the calculation error of the CM effect phase.
Elliptical modulation enables the direct measurement of {\delta}_{\rm{CM}} through the phase difference of the Stokes parameters. In this section, we will discuss the polarization optical components and their configurations in the polarization analyzer system. From equation (16), both s_2 and s_3 contain terms modulated with respect to ({\text{Δ}} \omega t) .
The intensity measurement of s_2 can be performed without the presence of a waveplate. In principle, it is feasible to measure s_3 as well. However, for s_3 measurement, a quarter-wave plate needs to be installed in the optical path, requiring prior measurement of the waveplate’s absorption factor which involving additional effort [21]. Therefore, we ultimately decide to measure s_2 .
As shown in figure 18, the micro-displacement platform is used to adjust the length of the resonant cavity, ensuring that the frequency difference between the two laser sources is equal to the desired beat frequency. Modulating the orthogonal light generates an intermediate frequency signal IF \sim{\cal{O}}\left(1 \enspace{\rm{MHz}}\right) , facilitating heterodyne measurement. One laser source emits linearly polarized light, which is passed through a half-wave plate to be orthogonal to the polarized light emitted by the other laser source. The beams are then combined, adjusted to a collinear state, and guided through an optical waveguide to the top of EAST device, where they are incident on the plasma.
Direct measurement cannot determine the value of a specific component. The power measured by the detector I_{\rm{D}} , is the weighted sum of the individual components of the Stokes parameters [8].
| \begin{array}{l} I_{\rm{D}}\left(\theta\right)=\dfrac{1}{2}I_{0}\left[1+s_{1} \cos 2 \theta+s_{2} \sin 2 \theta\right], \end{array} | (33) |
where, I_0 represents the intensity before passing through the polarizer, \theta is the angle between the transmission axis of the polarizer and the X -axis. We set \theta=45{\text{°}} to ensure that the intensity only contains information about s_2 ,
| \begin{array}{l} I_{\rm{D}}(45{\text{°}} )=I_{0}\dfrac{1+s_{2}}{2}. \end{array} | (34) |
Under the assumption of a constant I_0 , there is a linear relationship between the intensity and s_2 , with s_2 having the highest coefficient, thus the phase measurement of s_2 can be transformed into a phase measurement of the intensity. Due to the difference in refractive index between the two beams in the plasma, the phase shift difference between the ordinary wave and the extraordinary wave is compared using a mixer and a phase discriminator. This allows for the determination of the chord-integrated plasma chord density.
When the phase difference of the CM effect is only about 15 degrees, reducing phase noise becomes an important issue. Additionally, when the CM effect is significant, heterodyne measurement methods can introduce an approximate 10 \% measurement error. Now, we will demonstrate that an accurate analytical solution for the ellipticity \chi is feasible under larger Cotton-Mouton (CM) effects.
Stokes parameters are observable, but they are not as intuitive as the polarization ellipse parameters. To gain a more comprehensive understanding of the changes in the polarization ellipse within the plasma, it is advantageous to convert the Stokes equations into expressions based on the parameters of the polarization ellipse.
To express the differential interaction between \chi and \psi [22]
| \frac{\rm{d} \chi}{{\rm{d}} z}=\frac{1}{2} \frac{{\rm{d}}{\delta}_{\rm{CM}}}{{\rm{d}} z} \sin 2 \psi , | (35) |
| \frac{\rm{d} \psi}{{\rm{d}} z}=\frac{{\rm{d}}\psi_{\rm{FR}}}{{\rm{d}} z}-\frac{1}{2} \frac{\rm{d}{\delta}_{\rm{CM}}}{{\rm{d}} z} \tan 2 \chi \cos 2 \psi, | (36) |
where,
| \left\{\begin{split} \psi \in\left(-\dfrac{{\text{π}}}{2}, \dfrac{{\text{π}}}{2}\right] \\ \chi \in\left[-\dfrac{{\text{π}}}{4}, \dfrac{{\text{π}}}{4}\right] . \end{split} \right. | (37) |
During the propagation of light in a plasma, the presence of the Faraday rotation (FR) effect causes the azimuthal angle to vary continuously. Consequently, \sin2\psi is not a constant. By equation (35), \psi is coupled explicitly to \chi . The FR effect, although not acting directly on \chi , couples implicitly to \chi by changing \psi through equation (36) [23].
If one of the FR and CM effects can be neglected, the other effect, no matter how strong, can be written as an explicit expression and can theoretically be measured [20]. When the CM effect can be ignored, the FR effect can be measured no matter how strong it is, and this conclusion was verified in JET [24]. This conclusion is obviously. When {\delta}_{\rm{CM}}=0 , equations (35) and (36) are written as
| \begin{array}{l} {\rm{d}} \chi= 0\\ {\rm{d}} \psi= {\rm{d}}\psi_{\rm{FR}}. \end{array} | (38) |
Conversely, when the FR effect can be neglected and the CM effect plays a dominant role, the problem can be considered as a generalized Volterra-like problem with non-constant coefficients [25].
| \frac{{\rm{d}} \chi}{{\rm{d}} z}=\frac{1}{2} \frac{\rm{d}{\delta}_{\rm{CM}}}{{\rm{d}} z}\sin 2 \psi, | (39) |
| \frac{\rm{d} \psi}{{\rm{d}} z}=-\frac{1}{2} \frac{{\rm{d}}{\delta}_{\rm{CM}}}{{\rm{d}} z}\tan 2 \chi \cos 2 \psi, | (40) |
equation (40) is divided by equation (39) to obtain
| \begin{array}{l} \dfrac{\rm{d} \psi}{\rm{d} \chi}=-\tan 2 \chi /\tan 2 \psi, \end{array} | (41) |
| \int {\rm{d}} \psi(\chi) \tan 2 \psi(\chi) =\int-\tan 2 \chi {\rm{d}} \chi. | (42) |
Because the indefinite integral
| \int \tan x {\rm{d}} x=-\ln |\cos x|+{{ C}}, | (43) |
where x is an arbitrary function and {{ C}} is a constant.
Therefore, integrating equation (42) to obtain,
| \begin{split}& -\frac{1}{2} \ln (\mid\cos 2 \psi\mid)+\frac{1}{2} \ln \left(\mid\cos 2 \psi_{0}\mid\right) =\\&\frac{1}{2} \ln (\cos 2 \chi)-\frac{1}{2} \ln \left(\cos 2 \chi_{0}\right), \end{split} | (44) |
where \chi_{0} and \psi_{0} are known constant values representing the polarization parameters of the incident light.
When \left|\cos 2 \psi_{0}\right| \cdot \cos 2 \chi_0\neq0 , it satisfies
| \begin{array}{l} \ln \left(\mid \cos 2 \psi \mid \cdot \cos 2 \chi\right)=\ln \left(\left|\cos 2 \psi_{0}\right| \cdot \cos 2 \chi_{0}\right), \end{array} | (45) |
| \begin{array}{l} |\cos 2\psi| \cdot \cos 2 \chi=\left|\cos 2 \psi_{0}\right| \cdot \cos 2 \chi_0. \end{array} | (46) |
Furthermore,
| \begin{array}{l} \cos ^{2} 2 \chi \cdot \cos ^{2} 2 \psi=\left(\cos 2 \psi_{0} \cdot \cos 2 \chi_{0}\right)^{2}, \end{array} | (47) |
define Const as
| \begin{array}{l} {\rm Const} =\left(\cos 2 \psi_{0} \cdot \cos 2 \chi_{0}\right)^{2}, \end{array} | (48) |
therefore,
| \begin{array}{l} \psi(\chi)=\pm\dfrac{1}{2} \arccos \left( {\rm{Const}} \cdot\sec 2 \chi\right). \end{array} | (49) |
According to the properties of trigonometric and inverse trigonometric functions, the constant {\rm{Const}} \in(0,1] in order to ensure that the value domain of \psi remains within the set of real numbers.
Figure 19 depicts the relationship between \psi and \chi obtained from equation (49), where the arrows indicate the direction of phase space trajectories influenced by the CM effect. Points p _{1} and p _{3} represent special cases at \chi=0 , where \psi reaches extremum. Similarly, points p2 and p4 correspond to \psi=0 , where \chi attains extremum.
If the initial polarization parameters ( \chi_0 , \psi_0 ) are given and the end-state \chi is measured, the phase shift of the CM effect can be written as a function \rm{d}\delta_{\rm{CM}}=\rm{d}\delta_{\rm{CM}}(\chi_0,\ \psi_0,\ \chi) , which achieves the decoupling of \chi from \psi .
Noting that at the singularity where \psi=0 , the denominator on the right-hand side of equation (41) becomes 0 in the improper integral. EAST satisfies that B_{z} is homogeneous along the range in the z direction and \psi varies monotonically along the range. Without loss of generality, \psi is assumed to be monotonically decreasing. We classify and discuss two distinct scenarios for the phase space trajectories based on the positive and negative values of \psi . The first scenario involves trajectories moving from p _{4} to p _{1} and then to p _{2} , while the second scenario entails trajectories moving from p _{2} to p _{3} and then to p _{4} .
(1) \psi in the range greater than 0.
Equation (47) is written as
| \begin{array}{l} \cos ^{2} 2 \chi \cdot \cos ^{2} 2 \psi= {\rm{Const}} >0, \end{array} | (50) |
substituting equation (35) into equation (50),
| \begin{array}{l} \cos ^{2} 2 \chi \cdot\left(1-\sin ^{2} 2 \psi\right) =\cos ^{2} 2 \chi \cdot\left(1-\left(\dfrac{2 {\rm{d}} \chi}{\rm{d}{\delta}_{\rm{CM}}}\right)^{2}\right) = {\rm{Const}} , \end{array} | (51) |
because
| \begin{array}{l} \dfrac{\cos ^{2} 2 \chi}{ {\rm{Const}} }=\dfrac{1}{\cos ^{2} 2 \psi}\geqslant1, \end{array} | (52) |
hence
| \begin{array}{l} \cos ^{2} 2 \chi\geqslant {\rm{Const}} . \end{array} | (53) |
According to equation (51), the integrand function can be computed as follows:
| {\rm{d}}{\delta}_{\rm{CM}}=\frac{2 \cos 2 \chi}{\sqrt{\cos ^{2} 2 \chi- {\rm{Const}} }} {\rm{d}} \chi,\quad(\psi>0). | (54) |
The definite integral is
| \begin{array}{l} {\delta}_{\rm{CM}}=\left.\arctan \left(\dfrac{\mid\sin 2 \chi\mid}{\sqrt{- {\rm{Const}} + \cos ^{2} 2 \chi}}\right)\right|_{\chi_{0}}^\chi,\quad(\psi>0). \end{array} | (55) |
The integral calculation procedure of equation (55) is shown in Appendix B.
(2) \psi in the range less than 0.
If the polarized light propagates such that \psi=0 , i.e., (\cos ^{2} 2 \chi- {\rm{Const}} )=0 , the singularity of this elliptic degree \chi can be defined as
| A=\frac{\arccos( \sqrt{ {\rm{Const}} })}{2}, | (56) |
by equation (42),
| \int_{0}^{\psi} \tan 2 \psi {\rm{d}} \psi=-\int_{A}^{\chi} \tan 2 \chi {\rm{d}} \chi, | (57) |
from equation (35), when \psi<0 , then \sin 2\psi<0 . Implying d \chi<0 , \chi is monotonically decreasing, i.e., \chi<A . When 0<\chi<A is satisfied,
| \int_{ }^{ }\tan(A-\chi){\rm{d}}\chi=\ln[\cos(A-\chi)]+C, | (58) |
holds true, where C is an arbitrary constant. Introducing the term s = A - \chi > 0 as the difference between the final state of ellipticity and the value of A , we can express d \chi =−d s . Substituting equation (58) into equation ( 57) results in:
| -\frac{1}{2} \ln (\cos 2 \psi) =\frac{1}{2} \ln (\cos (2A-2s))-\frac{1}{2} \ln \left(\cos 2 A\right), | (59) |
| \begin{array}{l} \cos (2 \psi) \cdot \cos (2 A-2 s)=\cos (2 A), \end{array} | (60) |
| \begin{array}{l} \left[({\rm{d}}{\delta}_{\rm{CM}})^{2}-[2(-{\rm{d}} s)]^{2}\right] \cdot \cos ^{2}(2 A-2 s)=\cos ^{2} 2 A \cdot ({\rm{d}}{\delta}_{\rm{CM}})^{2}, \end{array} | (61) |
| \begin{array}{l} ({\rm{d}} {\delta}_{\rm{CM}})^{2}\left[\cos ^{2}(2 A-2 s)-\cos ^{2}(2 A)\right]=(2 {\rm{d}} s)^{2} \cdot \cos ^{2}(2 A-2 s). \end{array} | (62) |
The integrand function can be computed as follows:
| \begin{split} {\rm{d}} {\delta}_{\rm{CM}}=& \frac{2 \cos (2 A-2 s)}{\sqrt{\cos ^{2}(2 A-2 s)-\cos ^{2}(2 A)}} {\rm{d}} s\\ = & -\frac{2 \cos 2 \chi}{\sqrt{\cos ^{2} 2 \chi- {\rm{Const}} }} {\rm{d}} \chi,\quad(\psi<0). \end{split} | (63) |
The definite integral is
| {\delta}_{\rm{CM}}=-\arctan \left.\left(\frac{\mid\sin 2 \chi\mid}{\sqrt{\cos ^{2} 2 \chi- {\rm{Const}} }}\right)\right|_{\chi_{0}}^\chi,\quad(\psi<0). | (64) |
According to equations (54) and (63), the phase space trajectories of the integral function family for different values of Const are illustrated in figure 20, where the ordinate represents the dimensionless quantity “Integrand”. The vertical dashed line represents the inflection point at \psi=0 , denoted as \chi=A . Following the assumptions stated earlier, when \chi>0 , \psi monotonically decreases, resulting in clockwise phase space trajectories.
Based on the aforementioned analysis, the following conclusions can be inferred:
(1) The area enclosed by the “Integrand” curve and the Y -axis ( y=0 ) represents the definite integral of the CM effect cumulating along the trajectory. This area has a maximum range of {\text{π}} and is computed as the definite integral with the limits of \chi ranging from -A to A .
(2) We will now explore certain scenarios with distinct initial values of Const.
(i) In the process of solving the system of differential equations, Const \neq0 was employed. Specifically, the values of (\chi_0,\; \psi_0) are set such that Const = 0, resulting in a simplification of equation (55) as follows:
| \begin{array}{l} {\delta}_{\rm{CM}}=2 \chi. \end{array} | (65) |
Based on figure 20, it is evident that in this particular case, the domain of \chi exhibits the widest range.
(ii) In the scenario where Const = 1, the domain of ( \chi_0 , \psi_0 ) is restricted to a single isolated point at (0, 0), indicating that the incident laser is solely a horizontally polarized light. This isolated point, illustrated in figure 19, corresponds to point A coinciding with \chi=0 , which cannot be depicted in figure 20. Consequently, this confirms the inference that measuring X-mode or O-mode alone is insufficient.
(3) When Const \neq1 , the CM effect can be explicitly expressed in terms of the initial and final states of elliptical parameters. Notably, assuming the FR effect can be disregarded, this decoupling occurs independently of the final state solution for \psi .
Now, we consider the relationship between non-uniform power emission superposition modulation and ellipticity measurement from the perspective of Poincaré sphere. The simultaneous combination of the definition equation (48) for Const and the definition equation (8) for Stokes parameters yields,
| \begin{array}{l} {\rm{Const}}\equiv s_1^{2},\quad{\rm{as}} \quad z=0. \end{array} | (66) |
This indicates that both homodyne measurement and heterodyne measurement of orthogonal light with arbitrary power ratios share a common characteristic of s_1(0) remaining constant over time. This can be derived from equation (9) [20].
| \begin{array}{l} {\delta}_{\rm{CM}}= \left.\arctan\dfrac{s_{3}}{\sqrt{-s_{1}^{2}(0)+1-s_{3}^{2}}}\right|_{\chi_{0}}^\chi\\ \;\;\;\;\;\;\;\;= \left.\arctan\dfrac{s_{3}}{s_{2}}\right|_{\chi_{0}}^\chi. \end{array} | (67) |
When the modulation process crosses the position of s_2=0 , the monotonicity of \chi changes, which corresponds to the singularities mentioned in section 4.3. If s_{1}^{2}=0 , the modulation process is represented by the purple dashed line in figure 15, and the value range of the measured parameter s_2 is maximized. This implies the highest resolution for equal-power transmission of orthogonal light. If s_{1}^{2}=1 , the modulation process degenerates into an isolated point H as shown in figure 15, where s_2\equiv0 , meaning that there is no vertically polarized light emitted, making it impossible to measure the CM effect.
In order to simulate phase shift measurements under higher parameter conditions, a larger Cotton-Mouton (CM) effect than that of existing tokamak devices is considered. By comparing the definition equation (5), we verify the accuracy of the nonlinear calculation formula equation (55) for the CM phase shift {\delta}_{\rm{CM}} in ellipticity measurements and demonstrate its superiority over the previous linear approximation formula equation (65).
Figure 21 presents the results for the same shot (shot \# 98958) at the same moment (21 s), where the toroidal magnetic field B_{\rm{t}} is multiplied by \sqrt{3} , effectively amplifying the CM effect by a factor of 3. The vertical axis represents the phase shift, and the horizontal axis represents the incident position x_0 . Different incident elliptical polarization conditions, referred to as State 1, State 2, and State 3, are considered to simulate a more general initial s_1 . State 1 corresponds to the initial polarization parameters ( \chi_0 , \psi_0 ) of \left(0,\; \dfrac{{\text{π}}}{4}\right) , State 2 corresponds to \left(0,\; \dfrac{{\text{π}}}{5}\right) , and State 3 corresponds to \left(\dfrac{{\text{π}}}{20}\right. , \left.\dfrac{{\text{π}}}{5}\right) . In State 1, State 2, and State 3, the values of s_1 are 0, 0.309, and 0.294, respectively.
The error is defined as
| \begin{array}{l} {\text{Δ}}\phi=\dfrac{{\delta}_{\rm{CM}}-\phi}{{\delta}_{\rm{CM}}}\cdot100\%, \end{array} | (68) |
where \phi represents the phase shift.
When the polarization parameters ( \chi_0 , \psi_0 ) are (0, \dfrac{{\text{π}}}{4} ), which corresponds to s_1=0 , the error of the approximation formula is almost zero, whereas for other cases, the error becomes more pronounced. The error reaches 7.32 \% for State 2 and 11.55 \% for State 3. The computational equation is perfectly fit with the phase shift calculation of the defined equation in either incidence state, and the error in the numerical calculation is small, about 0.5 \% , which is basically negligible.
In summary, the three subplots in figure 21 correspond to the scenarios in equation (18), where the plasma Mueller matrix {\boldsymbol{M}}(z) is the same, but the initial Stokes parameter {\boldsymbol{s}}(0) is different. Since the plasma state remains the same throughout the passage, if the nonlinear computational model given by equation (55) is correct, the resulting {\delta}_{\rm{CM}} should be consistent with the computed results obtained from equation (5). The nonlinear equation makes polarization measurements without having to reduce the wavelength and sacrifice resolution for the applicability of the linear model, ensuring the measuring accuracy of large CM effect.
Next, we will discuss the second case mentioned in section 4.3, which corresponds to the situation when \psi crosses the zero point, i.e., across p _{2} and p _{4} (as shown in figure 20). In this case, the change in \chi does not exhibit monotonicity, resulting in the integrand function of {\rm{d}} {\delta}_{\rm{CM}} first being greater than 0 and then becoming less than 0 after crossing \chi=A . This indicates that {\delta}_{\rm{CM}} is no longer a single-valued function with respect to \chi . Therefore, correction is needed.
Figure 22 has the same plasma discharge parameters and toroidal magnetic field settings as figure 21. The only difference lies in the initial polarization parameters ( \chi_0 , \psi_0 ) set as \left(\dfrac{{\text{π}}}{12}\right. , \left. \dfrac{{\text{π}}}{12} \right). The code demonstrates that, when the CM effect is sufficiently strong, it has the ability to reverse the handedness of a wave that is initially launched with circular polarization. A noticeable alignment between the corrected results and the definition is evident, leading to a reduction in error from 48.34 \% (when x_0=-0.225 \enspace {\rm{m}} ) to approximately 0.5 \% . When x_0 > 0 , the sign consistency of \psi before and after passing through the plasma ensures that it does not enter the domain of equation (64), thus no correction is needed.
Although the nonlinear expression is precise, its limitation lies in the complex square root operation required for signal demodulation. Moreover, ensuring the consistency of the sign of \psi during the measurement process is challenging, and neglecting the Faraday rotation (FR) effect will lead to a significant increase in the calculation error of \chi . This makes it difficult to use as a diagnostic tool for real-time feedback of electron density during operation and control of fusion reactors. In the balance between accuracy and real-time capability, practical plasma discharges often do not adopt ellipsometry measurements.
This study combines theoretical derivation and simulations to systematically investigate the feasibility of measuring electron density using the Cotton-Mouton (CM) effect on EAST. For laser wavelength of 432.6 \mu m, fringe jumps due to phase shift over 2 {\text{π}} can be avoided to provide reliable density measurement for machine operation. The study clarifies that elliptical modulation enables the direct measurement of the phase variation of the second component s_2 of the Stokes parameters, providing information about {\delta}_{\rm{CM}}=C_{\rm{CM}} \lambda^{3} \int B_{\perp}^{2} n_{{\rm{e}}} {\rm{d}}l . The measurement principle for heterodyne detection in the case of small CM effects is presented, along with the design of an optical setup. In a laboratory environment, the power of the two orthogonal laser beams received by the detector may not be strictly equal. It is shown that non-equal power radiation only affects the DC offset of the light intensity, and does not affect the phase variation of measured signals. It is further confirmed that the higher-order term errors introduced by the Faraday rotation effect on CM effect are small (about 1 \% ) for current parameters of EAST. However, in the case of large CM effects ( {\delta}_{\rm{CM}}\sim{\cal{O}}\left(1\right) ), the higher-order terms of the Mueller matrix will introduce significant errors.
Current plasma density diagnostics based on polarimeter interferometers consider the ellipticity parameters as linear functions of phase shifts. As future fusion devices improve in their parameters, this approach will introduce increasingly large errors. This paper reveals the coupling between the ellipticity parameters \chi and \psi , the potential errors they may introduce in plasma density diagnostics based on the CM effect. The study provides an explicit expression for the CM effect phase shift {\delta}_{\rm{CM}} as a function of \chi , decoupled from the azimuthal angle \psi , when the Faraday rotation effect can be neglected.
In summary, for EAST experiment parameters, a vertical-view Cotton-Mouton polarimeter interferometer is capable of providing reliable density measurement with excellent time resolution (a few MHz).
The following section will demonstrate the analytical expression for the first-order expansion of {s}_{2}(z) when W \sim {\cal{O}}\left(1\right) . When {\boldsymbol{s}}(z=0) is modulated with a significant CM effect, numerical integration becomes inconvenient.
In such cases, the zeroth-order Mueller matrix differential equation is introduced to facilitate the analysis [20]
| \begin{array}{l} \dfrac{{\rm{d}} {\boldsymbol{M}}^{(0)}}{{\rm{d}} z}={\boldsymbol{A}}_{0}(z) \cdot {\boldsymbol{M}}^{(0)}(z), \end{array} | (69) |
where,
| \begin{array}{l} {\boldsymbol A}_{0}(z)=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -\Omega_{1} \\ 0 & \Omega_{1} & 0 \end{array}\right). \end{array} | (70) |
Given the initial condition {\boldsymbol{M}}^{(0)}(z = 0) = {\boldsymbol{E}} , where \boldsymbol{E} denotes the identity matrix, the integration yields:
| \begin{array}{l} {\boldsymbol{M}}^{(0)}(z)=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos W_{1} & -\sin W_{1} \\ 0 & \sin W_{1} & \cos W_{1} \end{array}\right). \end{array} | (71) |
Based on the reference [17], it can be derived that the first-order terms of Mueller matrix do not affect the phase of {s}_{2}(z) .
| \begin{split} &{{\boldsymbol{M}}^{(1)}(z)}=\\&\left( \begin{array}{ccc} {0} & {\left(P_{1}^{(0)} \sin W_{1}-Q_{1}^{(0)} \cos W_{1}\right)} & {\left(P_{1}^{(0)} \cos W_{1}+Q_{1}^{(0)} \sin W_{1}\right)} \\{ Q_{1}^{(0)} }& { 0} & {0} \\ {-P_{1}^{(0)}} & {0 }& {0} \end{array} \right), \end{split} | (72) |
where,
| \begin{split} P_{1}^{(0)} =\int_{0}^{Z} \;{\rm{d}} z^{\prime}\left(\Omega_{2}^{\prime} \cos W_{1}^{\prime}-\Omega_{3}^{\prime} \sin W_{1}^{\prime}\right) \\ Q_{1}^{(0)} =\int_{0}^{Z} \;{\rm{d}} z^{\prime}\left(\Omega_{2}^{\prime} \sin W_{1}^{\prime}+\Omega_{3}^{\prime} \cos W_{1}^{\prime}\right) . \end{split} | (73) |
Similarly to equation (26), the second component s_2 of the Stokes parameters is given by
| \begin{split} {s}_{2}(z) =&Q_{1}^{(0)} \cos 2 \alpha+\cos W_{1}\sin 2 \alpha \cos \left({\text{Δ}}\omega t\right)\\&-\sin W_{1} \sin 2 \alpha\sin \left({\text{Δ}}\omega t\right) \\ =&Q_{1}^{(0)} \cos 2 \alpha+\sin 2 \alpha \cdot \cos \left[\left({\text{Δ}}\omega t\right)+W_{1}\right]. \end{split} | (74) |
Namely, for even values of n, {\boldsymbol{M}}^{(n)} has the same distribution of zeros as {\boldsymbol{M}}^{(0)} and, for odd values of n, the same as {\boldsymbol{M}}^{(1)} . It can be observed that the non-zero elements {{M}}_{22} and M_{23} of the second-order matrix M^{(2)} will affect the phase of {s}_{2}(z) [14].
Now, the indefinite integral of equation (55) is to be calculated. From equation (54),
| \begin{split} {\rm{d}} {\delta}_{\rm{CM}} =& \frac{\cos 2 \chi}{\sqrt{ \cos ^{2} 2 \chi- {\rm{Const}} }} {\rm{d}}(2 \chi) \\ =&\frac{1}{\sqrt{ 1- {\rm{Const}} -\sin ^{2}(2 \chi) }} {\rm{d}}(\sin 2 \chi). \end{split} | (75) |
Let \sin 2 \chi=s .
| \begin{array}{l} {\rm{d}} {\delta}_{\rm{CM}}=\dfrac{1}{\sqrt{ 1- {\rm{Const}} -s^{2} }} {\rm{d}} s, \end{array} | (76) |
let
| u=\frac{1}{\sqrt{ 1- {\rm{Const}} -s^{2} }}, | (77) |
solve s
| s= \pm \frac{\sqrt{(1- {\rm{Const}} ) u^{2}-1}}{u}. | (78) |
The positive sign is taken when \sin 2 \chi>0 and the negative sign is taken when \sin 2 \chi<0 .
Derivation of equation (77)
| \begin{array}{l} {\rm{d}} u=\dfrac{s}{ (- {\rm{Const}} -s^{2}+1) ^{\frac{3}{2}}} {\rm{d}} s, \end{array} | (79) |
substituting equation (79) into equation (76),
| \begin{split} {\rm{d}} {\delta}_{\rm{CM}} &=\frac{ - {\rm{Const}} -s^{2}+1 }{ s} {\rm{d}} u \\ & =\frac{u \cdot \frac{1}{u^{2}}}{ \pm \sqrt{(1- {\rm{Const}} ) u^{2}-1}} {\rm{d}} u\\ & =\frac{1}{ \pm u \sqrt{(1- {\rm{Const}} ) u^{2}-1}} {\rm{d}} u, \end{split} | (80) |
let
| \begin{array}{l} t=\sqrt{(1- {\rm{Const}} ) u^{2}-1}, \end{array} | (81) |
| \begin{array}{l} t^{2}=(1- {\rm{Const}} ) u^{2}-1, \end{array} | (82) |
solve u^{2}
| \begin{array}{l} u^{2}=\dfrac{t^{2}+1}{1- {\rm{Const}} }, \end{array} | (83) |
derivation of equation (81) yields
| {\rm{d}} t=\frac{(1- {\rm{Const}} ) u}{\sqrt{(1- {\rm{Const}} ) u^{2}-1}} {\rm{d}} u, | (84) |
substituting equation (84) into equation (80),
| \begin{split} {\rm{d}} {\delta}_{\rm{CM}} =&\frac{1}{(1- {\rm{Const}} ) u^{2}} {\rm{d}} t \\ =& \pm \frac{1}{1+t^{2}} {\rm{d}} t, \end{split} | (85) |
| \begin{split} {\delta}_{\rm{CM}} = & \pm \int \frac{1}{1+t^{2}} {\rm{d}} t \\ = & \pm \arctan t+{{C}}. \end{split} | (86) |
Also because
| \begin{split} t &=\sqrt{(1- {\rm{Const}} ) u^{2}-1} \\ & =\sqrt{\frac{1- {\rm{Const}} }{ 1- {\rm{Const}} -s^{2} }-1} \\ &=\sqrt{\frac{s^{2}}{ 1- {\rm{Const}} -s^{2} }}, \end{split} | (87) |
| \begin{array}{l}\delta_{\rm{CM}}=\pm\arctan\left(\dfrac{\sin2\chi}{\sqrt{-\rm{Const}-\sin^2(2\chi)+1}}\right)+{C}.\end{array} | (88) |
When \chi_{0},\; \psi_{0} are known, the constant C can be found. The sign of {\delta}_{\rm{CM}} depends on the positivity or negativity of \sin 2 \chi , which is taken to be positive when \sin 2 \chi>0 and positive when \sin 2 \chi<0 is taken as negative.
| [1] |
Brombin M et al 2008 Rev. Sci. Instrum. 79 10F701 doi: 10.1063/1.3001671
|
| [2] |
Liu H Q et al 2014 Rev. Sci. Instrum. 85 1939 doi: 10.1063/1.4889777
|
| [3] |
Hutchinson I H 2002 Principles of Plasma Diagnostics 2nd (Cambridge: Cambridge University Press
|
| [4] |
Akiyama T et al 2006 Rev. Sci. Instrum. 77 10 doi: 10.1063/1.2229275
|
| [5] |
Donné A J H 1995 Rev. Sci. Instrum. 66 3407 doi: 10.1063/1.1145516
|
| [6] |
Boboc A 2006 Rev. Sci. Instrum. 77 10F324 doi: 10.1063/1.2229169
|
| [7] |
Akiyama T et al 2015 Nucl. Fusion 55 093032 doi: 10.1088/0029-5515/55/9/093032
|
| [8] |
Zhang J 2013 Study of internal magnetic field via polarimetry in fusion plasmas PhD Thesis University of California, Los Angeles, USA
|
| [9] |
Donné A J H et al 2004 Rev. Sci. Instrum. 75 4694 doi: 10.1063/1.1804372
|
| [10] |
Chipman R A et al 2018 Polarized light and optical systems (CRC press
|
| [11] |
Bergerson W F et al 2012 Rev. Sci. Instrum. 83 4694 doi: 10.1063/1.4731757
|
| [12] |
Born M et al 2013 Principles of optics (Elsevier
|
| [13] |
Liu H Q et al 2013 J. Inst. 8 C11002 doi: 10.1088/1748-0221/8/11/C11002
|
| [14] |
Segre S E 2006 Plasma Phys. Control. Fusion 41 339 doi: 10.1088/0741-3335/48/3/001
|
| [15] |
Roy-Brehonnet F L et al 1997 Prog. Quant. Electr. 21 109 doi: 10.1016/S0079-6727(97)84687-3
|
| [16] |
Huard S et al 1997 Polarization of Light (Chichester: Wiley
|
| [17] |
Segre S E 1999 Plasma Phys. Control. Fusion 41 R57 doi: 10.1088/0741-3335/41/2/001
|
| [18] |
Azzam R M A et al 1979 Ellipsometry and Polarized Light (Amsterdam: North-Holland
|
| [19] |
Guenther K et al 2004 Plasma Phys. Control. Fusion 46 1423 doi: 10.1088/0741-3335/46/9/006
|
| [20] |
Segre S E 1995 Physics of Plasmas 2 2908 doi: 10.1063/1.871190
|
| [21] |
Goldstein D et al 2003 Polarized Light, Revised and Expanded (CRC Press
|
| [22] |
Imazawa R et al 2012 Plasma Phys. Control. Fusion 54 055005 doi: 10.1088/0741-3335/54/5/055005
|
| [23] |
Segre S E 2000 J. Opt. Soc. Am. A 17 95 doi: 10.1364/JOSAA.17.000095
|
| [24] |
O’Rourke J 1984 Plasma Phys. Control. Fusion 26 1139 doi: 10.1088/0741-3335/26/9/010
|
| [25] |
Orsitto F P et al 2011 Plasma Phys. Control. Fusion 53 035001 doi: 10.1088/0741-3335/53/3/035001
|
| [1] | Yuqiang ZHANG, Xingang YU, Zongbiao YE. Particle-in-cell simulations of EUV-induced hydrogen plasma in the vicinity of a reflective mirror[J]. Plasma Science and Technology, 2024, 26(8): 085503. DOI: 10.1088/2058-6272/ad48d0 |
| [2] | Yulei WANG (王雨雷), Feng YUAN (袁丰), Jian LIU (刘健). A relativistic canonical symplectic particle-in-cell method for energetic plasma analysis[J]. Plasma Science and Technology, 2020, 22(6): 65001-065001. DOI: 10.1088/2058-6272/ab770e |
| [3] | ZHANG Ya (张雅), LI Lian (李莲), JIANG Wei (姜巍), YI Lin (易林). Numerical Approach of Interactions of Proton Beams and Dense Plasmas with Quantum-Hydrodynamic/Particle-in-Cell Model[J]. Plasma Science and Technology, 2016, 18(7): 720-726. DOI: 10.1088/1009-0630/18/7/04 |
| [4] | S. CORNISH, J. KHACHAN. The Use of an Electron Microchannel as a Self-Extracting and Focusing Plasma Cathode Electron Gun[J]. Plasma Science and Technology, 2016, 18(2): 138-142. DOI: 10.1088/1009-0630/18/2/07 |
| [5] | ZHOU Weimin (周维民), GU Yuqiu (谷渝秋), WU Fengjuan (吴凤娟), ZHANG Zhimeng (张智猛), SHAN Lianqiang (单连强), CAO Leifeng (曹磊峰), ZHANG Baohan (张保汉). Focusing of Intense Laser via Parabolic Plasma Concave Surface[J]. Plasma Science and Technology, 2015, 17(12): 996-999. DOI: 10.1088/1009-0630/17/12/03 |
| [6] | ZHU Zhenni(朱珍妮), WU Zhengwei(吴征威), LI Chunhua(李春华), YANG Weihong(杨维纮). Electron Acoustic Solitary Waves in Magnetized Quantum Plasma with Relativistic Degenerated Electrons[J]. Plasma Science and Technology, 2014, 16(11): 995-999. DOI: 10.1088/1009-0630/16/11/01 |
| [7] | XIA Xiongping (夏雄平), YI Lin (易林). Relativistic Filamentation of Intense Laser Beam in Inhomogeneous Plasma[J]. Plasma Science and Technology, 2012, 14(12): 1054-1058. DOI: 10.1088/1009-0630/14/12/04 |
| [8] | JI Liangliang (吉亮亮), SHEN Baifei (沈百飞), ZHANG Xiaomei (张晓梅), WANG Wenpeng (王文鹏), YU Yahong (郁亚红), WANG Xiaofeng (王晓峰), YI Longqing (易龙卿), SHI Yin (时银), et al. Plasma Approach for Generating Ultra-Intense Single Attosecond Pulse[J]. Plasma Science and Technology, 2012, 14(10): 859-863. DOI: 10.1088/1009-0630/14/10/01 |
| [9] | WANG Zhen (王振), WANG Ninghui (王宁会), LI Tie (李铁), CAO Yong (曹勇). 3D Numerical Analysis of the Arc Plasma Behavior in a Submerged DC Electric Arc Furnace for the Production of Fused MgO[J]. Plasma Science and Technology, 2012, 14(4): 321-326. DOI: 10.1088/1009-0630/14/4/10 |
| [10] | DENG Yongfeng(邓永锋), TAN Chang(谭畅), HAN Xianwei(韩先伟), TAN Yonghua(谭永华). Numerical Simulation of the Self-Heating Effect Induced by Electron Beam Plasma in Atmosphere[J]. Plasma Science and Technology, 2012, 14(2): 89-93. DOI: 10.1088/1009-0630/14/2/01 |
| 1. | Ren, B., Zhang, T., Wu, Z. et al. Low-Temperature Oxidation of Diesel Particulate Matter Using Dielectric Barrier Discharge Plasma. Plasma Chemistry and Plasma Processing, 2024. DOI:10.1007/s11090-024-10492-6 |
Figures(22)
Supported by: Beijing Renhe Information Technology Co., Ltd. 
DownLoad:
