Plasma propagation in single-particle packed dielectric barrier discharges: joint effects of particle shape and discharge gap

1.   Introduction

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 $\mu$m is chosen, effectively compensating for the disadvantage of fringe jumps when the phase difference exceeds 2${\text{π}}$ 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 $\mu$m 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 $\chi$ for phase difference not much less than one. Section 5 provides a brief summary.

2.   The propagation of electromagnetic waves on EAST tokamak

2.1   EAST plasma

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 $R_0\in(1640 \enspace{\rm{mm}},\enspace2000\enspace{\rm{mm}})$.

Figure  1.  Diagram of the definition of the coordinate system. Blue line represents the flux surface. Red dashed line represents the last closed magnetic surface.

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$x_0$ is defined as

The vertical channel passing through the magnetic axis is defined as $x_0 = 0$, which corresponds to the range of $x_0\in(-270\enspace{\rm{mm}},\enspace90\enspace{\rm{mm}})$ in the minor radial direction. Electron density ($n_{\rm{e}}$) and toroidal magnetic field distribution ($B_{\rm{t}}$) for EAST shots $\#$71320 at 3.3 s and $\#$98958 at 21 s are shown in figures 2 and 3, respectively.

Figure  2.  The typical distribution of $n_{\rm{e}}$ in the poloidal section for EAST shots $\#$71320 and $\#$98958, where $\rho$ represents the normalized magnetic flux.

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Figure  3.  The typical distribution of $B_{\rm{t}}$ in the poloidal section for EAST shots (a) $\#$71320 and (b) $\#$98958. Display only the portion of normalized magnetic flux that is less than or equal to 1. The red arrow indicates the approximate location of viewing chords.

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2.2   Elliptical polarization of laser beam [8]

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.

Figure  4.  Schematic representation of $\alpha$, $\delta$, $\chi$ and $\psi$ in the polarization ellipse.

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In the electric field coordinate system ${X}$-${Y}$, $\alpha$ is the arctangent of the ratio of electric field amplitude in the $Y$ direction to that in the $X$ direction.

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 $X'$-$Y'$, $\chi$ is the ellipticity, referred to as the arctangent of the ratio of the short axis of the ellipse to the long axis

$\psi$ is the azimuthal angle of the polarization plane, referred to as the angle between the long axis of the ellipse ${X}^\prime$ and the $X$-axis in the electric field coordinate system. $\psi_{\rm{FR}}$ is the change in the angle of the polarization plane due to Faraday effect

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.

Figure  5.  A schematic representation of polarized light entering a birefringent medium ‒ a linear half-wave plate, depicting the phase delay. Three states of polarized light from top to bottom are the the O-mode, X-mode, and their combined light.

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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].

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${\text{π}}$.

Figure  6.  Phase shift of CM effect in the $X$-direction channels, where the horizontal axis represents the position of the incident laser. The solid line represents shot $\#$98958, and the dashed line represents shot $\#$71320.

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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.

Figure  7.  The relationship between the phase shift of the CM effect along the central chord and the wavelength of far-infrared laser. The black dot represents the selected wavelength of 432.6 $\mu$m.

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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 ${n}_{2}$:

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  8.  The relationship between phase shift and incident position under different laser wavelengths (shot $\#$71320).

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Figure  9.  The relationship between the incident position and the lateral offset under different laser wavelengths (shot $\#$71320).

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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.

Figure  10.  CM effect transformed with time, where the string integral of the density is given by a solid source interferometer arranged in the centre channel. The $B_{\rm{t}}$ approximation is considered equal along the course and is regarded as the magnetic field at the magnetic axis. The wavelength is taken to be 432.6 $\mu {\rm{m}}$ (shot 98958)

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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.

Figure  11.  The schematic diagram depicting the variation of the polarized ellipse along the central chord, where the abscissa represents $E_{x}$ and the ordinate represents $E_{y}$.

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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.

3.   Evolution of polarization state based on Stokes equation

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.

3.1   Description of the polarization state of electromagnetic waves based on the Poincaré sphere

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$\psi$.

Figure  12.  Stokes parameters map the polarization state to a point on the Poincaré sphere, which is a unit sphere in an abstract 3-D space.

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The normalized Stokes parameters can be defined as [15]

where,

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].

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.

Figure  13.  The variation of the Stokes parameters along the central chord when the central chord of the light beam varies along the negative direction of the $Z$-axis.

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3.2   Derivation of elliptical modulation based on Jones vectors

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],

The electric field resulting from the superposition of the X-mode and O-mode is

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

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

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.

Figure  14.  Addition of equal horizontally and vertically polarized beams with a 20$\%$ frequency difference yields a time-varying polarization state [10].

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From equation (8), the Stokes parameters of the synthesized light resulting from the superposition of horizontal and vertical polarization modulation can be obtained as

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:

Figure  15.  Schematic diagram of elliptical polarization modulation trajectory under different transmission ratio conditions by superimposing X-mode and O-mode.

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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.

3.3   Approximate analytic solutions of $s_2$

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]:

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].

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) $W_{1,2,3} \ll 1$

According to equation (18), ${\boldsymbol{A}}$ satisfies the following equation:

where now

where the components of the matrices ${{M}}^{(1)}, {{M}}^{(2)}, {{M}}^{(3)}\cdots$ are given by

where again repeated indices imply summation [20].

When the plasma effect is small, the Mueller matrix can be expanded to second order,

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],

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:

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 $s_2$.

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.

Figure  16.  Comparative analysis of the time modulation of $s_2$ (equation (27)) and the simulation results based on definition (equation (10)). The comparative analysis of the simulated results for time modulation of $s_2$. The left plot represents equal power ratio transmission, while the right plot represents non-equal power transmission.

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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 $W$

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,

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]

where,

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.

Figure  17.  In the case of a significant CM effect ($\times$3), comparative analysis of the time modulation of $s_2$ (equation (27)) and the simulation results based on definition (equation (10)). The left plot represents equal power ratio transmission, while the right plot represents non-equal power transmission.

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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.

3.4   Determination of Stokes vector

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.

Figure  18.  Schematics of the optical arrangement of the CM polarimeter.

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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].

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 $s_2$,

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.

4.   Measurement of Cotton-Mouton effect based on ellipticity $\chi$

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.

4.1   The system of differential equations involving $\psi$ and $\chi$

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]

where,

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].

4.2   The expression of $\psi(\chi)$

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

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].

equation (40) is divided by equation (39) to obtain

Because the indefinite integral

where $x$ is an arbitrary function and ${{ C}}$ is a constant.

Therefore, integrating equation (42) to obtain,

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

Furthermore,

define Const as

therefore,

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.

Figure  19.  Considering different values of Const, the phase space trajectories of $\chi-\psi$ and the extremal points p$_{1-4}$ are depicted. The arrows indicate the direction influenced by the CM effect.

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4.3   The explicit expression after decoupling ${\delta}(\chi)$

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

substituting equation (35) into equation (50),

because

hence

According to equation (51), the integrand function can be computed as follows:

The definite integral is

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

by equation (42),

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,

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:

The integrand function can be computed as follows:

The definite integral is

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.

Figure  20.  The integrand of ${\rm{d}} {\delta}_{\rm{CM}}$ (equations (54) and 63) under different Const, where $p_{1-4}$ is the same as defined in figure 19. The abscissa of the black vertical dashed line represents the inflection point $\chi=A$. Different line types correspond to different emission power ratios. The blue solid line represents equal power emission with Const=0; the green dashed line represents Const$=\dfrac{1}{2}$; and the red dotted line represents Const$=\dfrac{\sqrt{3}}{2}$. The case of Const $=1$, which corresponds to point A coinciding with $\chi=0$, cannot be depicted in this figure.

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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:

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 $\psi$.

4.4   Commonalities with heterodyne measurements

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,

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].

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.

4.5   Modeling results and discussion of ellipticity measurement

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.

Figure  21.  Comparison of CM phase shift calculation results between nonlinear computational formula and the definition formula under different incident elliptical polarization conditions. The results calculated using equation (5) are represented by dashed lines, the results calculated using the nonlinear calculation formula equation (55) are represented by “$\times$”, and the results calculated using the previous linear approximation formula equation (65) are represented by “+”.

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The error is defined as

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.

Figure  22.  Correction of computational equation. The results calculated using equation (5) are represented by dashed lines. Due to the consistency of the plasma discharge parameters and the toroidal magnetic field between the scenarios in figure 21, the ‘Definition’ curves in the two plots are identical. The result of the calculation solely using equation (55) is represented as ‘$\times$’ and the result of the correction using both equation (55) and equation (64) is represented as ‘+’.

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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.

5.   Conclusion

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).

Appendix A

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]

where,

Given the initial condition ${\boldsymbol{M}}^{(0)}(z = 0) = {\boldsymbol{E}}$, where $\boldsymbol{E}$ denotes the identity matrix, the integration yields:

Based on the reference [17], it can be derived that the first-order terms of Mueller matrix do not affect the phase of ${s}_{2}(z)$.

where,

Similarly to equation (26), the second component $s_2$ of the Stokes parameters is given by

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].

Appendix B

Now, the indefinite integral of equation (55) is to be calculated. From equation (54),

Let $\sin 2 \chi=s .$

let

solve $s$

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)

substituting equation (79) into equation (76),

let

solve $u^{2}$

derivation of equation (81) yields

substituting equation (84) into equation (80),

Also because

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.