Propagation in toroidal vacuum device

E.3 Propagation in toroidal vacuum device

E.3.1 Magnetic configuration

Figure 13: Toroidal projection of the ray trajectory of an electromagnetic wave in vacuum.

Figure 14: Poloidal projection of the ray trajectory of an electromagnetic wave in vacuum.

In vacuum, electromagnetic waves trajectories are straight lines, which offers an interesting benchmarking case for the curvilinear coordinate system used with C3PO ray-tracing. It is enforced that at ψ = ψ_a the wave is reflected like in a mirror, so that the ray remains confined in the machine. Circular concentric poloidal magnetic flux surfaces are considered with

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(180)

and the poloidal magnetic field is considered to be created by a virtual toroidal current I_p of 1.0 MA and a parabolic safety factor profile of the form

\text{[math]}

(181)

with q_min = 1 and q_max obtained from Ampère’s law.

Since rays are straight lines, it is possible to calculate the exact rays trajectories from geometrical arguments. Since ∥∇ ρ ∥ = 1, the initial value of the wave vector is k₀ = k_ρ0+ k_θ0+ k_ϕ0 with k_θ0 = m₀a_p∕ρ₀ and k_ϕ = n₀a_p∕R₀. k₀ is expressed in cartesian coordinates using the relation k₀^cartesian = M_cc (θ0,ϕ0) ⋅ k₀^curvilinear where

\text{[math]}

(182)

The initial position at launch in the cartesian coordinates -- -- --
(x0, y0,z0) normalized to a_p is given by the relations

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(183)

where (ρ0,θ0,ϕ0) are initial curvilinear coordinates. Since the ray is a straight line, its position is given by the parametric equation

\text{[math]}

(184)

where s (l) is the ray length starting from s (l = 0) = 0 at (x0,y0,z0) .Once ( )
x(l),y(l),z(l) known, it is possible to determine back (ρ,θ, ϕ) by inverting (??). From the determination of (ρ,θ,ϕ) , it is possible to evaluate k_l^curvilinearat all positions, using k_l^cartesian = k₀^cartesian and

\text{[math]}

(185)

Then m = k_θρ∕a_p and n = k_ϕR∕a_p

In order to keep the ray inside the toroidal chamber, a specular reflection is enforced at ψ_a.If the condition ρ > 1 is encountered, then the new direction of the ray k₀^cartesian of the i + 1 ray segment is deduced from k₀^cartesian according to the conditions

where ( )
^ρ,^θ, ^ϕ are used at the increment (l - 1) . It corresponds to the relation

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(189)

with

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(190)

and

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(191)

E.3.2 Wave initial conditions

In the example here given, the ray is launched from the position

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(192)

with spectral properties

\text{[math]}

(193)

so that the wave can propagate into vacuum region.

E.3.3 Results

As expected, ray trajectory is made of pieces of straight lines, which can be easily shown from the toroidal projection in Fig 13. Because of the toroidal topology, the poloidal projection in Fig. 14 does not give straight lines. This result highlights the fact that it is very difficult to have a correct picture of the ray path from this type of representation. Comparison with simple geometrical optics is excellent.

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