12th International Conf. on Mathem.& Num. Aspects of Wave Propag., Karlsruhe (Germany), July 20-24, 2015
Huygens-Fresnel’s wavefront tracing in non-uniform media
F.A. Volpe, P.-D. Létourneau, A. Zhao Dept Applied Physics and Applied Mathematics Columbia University, New York, USA
1 Motivation 1: validate unexpected EC full-wave results, and find more, but at ray tracing costs • Faster than full-wave, more informative than ray-tracing (diffraction, scattering, splitting, …) • Not replacing full-wave, but can guide it – (e.g. scan large spaces, select interesting cases) • Might replace ray tracing in real-time?
• Full-waves in EC range are rare – >106 cubic wavelength • But found unexpected effects, t.b.c.: – O couples with X at fundamental harmonic – High production of EBWs at UHR – Collisionless damping of EBWs at UHR • Concern for ITER: deposition in wrong place
• Fictitious, due to reduced wpe and wce?
2 Motivation 2: test old/new ideas too complicated for analytical treatment and too time-consuming for full-wave
• EBWs – Electromagnetic nature – Interband tunneling • oblique propagation, large Larmor radii, non-circular orbits, non-sine-waves
– Autoresonance (N||=1 but N100…) – Cherenkov radiation – Nonlinear
• Affecting ne and B in which they propagate • Mode Conversions – Must evanescent barrier be thinner than wavelength or skin depth? – Parametric decay – Chaotic trajectories – Auto-interference of coexisting O, X and B
– Self-scattering (e.g. of O off ne modulation by X or B) • Generic Wave Physics – Ponderomotive (density pump-out by ECH?) – Beam Broadening due to finite frequency bandwidth – Broadening by Scattering off “blobs”
3 Simple idea: numerically implement Huygens-Fresnel’s principle of diffraction • Huygens construction: “each element of a wavefront may be regarded as the center of a secondary disturbance which gives rise to spherical wavelets". “The position of the wavefront at any later time is the envelope of all such wavelets" [Traitė de la Lumiere, 1690] • Fresnel principle: Huygens' secondary wavelets interfere with each other. Interference pattern reveals locus of iso-phase points. • Light does not emit light, but wave from another domain is indistinguishable from wave appropriately excited at the boundary.
4 literature inspired by Huygens’ construction, e.g. in seismology, but rarely includes diffraction
5 From diffraction formulas to numerical problem
1 cos 휃 푒푖푘푟 In 3D 퐄 퐱′ = − 푖푘(1 + cos 휃) 퐄 퐱 푑푆, 4휋 푟 푟 푆 표푏푙푖푞푢푖푡푦 푛푒푎푟−푓푖푒푙푑 푑푖푝표푙푒
1 + cos 휃 푒푖푘푟 In 2D 퐄 퐱′ = 푒−푖휋 4 퐄 퐱 푑푙. 푙 2휆 푟
Given a surface S or line l, find a new set of points x’ where E has the same phase.
6 Let us discretize
푛 푚 푖푘 푒푖푘푟AB,ab,c In 3D 퐄푨푩풄 = − 1 + cos 휃AB,ab,c 퐄풂풃 Δ푆푎푏. 4휋 푟AB,ab,c 푏=1 푎=1 푚 푖푘푟Aa,c −푖휋 4 1 + cos 휃Aa,c 푒 In 2D 퐄푨풄 = 푒 퐄풂 Δ푙푎 휆 푟Aa,c 푎=1
Given a set of points 퐱풂풃 and associated 퐄풂풃, identify a set of points 퐱′푨푩풄 where all 퐄푨푩풄 have same phase. 풄 is an iteration index.
Without losing generality, will search for Re(퐄푨푩풄)=0. Can be generalized to any phase-advance by phase-shifting the sources.
7 Step 1: Discretize wavefront
• Allow for different intensities (color-coded). • Very first array can be at the antenna, phased array or last mirror, and can have complicated shape or phase-pattern. It can be placed at the mouth of a waveguide, and subsequent windows, mirrors, lenses be directly modeled in the WT, before the wavefront enters in the plasma.
8 Step 2: Define l/2 long search-interval for point forming next wavefront
• E.g. by displacing all points xa by same amounts Dx and Dx+l/2. • Consistent with each other (find points belonging to same WF). • l/2 for solutions to be unique • Avoid ∆푥 ≪ 휆 to discard dipole correction.
9 Step 3: repeatedly evaluate diffraction formula in various points until finding Re(E)=0
• For each search-interval do – While |Re 퐄 | > ε do
• Try improved x’ac , from bisection.
• Evaluate E in x’ac , by summing contributions from sources xa. Sums can be restricted to a in the neighborhood of A.
10 Step 4: loci of Re(E)=0 define new wavefront. Use Im(E) to ascribe amplitudes. Start over.
• Instead of Re(E)=0, we could have searched for arctan[Im(E)/Re(E)]-f=0. • Wavefront-to-wavefront Df is regulated by: – Choice for search intervals coarse adjustment, modulo p – Initial phases of sources (E is complex) fine adjustment
11 Planar waves in uniform medium
30
20
l
/ y
10
0 0 20 40 60 80 100 x/l
12 Diffraction around an obstacle
30
20
l
/ y
10
0 0 10 20 30 40 Polar grid, rather than Cartesian,
to be used for zero search in this region. 13 Refraction in slowly varying medium
20
2.0 l
/ 10 y 1.0
0 0.0 0 10 20 30 x/l N
14 Discontinuities require two Green’s functions
푒푖푘2|퐱2,푗−퐱푏,푖푗| 푒푖푘1|퐱푏,푖푗−퐱1,푖|
|퐱2,푗 − 퐱푏,푖푗| |퐱푏,푖푗 − 퐱1,푖|
For every pair of emitter x1,i and observer x2,j,
boundary point xb,ij is uniquely determined by Snell’s law.
15 Snell’s law
20
2.0 l
/ 10 y 1.0
0 0.0 0 10 20 30 N Corrugations, due to lack of ghost cells. Jumps due to misplaced search-interval. Both are amplified by discontinuity.
Need to add more ghost cells and better define search-intervals. 16 Modeling of wavefront passing through lens is encouraging for gyrotron beam crossing “blobs”
17 Gaussian beam, intensity
0 60 10
10-2 40
10-4
l
/ y
20 10-6
10-8 0 0 20 40 60 x/l
18 Gaussian beam, electric field
0 60 10
10-2 40
10-4
l
/ y
20 10-6
10-8 0 0 20 40 60 x/l
19 Strongly focused Gaussian beam, intensity
0 60 10
10-2 40
10-4
l
/ y
20 10-6
10-8 0 0 20 40 60 Phase-jump issues expected to be solved by improved (adaptive?) definition of interval for zero search by bisection method.
20 Strongly focused Gaussian beam, E
0 60 10
10-2 40
10-4
l
/ y
20 10-6
10-8 0 0 20 40 60 x/l
21 Estimated CPU time
• Reconstructing with 휆 /100 precision a wavefront at a distance 휆 from the previous one does not require 100 steps. Only log2100 = 6-7 bisections. • Wavefront distance not constrained by 휆. Only by non-uniformity scale 퐿푛.
• Estimated number of “point-to-point” operations (E contribution of single emitter to single observer) is the product of:
푁푊푇=푁휆⊥/푓 wavefronts 푁퐸 points in new wavefront 푁퐸 points in old wavefront contributing to each point in new wavefront
푁푏푖푠=log2(푓푁푝푝푤) “trial-points” to localize a point on new wavefront where the wavefront distance is a fraction 푓 of l. • For 푓 ≪ 1, wavefront tracing is not much faster than full-wave, and is less informative just use full-wave.
22 Wavefront tracing shares some advantages of Full-wave • They provide spatial information on the wave field. Reconstructing wave- fields from ray tracing requires higher orders, or “wave packet” methods, or semi-classical techniques from quantum mechanics, or combined solution of wave transport equation. • Diffraction • Interference • Nonlinear interaction, if coupled with fluid or kinetic equations • Partial mode conversions, resulting in ray splitting, (transmitted and reflected). A wavefront -locus of points of same phase- is more general, and can be a manifold comprising “forward” and “backward” waves. • Asymmetric absorption of a wavefront entering a damping region with a shallow angle, so that part of it enters the region and starts being damped sooner than the rest. • More comprehensive power balances, including inefficient mode conversions, power escaping through ports and heating of the resistive walls rather than of the plasma.
23 Wavefront tracing shares some advantages of Ray Tracing • Calculations only performed where needed: , rather than in entire computational domain. • Explicit in time. • Can advance the wavefronts by spatial steps of order l. In principle ray tracing (WKB-approximated) should adopt steps ≫ 휆, although in practice it is often used with steps ≈ 휆.
• Advantage over ray tracing: a ray is an abstraction defined as the normal to the wavefront. Hence, a ray is non-ambiguously defined only for plane waves. In fact, the RT treatment assumes locally plane wavefronts and locally straight rays, where ‘local’ means on the scale of l. In brief, rays are local abstractions, only possible in mildly inhomogeneous media; wavefronts, defined as loci of points with the same phase, are the real observables that can always be defined.
24 Modifying the integrand can include anisotropy, absorption, mode conversions… k original Huygens-Fresnel k=k(x). non-uniformity k=k(x, x’-x) anisotropy k=k(x, x’-x) group-phase velocity decoupling k=k(x, x’-x, mode) polarization, birefringence 퐤 ∈ ℂ3 absorption
′ ′ 푒푖푘|퐱 −퐱| 푒푖퐤⋅(퐱 −퐱) Huygens-Fresnel Airy caustics Airy, Bessel evanescence Airy, parabolic cyl.func. Mode conversions
25 Summary and future work
• Numerical deployment of Huygens’ construction [Traitė de la Lumiere, 1690] is surprisingly rare. • Some exceptions in seismic wave research, and e.m. wave outreach/teaching. • Started exploring applicability to plasmas, including diffraction. • Very simple cases successfully modeled in 2D. – Diffraction around obstacle. – Refraction in slowly varying medium. – Gaussian beams. – Discontinuities – Lenses • Minor issues of phase-jumps and corrugations currently being fixed. • Formulated extension to 3D anisotropic media such as magnetized plasmas. • Future work: (1) detailed study of cpu-time scaling, (2) anisotropy, (3) comparison with ray tracing, full-wave and PIC, (4) extension to 3D.
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