The 17th Electromagnetic and Light Scattering Conference, College Station, TX, USA, 06.03.2018
Advanced topics related to the volume integral equation formulation of electromagnetic scattering Maxim A. Yurkin1,2 and Michael I. Mishchenko3 1
Voevodsky Institute of Chemical Kinetics and Combustion, Novosibirsk, Russia
2
Novosibirsk State University, Novosibirsk, Russia
3
NASA Goddard Institute for Space Studies, New York, USA
This and other presentations are available at https://www.researchgate.net/profile/Maxim_Yurkin/contributions
Volume-integral equation (VIE)
The frequency-domain VIE for the electric field inside the scattering object
Known for > 50 years, and intended to be equivalent to the differential Maxwell equations + boundary conditions.
Used as basis for a number of “numerically exact” computational methods, e.g. the DDA.
The latter has been successfully used for all classes of scatterers, including those with sharp edges and internal interfaces.
Why would we further study VIE? 2
Remaining issues
Literature is grouped around two extremes:
accessible derivations with all complex issues swept under the rug (strong singularity of the integral kernel) Mishchenko, Electromagnetic Scattering by Particles and Particle Groups: An Introduction (2014). Sancer et al. IEEE Trans. Antennas Propag. 54, 1488–1495 (2006).
mathematically rigorous treatises, based on simplified assumptions on particle shape and constitutive parameters (avoiding sharp edges/vertices and internal interfaces) Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (1969). Kirsch, Inv. Probl. Imag. 1, 159–179 (2007). Costabel et al. J. Comput. Appl. Math. 234, 1817–1825 (2010).
Fragmentary conditions for the existence and uniqueness of the VIE solution, especially for an absorbing host medium. ⇒ Gray zones for VIE applications 3
The goal
Accessible and general derivation of the VIE
from the differential Maxwell equations, transmission boundary conditions, and locally-finite-energy condition. explicit treatment of the kernel singularity. applies to a group of multi-layered particles with piecewise smooth (intersecting) boundaries and internal interfaces. passive host medium.
Conjecture a general sufficient conditions for the existence and uniqueness of the solution.
Alternative derivation by means of a continuous transformation of the everywhere smooth refractive-index function into a discontinuous one. 4
Maxwell equations
𝜀2 𝐫
Isotropic non-magnetic materials
Passive host medium (0 ≤ arg 𝜀1 < 𝜋)
Time-harmonic Maxwell equations
∇×𝐄 ∇×𝐇 ∇×𝐄 ∇×𝐇
𝐫 𝐫 𝐫 𝐫
= i𝜔𝜇0 𝐇 𝐫 𝐫 ∈ 𝑉ext = −i𝜔𝜀1 𝐄 𝐫 = i𝜔𝜇0 𝐇 𝐫 𝐫 ∈ 𝑉int = −i𝜔𝜀2 𝐫 𝐄 𝐫
Boundary conditions 𝐧 × 𝐄1 𝐫 − 𝐄2 𝐫 = 0 𝐧 × 𝐇1 𝐫 − 𝐇2 𝐫 = 0
𝜀1
∇ × ∇ × 𝐄 𝐫 − 𝑘12 𝐄 𝐫 = 𝐣(𝐫), 𝐫 ∈ ℝ3 \𝑆int 𝐣 𝐫 ≝ 𝑘12 [𝑚2 𝐫 − 1]𝐄 𝐫
𝑘1 = 𝜔 𝜀1 𝜇0 , 𝑚 𝐫 = 𝜀 𝐫 , 1, 𝐫 ∈ 𝑉ext 𝜀 𝐫 ≝ 𝜀2 𝐫 𝜀1 , 𝐫 ∈ 𝑉int
𝐫 ∈ 𝑆int 5
Scattering problem
Incident (given) and scattered (unknown) field 𝐄 𝐫 = 𝐄inc 𝐫 + 𝐄sca 𝐫 Silver–Müller radiation condition 𝐫 × ∇ × 𝐄sca 𝐫 + i𝑘1 𝑟𝐄sca 𝐫
𝑟→∞
0
or a weaker condition (𝐿2 -convergence)
1 lim ∆→∞ ∆2
𝑆∆
d2 𝐫 𝐫 × ∇ × 𝐄sca 𝐫 + i𝑘1 𝑟𝐄sca 𝐫
2
=0
Wilcox, Commun. Pure Appl. Math. 9, 115–134 (1956)
6
Starting the VIE derivation
Dyadic Green’s function exp i𝑘1 𝑅 𝐑⊗𝐑 i𝑘1 𝑅 − 1 𝐑⊗𝐑 𝐆 𝐫, 𝐫′ ≝ 𝐈− + 𝐈−3 4𝜋𝑅 𝑅2 𝑅2 𝑘12 𝑅2 ∇ × ∇ × 𝐆 𝐫, 𝐫′ − 𝑘12 𝐆 𝐫, 𝐫 ′ = 𝐈𝛿 𝐫 − 𝐫 ′ 𝐫 × ∇ × 𝐆 𝐫, 𝟎 + i𝑘1 𝑟𝐆 𝐫, 𝟎
𝑟→∞
𝒪 1 𝑟
Maxwell equations imply ∇ × ∇ × 𝐄sca 𝐫 ⋅ 𝐆 𝐫, 𝐫 ′ − 𝐄sca 𝐫 ⋅ ∇ × ∇ × 𝐆 𝐫, 𝐫 ′ = 𝐣 𝐫 ⋅ 𝐆 𝐫, 𝐫 ′ − 𝐄sca 𝐫 𝛿 𝐫 − 𝐫 ′ Dyadic Green’s theorem (simple derivations wrongly use it for 𝑉 = 𝑉int )
d3 𝐫 ∇ × ∇ × 𝐚 ⋅ 𝐀 − 𝐚 ⋅ ∇ × ∇ × 𝐀 𝑉
d2 𝐫 𝐧 × 𝐚 ⋅ ∇ × 𝐀 + 𝐧 × ∇ × 𝐚 ⋅ 𝐀
= 𝜕𝑉
7
Excluding the strong singularity (𝐫 ′ ∈ 𝑉int )
Apply dyadic Green’s theorem for 𝑉 = 𝑉int \𝑉𝛿 and let 𝛿 → 0 lim
𝛿→0 𝑉 \𝑉 int 𝛿
=
d3 𝐫 𝐣 𝐫 ⋅ 𝐆 𝐫, 𝐫 ′ − 𝐄sca 𝐫 𝛿 𝐫 − 𝐫 ′
− lim 𝑆int
𝛿→0 𝑆 𝛿
d2 𝐫
𝐧 × 𝐄sca 𝐫
⋅ ∇ × 𝐆 𝐫, 𝐫 ′
+ 𝐧 × ∇ × 𝐄sca 𝐫
⋅ 𝐆 𝐫, 𝐫 ′
𝐗 𝐫,𝐫′
𝐗 𝐫, 𝐫 ′ is continuous when 𝐫 crosses 𝑆 (due to boundary conditions)
lim
∆→∞ 𝑆∆
Singularity integral
d2 𝐫 𝐗 𝐫, 𝟎 = 𝟎 (due to radiation condition)
lim
𝛿→0 𝑆 𝛿
d2 𝐫 𝐗 𝐫, 𝐫 ′ = −𝐄sca 𝐫 −
𝐋⋅𝐣 𝐫 , 2 𝑘1
d2 𝐫
𝐋≝ 𝑆𝛿
𝐧⊗𝐑 4𝜋𝑅3
A sphere or a cube centered around 𝐫 ′ ⇒ self-term 𝐋 = 𝐈 3 8
Multi-layered multi-particle scatterer
Volume partition and indicator function
𝑉int =
𝑗
𝑉𝑗 , 𝑆int =
𝜒𝑉ext 𝐫 +
1, 𝐫 ∈ 𝑉, 0, otherwise
𝜒𝑉𝑗 𝐫 = 1 − 𝜒𝑆int 𝐫
Interchange 𝐫 and 𝐫 ′
lim
𝛿→0 𝑉 \𝑉 𝑗 𝛿
𝑗
𝑖
𝑆𝑖 , 𝜒𝑉 𝐫 ≝
3 ′
′
′
d 𝐫 𝐣 𝐫 ⋅ 𝐆 𝐫 , 𝐫 − 𝜒𝑉𝑗
𝐋⋅𝐣 𝐫 𝐫 + 𝐄sca 𝐫 𝑘12
d2 𝐫 ′ 𝐗 𝐫 ′ , 𝐫
= 𝜕𝑉𝑗
Add all together with that for 𝑉ext 𝐄sca
𝐋⋅𝐣 𝐫 𝐫 = lim d 𝐫 𝐆 𝐫, 𝐫 ⋅ 𝐣 𝐫 − 2 𝛿→0 𝑉 \𝑉 𝑘 1 int 𝛿 3 ′
𝐄 𝐫 = 𝐄inc 𝐫 + 𝑘12 lim
𝛿→0 𝑉 \𝑉 int 𝛿
′
′
d3 𝐫 ′ [𝑚2 𝐫 ′ − 1]𝐆 𝐫, 𝐫 ′ ⋅ 𝐄 𝐫 ′ − [𝑚2 𝐫 − 1]𝐋 ⋅ 𝐄 𝐫 9
Sharp edges and vertices
Additional assumptions to keep the solution unique
Charges and currents localized at shape singularities are zero (they do not radiate any energy) lim
d2 𝐫 𝐧 ⋅ ℜ 𝐄 𝐫 × 𝐇 𝐫
∗
= 0,
𝛿→0 𝑆 𝛿 Jones, The Theory of Electromagnetism (1964)
Follows from (equivalent to?) locally finite energy of the electromagnetic field ⇔ 𝐄 𝐫 and 𝐇 𝐫 are locally square-integrable
This is natural for VIE (and implicitly assumed). First reason, why VIE is superior. 10
Excluding shape singularities
Exclusion volume 𝑉s ≝
𝛿 𝑖 𝑉𝑖
,
regularized volumes: 𝑉𝑖′ ≝ 𝑉𝑖 \𝑉s ,
′ 𝑉ext ≝ 𝑉ext \𝑉s
The same derivation leads to 𝐄sca 𝐫 = lim
𝛿→0 𝑉 \ 𝑉 ∪𝑉 int 𝛿 s
d3 𝐫 ′ 𝐆 𝐫, 𝐫 ′ ⋅ 𝐣 𝐫 ′ −
𝐋⋅𝐣 𝐫 + 𝑘12
𝑖 𝑆𝑖𝛿
d2 𝐫 ′ 𝐗 𝐫 ′ , 𝐫
The remaining condition follows from local square integrability of the fields (through the analysis of singularity orders) lim
𝛿→0 𝑆 𝛿 𝑖
d2 𝐫 ′ 𝐗 𝐫 ′ , 𝐫 = 0 11
Back to differential formulation
Maxwell equations using Reynolds transport theorem d3 𝐫 ′ 𝐚 𝐫, 𝐫 ′ =
∇× 𝑉int \𝑉𝛿 (𝐫)
𝑉int \𝑉𝛿 (𝐫)
d2 𝐫 ′ 𝐧 × 𝐚 𝐫, 𝐫 ′
− 𝑆𝛿 (𝐫)
Boundary conditions through replacing 𝑉𝛿 by finite 𝑉0 , consisting of two prisms ℎ × 𝑑 × 𝑑 (ℎ ≪ 𝑑)
𝐄sca 𝐫 = −
d3 𝐫 ′ ∇ × 𝐚 𝐫, 𝐫 ′
d3 𝐫 ′ 𝐆 𝐫, 𝐫 ′ ⋅ 𝐣 𝐫 ′ + 𝑉int \𝑉0
𝐋 𝜕𝑉0 , 𝐫 ⋅ 𝐣 𝐫 , 𝑘12
𝑉0
d2 𝐫′
𝐋(𝑆, 𝐫) ≝ 𝑆
for a single prism 𝐋 𝜕𝑉, 𝐫 =
d3 𝐫 ′ 𝐆 𝐫, 𝐫 ′ − 𝐆st 𝐫, 𝐫 ′
⋅ 𝐣 𝐫′
𝐧 ⊗ 𝐑′ 4𝜋𝑅3
𝐧 ⊗ 𝐧, 𝑥 ∈ 𝑉 0, 𝑥∉𝑉
⇒ 𝐋 𝜕𝑉0 , 𝐫 = 𝐧 ⊗ 𝐧 𝐄1 𝐫 − 𝐄2 𝐫 = 𝐧 𝐧 ⋅ 𝐣2 𝐫 − 𝐣1 𝐫
, 𝐫 ∈ 𝑆int , 12
Uniqueness and existence
For non-absorbing host medium, the sufficient condition is
∃𝑐0 > 0: ∀𝐫 ∈ 𝑉int , 0 ≤ arg 𝜀 𝐫
< 𝜋 − 𝑐0 and 𝜀 𝐫
> 𝑐0
For absorbing host medium – only fragmentary results
ℑ 𝜀2 𝐫
> 0 or ( ℜ 𝜀 𝐫
> 𝑐0 & ℑ 𝜀 𝐫
≥0)
Analysis of matrix spectrum suggests ℑ 𝜀 𝐫
> 𝑐0
Conjecture:
Kirsch, Inv. Probl. Imag. 1, 159–179 (2007). Cessenat, Mathematical Methods in
Electromagnetism: Linear Theory and Applications (1996). Budko & Samokhin, SIAM J. Sci. Comput. 28, 682–700 (2006).
13
Uniqueness and existence
The matrix spectrum is related to Conv 𝜀 ℝ3 𝜀 ℝ3 are also relevant
and the limiting points of
Conjecture: Conv 𝜀2 𝑉int ⊂ 𝑍2 ⇕ Conv 𝜀 ℝ3 \𝑆int ⊂ 𝑍
Examples:
14
Continuity with respect to 𝑚(𝐫)
Everywhere smooth 𝑚𝑛 𝐫 , such that lim 𝓂𝑛 = 𝓂 (discontinuous) 𝑛→∞
For 𝓂𝑛 the VIE 𝒜(𝓂𝑛 )ℰ = ℰinc is easy to derive and well-behaved with smooth solution (no boundary conditions) Costabel et al. J. Comput. Appl. Math. 234, 1817–1825 (2010). Kline, Commun. Pure Appl. Math. 4, 225–262 (1951). Jones, The Theory of Electromagnetism (1964).
Conjecture:
lim 𝒜−1 (𝓂𝑛 )ℰinc = 𝒜−1 (𝓂)ℰinc
𝑛→∞
Postulate that 𝒜−1 (𝓂)ℰinc is solution of the scattering problem for arbitrary irregular 𝓂 – second reason, why VIE is superior
15
Hints for the continuity conjecture
Continuity of uniqueness and existence condition Conv
𝑛
𝜀𝑛 ℝ3
= Conv 𝜀 ℝ3 \𝑆int
well-behaved for all 𝓂𝑛 ⇔ well-behaved for 𝓂
Continuity of the convex hull of the operator spectrum Conv 𝜎 𝒜(𝓂𝑛 )
𝑛→∞
Conv 𝜎 𝒜(𝓂)
related to numerical solution of the VIE
Equivalent conjecture – uniform convergence of the discretized solution −1 lim 𝒜𝑁 (𝓂)ℰinc = 𝒜−1 𝓂 ℰinc
𝑁→∞
−1 −1 lim 𝒜−1 (𝓂𝑛 )ℰinc = lim lim 𝒜𝑁 (𝓂𝑛 )ℰinc = lim 𝒜𝑁 (𝓂)ℰinc = 𝒜−1 (𝓂)ℰinc
𝑛→∞
𝑛→∞ 𝑁→∞
𝑁→∞
16
Conclusion
General derivation of the VIE for a very general type of scatterer in a passive host medium
Looked at the VIE from multiple perspectives
Conjectured general sufficient condition for existence and uniqueness.
Conjectured continuity of the inverse integral operator with respect to the refractive-index function ⇒ simpler derivation of VIE
Need to be proven
Remaining work:
Specify proper function spaces (smoothness requirements)
Extend analysis to anisotropic and magnetic materials
Yurkin & Mishchenko, Phys. Rev. A, submitted.
17
Thank you! Maxim A. Yurkin Voevodsky Institute of Chemical Kinetics and Combustion, Laboratory of Cytometry and Biokinetics, Institutskaya 3, 630090, Novosibirsk, Russia
[email protected] https://sites.google.com/site/yurkin/ 18
