Sum Rules and Physical Bounds in Electromagnetics

Transcription

1 Sum Rules and Physical Bounds in Electromagnetics Mats Gustafsson Department of Electrical and Information Technology Lund University, Sweden Complex analysis and passivity with applications, SSF summer school, (August 8, 5)

2 Outline Sum rules in EM Finite objects Conclusions 3 References Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden ()

3 Sum rules and physical bounds on passive systems General simple approach. Identify a linear and passive system.. Construct a Herglotz (or similarly a positive real) function h(z) that models the parameter of interest. 3. Investigate the asymptotic expansions of h(z) as z ˆ and z ˆ. 4. Use integral identities for Herglotz functions to relate the dynamic properties to the asymptotic expansions. 5. Bound the integral. Examples: Matching networks [, 3], Radar absorbers [], Antennas [6, 7, 4], Scattering [, ], High-impedance surfaces [9], Metamaterials [5],Extraordinary transmission [8],Periodic structures [] ¾ ext( )/¼a Al Ag Au Cu a=.¹m /¹m Cross sections of nano spheres Im Y.9 Re P (Y) w=.9` d=.8` High-impedance surface. D/Q/(k a) 3 physical bounds.. ` Chu bound, Antenna D/Q. e^ ` ka ` = =/ ` /` /d a - Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (3)

4 Integral identities for Herglotz functions Herglotz functions with the symmetry h(z) = h ( z ) (real-valued in the time domain) have asymptotic expansions (N and N ) N h(z) = a n z n + o(z N ) h(z) = n= N n= b n z n + o(z N ) as z ˆ as z ˆ Im z where ˆ denotes limits in the Stoltz domain < θ arg(z) π θ??. They satisfy the identities ( N n N ) lim lim ε + y + π ε ε Im h(x + iy) x n dx = a n b n = θ Re b n n < a b n = a b n = a n n > Bernland, Luger, Gustafsson, Sum rules and constraints on passive systems, J. Phys. A: Math. Theor.,. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (4)

5 Integral identities for Herglotz functions Common cases Known low-frequency expansion (a ): { a z as z ˆ h(z) b z as z ˆ that gives the n = identity (we drop the limits for simplicity) lim lim /ε ε + y + π ε Im h(x + iy) x dx def = π Im h(x) x dx = a b a Known high-frequency expansion (short times) (b ): { a /z as z ˆ h(z) b /z as z ˆ that gives the n = identity π Im h(x) dx = a b b. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (5)

6 Forward scattering sum rule ɛ µ ê ˆk ɛ(r) µ(r) ê ˆk Assumptions: From physics: Finite scattering object composed of a linear, passive, and time translational invariant medium. Incident linearly polarized plane wave. The propagation (wavefront) speed is limited by the speed of light. Optical theorem (energy conservation). Induced dipole moment in the static limit. Shadow scattering Mats Gustafsson, Department of Electrical and Information Technology, Lund (extinction University, Sweden paradox) (6) for the

7 Forward scattering sum rule ɛ µ ê ˆk ɛ(r) µ(r) ê ˆk Use the n = identity with a = γ = ê γ e ê + (ˆk ê) γ m (ˆk ê) and b =, i.e., π σ ext (k) k dk = ê γ e ê + (ˆk ê) γ m (ˆk ê) or written in the free-space wavelength λ = π/k π σ ext (λ) dλ = ê γ e ê + (ˆk ê) γ m (ˆk ê) Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (7)

8 Propagation speed limited by the speed of light e ^ E=e ^ (E i+e s) ^k t=-`/c t= t=-`/c t=`/c ² = r &=3.5/` Causality in the sense that the scattered field cannot precede the incident field in the forward direction. Causal impulse response h t (t). Analytic transfer function h(k), (Fourier, Laplace transform of h t (t), where k = πf/c denotes the free-space wavenumber.). Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (8)

9 Propagation speed limited by the speed of light ` t=-`/c t= t=-`/c t=`/c e ^ _ E s ` ² = r &=3.5/` Causality in the sense that the scattered field cannot precede the incident field in the forward direction. Causal impulse response h t (t). Analytic transfer function h(k), (Fourier, Laplace transform of h t (t), where k = πf/c denotes the free-space wavenumber.). Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (8)

10 Energy conservation (passivity) a) b) t=t V n^ V c E=E i Object Ei k^ Object E=E s A E=E i E=E i +E s= T ( W ext,τ = Ei (t, r) H s (t, r)+e s (t, r) H i (t, r) ) ˆn(r) ds dt. V simplify to T W ext = E(t)h t (τ t)e(τ) dt dτ for all E implying cf., the optical theorem. R Im h(k) = σ ext (k) for Im k > Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (9) Ei k^

11 Low-frequency asymptotic expansion a) b) r Ã= Ã= ½(x,y,z) e=z^ ^ Ã= z h(k) = ê γ e êk + O(k ) as k (Kleinman&Senior 986). Polarizability dyadic γ e. Induced dipole moment p = ɛ γ e E ê. Variational principles γ e γ (Jones 985, Sjöberg 9). High contrast polarizability dyadic γ. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden ()

12 High-frequency asymptote 4 ¾ext/A z^ k^ a ka.45a.55a.55a Shadow scattering (Peierls 979, Gustafsson etal 8). Im h(k) = σ ext (k) A on average as k. h(k) ˆ, i.e., for < δ < arg k < π δ. the extinction paradox. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden ()

13 Forward scattering sum rule ɛ µ ê ˆk ɛ(r) µ(r) ê ˆk Use the n = identity with a = γ = ê γ e ê + (ˆk ê) γ m (ˆk ê) and b =, i.e., π σ ext (k) k dk = ê γ e ê + (ˆk ê) γ m (ˆk ê) or written in the free-space wavelength λ = π/k π σ ext (λ) dλ = ê γ e ê + (ˆk ê) γ m (ˆk ê) Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden ()

14 Polarizability dyadic and induced dipole moment The induced dipole moment can be written p = ɛ γ e E a) E where γ e is the polarizability dyadic. Example (Dielectric sphere) + % - A dielectric sphere with radius a and relative permittivity ɛ r has the polarizability dyadic as ɛ r. γ e = 4πa 3 ɛ r ɛ r + I γ = 4πa 3 I b) equipotential lines + % - E Analytic expressions for spheroids, elliptic discs, half spheres, hollow half spheres, touching spheres,... equipotential lines Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (3)

15 Extinction cross sections for a = 5 nm spheres σ ext = σ a + σ s = P a + P s E i /η Sum of the scattered and absorbed powers divided by the incident power flux. Integrate over the free-space wavelength λ = π/k ¾ ext( )/¼a Al π Ag Au Cu σ ext (λ) dλ = ê γ e ê = 4πa 3 a=.¹m /¹m ¾ ext( )/¼a a=.¹m The areas under the curves are identical. Time-domain approach to the forward scattering sum rule, Proc.R.Soc.A, Ag a a=.¹m.8a Ag Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (4) /¹m

16 Forward scattering of antennas h/cm d=3.57mm 3 5Ω ¾ext=Im h `=5cm f/ghz 6 I I Forward scattering measurement of a dipole antenna. Loaded, short, and open circuit. Length 5 cm and.5 GHz to 6 GHz. sim: sim: meas: π π R k Rk k σext (k) k σext (k) k dk dk loaded short open Re h d=3.57mm ¾ext=Im h `=5cm f/ghz Re h h/cm d=3.57mm 5.mm `=5cm ¾ext=Im h 5 in cm3 γ 3 h/cm 4 I Re h Forward scattering of loaded and unloaded antennas, IEEE-TAP,. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (5) 4 5 f/ghz 6

17 Physical bounds on antennas Given a geometry, e.g., sphere, rectangle, spheroid, or cylinder. How does D/Q (directivity bandwidth product) depend on the geometry for optimal antennas? D Q ηk3 (ê γe ê+(ˆk ê) γ ) π m (ˆk ê) based on Passive materials Antenna forward scattering Identities for Herglotz function Physical limitations on antennas of arbitrary shape Proc R. Soc. A, , 7. Illustrations of new physical bounds on linearly polarized antennas, IEEE Trans. Antennas Propagat., 9. Absorption Efficiency and Physical Bounds on Antennas, Int. J. of Antennas and Propagat., , Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (6)

18 Circumscribing rectangles D/Q/(k a) 3 physical bounds Chu bound, e^ ` ka ` a. = =/ ` /`.. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (7)

19 Circumscribing rectangles D/Q/(k a) 3 physical bounds Chu bound, e^ ` ka ` a. = =/ ` /`.. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (7)

20 How can we measure the polarizability? Change of capacitance in a parallel plate capacitor. The polarizability in a parallel plate waveguide. The periodic polarizability for symmetric objects. Objects with increasing distance between the coins. Large separation of charge give a large polarizability. D. Lovrić, Theoretical and experimental studies of polarizability dyadics,. Kristensson, The polarizability and the capacitance change of a bounded object in a parallel plate capacitor, Physica Scripta,. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (8)

21 High-contrast polarizability dyadics: γ γ is determined from the induced normalized surface charge density, ρ, as ê γ ê = ê rρ(r) ds E V where ρ satisfies the integral equation ρ(r ) 4π r r ds = E r ê V n V a) b) + % - equipotential lines E E with the constraints of zero total charge V n ρ(r) ds= + % - Can also use FEM (Laplace equation). equipotential lines Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (9)

22 Removal of metal from a square plate /` 3 e^.8 e^.6.4. A/`.... Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden ()

23 Removal of metal from a square plate and circular disk a y a/ z x 6 a 3 (xx+yy) ^ ^ ^^ a 3(4.3 xx+4.5 ^^ yy) ^^ 3 ` `/ º ` `/5 a 3(4. xx+4.8 ^^ yy) ^^ `/ º 3.4 ` (xx+yy) ^^ ^^ 3 3 ` (.94 xx+.96 ^^ yy) ^^ ` (.5 xx+.93 ^^ yy) ^^ Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden ()

24 Sum rules for passive systems Extra ordinary transmission Forward scattering Artificial µ ɛ near Spherical modes Negative refraction Scattering Dispersion Antennas Sum rules for passive systems Matching Periodic structures Extra ordinary transmission Absorbers Highimpedance surfaces Blockage Cross section Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden ()

25 Sum rules for passive systems Re Y big Y () = Y Y () = Extra ordinary transmission Forward scattering Artificial µ Y ɛ near ( ) = Spherical modes Extra ordinary transmission Negative Y refraction Y ( ) = Γ Γ () = Scattering Dispersion Sum rules for passive systems Matching Γ Γ () = Periodic structures Y Y () = Antennas Γ Γ () = Absorbers Highimpedance surfaces Blockage Cross section Y Y () = Re Y big Y () = T T () = Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (3)

26 Basically four (or two) cases admittance want large Re Y (ω ) with Y () =. forward scattering (cross section) Use identity S-parameter want S(ω ) δ with S() =. absorber, matching, blockage, modes,... Use log+identity admittance want small Y (ω ) δ with Y () =. high impedance surface, temporal dispersion. Use pulse+identity S-parameter want S(ω ) with S() =. high impedance surface, extra ordinary transmission Use Cayley+pulse+identity Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (4)

27 Outline Sum rules in EM Finite objects Conclusions 3 References Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (5)

28 Conclusions Sum rules derived from integral identities for Herglotz functions. Physical bounds. Extinction cross section, antennas, extra ordinary transmission, transmission blockage, high-impedance surfaces, radar absorbers, temporal dispersion, perfect lenses, artificial magnetism. Why physical bounds for passive systems? Realistic expectations. Possible/impossible. Possible design improvements. Is it worth it? Figure of merit for a design. Use of active and/or non-linear systems ¾ ext( )/¼a Al Ag Au Cu a=.¹m /¹m Cross sections of nano spheres.. D/Q/(k a) 3 Chu bound, ka physical bounds Antenna D/Q. Im Y.9 e^ ` ` = =/ ` /` w=.9` d=.8` Re P (Y) High-impedance surface. ` a /d - Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (6)

29 References ( Passive systems and Herglotz functions A.H. Zemanian, Distribution Theory and Transform Analysis: An Introduction to Generalized Functions with Applications, 965 F.W. King, Hilbert Transforms, vol,, 9. A. Bernland, A. Luger, M. Gustafsson, Sum rules and constraints on passive systems, J. Phys. A: Math. Theor.,. Antennas M. Gustafsson, C. Sohl, and G. Kristensson. Illustrations of new physical bounds on linearly polarized antennas. IEEE Trans. Antennas Propagat., May 9. C. Sohl and M. Gustafsson. A priori estimates on the partial realized gain of Ultra-Wideband (UWB) antennas. Quart. J. Mech. Appl. Math., 8. M. Gustafsson, C. Sohl, and G. Kristensson. Physical limitations on antennas of arbitrary shape. Proc. R. Soc. A, 7. M. Gustafsson, Sum rules for lossless antennas, IET Microwaves, Antennas & Propagation,. M. Gustafsson, J. Bach Andersen, G. Kristensson, G. Frolund Pedersen, Forward scattering of loaded and unloaded antennas, IEEE-TAP,. Forward scattering C. Sohl, M. Gustafsson, and G. Kristensson, Physical limitations on broadband scattering by heterogeneous obstacles, J. Phys. A: Math. Theor., 7. C. Sohl, M. Gustafsson, and G. Kristensson, Physical limitations on metamaterials: Restrictions on scattering and absorption over a frequency interval, J. Phys. D: Applied Phys., 7. C. Sohl, C. Larsson, M. Gustafsson, and G. Kristensson, A scattering and absorption identity for metamaterials: experimental results and comparison with theory, J. Appl. Phys., 8. M. Gustafsson. Time-domain approach to the forward scattering sum rule Proc. R. Soc. A,. Temporal dispersion in metamaterials M. Gustafsson and D. Sjöberg, Sum rules and physical bounds on passive metamaterials, New Journal of Physics,. Periodic structures M. Gustafsson, C. Sohl, C. Larsson, and D. Sjöberg, Physical bounds on the all-spectrum transmission through periodic arrays, EPL Europhysics Letters, 9. M. Gustafsson and D. Sjöberg, Physical bounds and sum rules for high-impedance surfaces, IEEE-TAP,. M. Gustafsson, I. Vakili, S.E. Bayer Keskin, D. Sjberg, C. Larsson, Optical theorem and forward scattering sum rule for periodic structures, IEEE-TAP,. Extraordinary transmission M. Gustafsson. Sum rule for the transmission cross section of apertures in thin opaque screens. Opt. Lett., 9. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (7)

30 Outline Sum rules in EM Finite objects Conclusions 3 References Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (8)

31 References A. Bernland, M. Gustafsson, and S. Nordebo. Physical limitations on the scattering of electromagnetic vector spherical waves. J. Phys. A: Math. Theor., 44(4), 454,. H. W. Bode. Network analysis and feedback amplifier design, 945. Van Nostrand, 945. R. M. Fano. Theoretical limitations on the broadband matching of arbitrary impedances. Journal of the Franklin Institute, 49(,), and 39 54, 95. M. Gustafsson. Sum rules for lossless antennas. IET Microwaves, Antennas & Propagation, 4(4), 5 5,. M. Gustafsson and D. Sjöberg. Sum rules and physical bounds on passive metamaterials. New Journal of Physics,, 4346,. M. Gustafsson, C. Sohl, and G. Kristensson. Physical limitations on antennas of arbitrary shape. Proc. R. Soc. A, 463, , 7. M. Gustafsson, C. Sohl, and G. Kristensson. Illustrations of new physical bounds on linearly polarized antennas. IEEE Trans. Antennas Propagat., 57(5), 39 37, May 9. M. Gustafsson. Sum rule for the transmission cross section of apertures in thin opaque screens. Opt. Lett., 34(3), 3 5, 9. M. Gustafsson and D. Sjöberg. Physical bounds and sum rules for high-impedance surfaces. IEEE Trans. Antennas Propagat., 59(6), 96 4,. M. Gustafsson, I. Vakili, S. E. B. Keskin, D. Sjöberg, and C. Larsson. Optical theorem and forward scattering sum rule for periodic structures. IEEE Trans. Antennas Propagat., 6(8), ,. K. N. Rozanov. Ultimate thickness to bandwidth ratio of radar absorbers. IEEE Trans. Antennas Propagat., 48(8), 3 34, August. C. Sohl, M. Gustafsson, and G. Kristensson. Physical limitations on broadband scattering by heterogeneous obstacles. J. Phys. A: Math. Theor., 4, 65 8, 7. Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden (9)