Superluminal Velocities in Causal Media Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering University of Toronto University of Toronto Group velocity I. Introduction (background) exceeding c, (superluminal) II. Time-domain experiment Mohammad Mojahedi (UofT) III. Frequency-domain experiment George Eleftheriades (UofT) Omar Siddiqui (UofT) IV. Forerunners and Fronts: Why Einstein causality is Jonathan Woodley (UofT) not violated Kevin Malloy (UNM) V. Negative Group Velocities and Composite Raymond Chiao (UC Berkeley) Medium with Negative Index of Refraction Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering University of Toronto University of Toronto • Early History * Maxwell Eqs. (1865); Hertz experiment (1888) * Hamilton first mention of group velocity (1839) * Rayleigh Generalization (1877) * Einstein special relativity (1905) UV Laser * Sommerfeld and Brillouin * x KDP > Phase velocity, group velocity, Energy velocity, Sommerfeld signal velocity x 1 x 2 g ≈ 1.7 c V > Sommerfeld forerunner (precursor), Brillouin forerunner (precursor) = t1 • Question of electron tunneling time * MacColl (1932): transmitted wave packet appears on the other side of the potential barrier almost instantaneously Trombone Pris m Co inc . * Wigner (1955): there is a finite time associated with the tunneling Co unt e r 1 DPC * Hartman (1962): for an opaque barrier the tunneling time is superluminal x 1 x 2 x * Variety of tunneling times has been proposed: local Larmor times, dwelling time, Buttiker- = t2 Landauer time, phase time, extrapolated phase time, etc. * Despite the numerous proposals, one can always provide an operational definition of the time-of- flight Dr. Mohammad Mojahedi, University of Toronto (KITP Quantum Optics Miniprogram 7/11/02) 1
Superluminal Velocities in Causal Media Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering University of Toronto University of Toronto TIME DOMAIN MEASUREMENTS • Traditionally it was thought that tunneling wave packets were distorted such that it rendered the Cut Waveguide + group velocity meaningless or unphysical Directional Coupler • Single microwave C H A pulse 10 ns long B W O M C “When considerable absorption occurs. The group velocity can not be used, since in an absorbing medium (FWHM) wave packets are not propagated but rapidly ironed out” (Landau and Lifshitz, Electrodynamics of • Centered at 9.68 GHz with continuous media, pp. 285) D e te c to r + Rigid Coaxial 100 MHz cable A tte n u a to r “In particular, in regions of anomalous dispersion the group velocity may exceed the velocity of light or bandwidth F a s t T r ig g e r SCD 5000 SCD 5000 (FWHM) Tektronix Tektronix become negative, and in such cases it has no longer any appreciable physical significance” (Born and Wolf, Principles of optics , pp. 75) “…if absorption also occurs, a (the wave number) becomes complex or imaginary and the group velocity ceases to have a clear physical meaning” (Brillouin, Wave propagation in periodic structure , pp. 75) • J. D. Jackson had considered superluminal group velocity as “…just not a useful concept” ( Classical "Center" Electrodynamics , pp. 302), however this has been revised in 1998 edition. "Side" Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering University of Toronto University of Toronto 1 "side"; free-space "center"; Tunneling Average of five shots. 1 Signal Amplitude [A.U.] 0.8 Every third data point is shown. Average of five shots Signal Amplitude [A.U.] 0.8 "side" 0.6 "center" "side" raw data 0.6 "center" raw data 0.4 • The 1DPC is 0.4 inserted along • 1DPC with five 0.2 “center” path polycarbonate 0.2 dielectric slab • 440 ± 20 ps shift to 0 0 0 8 16 24 32 40 48 earlier times 0 8 16 24 32 40 48 • The delay between Time [ns] Time [nano-second] “center” and “side” is • Group velocity 1.1 adjusted so the peaks 1.1 440 ps Average of five shots (2.38 ± 0.15) c arrive a at the same 1 Average of five shots "side" time Signal Amplitude [A.U.] Signal Amplitude [A.U.] 1 "center" • Solid curves are the 0.9 weighted-curve-fit 0.9 (nonlinear least 0.8 square fit) 0.8 0.7 0.7 "side" ; free-space "center" ; Tunneling 0.6 0.6 15 16 17 18 19 20 21 22 15 16 17 18 19 20 21 22 Time [nano-second] Time [ns] Dr. Mohammad Mojahedi, University of Toronto (KITP Quantum Optics Miniprogram 7/11/02) 2
Superluminal Velocities in Causal Media Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering University of Toronto University of Toronto Dispersion consideration “… with anomalous dispersion, due to the strong absorption which destroys the significance of a characteristic wavelength after a short path, one can no longer sharply define the velocity of S H A S H A propagation of the energy” (L. Brillouin, Wave propagation and group velocity , pp. 22) “…but if absorption also occurs, a (the wave number) becomes complex or imaginary and the L e n s group velocity ceases to have a clear physical meaning” (L. Brillouin, Wave propagation in periodic structures , pp. 75) 1 Average of five shots Free space pulse is advanced in time 0.8 "center" ; Tunneling Signal Amplitude V e c t o r N e tw o r k "center" ; free-space A n a ly z e r H P 8 7 2 2 D 0.6 200 1.0 0.4 Measured phase Transmission Phase (Degree) 150 Calculated phase Transmission amplitude 0.2 0.8 100 Measured amplitude Calculated amplitude 0 50 0.6 0 0 8 16 24 32 40 48 0.4 Time [nano-second] -50 -100 0.2 • FWHM for free-space wave packet is 9.11 ns -150 1.5 % increase • FWHM for tunneling wave packet is 9.246 ns -200 0.0 20.0 20.5 21.0 21.5 22.0 22.5 23.0 Frequency (GHz) Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering University of Toronto University of Toronto Best nonlinear least square fit based on Levenberg- -200 Marquardt algorithm Fitting, N = 3 N = 2 Fitting, N = 1 Mea sured Unwrapped Phase (Degree) n i = 1 -400 N =1 d i 1.794 cm 1.825 cm 1.76 cm n j = 3.4 − i 0.002 -600 2 d j 1.399 cm 1.366 cm 1.396 cm 1.33 cm d i = 1.76 cm ′ n 3.216 3.288 3.245 3.40 -800 3 j d j = 1.33 cm n ′ ′ 0.002 0.002 0.002 0.002 j -1000 measurement Theory -1200 20.0 20.5 21.0 21.5 22.0 22.5 23.0 2.5 Fitted measurement Frequency (GHz) Theory N =3 2.0 V g = L PC τ g = 2.5 Measurement ( ) N = 4 L PC − ∂φ ∂ω Theory τ = − ∂ φ ∂ ω 1.5 g 2 L PC : Light Line the physical 2 /c Group delay (ns) g V φ ∴ transmissi on phase 1.0 thickness of the 1.5 1DPC 1 1 0.50 3 0.5 2 0.0 1 0 20.0 20.5 21.0 21.5 22.0 22.5 23.0 20 20.5 21 21.5 22 22.5 23 Frequency (GHz) Frequency (GHz) Dr. Mohammad Mojahedi, University of Toronto (KITP Quantum Optics Miniprogram 7/11/02) 3
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