How a wave packet propagates at a speed faster than the speed of - PDF document

How a wave packet propagates at a speed faster than the speed of light A novel superluminal mechanism with high transmission and broad bandwidth Tsun-Hsu Chang ( 張存續 ) Department of Physics, National Tsing Hua University Claim: The phenomena we present here do not violate the special relativity, which is a cornerstone of the modern understanding of physics for more than a century. 1 Outline  Introduction (evanescent wave)  Matter wave and electromagnetic wave  Modal analysis (a 3D effect)  New superluminal mechanism (propagating wave)  Manipulating the group delay  Conclusions  Acknowledgement 2

The Fastest Person Usain Bolt is a Jamaican sprinter widely regarded as the fastest person ever. 100 m in 9.58 s, Speed ~ 10 m/s . [ 3 Top Speed of Racing Car: Formula 1 The 2005 BAR-Honda set an unofficial speed record of 413 km/h at Bonneville Speedway. Speed ~ 115 m/s . [ 4

Flight Airspeed Record: SR-71 Blackbird The SR-71 Blackbird is the current record-holder for a manned air breathing jet aircraft. 3530 km/h ~ 980 m/s 5 Controlled Flight Airspeed Record: Space Shuttle Fastest manually controlled flight in atmosphere during atmospheric reentry of STS-2 mission is 28000 km/h ~ 7777 m/s. 6

Highest Particle Speed: LEP Collider The Large Electron–Positron Collider (LEP) is one of the largest particle accelerators ever constructed. The LEP collider energy eventually topped at 209 GeV with a Lorentz factor γ over 200,000. LEP still holds the particle accelerator speed record. 10 1 1 v β = = (1 − ) = 0.999999999988 2 2 c γ just millimeters per second slower than . c m 2 0 = E c 2 1 − β Matter cannot exceed the speed of light in vacuum. How about wave? 7 Superluminal Mechanism: Anomalous dispersion ck The index of refraction n ( ω ) is a function of frequency. ( ) = n k ω ( ) k ω ( ) k c Phase velocity: ≡ = (7.88) v p ( ) k n k ω d c ≡ = Group velocity: (7.89 ) v g ω + ω ω ( ) ( ) dk n dn d φ ( ) d d kL L Grou p delay: τ ≡ ≈ = g ω ω d d v g See waves in a dielectric medium [ Jackson Chap. 7 ] 8

Anomalous Dispersion: Waves in a dielectric medium 2 f 2 Ne f Ne j 0 ε = ε + + (7.51)  i m 0 2 2 m ω γ ( − ω ) ω − ω − ωγ i (bound) i j 0 j j     Properties of ε : negligible (  0 or very small) = f 0 When ω is near each ω j (binding frequency of the j th group of electrons), ε exhibits resonant behavior in the form of anomalous dispersion and resonant absorption. Re ε Im ε 0 ω PA: Polyamides are semi-crystalline polymers. The data was measured with a THz-TDS system. 9 The tunneling effect V E − 2( ) E V = = ? v V 0 m I II III The microwave propagating in a waveguide system seems to be analogous to the behavior of a one-dimensional matter wave. L Comparing with the matter wave, the electromagnetic wave is much more easier to implement in experiment. 10

Summary #1  Anomalous dispersion and tunneling effect are the two major mechanisms for the superluminal phenomena.  Both mechanisms involve evanescent waves, which means waves cannot propagate inside the region of interest. 11 Part II. Analogies Between Schrödinger ’ s Equation and Maxwell ’ s Equation 12

Analogies Between Schrodinger and Maxwell Equations Maxwell ’ s wave equation Time-independent for a TE waveguide mode Schrodinger ’ s equation 2 2 2 ∂ ω ω 2 ∂ 2 2 m m − µε + µε = ( ( ) ) 0 − + ϕ = c [ ( ) ] ( ) 0 z B V z E z 2 2 2 z ∂ 2 2 2 ∂   z c c z ω 2 2 m µε ( ) c ( ) z V z 2 2 c  2 ω 2 m 2 2 µε = = k E k 2 z 2 z  c Anything else?  Transmission and reflection coefficients  Probability and energy velocities  Group and phase velocities 13 Transmission for a Rectangular Potential Barrier   2 ( − ) − 1 1 2 ( ) V V m V E 2 2 0   < : = + 1 sinh (2 κ ), where κ = E V QM a   2 4 ( − )( − ) T V E E V    0 By analogy, the transmission parameter of an electromagnetic wave can be expressed as   2 2 2 2 2 ω − ω ω − ω 1 1 ( ) ( ) 2 2 0  c c  c ω < ω : = + 1 sinh (2 κ ), where κ = EM a   c 2 2 2 2 2 4 ( T ω − ω )( ω − ω ) c   0 c c 14

Analogies Between Probability and Energy Velocities Quantum Mechanics: Electromagnetism: Probability velocity Energy Velocity   J ˆ = ( ⋅ ) P P S da e z = x v v E = A prob 2     1 ψ  U = ⋅ + ⋅ ( ) U E D B H da 16 π A * ω 2 Γ * 2 ( − ) 2 Im( Γ ) 2 Im( ) c V E − 1 c 2 2 2 κ − 2 κ ω [ x + Γ x + 2 Re( Γ )] µε 2 κ − 2 κ + Γ + Γ m e e ( z z ) 2 Re( ) e e ω < ω E < V c Can we use EM wave to study a long-standing debate in QM, i.e. the tunneling time? 15 2 a dx  Δ = t E < QM: Tunneling Time Calculation V v 0 prob 2 a 1 m  2 2 κ − 2 κ Δ = [( z + Γ z ) + 2 Re( Γ )] t e e dz * − Γ 2 ( ) 2 Im( ) V E 0   1 1 m 2 4 κ − 4 κ = (( a − 1 ) − Γ ( a − 1 )) + 4 Re( Γ ) e e a   * 2 ( − ) 2 Im( Γ )  2 κ  V E 2 a dx  Δ = ω < ω t EM: Tunneling Time Calculation c v 0 E 2 2 a µεω 1 2 κ − κ  2 2 z z Δ = + Γ + Γ [( ) 2Re( )] t e e dz 2 2 * ω − ω 2 Im( Γ ) c c 0 2 µεω 1  1 ((  2 4 κ − 4 κ a a = − 1) − Γ ( − 1)) + 4 Re( ) Γ e e a   2 2 *  2 κ  ω − ω 2 Im( Γ ) c c 16

Summary #2  Superluminal effect is common to many wave phenomena.  The matter wave and the electromagnetic wave share many common characteristics. The moment of truth: Put the idea to the test in a 3D-EM system. 17 Part III. Modal Analysis: Effect of high-order modes on tunneling characteristics H. Y. Yao and T. H. Chang , “ Effect of high-order modes on tunneling characteristics" , Progress In Electromagnetics Research, PIER, 101 , 291-306, 2010. 18

Geometric and material discontinuities (B) (A) µ = ε = 1 for all regions µ = ε = 1 ; 1 for I and III For TE 10 mode r r r r π 1 1 π c c 2 2 = ω 2 − ω , ω = 2 a a = ω − ω a , ω a = k k 1 1 c c c c c a c a π 1 µ = 1 ; ε ≠ 1 for I and III c 2 2 c c = ω − ω , ω = k r r 2 c c c c 2   1 ω a   What is the difference between (A) and (B)? 2 = ω − c k   2 ε v ω   c ω r c ω ω a c ik z ik z e 1 Be 2 κ ik z z e Be 1 2 ik z ik z De De 1 1 Reduce to 1-D case − − − ik z ik z ik z Ae Ce − κ 1 2 Ae 1 z Ce ω a 2 ω c Potential-like diagram c c 19 Region I Region II Region III Region I Region II Region III Transmission amplitude for two systems − 2 ik L k k e 1 = 1 2 Transmission * D ≡ × T D D 2 2 2 cos( ) − ( + ) sin( ) k k k L i k k k L amplitude 1 2 2 1 2 2 (A) (B) ε > 1 r Disagree! Why? ε < 1 r 20

Group delay for two systems δφ   2 2 L d ( + ) tan( ) k k k L τ = = − 1 δφ = tan 1 2 2   g ω v d 2 k k   g 1 2 (B) (A) ε > 1 r Disagree! Why? ε < 1 r 21 Modal Effect L (c) (a) Region III Region II Region I E V 0 B e i k 2 z e i k 1 z D e i k 1 z (d) A e - i k 1 z C e - i k 2 z L V Region III Region II Region I (b) ω ω c a Σ B n e i k n z e i k 1 z Σ D n e i k n z Σ A n e - i k n z Σ C n e - i k n z (e) L ω c b It is a 3-D problem. Modal effect should be considered. 22