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Energy storage in superluminal barrier tunneling: Origin of the “Hartman effect”

Herbert G. Winful

Open Access Open Access

Abstract

We show that the anomalously short delay times observed in barrier tunneling have their origin in energy storage and its subsequent release. The observed group delay is proportional to the energy stored. This delay is not a propagation delay and should not be linked to a velocity since evanescent waves do not propagate. The “Hartman effect”, in which the group delay becomes independent of thickness for opaque barriers, is shown to be a consequence of the saturation of stored energy with barrier length.

©2002 Optical Society of America

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Figures (1)
Fig. 1
Fig. 1 Schematic of a photonic bandgap structure (PBG).

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Equations (22)

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n = n 0 + n 1 cos ⁡ ( 2 β 0 z ) ,
E z t = Re [ E F z t e i ( β 0 z − ω 0 t ) + E B z t e − i ( β 0 z + ω 0 t ) ]
∂ E F ∂ z + 1 v ∂ E F ∂ t = iκ E B
∂ E B ∂ z − 1 v ∂ E B ∂ t = − iκ E F ,
∂ 2 E F ∂ z 2 − 1 v 2 ∂ 2 E F ∂ t 2 = κ 2 E F .
K 2 = ( Ω 2 − Ω c 2 ) v 2 ,
exp ( − γz ) exp ( − i Ω t )
d E F dz = iκ E B + i ( Ω v ) E F ,
d E B dz = − iκ E F − i ( Ω v ) E B .
E F ( z ) = E 0 [ γ cosh ⁡ γ ( z − L ) + i ( Ω v ) sinh γ ( z − l ) ] g ,
E B ( z ) = − i [ E 0 κ sinh ⁡ γ ( z − L ) ] g ,
ϕ t = tan ⁡ − 1 [ ( Ω γv ) tanh ⁡ γL ] .
ϕ r = ϕ t + π 2 .
〈 U 〉 = 1 2 ε ∫ vol [ ∣ E F ∣ 2 + ∣ E B ∣ 2 ] dv ,
〈 U 〉 = ( 1 2 ε E 0 2 A ) [ κ 2 γ 2 tanh ⁡ γL γ − L ( Ω γv ) 2 sech ⁡ 2 γL ] cos ⁡ 2 ϕ t ,
τ g = d ϕ t d Ω .
τ g = 1 v [ κ 2 γ 2 tanh ⁡ γL γ − L ( Ω γv ) 2 sech ⁡ 2 γL ] cos ⁡ 2 ϕ t .
〈 U 〉 = ( 1 2 ε E 0 2 Av ) τ g .
〈 U 〉 = P in τ g .
τ D = 〈 U 〉 P in
τ D = ( ∣ R ∣ 2 d ϕ r d Ω + ∣ T ∣ 2 d ϕ t d Ω ) ,
τ g = τ D = d ϕ d Ω ,
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