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and “Stopped Light” “Faster than Light” Propagation of Laser Pulses Phillip Sprangle Naval Research Laboratory Washington, DC Northeastern University Boston February 7, 2003 Collaborators : J. R. Pe ñ ano and B. Hafizi Outline Introduction Phase and Group Velocities

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and “Stopped Light”

“Faster than Light” Propagation of Laser Pulses

Phillip Sprangle

Naval Research Laboratory

Washington, DC

Northeastern UniversityBostonFebruary 7, 2003

Collaborators : J. R. Peñano and B. Hafizi


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Outline

  • Introduction

  • Phaseand Group Velocities

  • “Faster than Light” Experiment

  • Interpretation of “Faster than Light” Experiments

  • “Stopped” Light

  • Summary


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”Faster than Light”

Propagation of Laser Pulses

  • In a recent Nature article [1], researchers reported observing

  • superluminalpropagation of a laser pulse in a gain medium

  • by a new mechanism

  • The Nature article received a great deal of attention in both

  • the scientific community and the world press

[1] L. J. Wang, A. Kuzmich, and A. Doggariu, Nature (London)406, 277 (2000)


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World Wide Popular Press Releases

  • “The speed of light is exceeded in lab”, Washington Post, 20 July, 2000

  • “Speed of light may not be the last word”, The Boston Globe, 20 July, 2000

  • “Faster than light, maybe, but not back to the future”, New York Times, 30 July, 2000

  • “A pulse of light breaks the ultimate speed limit”, Los Angeles Times, 20 July, 2000

  • “Light goes backwards in time”, The Guardian, 20 July, 2000

  • “It’s confirmed: speed of light can be broken”, India Today, 21 July, 2000

  • “Ray of light for time travel”, South China Morning Post, 21 July, 2000


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signal

velocity

u

Causality

  • Causality requires that the effect appears after the cause

  • According to special relativity, in a frame moving

  • with velocity v < c, the time interval between events is

  • Special relativity requires that the signal velocity be less than c

  • Causality is violated if signal velocity u is greater than c


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Cesium gas

T = 3.7sec ( 3700 ft)

= 6cm

laser frequency

is between

two gain lines

“Faster than Light” Experiment

Wang, et al., Nature, 406, 277 (2000)

Time delay ׃


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Experimentally observed advancement of a laser pulse

in a gain doublet[Wang, et al., Nature, 406, 277 (2000)]

The Nature article claims that

1) “differs from previously studied anomalous dispersion associated with an

absorption or a gain resonance”

2) “ the shape of the pulse is preserved”

3) “the argument that the probe pulse is advanced by amplification

of its front edge does not apply”


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Laser Pulse

vg

vph

Phase and Group Velocity of Laser Pulses

Phase Velocity is the velocity of the phase

Group Velocity is the velocity of the pulse envelope

  • If the pulse envelope does not distort then the group velocity is the

  • velocity of energy flow


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Two Level Atom, α > 0

Refractive Index

  • The refractive index represents the medium’s response to the fields

  • Using the Classical Lorentz or Quantum Two Level Atom model

where α denotes population inversion, N is the density of atoms,

Ω is the atomic binding frequency, Γ is the damping rate ,

q is the electronic charge and m the electronic mass


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where

  • Group velocity:

Phase and Group Velocities

  • There are conflicting interpretations of how pulses propagate

  • when vg is “abnormal” , i.e., vg < 0 or vg > c



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Abnormal Group Velocities

Can have abnormal group velocities, vg > c or vg < 0, in a

gain medium

and/or

loss medium


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Two Level Atom , α < 0

  • Index of refraction:

  • For low damping

(amplifying)

Superluminal Group Velocity

(“Optical Tachyons”)


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Absorptive Medium

Excited Medium

Im(Dn)103

Im(Dn)103

Abnormal Group Velocities


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Laser electric field ׃

,

Interpretation of “Faster than Light”

Experiments

Sprangle, Peñano and Hafizi, Phys. Rev. E, 64, 026504 (2001)

Pulse Envelope equation ׃


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distorted

envelope

differential

gain

higher order

differential gain

initial

envelope

vg = c/(1 + c1)

Solution of Envelope Equation

Laser pulse envelope:

  • Front pulse propagates at the speed of light and undergoes distortion

  • Front is preferentially amplified when k1 < 0, i.e., differential gain.


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Pulse Distortion in a Gain Medium

differential gain

Nature Article

|A(z = L,t)|

vacuum

(v = c)

t - z/c

Front of

Pulse


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Group Velocity

Laser

Spectrum

vg(w)/c

Laser Envelope

Gain Spectrum

Laser

Spectrum

A(z,t)/A0

-Im(Dn)

Apparent Superluminal Propagation

of a Laser Pulse

W= 0.08 ωo , G = 0.08 ωo


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Differential Gain Interpretation

Wang, et al.

DT = 62 nsec

pulse peak

pulse amplitude

Time (msec)

Comparison with Experiment

Using the reported experimental parameters, the differential gain effect can account for the observed 62 nsec pulse advancement

Pulse advancement occurs by amplification of the leading edge


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Superluminal Laser Pulses

  • Wang et al. interpreted experimental results as superluminal

  • propagation without pulse distortion

  • Analysis shows that pulse distortion was present and

  • responsible for the apparent superluminal propagation

  • The front of the distorted pulse propagates at speed of light

  • while the peak was superluminal


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“Stopped Light” Pulses

  • Quantum interference effects can lead to

  • Electromagnetically Induced Transparency* (EIT)

  • EIT results in

  • Extremely low group velocities

  • High transparency

  • Refractive index is unity

* S. E. Harris , Phys. Rev. Lett. 72, 52 (1994)


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3

Strong coupling

beam

Highly damped

level

2

Probe pulse

1

>> 1

= 1

Electromagnetically Induced Transparency

3 Level Atom

  • Quantum interference effects result in state 3 being unpopulated,

  • no interaction with probe beam


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Probe pulse

Ultra cold gas of

Sodium atoms

Bose-Einstein

Condensate

Strong

coupling

beam

Giant Kerr nonlinearity ׃

“Stopped Light” Experiment

S.E. Harris Phys. Today 50, 36 (1997), L. V. Hau, et.al., Nature 397, 594 (1999)

Probe pulse velocity

reduced to 38 mph


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  • Envelope of the probe pulse at resonance

: Rabi frequency of coupling beam

Probe Pulse has Extremely Low

Group Velocity and Losses

  • The group velocity of the probe pulse is

  • No losses in probe pulse


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Highly Opaque Material (no coupling beam)

Wc/Wp = 0, Dw /Wp= 1, G3/Wp = 10


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Ultra Slow Light (with coupling beam)

Probe pulse compressed by the factor c/vg >> 1

The excited states are not populated

The probe pulse energy is converted into the coupling beam energy

Wc /Wp = 1/40, Dw = 0


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  • - extremely low group velocities, vg ~ 38 mph

  • extremely high transparencies

Summary

  • Recent experiments have been interpreted as superluminal laser pulse

  • propagation [Wang, et al., Nature, 406, 277 (2000)]

  • The apparent superluminal propagation is actually due to a

  • differential gain mechanism [Sprangle, et al., Phys. Rev. E, 64,026504 (2001)]

  • which can give the appearance of faster-than-light propagation

  • The leading edge of laser pulse travels at speed of light in vacuum

  • Differential pulse distortion can occur in a gain or a loss medium

  • Potential applications include : optical switching and quantum computing



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Laser Envelope Equation

Wave Equation

bound

electron

Representation of electric and polarization fields, 1-D

Need relationship between A(z,t) and B(z,t)

Causal relationship between P and E through susceptibility


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Electromagnetic

Energy flow

z

laser pulse

envelope

where

׃carrier frequency, ko׃ carrier wavenumber and

unit vector

Electromagnetic Wave Equation

Wave Equation:

where P is the polarization field

representing the medium response

Electric Field:


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Group Velocity

vg

dispersive medium

z

: refractive index

Expand wavenumber about carrier frequency

where

is the group velocity


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  • The real part of the refractive index near the

  • resonance frequency,

  • If the medium is amplifying ( < 0) then the

  • group velocity is superluminal ( vg > c), (“Optical Tachyons”)

Superluminal Group Velocity


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Laser Envelope Equation

In frequency domain

Using the representations for the fields

Since

is non zero for small positive values of


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Laser Envelope Equation

By definition

and

therefore

The refractive index

is given by


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Standard Fourier Transform Approach

In the standard Fourier Transform approach care must be taken in ordering the terms

For example:

Index is expanded to 1st order

(GVD is neglected)

Exponential factor contains contributions beyond the order of approximation


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Lowest Order Fourier Transform Solution

The lowest order solution ( neglecting k2 ) is

Pulse propagates undistorted with group velocity:

If vg > c, or < 0 ( abnormal ) the result is unphysical


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Gaussian Shaped Laser Pulses

Using the standard Fourier Transform approach, GVD effects , can be analyzed for specific pulse shapes (e.g., Gaussian)

For an initially Gaussian pulse:

where T is the initial pulse duration

Pulse envelope propagates at velocity vg and remains Gaussian, but with a different amplitude and width. [C.G.B. Garrett & D.E. McCumber, PRA, 1, 305 (1970), E.L. Bolda, et al., PRA, 49, 2938 (1994)]


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Gaussian Laser Pulse Solution

Input pulse is Gaussian:

The pulse at z is distorted (not Gaussian):


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Comparison with Experiments

Gain Medium

Susceptibility model describing a gain doublet [Wang, et al., Nature, 406, 277 (2000)] :

For small line width, i.e., ,

A gain medium (inverted population) implies that k1 < 0 , ( vg < 0 or > c )


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simulation

2nd order solution

vacuum solution

Exact Numerical Solution

front

Input pulse shape:

|A(z = L, t)|

peak

|A(z = L, t)|

vacuum

|A(z = L, t)|

back

|A(z = L, t)|

t / 2T

t / 2T


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Comparison with Experiment

Parameters

Numerical calculation

  • M1,2 = M = 0.18 Hz

  • f1 = 3.5 x 1014 Hz

  • f2 = f1 + 2.7 MHz

  • = 0.46 MHz

    L = 6cm

Wang, et al.

vp(w0) = -c/305


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Pulse Energy

For the example:


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Nonlinear Lorentz Model of Excited Atoms

Displacement of electronic distribution (Duffing Eq.)

Polarization field:


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Index of refraction

inverted population term

for

no homogeneous gain,

but can have differential gain


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Background

In a gain medium, superluminal propagation proceeds by the preferential amplification of the leading edge of the pulse.

C.G.B. Garrett & D.E. McCumber, PRA, 1, 305 (1970)

Propagation of Gaussian pulses.

R.Y. Chiao, PRA, 48, R34 (1993)

Off-resonance superluminal propagation

A.M. Steinberg & R.Y. Chiao, PRA, 49, 2071 (1994)

Dispersionless propagation in a gain doublet

E.L. Bolda, et al., PRA, 49, 2938 (1994)

(what’s new about this?)


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Background

In an attenuating medium, undistorted, superluminal, and causal propagation of pulses (not necessarily Gaussian) is possible under certain conditions. [M.D. Crisp, PRA 4, 2104 (1971)]


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Envelope Equation

Substituting E and P, and relationship between A(z,t) and B(z,t),

into wave equation yields an envelope equation

For narrow spectral pulse widths ( long pulses ) can keep terms

to order

( lowest order in GVD )

Envelope Equation


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Analysis

Consistent Solution:

Exponential is expanded to consistent order

Pulse propagates at the speed of light and undergoes distortion.

Front is preferentially amplified when k1 < 0, i.e., differential gain.


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Analysis

Integral solution:

Standard Approach:

Lowest order solution is obtained by neglecting k2 and carrying out the integral without expanding the exponential: [Refs.]

Analysis results in a pulse propagating undistorted at the group velocity:


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“Faster than Light” Propagation of Laser Pulses

Dr. Phillip Sprangle

Naval Research Laboratory, Washington D.C.

Abstract

Propagation of electromagnetic pulses at velocities greater than the speed of light, “superluminal” propagation, has been the source of interest in the scientific community for sometime. In a recent Nature article [1], researchers reported observing superluminalpropagation of a laser pulse in a gain medium by a new mechanism. This article received a great deal of attention in both the scientific community and the world press. In this talk we show that superluminal propagation is actually due to a differential gain mechanism [2] in which the front of the pulse is amplified more than the back. Differential gain or absorption can give the appearance of faster-than-light propagation.

[1] L. J. Wang, A. Kuzmich, and A. Doggariu, Nature (London) 406, 277 (2000)

[2] P. Sprangle, J. R. Peñano, and B. Hafizi, Physical Review E, 64, 026504 (2001)


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Absorptive Medium

Im(Dn)103

Re(Dn)103

(bp-1) )103

bg


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Excited Medium

Im(Dn)103

Re(Dn)103

(bp-1) )103

bg


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