Fast Light, Slow Light

1. Fast Light, Slow Light David Jun 3/8/05

2. Outline: • Section 1: Introduction • Definitions • Recent Studies • Motivations • Section 2: Working Principles • Dispersion • Wave Velocities (V_phase and V_group) • Group index of refraction • Section 3: Controversy/Debate • Einstein’s Theory of Special Relativity • Signal Velocity • Section 4: Conclusions and Reference

3. Section 1. Introduction • Definitions: • Fast light (superluminal light): vg > c or vg is negative • Slow light (subluminal light): vg << c • “Stored” or stopped light: vg ~ 0 • Recent Studies: • A group from Univ. of Rochester used an alexandrite crystal to reduce the speed of light to 91m/s and minus 800m/s (M. Bigelow et al. 2003 Science 301 200) • Connie J. Chang-Hasnain (UC Berkeley), Hailin Wang (Uoreg), Shun-Lien Chung (UIUC) slowed down the group velocity of light to about 6 mi/sec in semiconductors (Oct 1, 2004, Optics Letters) • A group from Harvard University stopped light particle in their tracks for 10~20 msec in rubidium gas

4. Section 1. Introduction • Motivations: - Motivated by uncovering new physical phenomena - Practical applications: • High performance communications • Controllable optical delay lines • Optical data storage • Optical memories • Devices for quantum information

5. Section 2: Working Principles - Dispersion • Dispersion: • All material media w/ the exception of vacuum is dispersive; meaning its index of refraction is frequency dependent • From Maxwell’s Equations and Wave equation: • Index of refraction:

6. Section 2: Working Principles - Dispersion • Assume magnetically “simple” (m~mo): • n subject to an applied electric field: • Polarization: • Different polarization results depending on the frequency of the incident electromagnetic wave

7. Section 2: Working Principles - Dispersion • Analytic expression for Dispersion: • Forced Oscillator model: Total force on an electron due to E(t) = Eo*coswt:

8. Section 2: Working Principles - Dispersion • Relative displacement between the (-) e-cloud and the (+) nucleus: Note: • w<wo : x(t) and E(t) in phase • w>wo: x(t) and E(t) out of phase

9. Section 2: Working Principles - Dispersion • Density of dipole moment (polarization): since (Dispersion Equation)

10. Section 2: Working Principles - Dispersion • Complications/Implications/Corrections: • Multiple natural frequency wo: • Absorption (damping term added): • Local electric field effect:

11. Section 2: Working Principles - Dispersion • Dispersion Eq: • woj^2>> w^2: n gradually increases w/ frequency (Normal Dispersion) • woj^2<< w^2: n gradually decreases w/ frequency (Anomalous Dispersion)

12. Section 2: Working Principles – Wave Velocities • Wave Velocities: • Phase Velocity (vp): speed at which any fixed phase or the shape of the wave is moving example: E(t,x) = Eo*cos(kx-wt) Vp = w/k

13. Section 2: Working Principles – Wave Velocities • Group Velocity (vg): speed of the overall shape (modulation envelop) of the wave’s amplitude example: consider two harmonic waves where k1>k2 andw1>w2 where

14. Section 2: Working Principles – Wave Velocities • Overall wave: where the former = carrier wave the latter = modulation envelop

15. Section 2: Working Principles – Wave Velocities If the frequency range Dw centered about w is small,

16. Section 2: Working Principles – Wave Velocities The relationship between Vp and Vg in a non-dispersive and dispersive system: In a non-dispersive system (vacuum): • Suppose two different traveling harmonic waves with the same phase velocity (v=v1=v2): vp=v=w/k Phase velocity independent of wavelength (dv/dk=0) therefore, vp=vg=v1=v2 In a dispersive system:

17. Section 2: Working Principles – Group Index of Refraction • Group index of refraction (ng): vg = c/ng where Consequences: If dn/dn is plus, then ng > 1 vg < c If dn/dn is minus, then ng < 1 vg > c If dn/dv is minus and large, then ng is (-) (-)vg

18. Section 2: Working Principles – Negative Group Velocity • Negative group velocity: • The peak of the emerging pulse occurs at an earlier time than the peak of the incident pulse. Example: Consider a pulse traversing a medium of length L: t_traverse in the medium=L/vg t_traverse in vacuum = L/c Delay time Dt = L/vg – L/c = (ng-1)L/c Negative velocity requires a (-) ng, then Dt < 0 (i.e. pulse arrives early)

19. Section 3: Controversy/Debate • Einstein’s Theory of Special Relativity: Nothing can travel faster than the speed of light (not exactly, examples: vg>c, motion of a spotlight projected on a distant wall) • With the Principle of Causality, the relativity theory says No signal (information) or energy can exceed the speed of light • Characteristics of a “signal”: A train of oscillations that starts from zero at some point and end at some point. (a pulse, not a simple periodic wave) Question: what about vg > c?

20. Section 3: Controversy/Debate A group velocity greater than c does not contradict relativity because group velocity is not in general a signal velocity.

21. Section 3: Controversy/Debate In anomalous dispersion: If vg > c, essentially the entire transmitted pulse is a “reconstruction” of a tiny, early-time tail of the incidence pulse. No new information is transferred.

22. Section 4: Conclusions and References • Conclusions: • A light pulse can propagate with a group velocity exceeding or below “c” due to dispersion • Faster-than-c group velocities do not violate Einstein’s Theory of Relativity because the group velocities do not represent the speed at which real information or energy is moving • References: • The speed of information in a “fast-light” optical medium, M.S. Bigelow et al., Science 301, 200-2 (2003) • Optics, Eugene Hecht, 4th Edition • Fast light, Slow light, Raymond Y. Chiao and Peter W. Milonni, Optics and Photonics News, June 2003 • Gain-assisted superluminal light propagation, L.J. Wang et al., Nature, Vol 406, July 2000