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  sec 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” (  ~  o): – 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*cos  t:

8. Section 2: Working Principles - Dispersion Relative displacement between the (-) e-cloud and the (+) nucleus: Note: –  <  o : x(t) and E(t) in phase –  >  o: 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  o: –Absorption (damping term added): –Local electric field effect:

11. Section 2: Working Principles - Dispersion Dispersion Eq:  oj^2>>  ^2: n gradually increases w/ frequency (Normal Dispersion)  oj^2<<  ^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-  t) Vp =  /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 and 1>2 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  centered about  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=  /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/d is plus, then ng > 1 vg < c If dn/d is minus, then ng 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  t = L/vg – L/c = (ng-1)L/c Negative velocity requires a (-) ng, then  t < 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, 4 th 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