1. Why Do CPO Lead to Slow Light When the Laser
Linewidth Exceeds the Width of the CPO Transparency Window?
Giovanni Piredda and Robert W. Boyd
University of Rochester

2. Coherent Population Oscillations
The technique of Coherent Population Oscillations (CPO) allows control of the velocity of light. 2 b a = T j b i ! p ! s b a = 1 T j a i s i g n a l m e a s u r b o p t i n d l y p u m m a t e r i l d u

3. CPO in a graph

4. CPO has been used with modulation bands ranging from Hz [1] to THz [2].
In some of these experiments, for example in (1), slow light is observed even if the bandwidth of the laser, on the order of 1 GHz, is much larger than the width of the CPO hole (37 Hz).
This is possible because CPO is an effect that is sensitive to light intensity, not field.
[1] M. S. Bigelow et al, PRL 90, (2003)
[2] M. van der Poel et al, Optics Express 13, 8032 (2005)

5. CPO from rate equationsd n t = I ( ) N 2 @ I x = ( N 2 n )
Assume an intensity of the form I = + 1 e x p ( i ! t )
and solve using the trial solution n = + 1 e x p ( i ! t )

6. CPO from rate equations
Neglect second order and collect terms with the same time dependence; the solutions are n = 1 2 N I s a t + n 1 = 2 ( N ) I s a t + i !
Using these solutions we obtain equations for I0 (the pump) and I1 (the modulation) @ I x = N 1 + s a t @ I 1 x = N + s a t [ i ! ( ) 2 ]

7. CPO from rate equations@ I 1 x = N + s a t [ i ! ( ) 2 ]
In the equation for I1 two features appear:
a dip in the absorption (the CPO hole)
a phase shift of the modulation
The phase shift leads to the slow light effect; it is related to the group velocity of light, because I1 represents a modulation

8. The comparison with experimental results is excellent@ I 1 x = N + s a t [ i ! ( ) 2 ]
Figures from M. S. Bigelow et al, PRL 90, (2003)

9. # CPO with non-collinear pump and signal
The delay that can be achieved using CPO is limited by pump absorption; in order to reach large fractional delays it is necessary that the pump intensity is uniform along the propagation direction of the signal.
A way to achieve this is to let pump and signal propagate in different directions s i g n a l # m a t e r i l d u p u m
Pei-Cheng Ku et al., OL 29, 2291; Su and Chuang, OL 31, 271

10. CPO with non-collinear pump and signal
The refractive index for the signal beam can be calculated using the density matrix equation of motion for the two-level atom 2 b a = T j b i ! p = b a ! s b a = 1 T j a i b a ( t ) = i T 2 ~ E p e ! + s [ ] b a = ( ) + e i t

11. CPO with non-collinear pump and signalb a = ( ) d c + e i t ( b a ) = 2 T 1 i +
population oscillation = 1 T + 4 2 j b a ~ E p s
power broadened linewidth = 2 T j b a ~ E p s
amplitude factor
the polarization drives
the wave equation r 2 E + 1 c @ t = 4 P

12. CPO with non-collinear pump and signal
The following term appears in the polarization: E p ( E s p ) e i ! t n / E s p E s
It represents the scattering of the pump beam off the population grating created by the beating of the pump with the signal; it influences the absorption and refractive index of the signal @ E s z = 2 ! c à N j b a T ~ 1 + 4 p i

13. CPO with non-collinear pump and signal@ E s z = 2 ! c à N j b a T ~ 1 + 4 p i

14. Conclusions
A laser with a 1 GHz bandwidth can burn a 37 Hz hole because CPO is an effect that is sensitive only to intensity modulations.
The CPO effect is the same for every angle between the propagation directions of pump and signal.

15. Acknowledgements
Prof Boyd’s research group, Aaron Schweinsberg in particular
DARPA and NSF for financial support