1. 1. Uniform dielectrics, not absorbing
VELOCITIES
1. Uniform dielectrics, not absorbing
2. Waveguides
Metallic
Fibers
3. In uniform, isotropic saturable gain media
4. In uniform, isotropic saturable absorbers
5. Crystal velocity
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2. E(t) Broadening and chirping time Electric field amplitude time
Pulse broadening, dispersion, in a uniform dielectric
E(t)
Broadening
and
chirping
time
Electric field
amplitude
time
z = v1t
(fast)
z = v2t
(slow)
z = ct

3. Solution of 2nd order equation
Group delay in uniform dielectric
Solution of 2nd order equation
Propagation through medium
No change in frequency spectrum z W
Z=0
To make F.T easier shift in frequency
Expand k value around central freq wl
Expand k to first order, leads to a group delay:

4. Group velocity dispersion
Expansion orders in k(W)--- Material property

5. In uniform lossless dielectrics
Phase velocity Group velocity (delay) Group velocity dispersion
Units mm/fs fs/mm fs2/mm
In fibers: ps/(km/nm)
Same definitions in transmission lines
Not the same in waveguides

6. VELOCITIES IN WAVEGUIDES
a) Metallic
b) Fibers

7. a) Velocities in metallic waveguides
TE: f=0 H
TM: f=p
Phase velocity:
Group velocity:

8. a) Velocities in fibers
Simple step index fiber
V = 2.405
Propagation constant b

10. 1. Uniform dielectrics, not absorbing
VELOCITIES
1. Uniform dielectrics, not absorbing
2. Waveguides
Metallic
Fibers
3. In uniform, isotropic saturable gain media
4. In uniform, isotropic saturable absorbers
5. Crystal velocity
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11. Pulse velocities in uniform, isotropic saturable gain media
k = k’ – i a
Phase velocity
Group velocity?
Usually described as
Wrong!
Numerous “slow light” papers dedicated to this concept!
Basov – Soviet Physics JETP 23:16-22 (1966)
Superluminal pulse velocity in an amplifier.
Pulse generated by Q-switching: exponential rise.
Pulse front amplified, pulse tail not because of saturation or depletion.

13. 1. Uniform dielectrics, not absorbing
VELOCITIES
1. Uniform dielectrics, not absorbing
2. Waveguides
Metallic
Fibers
3. In uniform, isotropic saturable gain media
4. In uniform, isotropic saturable absorbers
5. Crystal velocity
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14. t1 A B z t
t = z/c
t =- z/c

15. B A t2 t1 z t3 t4 t5
t =- z/c t
t = z/c

16. B A t1 z t2 t3
t =- z/c t
t = z/c

17. 1. Uniform dielectrics, not absorbing
VELOCITIES
1. Uniform dielectrics, not absorbing
2. Waveguides
Metallic
Fibers
3. In uniform, isotropic saturable gain media
4. In uniform, isotropic saturable absorbers
5. Crystal velocity
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18. Uniaxial crystals
ne does not depend on j

19. z Uniaxial crystals k E Walk-off angle: no ne (q) D q E x H = s b ne
Equation of the index ellipse:
Walk-off angle: z k E no E
ne (q) D q
E x H = s b ne
x, y or r

20. Orienting Uniaxial crystalsk z
x, y or r q b
ne (q) ne no L qc

21. z Velocities in Uniaxial crystals qc E k L E no vp = c/ne (q) q b ne
vr = ray velocity
x, y or r L