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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

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

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:

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

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

VELOCITIES IN WAVEGUIDES a) Metallic b) Fibers

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

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

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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

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.

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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

t1 A B z t t = z/c t =- z/c

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

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

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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

Uniaxial crystals ne does not depend on j

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

Orienting Uniaxial crystals k z x, y or r q b ne (q) ne no L qc

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