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1 Index of refraction varies significantly for broadband pulses Δt=10 fs Δλ =90nm index of refraction may vary by nearly 1% phase speed depends on n v φ (λ) = c n(λ) n phase relations will be lost as pulse propagates! fused silica visible spectrum

2 Chromatic dispersion From French < Latin < Greek khrōmatikos, from khrōma, khrōmat- color n 1 sin( θ 1 ) = n 2 sin( θ 2 ) thus refraction depends on speed of phase fronts thus refraction depends on index of refraction when the index of refraction depends on wavelength, it follows that colours of light will disperse through different angles this is chromatic dispersion spreading our according to colour Phase front matching leads to Snell s Law:

3 Group and phase speed Group and phase speeds may differ no dispersion in this example, envelope propagates true dispersion important in laser pulse formation (and esp. in solitons) here v φ > c, and v g <c information travels at v g, doesn t violate speed of light v φ is a relationship quantity, not a physical quantity

4 Decomposing pulses frequency chirping An example: WWI radio operators heard funny sliding tones on their radio sets, which they termed whistlers turned out to be the signal from lightning strikes in the opposite hemisphere the lightning strikes excited EM pulses in the ionosphere these EM waves interact with ionized gas trapped on magnetic field lines the phase speed depends strongly on wavelength the high-frequency parts of the pulse travel faster and arrive first, rest arrives later

5 Dispersion of acoustic pulses Boundary conditions also affect dispersion: through the boundaries, wave interact with themselves and then also can be attenuated these changes produce modal dispersion this applies for acoustic waveguides this also applies for optical waveguides much of the observed groupvelocity dispersion in fibers is due to modal dispersion fiber size can be designed to affect modal dispersion

6 Modal dispersion Different normal modes have different dispersion relations relation of ω and k z (also called β) sometimes called ω β curve ω = c k 2 z + m π 2 L x + n π L y Modes have a lowest cut-off frequency (except for (0,0) mode) these cut-offs are related to the dispersion experienced by a pulse in that mode the (0,0) mode has no cutoff, and no dispersion 2 ω (0,0) (0,1) (1,0) (1,1) (1,2) (2,1) (2,2)

7 Modal dispersion II For a given mode: phase speed v φ (ω) v φ (ω o ) ω o k(ω o ) group speed v g (ω) v g (ω o ) k(ω) ω ωo add span of broad bandwidth importance of dispersion in pulse formation (and solitons?) acoustic demos, whistlers, etc material and modal dispersion techniques of measurement 1 ω v φ v g (1,0)

8 Group velocity dispersion The broadband frequency components of an ultrafast pulse see different group velocities the pulse is teased apart by frequency

9 Pulse compression and stretching A stretched pulse can be compressed by the right sign of dispersion. In the fiber laser, two kinds of fiber stretch oppositely; in each segment the pulse compresses and stretches again. Kerr rotation happens twice.

10 Chirping of pulse Sonogram of acoustic pulse chirped in waveguide

11 Causes of group velocity dispersion Group velocity dispersion (GVD) may result from material dispersion: intrinsic dependence of the phase speed on frequency, in that material modal dispersion: dispersion of the overall or effective propagating wave resulting from the boundary conditions of a waveguide other interactions: multilayer mirrors, interactions with resonant waves cf. whistler waves interact with electron cyclotron motion These actually each come down to the Kramers-Kronig relation the real and imaginary parts of the index of refraction are tied e.g., absorption features in the ultraviolet produce real-index changes in the visible, in fused silica e.g., boundary conditions exclude certain frequencies of a waveguide; for these, waves are evanescent and Im[n]>0. this, the Re[n] of the propagating part changes From

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