CAVITIES AND STABILITY Typical cavity A D2 1

CAVITIES AND STABILITY

Typical cavity (A + D)/2 = (1 -2 L/R) = = Stability: SPECIAL CAVITIES Concentric Confocal Flat-flat L=R L = R/2 L=0 -1 -2 0 -1 1 0 0 -1 L 0 2 L 1 0 L L

Review The meaning of “stability” 2

Thin lens: renormalization q is a complex number that repeat itself in phase and amplitude for the stable solution.

For a departure from equilibrium: That means that the complex vector Dq rotates, and its components oscillate -i/r 1/q q 1/R Matrix multiplication, starting at w 0 = 0. 2 instead of 0. 5. w 0

For a departure from equilibrium: Undamped oscillations as a function of N How do you damp them? Add an imaginary part ie to q Modified ABCD matrix for the flat-curved cavity:

The meaning of “stability” Multiplying the matrices, we converge to the stable solution

Final project d = 1 cm Lens K: f = 6 m L d L=1 m M : Radius of curvature R = 3 m M K Part I: no aperture Beam size and curvature at K (analytic) Start with 80% of w at K, find w, R at every round-trip. Use a stabilizing e

L Part II: Aperture (slit) at A d A M K

GAUSSIAN PULSE PROPAGATION 1) Review of past approach 2) More expedient: use Fourier transform of the wave equation

Study of propagation from second to first order

From Second order to first order (the tedious way) (Polarization envelope)

Gaussian pulses Bandwidth limited: Pulse duration FWHM Fourier transform Chirped Gaussian Fourier Transform Bandwidth duration product

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

Fourier transform of Maxwell’s equation in TIME Plane wave F. T. solution Centered at w Shift k is the first order of the expansion (in frequency domain) Shift property:

First derivative of k(W): group velocity. Instead of we have In the Fourier domain: next order in the development of k(W)

Study of linear propagation Expansion orders in k(W)--- Material property

Retarded frame

Pulse broadening, dispersion Spectral phase Electric field amplitude w 5 w 4 w 3 w 2 0 w 1 w 3 w 5 W time w 1 Spectral phase W z = ct z = v 1 t (fast) z = v 2 t (slow)

Pulse broadening, dispersion E(t) Broadening and chirping Electric field amplitude time z = v 2 t (slow) time z = v 1 t (fast) z = ct

Pulse compression through chirping, E(t) Chirping and compression Electric field amplitude time z = v 2 t (slow) z = v 1 t (fast) time z = ct

PHASE MODULATION e E(t) = (t)eiwt-kz e(t, 0) n(t) or k(t) e(t, 0) eik(t)d d DISPERSION e(DW, 0) n(W) or k(W) e(DW, 0)e-ik(DW)z

Propagation in the time domain PHASE MODULATION Time lens: k(t)d = j(t) = (t 2/f. T 2) e E(t) = (t)eiwt-kz e(t, 0) n(t) or k(t) e(t, 0) eik(t)d

Propagation in the frequency domain DISPERSION e(DW, 0) n(W) or k(W) e(DW, 0)e-ik(DW)z Retarded frame and taking the inverse FT:

linear propagation of Gaussian pulses 0 w W

Pulse propagation through 2 mm of BK 7 glass -20 -10 0 10 20 1 0 -1 -6 -4 -2 0 2 4 6