Space-time analogy True for all pulse/beam shapes Paraxial approximation (use of Fourier transforms) Gaussian beams (q parameters and matrices) Geometric optics?? a dimensionless a Dimension of time

SPACE TIME Fourier transform in time Fourier transform in space

e(-r/M) e(r) Space-time analogy Geometric optics d1 d2 SPACE DIFFRACTION DIFFRACTION By matrices:

e(--t/M) e(t) Space-time analogy Geometric optics d1 d2 TIME DISPERSION DISPERSION By matrices: y length in time T = chirp imposed on the pulse

e(-r/M) e(r) Space-time analogy Gaussian optics d1 d2 SPACE DIFFRACTION DIFFRACTION By matrices:

e(--t/M) e(t) Space-time analogy Gaussian optics d1 d2 TIME DISPERSION DISPERSION By matrices: = chirp imposed on the pulse Find the image plane:

WHAT IS THE MEANING k”d? Lf Fiber L Prism Lg b Gratings d Fabry-Perot at resonance

e(-r/M) e(r) e(t) e1(t) TIME MICROSCOPE d1 d2 d2 d1 TIME LENS DIFFRACTION DIFFRACTION e(t) d2 d1 DISPERSION e1(t) TIME LENS DISPERSION

e1(t) e1(t)eiat ep(t) = eeiat w1 w1 + wp wp TIME LENS DISPERSED INPUT OUTPUT w1 + wp e1(t)eiat 2 wp CHIRPED PUMP ep(t) = eeiat 2

e e e e e e e e e e e (a) (a) (b) (b) x x d d d d image image object 1 1 2 2 image image object object e ( r ) e e e ( - r/M ) e e ( ( ) ) ( ( - - r/M r/M ) ) r r (a) (a) diffraction diffraction diffraction diffraction e ( t ) e e ( ( t t ) ) e ( - t/M ) e ( - t/M ) TIME TIME LENS LENS (b) (b) dispersion dispersion dispersion dispersion

FEMTOSECOND COMMUNICATION: Space-time analogy – application to fs communication FEMTOSECOND COMMUNICATION: Commercial fs lasers – a pulse duration of 50 fs. (20 THz) One can easily “squeeze” a 12 bit word in 1 ps

Propagation of time- multiplexed signals Time compressor Time EMITTER RECEIVER 1 ns 1 ns 1 ns 1 ns Time compressor Time stretcher time time, ps 4 3 2 1

Time “telescope” (reducing) Time “microscope” (expanding)

Time matrices – application to laser cavity

y’’ L n2

In terms of modes, this cavity is equivalent to g Gaussian mirror (localized gain) In terms of modes, this cavity is equivalent to

Chirp evolution using ABCD matrix in time Cavity in time Cavity in space Dispersion Kerr f = R/2

Damping effect Ti:Saph laser with wavelength at 795nm, beam waist of 211μm, cavity length d = 89cm and equivalent radius of curvature R = 92.5cm. The damping coefficient is = 0.01. Ti:Saph laser with wavelength at 770nm, pulse width about 100fs and pulse energy fixed around 27.5nJ. The Ti:Saph crystal is 3mm long with a Kerr coefficient of 10.5×10−16 cm2/W. The dispersion of the cavity is -800fs2. The damping coefficient is = 0.01.

Oscillating solution and damping If p1 starts with some departure from equilibrium: = Damping in a real laser results from the balance of gain and losses. Mathematically introduce a phenomenological damping coefficient ε Ti:Saph laser with wavelength at 770nm, pulse width about 100fs and pulse energy fixed around 27.5nJ. The Ti:Saph crystal is 3mm long with a Kerr coefficient of 10.5×10−16 cm2/W. The dispersion of the cavity is -800fs2. The damping coefficient ε = 0.01. (does not affect the oscillation period)

Saturable gain off-resonance: d Saturable gain off-resonance: g (localized gain) No more Gaussian No more damping required

y’’ Gaussian approximation no longer valid L Paraxial still valid F n2 (localized gain)

y’’ L F n2 L d g A (localized gain)

SPACE TIME Fourier transform in time Fourier transform in space