SPHERICAL ABERRATION Beam expander with spherical lenses Far

SPHERICAL ABERRATION Beam expander with spherical lenses Far field Experiment to measure: Michelson interferometer

Calculation of the diffracted pattern of a distorted wavefront Maxwell Paraxial approximation Fourier transform Solution: Fourier transform of measured at near field Inverse Fourier transform: Far field

Most general tool to solve a general propagation problem Maxwell Paraxial approximation Fourier transform Solution: That was our first approach to derive the Gaussian beam propagation Its Fourier transform is: The initial field does not need to be a Gaussian.

Example of application: find the beam profile of a laser. space I FT (lens) FT FT FT I FT (lens)

Space-time analogy SPACE TIME Fourier transform in time Fourier transform in space

Space-time analogy True for all pulse/beam shapes Paraxial approximation (use of Fourier transforms) Gaussian beams (q parameters and matrices) Geometric optics? ? a dimension of 1/v a dimensionless

Space-time analogy Geometric optics d 2 e(r) e(-r/M) d 1 SPACE DIFFRACTION By matrices:

Space-time analogy Geometric optics d 2 e(t) e(--t/M) d 1 TIME DISPERSION By matrices: y length in time T = chirp imposed on the pulse

Space-time analogy Gaussian optics d 2 e(r) e(-r/M) d 1 SPACE DIFFRACTION By matrices:

Space-time analogy Gaussian optics d 1 d 2 e(t) e(--t/M) TIME DISPERSION By matrices: = chirp imposed on the pulse Find the image plane:

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

TIME MICROSCOPE d 2 e(r) e(-r/M) d 1 DIFFRACTION e(t) d 1 d 2 DISPERSION e (t) 1 DISPERSION TIME LENS

TIME LENS DISPERSED INPUT e (t) 1 w 1 + wp TIME LENS OUTPUT e (t)e 1 wp e (t) = ee CHIRPED PUMP p iat 2

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 timemultiplexed signals EMITTER RECEIVER 1 ns o me ess Ti mpr co e r m Ti tche e str r time, ps 4 3 2 1 0

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

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