+ applications

{ Femtosecond Optics 1. Dispersion 2. Interferometer to measure linear dispersion 3. Wave front and energy front A general rule Grating Interface Pair of prisms Applications { Traveling wave amplifier Single shot autocorrelator Femtonitpicker Other diagnostic with pair of prisms

Wave front and energy front db tan g = l dl

b = q - q db tan g = l dl q q q q q q r i Tilting at an interface TAC = L sin qi /c q q Pulse n sin = sin r i TBD = L sin qr n/c TBD – TAC = -L sin qr /vg + L sin qi /c = L sin qr (n/c – 1/vg) = L sin qr l dn c dl DC = L cos q r A q i q i B C D’ g q r D

Consequences of an energy front tilt Focusing: expanded focal region, less intensity Cavity – pulse stretching across the transverse direction. Remedies: image reversal pairs of elements

Energy front & Phase front Fermat’s principle  OPL  Phase velocity Energy front (pulse propagation)  Group velocity

Wave front and energy front Group delay Wave front and energy front db tan g = l db g = tilt of energy front = angular dispersion dl dl Group velocity dispersion There is also a relation between GVD and angular dispersion

Whether considering group delays or group velocity dispersion (GVD), we will consider sufficiently broad beams, and sufficiently short propagation distances Lp behind the element, such that diffraction effects can be neglected. Q S0, SW S’0 S’W a P0 PW rW r0 L

Q S0, SW S’0 S’W a P0 PW rW r0 L

The most widely used optical devices for angular dispersion are prisms and gratings. To determine the dispersion introduced by them we need to specify not only a(W), but also the optical surfaces between which the path is being calculated. The ``dispersion'' of an element has only meaning in the context of a particular application, that will associate reference surfaces to that element. Indeed, we have assumed in the previous calculation that the beam started as a plane wave (plane reference surface normal to the initial beam) and terminates in a plane normal to the ray at a reference optical frequency $\omega_\ell$. The choice of that terminal plane is as arbitrary as that of the reference frequency $\omega_\ell$ (cf. Chapter 1, Section~\ref{Fourier-rep}). After some propagation distance, the various spectral component of the pulse will have separated, and a finite size detector will only record a portion of the pulse spectrum. 10

a h L A B L g C (R)

2

INTERFEROMETER TO MEASURE DISPERSION No dispersion CW interference Pulse  zero delay group velocity Fringes phase velocity Phase velocity

Interferogram without sample, and its Fourier Transform -6 -4 -2 2 4 6 -1 1 -20 -10 10 20 Delay (fs)

Interferogram for 2 mm of glass, spectrum and spectral phase. Reminder White light interferometry was for dispersion measurement 30 60 90 120 -1.0 -0.5 0.0 0.5 1.0 Normalized Intensity (a.u.) Relative Delay ( m m) 100 200 300 400 Relative Delay (fs) m Wavelength ( m) 1.2 1 0.8 0.6 5 1.0 Spectrum Phase 0.8 Polynomial Fit -5 Spectral Amplitude (a.u.) 0.6 Spectral Phase (rad) -10 0.4 -15 0.2 -20 0.0 -0.5 0.0 0.5 1.0 W (1/fs)

{ Femtosecond Optics 1. Dispersion 2. Interferometer to measure linear dispersion 3. Wave front and energy front A general rule Grating Interface Pair of prisms Applications { Traveling wave amplifier Single shot autocorrelator Femtonitpicker Other diagnostic with pair of prisms

I don’t have a fs detector… There’s a hole in my bucket, dear Liza, dear Liza There’s a hole in my bucket, dear Liza, a hole Then fix it, dear Henry, dear Henry, dear Henry Then fix it, dear Henry, dear Henry, then fix it With what shall I fix it, dear Liza, dear Liza? With some straw… The straw is too long, dear Liza, dear Liza Then cut it…. In what shall I get it, dear Liza, in what? In a bucket dear Goofy, dear Henry, dear Henry

E(W)eif(W)

td f(W) Either accurate spectral amplitude OR accurate spectral phase Cross-correlator td f(W)