# What does gating problem mean

## Characterization of ultrashort light pulses

• Characterization of ultrashort light pulses By Tobias Caspers, supervised by Dr. Frank Noack, As part of the training seminar “Short Pulse Lasers and Applications” with Prof. Dr. Ingolf Hertel.

• General: measurement of short events

• 0. Topic overview • Classification of the achievable parameters • “Simple” measuring devices • Correlation • Measurement of the pulse course: FROG and SPIDER

• 1. Classification of the achievable parameters • Pulse duration: 3fs • Bandwidth: a few hundred THz • Power: up to 1000 TW • Intensity: up to

• 2. "Simple" measuring devices • Energy: Pyroelectric Detector • Power: Photodetector • Intensity vs. Time: Photodiode • Intensity vs. Time: Streak Camera

• 2.1 Pyroelectric detector Basically: ion crystals with structure-related spontaneous polarization, e.g. tourmaline (aluminum borosilicate). Effect: When heated, a voltage builds up proportional to the rise in temperature. Compare piezo crystal!  Use for energy measurement. • Typical data: • Energy measurement up to a few mJ possible. • Response time: a few ms • Electrical response: a few V / mJ

• 2.2 The photodetector Basically: Thermocouples consist of two metal contact points, one of which is kept constant at a reference temperature and the other is brought to the temperature to be measured. Caution: Only to measure the average power Some data:

• 2.3 The photodiode Basically: pn junctions operated in reverse direction reverse current proportional to the illuminance. Some data:

• 2.4 The streak camera Structure: temporal Resolution: Approx. 1 ps

• Reconciliation Finding: There is the following problem: The measuring devices available to us have far too long response times. Solution?

• Reconciliation Finding: There is the following problem: The measuring devices available to us have far too long response times. Solution: CORRELATION FUNCTIONS! Because: If known, it can be determined from (measurably!).

• 3. Something about correlation functions Definitions: Fourier transformation: Cross correlation function: Convolution: Autocorrelation function:

• 3. Some properties of correlation functions Correlation or convolution theorem: Let and Then: and Parseval's theorem:

• 3. Some examples of autocorrelation functions

• 3. Be careful! The autocorrelation is not objective Illustration!

• Transition: Experimental Realization Statement: “normally” E-fields are superimposed additively according to the superposition principle, but we need multiplicative superimposition! Solution?

• Transition: Experimental Realization Statement: “normally” E-fields are superimposed additively according to the superposition principle, but we need multiplicative superimposition! Solution: The prerequisite for the validity of the superposition principle is the linearity of the E-field • NON-LINEAR OPTICS! This means that higher order terms follow:

• 4. FROG & SPIDER • Prerequisite: non-linear optics • Objective • Preliminary stages: Cross correlation and autocorrelation • FROG • SPIDER

• 4.1 Non-linear optics Wave equation: linear case: non-linear case: with gives: ....

• 4.1 Non-linear optics Nonlinear case: With results in: .... SHG SFG DFG

• 4.1 Non-linear optics The terms are written out again: SHG: = Second Harmonic Generation SFG: = Sum Frequency Generation DFG: = Difference Frequency Generation Correspondingly, higher-order terms include: THG: = Third Harmonic Generation ....

• 4.1 Non-linear optics Other important third-order effects: SD: = Self Diffraction

• 4.1 Non-linear optics Further important third-order effects: PG: = Polarization Gating

• 4.1 Non-linear optics Summary: In non-linear media, different rays can influence each other and create new rays. The effects required here are: SHG: PG: SD: THG: 2nd order effects 3rd order effects

• 4.2 Objective ?: Measurement of the actual signal course !:

• 4.2 Objective How do you measure something like that? If E (t) is the waveform to be measured, the spectrogram is: See Fourier Trafo: where g (t-t) is the variable-delayed gate function and t is the delay. Without g (t-t), SpE (w, ) would simply be the spectrum.

• Spectrogram

• 4.3a preliminary stage: cross-correlation Apart from the square, it is comparable to a cross-correlation. The prerequisite, however, is that a correspondingly short gate function is available! (Reminder):

• Reconciliation But what do you do when the signal to be measured is already the shortest that can be generated?

• Reconciliation But what do you do when the signal to be measured is already the shortest that can be generated? You measure it with yourself!  autocorrelation

• 4.3.b autocorrelation Crossing beams in an SHG crystal, varying the delay between them, and measuring the second-harmonic (SH) pulse energy vs. delay yields the intensity Autocorrelation: Input pulse aperture eliminates input pulses and also any SH created by the individual input beams. Mirror Beam-splitter SHG crystal Slow detector E (t) Mirrors E (t – t) Lens Delay The Intensity Autocorrelation:

• 4.3.b Autocorrelation in single shot mode It is often difficult to carry out a new measurement for each delay. • Is it possible to query different delays when measuring a pulse? Yes you can:

• 4.3.b Autocorrelation in single shot mode Again in detail:  Time delay manifests itself on the local axis via:

• 4.3.b autocorrelation some examples

• 4.3.b autocorrelation some examples

• 4.3.b autocorrelation some properties • Symmetrical around • Maximum value at • Not unique! Loss of information • The autocorrelation provides information about the pulse duration: And for convolutions, the following applies: And thus: Caution! dependent on pulse shape.

• Summary: We can measure: • Energy • Power • Spectrum • Pulse duration (with restrictions) Is that enough?

• Summary: We can measure: • Energy • Power • Spectrum • Pulse duration (with restrictions) Is that enough? No! We have no information about the phase!

• Cutscene: The Importance of the Phase The pulses shown here only differ in phase:

• Cutscene: The Importance of the Phase The phase contains the time (or here: the location) information. Clear: frequency information from top left, phase information from top right. Frequency info from top right, phase info from top left.

• 4.4.a FROG Frequency Resolved Optical Gating Problem: The complete signal information cannot be obtained from the autocorrelation, since the one-dimensional phase restoration problem cannot be solved! Solution: The two-dimensional phase restoration problem can be solved (except for trivial ambiguities)! (Reason :) The fundamental theorem of algebra is valid in one, but not in two dimensions.

• 4.4.a FROG This 2-dimensional problem arises as follows: Simultaneous measurement of the spectrum and the delay. (Delay: as with one shot autocorrelation spectrum: perpendicular to this, the spectral distribution of the signal) This results in the so-called FROG trace: With gives:

• 4.4.a FROSome FROG traces Frequency Time Frequency Delay

• 4.4.a FROSome FROG traces, somewhat more complex Intensity Frequency Time Frequency Frequency Delay Delay

• SHG FROG Measurements of a 4.5-fs Pulse! Baltuska, Pshenichnikov, and Weirsma, J. Quant. Electron., 35, 459 (1999).

• Single-shot polarization gate FROG Kane and Trebino, Opt. Lett., 18, 823 (1993).

• FROG geometries: Pros and Cons Second-harmonic generation most sensitive; most accurate Third- harmonic generation tightly focused beams useful for UV & transient-grating experiments Transient- grating simple, intuitive, best scheme for amplified pulses Polarization- gate Self- diffraction useful for UV

• 4.4.b SPIDER functional principle Chirped pulse t This pulse sums with the green part of the chirped pulse. This pulse sums with the blue part of the chirped pulse. t t t SFG The output pulse is split into two partial pulses: One experiences a frequency chirp, the other is delayed. Phase information results from the superposition!

• 4.4.b SPIDER spectral phase interferometry for direct electric-field reconstruction Structure: Participating filters: Temporal Phase Modulator Spectral Phase Modulator

• 4.4.b SPIDER The following results for the combined signal: If the response function of the spectrometer is assumed to be delta-shaped, the result is: