Multiphoton Intrapulse Interference Phase Scan - Theory

Theory

A MIIPS-based device consists of two basic components controlled by a computer: a pulse shaper (usually a liquid crystal based spatial light modulator - SLM) and a spectrometer. The pulse shaper allows manipulation of the spectral phase and/or amplitude of the ultrashort pulses. The spectrometer records the spectrum of a nonlinear optical process such as second harmonic generation produced by the laser pulse. The MIIPS process is analogous to the Wheatstone bridge in electronics. A well-known (calibrated) spectral phase function is used in order to measure the unknown spectral phase distortions of the ultrashort laser pulses. Typically, the known superimposed function is a periodic sinusoidal function that is scanned across the bandwidth of the pulse.

MIIPS is similar to FROG in that a frequency trace is collected for the characterization of the ultrashort pulse. In Frequency-resolved optical gating, a FROG trace is collected through scanning the ultrashort pulse across the temporal axis, and detecting the spectrum of the nonlinear process. It can be expressed as

I(\omega,\tau)=\left|\int{E(t)g(t-\tau)e^{i\omega t}\mathrm{d}t}\right|^2

In MIIPS, instead of scanning on the temporal domain, a series of phase scan is applied on the phase domain of the pulse. The trace of the MIIPS scan is consisted of the second-harmonic spectra of each phase scan. The signal of MIIPS can be written as

I(2\omega)=\left|\int{|E(\omega)|^2e^{i\phi}\mathrm{d}\phi}\right|^2

The phase scan in MIIPS is realized with introducing a well-known reference function, by the pulse shaper to locally cancel distortions by the unknown spectral phase, of the pulse. The sum of the unknown phase and the reference phase is given by . Because the frequency doubled spectrum of the pulse depends on, it is possible to accurately retrieve the unknown .

The phase modulation procedure of the physical process is generally a continuous function. Thus, the SHG signal can be expanded with a Taylor expansion around :

I(\omega)= \left| \int| E(\omega+\Omega)| |E(\omega-\Omega)| \times \text{exp} \{i\} \mathrm{d}\Omega \right|^2

And

\phi(\omega+\Omega)+\phi(\omega-\Omega)=2\phi0+\phi''(\omega)\Omega^2+...+\frac{2}{(2n)!}\phi^{2n'}(\omega)\Omega^{2n}

According to this equation, the SHG signal reaches maximum when is zero. This is equivalent to . Through scanning of, the can be decided.

The frequency doubled spectrum recorded for each full scan of the reference phase results in two replicas of the MIIPS trace (see Figure 1, four replicas shown). From this data, a 2D plot for SHG is constructed where . The second harmonic spectrum of the resulting pulse has a maximum amplitude at the frequency where the second derivative of the pulse has been compensated. The lines describing are used to obtain analytically the second derivative of the unknown phase. After double integration the phase distortions are known. The system then introduces a correction phase to cancel the distortions and achieve shorter pulses. The absolute accuracy of MIIPS improves as the phase distortions diminish, therefore an iterative procedure of measurement and compensation is applied to reduce phase distortions below 0.1 radian for all frequencies within the bandwidth of the laser.

When all phase distortions have been eliminated, the pulses are the shortest they can be and are considered to be Bandwidth-limited-pulse|transform limited (TL). The MIIPS trace corresponding to TL pulses shows straight parallel lines separated by . Once spectral phase distortions have been eliminated, the shaper can be used to introduce calibrated phases and amplitudes to control laser induced processes.

MIIPS technology has been applied successfully in selective excitation of multiphoton imaging and femtosecond light-mass interaction study.

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