Spectral Phase Interferometry For Direct Electric-field Reconstruction - Theory

Theory

The intenesity of the interference pattern from two time-delayed spectrally sheared pulses can be written as

\begin{align}S(\omega) &= |E(\omega) + E(\omega-\Omega)e^{i\omega\tau}|^2\\ &= I(\omega) + I(\omega-\Omega) + 2\sqrt{I(\omega)I(\omega-\Omega)}\cos\end{align},

where is the analytic signal representing the uknown (upconverted) field being measured, is the spectral shear, is the time delay, is the spectral intensity and is the spectral phase. For a sufficiently large delay (from 10 to 1000 times the Fourier transform limited pulse duration), the interference of the two time-delayed fields results in a cosine modulation with a nominal spacing of ; any dispersion of the pulse resulting in minor deviations in the nominal fringe spacing.

The unknown spectral phase of the pulse can be extracted using a simple, direct algebraic algorithm first described by Takeda. The first step involves Fourier transforming the interferogram into the pseudo time domain:

\begin{align}\widetilde{S}(\widetilde{t}) &= \mathfrak{F}\\ &= \widetilde{E}^{dc}(\widetilde{t}) + \widetilde{E}^{ac}(\widetilde{t}-\tau) + \widetilde{E}^{-ac}(\widetilde{t}+\tau)\end{align},

where is a 'direct current' (dc) term centred at with a width inversely proportional to the spectral bandwidth, and are two 'alternating current' (ac) sidebands resulting from the interference of the two fields. The dc term contains information about the spectral intensity only, whereas the ac sidebands contain information about the spectral intensity and phase of the pulse (since the ac sidebands are Hermitian conjugates of each other, they contain the same information).

One of the ac sidebands is filtered out any inverse Fourier transformed back into the frequency domain, where the interferometric spectral phase can be extracted:

\begin{align}D(\omega, \Omega) &= \mathfrak{F}^{-1}\\ &= \sqrt{I(\omega)I(\omega-\Omega)}e^{i}e^{-i\omega\tau}\end{align}.

The final exponential term, resulting from the delay between the two interfering fields, can be obtained and removed from a calibration trace, which is achieved by interfering two unsheared pulses with the same time delay (this is typically performed by measuring the interference pattern of the two fundamental pulses which have the same time-delay as the upconverted pulses). This enables the SPIDER phase to be extracted simply by taking the argument of the calibrated interferometric term:

\begin{align}\theta(\omega) &= \angle\\ &= \phi(\omega-\Omega) - \phi(\omega)\end{align}.

There are several methods to reconstruct the spectral phase from the SPIDER phase, the simplest, most intuitive and commonly used method is to note that the above equation looks similar to a finite difference of the spectral phase (for small shears) and thus can be integrated using the trapezium rule:

.

This method is exact for reconstructing group delay dispersion (GDD) and third order dispersion (TOD); the accuracy for higher order dispersion depends on the shear: smaller shear results in higher accuracy.

An alternative method us via concatenation of the SPIDER phase:

\begin{align}\phi(\omega_0 + N|\Omega|) &= \begin{cases} -\sum^N_{n=1} \theta(\omega_0 + n\Omega) &\text{if}\, \Omega>0\\ \sum^{N-1}_{n=0} \theta(\omega_0 + n|\Omega|) &\text{if}\, \Omega<0 \end{cases}\end{align}

for integer and concatenation grid . Note that in the absence of any noise, this would provide an exact reproduction of the spectral phase at the sampled frequencies. However, if falls to a sufficiently low value at some point on the concatenation grid, then the extracted phase difference at that point is undefined and the relative phase between adjacent spectral points is lost.

The spectral intensity can be found via a quadratic equation using the intensity of the dc and ac terms (filtered independantly via a similar method above) or more commonly from an independent measurement (typically the intensity of the dc term from the calibration trace), since this provides the best signal to noise and no distortion from the upconversion process (e.g. spectral filtering from the phase matching function of a 'thick' crystal).

Read more about this topic:  Spectral Phase Interferometry For Direct Electric-field Reconstruction

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