Ogilvie Group
Pulse-shaping tutorial: Part 2
In a nonlinear interaction such as two-photon absorption, which was employed in our recent fluorescence-resonance energy transfer (2P-FRET) microscopy experiment (read about it here), two near-IR photons from the pulsed laser beam are absorbed to bridge an energy gap in a fluorescent protein and transfer an electron from the ground state manifold to an excited state. The protein eventually reverts back to the ground state and emits a visible photon as fluorescence, which is detected with sensitive detectors (such as photomultiplier tubes) to generate an image. This phenomenon is shown in a highly simplified fashion on the right hand side of the figure below: the solid horizontal lines represent the energy level states of the fluorescent protein, the arrows pointing up represent the absorbed photons, while the wiggly arrow pointing downwards denotes the fluorescence emission. Note that the energies of the absorbed and emitted photons differs greatly as will be discussed below.
During the absorption process, only the combined photon energies \(\hbar (\omega_1 + \omega_2)\) have to match the energy gap between the ground and excited states \(\hbar\omega_g\). Thus in addition to two photons of energy \(\hbar \omega_g/2\), any pair of photons (\(\hbar\omega_g/2-\delta,\hbar\omega_g/2+\delta\)) will do the trick, where \(\delta\) is some energy difference. This is shown on the left hand side of the figure above: the spectrum of the pulse responsible for exciting the fluorescent protein (sometimes called the "pump beam") is shown with the photon-pairs labelled. In fact, for a broadband laser, all such photon-pairs present in the pulse spectrum participate in the absorption process and we can mathematically write the two photon absorption \(TPA (\omega_g)\), at frequency \(\omega_g\) as :
$$ TPA(\omega_g) \propto \left\lvert \int_{-\infty}^{\infty} |E(\omega)| |E(\omega_g-\omega)| \mathrm{e}^{i(\phi(\omega)+\phi(\omega_g-\omega))} \mathrm{d}\omega \right\rvert^2 $$
where, \(E(\omega)\) and \(\phi(\omega)\) are the electric field and phase at frequency \(\omega\) respectively. The notation is slightly different from the figures above, with \(\omega\) substituted for \((\omega_g/2-\delta)\) to improve the equation readability. Here, we start to see the power of being able to manipulate spectral phase. Each pair of frequencies \((\omega, \omega_g-\omega)\) contributes an absorption term proportional to the product of their electric fields. But, the absorption term also carries a phase equal to the sum of their original phases. The integral denotes that each such pair makes a contribution to the absorption and the total absorption is equal to the sum of all such contributions. To get a better idea about the process, look at the visualization below:
On the left-hand side, the spectral field (blue) and the corresponding phase values (green) are approximated by about 60 vertical bars. The black line is \(\omega_g/2\), the central frequency of the pump whose two-photon absorption at \(\omega_g\) is being investigated. The reader may choose a different central frequency by clicking on the corresponding vertical blue bar. On the right hand side, is a phasor diagram showing the magnitude(denoted by the length) and phase(denoted by the angle) of each contributing component of absorption (gray) and the resulting total absorption (orange). By hovering the mouse over the spectrum, each pair of frequencies contributing to two-photon absorption at \(\omega_g\) can be highlighted (red) and the contribution of that pair to the total absorption is highlighted on the right (red). The reader is encouraged to explore the effect of shifting the phase offset on the total absorption. At what phase offsets is the absorption maximum? When is it minimum?
Notice that a phase shape that is symmetric about the central frequency (eg. when phase offset is zero), the resulting absorption is at a minimum because a each absorption component has a different phase angle resulting in a spread. Whereas, when the phase shape is anti-symmetric, the absorption components all align along the same phase and the total absorption is at its maximum. Thus, if the individual absorption contributions can be arranged evenly about the unit circle, the total absorption can be entirely eliminated. This can be seen more easily when we switch to a random phase distribution. The phase distribution generated by a random function distributes the individual components such that on an average they cancel out. Because of the random nature of the distribution, the absorption may not cancel every time, but it tells us that by carefully choosing a (semi-random) distribution about a frequency \(\omega_g/2\), the absorption at frequency \(\omega_g\) can be eliminated, whereas having a constant or anti-symmetric phase can promote full absorption.
Of course, arranging the phases for one particular frequency can give rise to unwanted features in absorption at other wavelengths. So, to achieve the results one wants, the spectral phase has to be designed taking into consideration the entire bandwidth and what particular absorption structure is desired. In the visualization below, an excitation pulse that is Gaussian in frequency is shown on the left. This pulse is manipulated by a 640-pixel SLM generated phase-mask as can be found on real-life pulse-shaping devices. By combining the the principles of a flat phase promoting absorption and a random phase eliminating it, a phase function, designed such that it can let us select the part of spectrum over which absorption is to take place, is shown below it. On the right is the relative strength of the two-photon absorption resulting from the phase mask as a function of the wavelength - the higher the number, the better is the probability of exciting a fluorophore at that wavelength. Of course, the actual excitation also depends on the absorption strength of the fluorophore at that wavelength. This technique was first demonstrated as an application for selective second harmonic generation - read about it here.
This selective excitation is particularly useful when the user wants to excite a single fluorophore at a time out of many in a sample under consideration. This was demonstrated in a recent experiment in our group which you can read about here. By changing the position of the flat phase and its width, the absorption can be altered in milliseconds. By changing the width of the flat phase section and the center wavelength about which it is implemented, the reader can investigate the effect of the applied phase on the two-photon absorption probability of the pump beam. Note that by changing the phase amplitude to 0, the user can get an idea of the full two-photon absorption potential of the excitation pulse.
Of course, this method works well when the spectrum is a perfect Gaussian and the frequency selection is quite broad. The reader will have realized that when the flat phase section is narrowed, the overall strength of two-photon absorption also decreases and that narrow selective excitation can be difficult with this technique. The visualization below is the same as the one above, but it implements the technique for a real broadband laser spectrum (as measured in our lab on the Venteon Pulse:One 6-fs Titanium Sapphire laser).
In order to perform selective excitation over narrower regions, or to make more complicated selection structures such as multiple sections of two photon absorption, another technique is necessary. The problem is easier if only a narrow pump bandwidth is involved or one is concerned about a single excitation wavelength. But for a large bandwidth combined with a narrow selection, the phase applied needs to work out equally well over the whole bandwidth and it turns out to be an optimization problem with a large number of variables (n pixels and m levels of phases). One way to solve this problem is to implement a genetic algorithm with a target excitation function. It starts with a large population of phase masks (flat + random) described above and makes small changes to them in order to arrive at a more optimized solution. The successful implementation of this technique was essential to realizing our 2P-FRET microscopy experiment.
This tutorial is authored by Amar Bhagwat in HTML and Javascript and is powered by the data visualization library D3.js and the numerical calculation library Numeric Javascript. The equations are typeset in MathJax.
