leads to a spatial modulation of the shaper output. As simple example, the effect of a linear spectral phase (which corresponds to a wedge-shaped phase object in the Fourier domain - just like a thin glass prism) is demonstrated in Fig. 4.4.
Fig. 4.4: Space-time coupling in 4f-pulse shaping. a) 4f-setup without applied phase function. b) Shaper with linear spectral phase modulation, which corresponds to a wedge-shaped phase object (i.e. a “glass“wedge). This leads to a spatial displacement of the output beam (solid versus dashed lines), which is proportional to the slope of the spectral phase, or in other words to the temporal delay introduced by the shaping.
Fig. 4.5: Closed-loop optimization for compression of femtosecond laser pulses without a priori knowledge. An initial guess (e.g. the uncompressed pulse field) is fed into an evolutionary algorithm, which uses a given parameterization to generate a series of randomized variants of this initial field. Each variant is than realized by a pulse shaper and the resulting nonlinear signal measured. Based on the best trial solutions with the largest feedback signal, a new generation of variants is created, and the loop repeated, ideally until convergence is achieved.
Having said this, the choice of phase parameterization of the optimal learning is important.
Classically, one would simply use a Taylor-expansion of the spectral phase, which usually allows expressing typical material dispersion (the quadratic phase in first order) very efficiently [Eq. (2.5)]. If more complex phase patterns arise, as is the case e. g. for self-phase modulation in photonic crystal fibers (chapter 2.2.2), non-polynomic phase parameterizations will often perform better. An efficient parameterization is able to render a set of compensation phases with suitable shapes with a minimum of parameters. The more parameters are used, the longer an optimization will take to converge. Therefore, a parameterization based on Bézier curves was developed in this thesis. Bézier curves allow rendering smooth functions which are “bent” into shape by placing a number of nodes (Fig.
4.6).
Fig. 4.6: Schematic Bézier curve defined by 5 control nodes. It can be seen that the curve starts in the first node (P0) and ends in the last one (P4), while the intermediate nodes exert “attractive forces” on the dotted curve itself.
Therefore, by placing the nodes the shape of the curve can be altered, while smoothness is always ensured.
JG
where CJG( )ζ is the (vectorial) curve dependent on a single parameter ζ, which describes any position on the curve from its beginning at ζ = 0 to its end ζ = 1. The set of n + 1 control nodes is given by the vectors JGPi, and Bi,n(ζ) are the Bernstein polynomials:
( )ζ =⎛ ⎞⎜ ⎟⋅ ⋅ −ζ ( ζ) −
, ⎝ ⎠ i 1 n i
i n
B n
i . (4.4)
As can be seen in Fig. 4.6, generic Bézier curves in a two-dimensional setting (in this case the two axes will be corresponding to phase vs. frequency or pixel in order to represent compensation phase functions for the shaper) generally are not unique functions of one of the axes (e. g. frequency), as they can include loops. To restrict Bézier curves to unique functions in the frequency or pixel axis, the set of nodes JGPi has to be sorted such that PJGi have ascending abscissa coordinates. This is shown as an example in Fig. 4.7, where eight nodes have been created with arbitrary ordinate phases and ascending abscissa pixel values.
The flexibility of the Bézier curve upon variation of the ordinate value of the fourth node is also indicated.
Fig. 4.7: Example of a Bézier curve based on nodes with ascending abscissa coordinates, which ensures unique phase functions. The ordinate coordinate of the fourth node has been varied from -4.5π to 11π, demonstrating the flexible “bending” of the curve. Like this, a limited number of parameters allowing generation of a very flexible set of smooth, realistic trial phase functions.
The experimental realization of closed-loop pulse compression (Fig. 4.5) in a multiphoton microscope using Bézier phase parametrization only requires the broadband laser source described in chapter 2.2, a pulse shaper, the microscope setup with a nonlinear medium for generating the feedback signal and detection (Fig. 4.8). In this work, second harmonic generation was chosen as feedback signal, which is easily implemented with a small
Fig. 4.8: Experimental setup for in situ broadband pulse compression using a closed-loop optimization. The setup for broadband laser generation is the same as described in chapter 2.3. Briefly, a standard 100 fs-oscillator is coupled into a photonic crystal fibre (PCF) to generate broadband continuum. This light is then sent through a 4f pulse shaper (f = 200 mm), consisting of a pair of gratings (G1, G2 with 1200 grooves/mm), two spherical mirrors (SM1, SM2 with f = 200 mm) and a 640 pixel double liquid crystal spatial light modulator (SLM). The shaped pulses can be diagnosed either with an external frequency-resolved optical gating (FROG) setup, or second harmonic generation (SHG) can be performed with a crystal in situ in the microscope.
Therefore, the first step towards in situ compression in the microscope was a closed-loop optimization run on the total SHG signal in the FROG, which allows a direct characterization of the ultrashort pulses before and after performing the optimization. The SHG-FROG method can be understood simply as a spectrally resolved autocorrelation:
integrating the FROG trace for each delay, the autocorrelation of the measured pulse is obtained. In the two-dimensional FROG traces, however, the exact shape of the pulse is contained in the symmetric pattern. A suitable reconstruction algorithm can use this pattern to compare it to trial solutions, and thus find the pulse shape which is responsible for creating such a FROG trace.[89, 90]
Fig. 4.9: Typical learning curve for a closed-loop pulse compression in situ in
The optimization was parameterized with a Bézier curve based on 12 nodes. Convergence of the evolutionary optimization typically and reproducibly occurred after ~30 generations, in about 30 min. A typical learning curve showing the average and best relative SHG intensity for each generation of the optimization can be seen in Fig. 4.9.
With the FROG setup, the uncompressed pulses have been characterized (Fig. 4.10a) and reconstructed (512×512 grid, FROG error 0.0061) to yield the complex spectral phase of the PCF output (Fig. 4.10c, solid curve), which due to additional fibre and experimental dispersion differs from the symmetric spectral phase expected for pure self-phase modulation, seen e.g. in Fig. 2.4, chapter 2.2.1. In Fig. 4.10c, the phase on the pulse shaper (dashed curve) shows that no modulation has been applied, hence it is a horizontal line.
After optimization, with the optimal trial Bézier phase function found (Fig. 4.9) and applied in the shaper (Fig. 4.10d, dashed curve), the FROG trace looks markedly different, exhibiting by far most intensity centred around zero delay. The now reconstructed phase (512×512 grid, FROG error 0.0066, Fig. 4.10d - solid curve) is much flatter and only contains minor remaining “wiggles”. Consequently, the optimal phase ϕcorr found by the evolutionary algorithm directly corresponds to the smoothed, sign-inverted reconstructed phase of the uncompressed pulse (Fig. 4.10c, solid curve).
Fig. 4.10: Measurement of evolutionary pulse compression with frequency-resolved optical gating (FROG). The measured traces (a, b) can be reconstructed to yield the pulse spectrum and phase (c, d). Note that in the compressed case (d) the reconstructed phase (solid curve) is much flatter than in the uncompressed case (c), and the applied compensation phase
algorithm is capable of compressing the PCF continuum, although so far only in the FROG setup. If the same procedure is followed for a flipped mirror in order to send the shaped laser beam into the microscope, very similar evolutionary optimization of SHG signal can be performed. At this stage, however, the pulses cannot be characterized directly as is possible in the FROG setup. Comparing the compensation phases found in the microscope with the phase ϕcorr of Fig. 4.10d (dashed curve), the same functional shape is found with a strong additional quadratic term of ϕ” = -3800 fs2 to compensate the dispersion of the microscope objective.
At this point, with the FROG pulse characterization available, it can also be shown how, moreover, compressed pulses can be shaped by adding an arbitrary phase modulation on top of the compression phase ϕcorr. This shall briefly be demonstrated in Fig. 4.11 for a supplementary sinusoidal modulation ϕmod = a × sin(bω + c) with a = 1.23, b = 150 fs and c = 0 to yield a shaper phase ϕSLM = ϕcorr + ϕmod.
Fig. 4.11: FROG measurement (a) and accordingly applied phase ϕSLM (b) of a compressed PCF continuum with an additional sine modulation, leading to the splitting of the compressed pulse into a multipulse sequence. The intra-pulse distance b was chosen to be 150 fs, and the FROG trace shows the characteristic interferences expected for such pulse shapes.
As can be seen from this example, closed-loop optimization is able to successfully perform in situ compression for pulse shaping applications in microscopy. The drawbacks of the very simply implemented method are, however, that it is rather slow (~30 min) and does not reproduce all details of complex phase distortions (see remaining “wiggles” in Fig. 4.10d).
conventional FROG, although FROG variants can also be developed for collinear beam arrangement (see e. g. XFROG implementation later in this text).