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between wavelength components increases. As a consequence, the pulse loses its original temporal coherence, leading to a longer pulse duration. These results emphasise the need to incorporate an optical compensation system in DMD-based setups to effectively mitigate these diffraction-induced effects. Without such compensation, the temporal width may increase significantly, potentially compromising system performance in ultrashort pulse experiments. Additionally, the total optical diffraction efficiency of the DMD system depends on the fill factor of the micromirror array, the surface reflectivity of the micromirror, the transmission of the window and the diffraction efficiency. For the DLP7000 DMD used in our experiment, the central wavelength of the laser beam, and the aperture of the collection system, these values are nominally 92 %, 88 %, 95 %( single pass), and 86 %, respectively [ 36 ]. Taking all this factors into account, the total optical efficiency of the DMD system is 63 %.
In conclusion, the DMD introduces modifications in the spatiotemporal structure of a pulsed laser. The GDD introduced by the DMD represents a minor effect for conventional pulse durations. However, since the DMD acts as a diffraction grating, the spatiotemporal propagation of the pulse is strongly affected by its diffractive nature. Thus, for short pulse durations, the implementation of adequate dispersion compensation must be carefully considered [ 43, 44 ].
4 Beam characterization through single-pixel imaging
SLM such as the DMD are key components in single-pixel imaging techniques( SPI). Based on the study presented in Section 2, it is possible to obtain images of ultrashort light pulses using a DMD without significantly reducing the beam energy or not reducing any. This imaging modality operates by sampling the object with a sequence of structured masks, while the total light intensity transmitted or reflected by each mask is recorded by a single-pixel( bucket) detector [ 45 ]. The final image is then numerically reconstructed from the photocurrent signal using computational algorithms. The simplicity of the sensing device enables efficient operation under low-light conditions unlike unconventional imaging techniques, which are rely primarily on pixelated sensors. Furthermore, this approach simplifies the measurement of the spatial distribution of multiple optical properties of the sample, such as its spectral content, in a direct approach. Moreover, single-pixel detectors can operate over a much broader spectral range than conventional cameras. The imaging process can be described by the following matrix – vector expression [ 46 ]:
y ¼ Ax þ; ð7Þ
where x represents the image to be reconstructed in a onedimensional vector form, A is the measurement matrix that encodes the sequence of sampling masks in its rows, constructed with an orthonormal set of functions, and y is a vector containing the corresponding measurement sequence. The vector e represents the noise present in the measurement process.
Given an object of dimension N( where N is the number of pixels), an appropriate mathematical basis, such as the Walsh-Hadamard basis, with this N dimensional space must be selected as the measurement basis. The inverse of the measurement matrix can then be computed for the recovery of the object, assuming low values of noise( e ~ 0):
x ¼ A �1 y: ð8Þ
Certain bases are associated with fast and efficient digital transformation algorithms, making image reconstruction practical even for large image sizes. In particular, the Hadamard basis is well suited to DMD-based single-pixel imaging techniques because its binary nature allows full utilisation of the DMS’ s refresh rate. The Hadamard basis functions are orthogonal and take binary values of + 1 or �1. The negative values can be encoded by considering that the measurement process in equation( 8) is linear and that the Hadamard matrix can be expressed as H = H + � H �, where H + is the Hadamard matrix with all entries of the value �1 changed to 0, and H� is its complementary matrix. With these properties, the measurement can be expressed as a two-step process: the H + pattern corresponding to each element of the A i basis is measured, followed by its complementary H � pattern; then, the two measured values are subtracted to measure each individual coefficient of y i. This differential measurement process suppresses parasitic signals, such as ambient light or slow fluctuations of the light source.
For the experimental implementation, the pulsed laser beam from the Ti: Sapphire laser( PRO-Compact) was incident on a DMD, where the sequence of Walsh-Hadamard patterns defined by matrix A was encoded. The scanning patterns were encoded as binary images with a resolution of 128 128 pixels, using 256 256 DMD micromirrors, covering an area of 3.5 3.5 mm 2. To properly scan the laser beam profile, the laser beam was aligned to propagate normal to the DMD surface, as shown in Figure 6a. For each scanning pattern, the reflected light, emerging at an angle of 24 º from the DMD, was collected by a pair of lenses in a close-to-afocal configuration with focal lengths of 125 mm and 50 mm. As there is an image relationship between the DMD plane and the photodiode plane, and a demagnified image of the light spot at the DMD is projected onto the active area of the photodiode, it is not necessary to take into account the spatial chirp introduced by the DMD. The light beam was focused onto a photodiode( DET36A2 Thorlabs). The photocurrent signal was digitised by a data acquisition board, which generated the measurement vector y. The irradiance map of the light beam was reconstructed by applying a fast inverse Hadamard transform, according to equation( 8). To reduce the noise arising from the laser fluctuations, several measurements were averaged. The final image was obtained using 32,768 patterns, with the DMD operated at a frequency limited by the laser repetition rate of 1 kHz, which obtained one pattern measurement for each pulse. The reconstructed irradiance