J. Eur. Opt. Society-Rapid Publ. 22, 46( 2026) 457
Figure 2. Schematic diagram of the optical system used to irradiate the DMD. From left to right: femtosecond laser, pair of linear polarisers, motorised shutter, focussing lens, and DMD mounted in a motorised XYZ translation stage.
as illustrated in the first column of images corresponding to fluence of 0.07 J / cm 2( see green point on the graph). In Zone II, the images show apparent damage to the micromirror coating. However, these regions were not considered in the area count. An example is shown in the second column of images, corresponding to a fluence of 0.09 J / cm 2( see red point on the graph). Finally, Zone III refers to the region containing missing micromirrors. The damaged area was quantified by counting the number of missing micromirrors, as shown in the third to fifth columns. The parameters associated with these images are indicated by blue points, corresponding to fluences of 0.11 J / cm 2, 0.12 J / cm 2, 0.18 J / cm 2 of respectively.
3 Temporal effects
In this section, we analyse the behaviour of ultrashort pulses after interacting with a DMD, and how these effects influence the temporal duration of the pulse. To do this, we first measure the intrinsic dispersion introduced by the DMD materials. Then we theoretically analyse the propagation of the pulsed laser beam after it strikes the DMD surface.
3.1 GDD by DMD
To evaluate the effect of the DMD materials on the laser spectrum, we employed a Ti: Sapphire amplifier( ASTRELLA, Coherent), with a central wavelength of approximately 800 nm, a measured bandwidth of 30 nm, a 31.6 fs pulse at the transform Fourier limit, and a repetition rate of 5 kHz. Also, a spectrometer( SC2000 Ocean Optics) was used to measure the spectral distribution of the pulsed beam before and after interaction with the DMD, as show in Figure 4a. The graph presents the normalised intensity( in arbitrary units) as a function of wavelength for both cases. Although the DMD has a negligible impact on the bandwidth because it remains unchanged, the pulse duration increases after passing through the DMD, this behaviour can be attributed to the dispersive characteristics of its constituent materials. The influence of each DMD component on the pulse temporal width was analysed, and the group velocity dispersion( GVD) introduced by the DMD materials was evaluated by comparing the pulse duration before and after reflection. The aluminium of the micromirrors contributes to a reported GVD of 10 fs 2 [ 38 ]. The protective window is then made of alkali borosilicate glass( type 7056) [ 36 ], similar to N-BK7 glass, which has a GVD of 44.65 fs 2 [ 39 ]. Consequently, given its thickness of 2 mm the total GDD introduced by the DMD is approximately 190 fs 2. To verify this estimation, the temporal width of the fs pulses( generated by before and after reflection from the DMD( in the“ off” state) was measured using an autocorrelator( Femtolasers Productions GmbH). The pulse duration before reflection was measured to be( 33.1 ± 1.6) fs. After reflection, it was ð36:3 1:6Þfs; according to Figures 4b and 4c respectively. These results were obtained with the DMD micromirrors in the off state. This result is consistent with the estimated GDD of 190 fs 2. FromtheGDD, wetheoretically estimate the pulse duration using the pulse elongation equation [ 40 ].
3.2 Theoretical and experimental spatiotemporal dispersion aspects
We also introduce a theoretical model to analyse the spatiotemporal dynamics of the electric field of an ultrashort pulse after its interaction with a digital micromirror device( DMD), based on the general principles of the Huygens – Fresnel integral framework. For simplicity, we initially considered only the X coordinate, assuming symmetry along the Y axis, and a single spectral component with frequency x within the spectrum. The input electric field profile, is assumed to be Gaussian beam in space and frequency,
described by the function U in ðx; xÞ ¼ exp �x 2 = 4r 2 x exp �x 2 r 2 t, whererx and r t are the width of spatial and temporal irradiance profiles, respectively, at 1. TheDMD e
is modelled as a reflective diffraction grating 2 with period p. For the mth diffraction order, under the paraxial approximation, the phase introduced by the DMD can be
h expressed as exp i 2p m x0 p i, as reported in [ 40 ]. The Huygens – Fresnel diffraction propagation of the beam over a distance L, under the Fraunhofer approximation – that is, considering the far-field diffraction of the DMD – can then be expressed as [ 41 ]: