J. Eur. Opt. Society-Rapid Publ. 22, 46( 2026) 459
Figure 4.( a) Normalised spectra of the pulsed laser before( blue) and after( red) reflection from the DMD. Both have a wideband of( 30.0 ± 0.6) nm at FWHM. Analogously,( b) shows the fringe-resolved autocorrelation( FRAC) trace of the input laser pulse, corresponding to a pulse duration of( 33.1 ± 1.6) fs, while( c) presents the FRAC autocorrelation of the pulse reflected from the DMD, resulting in a pulse duration of( 36.3 ± 1.6) fs.
is valid since the pulse bandwidth is a small fraction of the carrier frequency. Under this approximation, the phase term can be expressed as, 2pcmL ffi 2pcmL px px 0 þ 2pcmL x~ px 0 x 0
¼ d 0 þ d 1 x~ and r 2 xx r2 xx 0
. Furthermore, a first-order Taylor series�expansion is applied to the term xL, defined as c a i ¼ @ i xL
@ x i c x¼x0
, which yields xL = c ¼ a 0 þ a 1 x ~, with x~ ¼ x � x 0, simplifying the first exponential term. By substituting all these approximations into equation( 4), we obtain the following expression:
U out ðx; xÞ exp ½ i ða 0 þ a 1 x~ ÞŠ exp � x~ 2 r 2 t
" # ð exp � x � d 0Þ 2 þ 2d 1 ðx � d 0 Þþd 2 1 x~ 2: ð5Þ
4r 2 xx 0
It can be observed from equation( 5) that the term a 0 does not contribute to the irradiance field. The linear terms in x~ give rise to a group delay( GD), where the coefficient a 1 produces a uniform temporal shift in pulse arrival time. Also, there is a spatial broadening induced by angular dispersion. The resultant instantaneous irradiance distribution I out ðx; tÞ, obtained as the modulus square of the inverse Fourier transform, can be expressed as follows:
" # " # ð I out ðx; tÞ exp � t � a 1Þ 2 ð exp � x � d 0Þ 2: ð6Þ
2r 0 2
2r 0 2
With r 0 2 t
¼ r 2 t
1 þ 4 r 2 x p2 m 2
t
r 2 t p2 2 x 0 x
and r 0 2 x
¼ r 2 xx 0
ð1 þð4r 2 x p2 m 2 Þ = ðr 2 t p2 x 2 0 Þ). Since the irradiance remains as a product of two expanded Gaussians functions, there is a temporal shift determined by a 1, and a spatial shift governed by d 0. This spatial shift increases with the diffraction order m, and depends on the DMD micromirror spacing, p. As a result, a spatiotemporal deformation arises, characterized by a spatial broadening factor of r 2 x = r2 xx 0 and a temporal broadening factor of r 0 2 t
= r 2 t. Thisresultisconsistent with previous reports in the literature [ 42 ]. It should be emphasized that, in this theoretical treatment, non-linear phenomena inside the DMD – such as self-phase modulation occurring in the protective window – have not been considered, since under standard operating conditions these effects are negligible.
To experimentally observe this spatial broadening, we recorded the spatially dispersed beam using an infrared camera placed in the far-field of the DMD. In this configuration, the DMD acts as a diffraction grating, and the propagation to the far-field( Fraunhofer regime) performs the spatial mapping of the spectral components. The laser beam struck the DMD at normal incidence, with all micromirrors set to the ON state as shown in Figure 5a. To clearly highlight the angular dispersion for different wavelengths, three spectral bandpass filters with central transmissions of 780 nm, 800 nm, and 820 nm were placed between the DMD and the screen( Fig. 5a). Initially, an image of the full spectrum of the light beam after being reflected by the DMD was obtained for comparison, as shown in Figure 5b( i). Then, the corresponding spectral traces on the screen were recorded, as shown in Figures 5b( ii)– 5b( iv). The spectral bands exhibit similar spatial distributions but are clearly displaced relative to each other.
These findings reveal that the various wavelength components of the light are diffracted at different angles because of the wavelength-dependent diffraction properties of the micromirrors. This spatial separation of the spectral components results in lateral chromatic aberration. For ultrashort pulses, spatial wavelength dispersion directly affects the pulse’ s temporal characteristics. Because the spectral components are no longer confined to a single spatial point, their temporal overlap is greatly reduced, effectively elongating the pulse duration. This spectral-to-temporal mapping implies that, for Fourier transform-limited pulses, the pulse duration increases as a result of the chromatic dispersion introduced by diffraction. The temporal broadening becomes more pronounced as the diffracted pulse propagates over longer distances because the angular separation