Focusing ultrashort laser pulses with achromatic doublets

Ultrashort pulse lasers can be used to concentrate energy into a short time interval of only a few femtoseconds (corresponding to a few optical cycles in the visible wavelength range). When these ultrashort pulses are focused by an optical system, it is possible to achieve an extremely high peak power density (required for many scientific and industrial applications). The smaller the focused spot, or the shorter the pulse duration, the greater the peak-power density. Some applications that require pulses to be kept short (in time and space) near—or at—the focus include ultrafast laser direct writing,¹ material processing,² and nonlinear microscopy.³ Effects that can modify the spatial-temporal distribution of a pulse beam after spreading through a lens, however, have previously been reported and studied.^4–9

The three effects that can increase pulse duration and spot size, and thus cause pulse intensity to decrease, are group velocity dispersion (GVD), propagation time difference (PTD), and lens aberrations. The GVD effect causes temporal spreading in the pulse as it propagates through a dispersive material. The PTD effect is generated by the chromatic aberration of the lens, and may also introduce temporal spreading to the pulse. The GVD introduced by achromatic doublets can be compensated, however, with the use of an external second-order GVD compressor (based on prisms).

We have theoretically and experimentally investigated focusing of ultrashort pulses by lenses and mirrors. Our main aim is to understand how to control and improve focusing of pulses. We can thus optimize pulse characterization techniques and improve the design of the lasers we are currently constructing. In our work, we have used an achromatic doublet with a low numerical aperture (NA)—the dimensionless number that characterizes the angles over which a system can accept or emit light—to study the focusing of pulses. This doublet has a focal length of 30mm, an NA of 0.2, and is designed to operate in the near-IR between 750 and 1100nm. We conducted our experiments with 200fs pulses at 810nm, which were generated using a commercial Coherent MIRA 900 laser. We have also used our own titanium-sapphire laser—with an external compressor—which generates pulses as short as 20fs at 810nm.

With our in-house experimental setup, we were therefore able to measure pulses with a temporal duration of 20fs at the focus of an achromatic doublet. We compensated the GVD introduced by this doublet with the use of an external second-order GVD compressor (based on prisms). In addition, our theoretical calculations indicate that there is no PTD effect in this doublet.¹⁰ These calculations also show, however, that for pulses shorter than 20fs—or for larger NA lenses—the secondary color in the achromat introduces PTD that needs to be corrected (i.e., apochromatic lenses should be used instead). We have thus designed an apochromatic doublet to show that this lens increases the pulse-peak intensity produced by the equivalent achromatic lens.¹¹

We have also theoretically and experimentally investigated how monochromatic aberrations modify the spatial-temporal distribution of pulses at the focus of our apochromatic doublet.^{12, 13} Importantly, we were able to verify with our experiments that large monochromatic aberrations, which are induced when the achromat is rotated, do not change the pulse duration for pulses as short as 20fs at 810nm (see Figure 1). The monochromatic aberrations, however, introduce a large amount of spatial spreading. This causes the peak intensity to be reduced as the aberrations increase, until the autocorrelation signal is lost.

Figure 1. Theoretical (theo) and experimental (exp) normalized intensity autocorrelations (in time) of ultrashort laser pulses at the focal plane of an achromatic doublet. The results for a 20fs pulse at 810nm, for three rotation angles of the lens, with respect to its optical axis, are shown. Monochromatic aberrations are induced by rotating the doublet in this way. t: Time. T_int: Input pulse intensity full-width.

Our results also show that the diffraction of the beam by the aperture of the mirror will produce non-negligible temporal widening of optical pulses for a few cycles, even when all three spatial-temporal modification effects are corrected (e.g., with the use of an off-axis parabolic mirror to focus collimated light). The diffraction of the beam generates a spatial chirp in the focused pulse, which produces the temporal widening. Changes to the spectrum at each point in the diffraction pattern, at the focus of an ideal mirror, are illustrated in Figure 2 for two different cases. In the first case it is assumed that the bandwidth of the pulse is smaller than the frequency of the carrier. In the second case, however, this approximation is not made (i.e., for a few optical cycle pulses when the approximation is invalid). We used three different approaches to verify these results, i.e., that the pulse duration (τ) at the focus increases up to 3.47fs for a one-cycle optical pulse (when the initial incident pulse duration, τ₀, is 2.7fs at 810nm and there is uniform illumination on the aperture).^14–16 We have also extended our analyses to high-NA lenses.¹⁶ The results from these analyses show that for a 2.7fs unchirped incident pulse (at 810nm), the τ/τ₀ ratio increases to 1.367 for a lens with an NA of 1. We have also demonstrated that the τ/τ₀ ratio for the time-width of the focused spot pulse relative to the input pulse for a one-cycle pulse, at different carrier frequencies, is about 1.3 for uniform illumination.¹⁵ For Gaussian illumination, τ decreases, depending on the ratio between the incident Gaussian radius and the mirror's aperture diameter. As this ratio decreases toward zero, τ tends to 2.84fs, i.e., the shortest pulse duration that can be achieved at the focus is limited by diffraction.

Figure 2. Intensity spectra versus radius in a diffraction pattern, at the focus of an ideal mirror when (top) the bandwidth of the pulse is assumed to be smaller than the frequency of the carrier, and (bottom) when this assumption is not made. The incident pulse is for 2.7fs at 810nm (i.e., a one optical cycle pulse).

We have conducted theoretical calculations and experimental studies to investigate effects that cause spatial-temporal spreading of ultrashort laser pulses. We have used achromatic doublets to successfully focus the pulses from both commercial and in-house lasers. We have thus been able to demonstrate experimentally that monochromatic aberrations do not produce temporal spreading for pulses as short as 20fs at 810nm. We are currently constructing new lasers in our laboratory. These will allow us to verify some of the theoretical work for other carrier wavelengths and for broader bandwidth pulses (i.e., when spectral effects become more difficult to correct).

We gratefully acknowledge sponsorship of this work from the General Directorate of Academic Staff Affairs at the National Autonomous University of Mexico (DGAPA-UNAM), project PAPIIT-IG100615.