1. Introduction

Multifocusing of a laser beam with a diffractive beam splitter (DBS) is a versatile technique, which, for example, enables parallel laser processing with a high throughput, benefitting manufacturing industries. DBSs are extensively used with nanosecond and picosecond pulse lasers, as well as continuous wave lasers, to deliver laser power efficiently to multiple points on a workpiece [1–3]. However, this is not necessarily the case with ultrashort pulse lasers. A diffracted ultrashort pulse exhibits spatio-temporal distortions, i.e., chromatic aberrations and temporal stretches, owing to the finite spectral bandwidth of the pulse. The diffracted beam is elliptically distorted in space, and hence the spectrum is narrowed to stretch the pulse in time. (A visualization of the spatio-temporal coupling in multifocusing ultrashort pulses is reported in Ref. [4].) The pulse is also stretched when passing through bulky glassy substrates. As a consequence, the focusing resolutions of the optical setup are heavily affected both in space and time. Such pulse distortions can be ignored when splitting and focusing ∼100-fs pulses in the vicinity of the optical axis, yet processing applications are restricted [5–10]. The working area is further limited with shorter pulses.

Although several efforts utilizing static and dynamic diffractive optical elements have been made to correct spatio-temporal pulse distortions, temporal pulse broadening remains uncorrected, and the arrangement of split beams lacks flexibility [11–13]. Minguez-Vega et al. demonstrated multifocusing ultrashort pulse beams using a DBS, refractive achromat, and pair of diffractive lenses [14,15]. Multifocusing performance of this triplet was evaluated in optical experiments [16]. According to their reports, the spatial distortions were removed by the triplet while some temporal distortions were left, affecting the pulse peak intensities at outer diffraction angles. We have recently developed an achromatic cascade optical system with a DBS, which we call the prototype [17], and confirmed its achromaticity using 20-fs pulse beams. Meanwhile, the transmitted pulse was stretched to be 35 fs, independent of the diffraction angles. Pulse front distortions were detected in the transmitted pulses, which we found were due to the beam-radius-dependent group delays left in the prototype. Pulse front distortions can be reduced by narrowing the beam diameter but sacrificing the spatial focusing resolution.

In this report, we propose the inclusion of diffractive-refractive hybrid lenses into the prototype to perform an optimal management of dispersions in the system. We develop a simple formulation to design a system, which explicitly describes the contributions from the hybrid lenses to the removal of chromatic aberrations and parabolic pulse stretch, and evaluate the designed system with 20-fs pulses by characterizing the transmitted pulses. We demonstrate that a tailored combination of diffractive and refractive surfaces is crucial for distortion-free multifocusing of ultrashort pulses.

2. Pulse delivery concept

Figure 1 shows the schematic of the proposed pulse delivery system. The diffractive subsystem consists of a DBS and a diffractive focusing lens (DFL), which correct lateral chromatic aberrations. The afocal subsystem has a pair of diffractive-refractive hybrid lenses (HL₁, HL₂), which correct longitudinal chromatic aberrations and compensate parabolic pulse stretches. The non-parabolic component of pulse broadening can be cancelled by pre-chirping the input pulse. These two subsystems are connected in cascade. In the layout, F₁ is an intermediate focal plane, and F₂ is the exit focal plane or the working area. Upon entering the system, the ultrashort pulse beam is divided by the DBS, experiencing a large angular dispersion, which causes chromatic aberrations and temporal pulse broadening. When passing through the optical system, the pulse spectral components suffer group delay dispersions (GDDs), which depend on the beam radius. Thus, all these dispersions should be managed to obtain distortion-free arrayed pulse beams.

 

Fig. 1. Schematic diagram of the achromatic cascade optical system for multifocusing ultrashort pulses. DBS, diffractive beam splitter; DFL, diffractive focusing lens; HL₁ and HL₂, hybrid lenses; F₁ and F₂, intermediate and exit focal planes, respectively. The temporal profiles of pulses are illustrated.

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The aforementioned hybrid lens is composed of a positive refractive lens and a negative diffractive lens. As is well known, the dispersive properties of these lenses are opposite. After passing through a refractive lens, shorter wavelengths are focused closer to the lens than longer wavelengths, and a diffractive lens reverses this event [18]. In addition, after a refractive lens, the pulse front delays from the phase front, whereas after a diffractive lens, the pulse front advances the phase front [19,20]. Given these distinct natures, an appropriate combination of the two types of lenses enables simultaneous shaping of both phase and pulse fronts. A hybrid achromat to focus ultrashort laser pulses was investigated theoretically by Piestun et al. [21] and was verified experimentally by Fuchs et al. [22]. An adaptive control of phase and pulse fronts of ultrashort pulses was demonstrated by Sun et al. using a liquid-crystal spatial light modulator and a deformable mirror [23].

3. Formula derivation

To design the proposed optical system, we derive a set of simple formulas that couple space and time using a ray transfer matrix [24] and an analytical model of pulse propagation [19] by assuming small diffraction angles in the paraxial region. According to the ray-matrix method, beam propagation is described with the ray height y and the ray angle u, and hence the axial position z is expressed as z = ‒y / u. By approximating y and z to the first term in their Taylor series expansions, the lateral chromatic aberration Δy is defined as $\Delta y = ({\partial y/\partial \lambda } )\Delta \lambda $, and, similarly, the longitudinal chromatic aberration Δz is expressed as Δ$z = ({\partial z/\partial \lambda } )\Delta \lambda $, where Δλ = λ ‒ λ₀ and λ and λ₀ are an arbitrary wavelength and the central wavelength of the pulse, respectively. Hereafter, f(λ), F(λ), and MF(λ) denote the wavelength-dependent focal lengths of the DFL and hybrid lenses HL₁ and HL₂, respectively, and M is the image magnification of the afocal subsystem. The corresponding focal lengths at λ₀ are f₀, F₀, and MF₀, respectively.

In the diffractive subsystem, the DFL is located at a distance s from the DBS, and F₁ is located at a distance f₀ from the DFL. The lateral chromatic aberrations caused by the DBS can be corrected by appropriately positioning the DFL [11]. Using the ray matrix of the diffractive subsystem, the lateral chromatic aberration Δy at F₁ is formulated as

To connect the diffractive and afocal subsystems in cascade, the distance between the DFL and HL₁ is set to f₀ + F₀, and that between HL₁ and HL₂ is set to F₀ + MF₀. From the ray matrix of the afocal subsystem (See Appendix A), we find the following expression for the longitudinal chromatic aberration Δz at F₂,

By applying the formalism presented by Bor [19,20] to the hybridized cascade system, we describe the propagation time difference τ between the pulse and phase fronts at λ₀ as (see Appendix B)

These formulas can be applied to off-axis beams as well as on-axis beams under the first-order approximation. The principal rays at λ₀ impinge upon plane F₁ at normal incidence, and as they pass through the afocal subsystem, each of those rays has the same optical path length, which is used to formulate the time delay τ. The formulation above covers the prototype design with a pair of refractive singlets in the afocal subsystem [17].

With large chromatic aberrations introduced by the DBS, two conditions, Δz = 0 and Δτ = 0, cannot be fulfilled simultaneously for refractive lenses made from commercially available glasses, because of their dispersion properties. A stringent constraint exists between $dn/d\lambda {\; }$and ${d^2}n/d{\lambda ^2}$, which is described as $\lambda {d^2}n/d{\lambda ^2} ={-} 3{\; }dn/d\lambda $ [26], indicating that if ${\; }dn/d\lambda $ is large, ${d^2}n/d{\lambda ^2}$ should be large. This constraint can be avoided by incorporating a diffractive lens made from a low-dispersion glass. The material-independent Abbe number of diffractive lenses is ‒3.452 [18], whereas the Abbe numbers of ordinary lens glasses are ranged, for instance, from 17 to 95 [27]. A hybridization of diffractive and refractive surfaces enables the resultant dispersion properties to be varied in a wide range, as demonstrated by Harm et al. using a combination of a diffractive Moire lens and a liquid-filled refractive polymer lens [28].

Provided that the focal lengths of the refractive and diffractive lenses are f₁(λ) and f₂(λ), respectively, the focal length of the hybrid lens F(λ) can be expressed as F = f₁f₂/(f₁ + f₂), where the two lenses are assumed to be sufficiently thin, and f₁(λ) and f₂(λ) are given by

4. System design and fabrication

Using the derived formulas, we designed the achromatic cascade system and fabricated the optical components by assuming a Gaussian pulse beam with a pulse duration of t_p = 20 fs (FWHM), central wavelength of λ₀ = 780 nm, beam diameter of 5.0 mm, and spectral bandwidth of Δλ = 45 nm (FWHM). For a transform-limited pulse, Δλ is given by $\Delta \lambda = 0.441{\lambda _0}^2/c{t_\textrm{p}}$.

Assuming f₀ = 50 mm, the splitter-lens distance s was set to 50 mm from Eq. (1). Owing to the limited pulse energy available in the current study, we used a +1-st-order diffraction beam from a blazed phase grating instead of arrayed beams for pulse characterization. The diffraction angles set for the characterization were 1.7° and 2.9°, corresponding to beam heights of 1.5 mm and 2.5 mm from the optical axis, respectively.

The DFL with blazed surface-relief structures and the blazed phase gratings were both patterned in 1.0-mm-thick fused silica substrates using photolithography. The diffraction efficiencies > 90% were available from these diffractive optical elements, both of which were designed to function in the Raman–Nath regime with Q factors < 0.1, to offer high efficiencies over a wide wavelength range regardless of the polarization status [29]. The efficiency of a blazed phase grating in that scalar region is given by $\eta = \textrm{sin}{\textrm{c}^2}({1 - {\lambda_0}/\lambda } )$, and thus the spectral amplitude $\sqrt \eta $ varies from 0.997 to 1.000 as a function of λ for a 20-fs pulse. This spectral profile modulation is too small to cause a significant temporal pulse stretch. The refractive index dispersion is negligible over the pulse spectral range.

An example baseline design of the hybrid lens is shown in Fig. 2, where fused silica with an Abbe number of 69 was selected as the lens material. Choosing the hybrid lens focal length as F₀ = 150.0 mm with M = 1, the longitudinal chromatic aberrations were removed in theory for f₁₀ = 132.6 mm and f₂₀ = ‒1143.5 mm by setting Δz = 0 in Eq. (2). The dispersive properties of the hybrid lens were tailored as $dF/d\lambda = 3.2 \times {10^4}$ and ${d^2}F/d{\lambda ^2} ={-} 3.1 \times {10^9}/m.$ On the other hand, the dispersive properties of the refractive singlet in the prototype [17], which was made from highly dispersive glass E-FDS3 (HOYA) with an Abbe number of 16 [27], were $dF/d\lambda = 3.2 \times {10^4}$ and ${d^2}F/d{\lambda ^2} ={-} 1.2 \times {10^{11}}/m.$ As expected, ${d^2}\textrm{F}/d{\lambda ^2}$ of the hybrid lens was smaller by two orders of magnitude than that of the refractive singlet, whereas no difference was found in $dF/d\lambda $ between the two lenses. With these tailored dispersions, the pulse front distortion was computed from Eq. (3) as Δτ = 5.0 fs for 2R = 15 mm, and thus the pulse width at F₂ was estimated as 20.6 fs, suggesting that the transmitted pulse at F₂ remains as short as the input pulse. Using Eq. (6), the effective Abbe number V_eff of the designed hybrid lens was calculated as V_eff = 18, indicating that a large amount of chromatic aberrations produced by the DFL can be canceled owing to the action of the diffractive surface, even though a low-dispersion glass such as fused silica was selected for the hybrid lenses. The pulse front distortion can be further reduced by selecting a type of glass that is less dispersive than fused silica, for example, FCD100 (HOYA) with an Abbe number of 95 [27]. Nevertheless, we preferred to adopt fused silica because of its compatibility with dry-etching processes that we utilized for forming blazed corrugations on the diffractive surface.

 

Fig. 2. (a) Chromatic aberration and (b) pulse front distortion plotted against f₁₀ at λ₀, the focal length of the refractive lens, a component of the hybrid lens to be designed.

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However, the designed baseline configuration needs to be slightly modified, considering a scheme for fabricating the hybrid lens, which will soon be explained. After the modification, the focal lengths transformed to F₀ = 150.3 mm, f₁₀ = 132.2 mm, and f₂₀ = ‒1062.2 mm. This layout modification hardly changed the chromatic properties of the hybrid lens because we arranged the diffractive and refractive surfaces as close to one another as possible. Figures 3 and 4 present numerically simulated chromatic properties for the optical system with the modified hybrid lenses, whose ray matrix is given in Appendix C. In Fig. 3, the heights of the chief rays diffracted at 2.9°, as an example, are plotted against the wavelength, where the height for λ₀= 780 nm is 2.505 mm. The broken line shows the ray heights at F₁, and the solid line shows those at F₂. As observed, the spectral components walk off from the center of the beam spot in F₁, 2.2 µm at the edges of the pulse spectral range, 780 nm ± 23 nm, whereas they stay at the center in F₂. Walk offs of different spectral components in F₁ are corrected by a pair of the designed hybrid lenses. (In the first-order approximation, the chief rays are concentrated at the beam center y₀ = f₀u₀ in F₁. If we incorporate the second term in the Taylor series expansion of ray height y, the height f₀u₀ is modified by a factor of $1 - {({\Delta \lambda /{\lambda_0}} )^2}$). In Fig. 4, longitudinal chromatic aberrations for the on-axis beam are plotted against the wavelength. The broken line shows the longitudinal chromatic aberrations at F₁, and the solid line shows those at F₂. A large amount of aberrations at F₁ is corrected at F₂ by propagating the aberrated wave fronts through the afocal subsystem. No significant differences in the computed chromatic responses were observed between the on- and off-axis beams. Figure 5 shows pulse front distortions plotted as a function of the beam radius R₀ for t_p = 20 fs (Δλ = 45 nm) and t_p = 100 fs (Δλ = 9.0 nm), which were computed by using Eq. (4). For t_p = 20 fs, the distortion at R₀ = 7.5 mm was 5.0 fs. The distortion would be significantly smaller for 100-fs pulses.

 

Fig. 3. Heights of chief rays diffracted at 2.9° plotted against the wavelength. Broken line is at the intermediate plane F₁ and, solid line is at the exit plane F₂ in Fig. 1.

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Fig. 4. Chromatic aberrations plotted against the wavelength. Broken line is at intermediate plane F₁, and solid line is at exit plane F₂ in Fig. 1.

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Fig. 5. Pulse front distortions at F₂ in Fig. 1 as a function of the beam radius at the hybrid lenses. Solid line is for a 20-fs pulse, and broken line is for a 100-fs pulse.

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In the fabrication of the hybrid lenses, a diffractive lens and a refractive lens were prepared separately, and then they were integrated in a metal cylinder with a small air gap disposed between them. The diffractive lens with a negative focal length of ‒1062.2 mm was patterned with multilevel blazed corrugations on a fused silica substrate. The refractive lens with a positive focal length of 132.2 mm was formed into a fused silica plano-convex lens, 40 mm in diameter and a thickness of 5.93 mm at the center. We found it more difficult to control the refractive focal length than the diffractive focal length. A set tolerance of the refractive focal length was +0.0 mm and ‒0.1 mm. The air gap between the corrugations of the diffractive lens and the rear side of the convex lens was adjusted to 0.10 mm. The air gap distance was kept stable with a spacer. All air/glass interfaces were anti-reflection-coated to exclude stray light and hence improve the light use efficiency. Figure 6 presents a photograph of the fabricated hybrid lens and a schematic drawing of its cross-sectional view: the inset sketches continuous blazed corrugations for simplicity.

 

Fig. 6. Fabricated hybrid lens: (a) a photograph of the lens and (b) a sketch of its cross section.

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For pre-chirping the input pulse, we used a pair of commercially available chirped mirrors. The pre-chirp was initially set to ‒840 fs² to cancel a GDD, which yields non-parabolic pulse broadening common to the on- and off-axis beams, by considering the thicknesses of the optical components and the material dispersions of fused silica that provides a GDD of 35 fs²/mm. The initial pre-chirp would be adjusted for pulse characterization at individual diffraction angles by considering GDDs through angular dispersions. Given that the diffractive subsystem in Fig. 1 operates like a grating-pair stretcher [30], where the DBS diffracts a pulse beam and the DFL re-collimates and focuses it, by setting s = f₀, we find an expression for angle-dependent-GDDs in the paraxial approximation

5. System evaluation and results

The designed pulse delivery system was constructed on an optical bench with a Kerr-lens mode-locked Ti:sapphire laser (TFS-1, NEW FEMTO). The oscillator operated at an 80-MHz repetition rate and delivered 22-fs, 4-nJ optical pulses at λ₀= 780 nm and Δλ = 44 nm (FWHM). The pulsed beam from the oscillator was TEM₀₀, linearly polarized, and 4.5 mm (1/e²) in diameter at the chirped mirrors, that is, the beam diameter R₀ = 13.5 mm in the afocal subsystem. The beam diameter was adjusted to exclude spherical aberrations on the pulse duration [31].

The on- and off-axis pulse beams were characterized at and across exit plane F₂. The intensity distributions of the focused pulse beams were captured using a beam profiler (SP620U, Ophir-Spiricon) with a spectral range of 190–1320 nm and a pixel size of 4.4 µm × 4.4 µm. For the temporal characterization of the pulses, we built an interferometer upstream of the cascade system and measured the interferometric autocorrelations in the collinear second-harmonic generation geometry. The dispersion compensation accuracy in this experiment was limited to ±40 fs² in terms of GDD, corresponding to ±0.7 fs in the pulse duration, due to discrete adjustment of the pre-chirp. The second-harmonic emission from a 10-µm-thick barium borate (BBO) crystal was detected by a GaP photodiode. A bandpass filter and a small aperture were placed before the photodiode. The experimental setup is illustrated in Ref. [17].

5.1 Focused beam width

Figure 7 shows the focused beam widths w (1/e²) plotted against the detection position Δx for ±2.0 mm range around F₂, where u₀ is the diffraction angle at λ₀ and hybrid lens HL₂ lies in the negative region of Δx. In this figure, solid symbols show the vertical beam widths, and hollow symbols show horizontal beam widths. The pulse beams were diffracted horizontally. The solid line represents the computed beam widths for a Gaussian beam with no aberrations. The blazed phase gratings were in place for the off-axis measurements (u₀ = 1.7° and 2.9°), and no grating was in place for the measurements on the axis (u₀ = 0.0°).

 

Fig. 7. Focused beam widths around exit plane F₂ at the diffraction angles of u₀ = 0.0°, 1.7°, and 2.9°. The solid line represents a theoretical prediction.

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As shown in Fig. 7, for u₀ = 0.0°, the vertical and horizontal beam widths were both minimized at F₂, where the minimum spot size was found to be 13 µm, comparable to the diffraction limit at λ₀ = 780 nm, which was estimated as 2λ₀F₀/πR₀ = 11 µm. For u₀ = 1.7°, the vertical beam width was minimized at F₂, whereas the horizontal beam width was minimized ∼0.1 mm before F₂ owing to the astigmatism introduced by the convex surface of hybrid lens HL₂, and a circular beam spot (circle of least confusion) was found to be 18 µm in diameter. That astigmatic difference was smaller than the Raleigh depth, which was calculated as λ₀F₀²/πR₀² = 0.12 mm. Therefore, one can find the best image plane, that is, a plane perpendicular to the optical axis in which two beam spots at u₀ = 0.0° and 1.7° become circular with an equal diameter. For u₀ = 2.9°, however, the astigmatic difference increased to be ∼0.5 mm, and the minimum circular spot was found to be 30 µm in diameter. Note that astigmatism decreases by increasing the focal length F₀: astigmatic difference is proportional to ∼y²/F₀ [32].

5.2 Temporal pulse width

Figure 8 shows the interferometric autocorrelation traces obtained from the focused pulses. The trace from the input pulse in (a) and the traces obtained in the current system at and around the circle of least confusion for u₀ = 0.0°, 1.7°, and 2.9°, in (b), (c), and (d), respectively. In these measurements the pre-chirps were adjusted by taking account of GDDs through angular dispersions, which were 0 fs² for u₀= 0.0°, ‒61 fs² for u₀ = 1.7°, and ‒177 fs² for u₀ = 2.9°, respectively. To determine the pulse durations from these traces, we assumed a linearly chirped Gaussian pulse having a complex envelope $E(t )= {E_0}\textrm{exp}[{ - ({2\textrm{ln2}} )({1 + ia} ){t^2}/t_\textrm{p}^2} ]$ [33], in order to find the values of the chirp parameter a and t_p that yield the best fit of the computed trace to the measured trace. We employed the linear chirp model because we realized in this optical system GDDs would contribute much more to temporal pulse broadening than third-order-dispersions (TODs). About phase delays through angular dispersion, for instance, we estimated the magnitudes of GDD and TOD with the analytical model built for Eq. (7) to find that GDD would be roughly 30 times larger than TOD: $|{TOD/GDD} |\approx \Delta \lambda /{\lambda _{_0}}{\; }$, where Δλ was 23 nm and λ₀ was 780 nm. For the detection of GDD and/or TOD, a useful pulse characterization tool is presented in Ref. [34].

 

Fig. 8. Interferometric autocorrelation traces: (a) from the input pulse; (b) from the transmitted pulse through the hybridized system at u₀ = 0.0° (on axis), (c) u₀ = 1.7°, (d) and u₀ = 2.9°; and (e) from the transmitted pulse through the prototype at u₀ = 0.0°.

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All these traces in Fig. 8(a)‒(d) were well fit by assuming a transform-limited Gaussian pulse with a = 0.0 and t_p = 22 fs regardless of the diffraction angles, i.e., these focused pulses were nearly transform-limited, 22 fs in pulse duration. Fine structures in the traces differed very slightly, as can be seen. The trace in Fig. 8(a) was obtained by focusing the pulse beam with a 50-mm-focal-length silver-coated parabolic mirror, generating no chromatic aberrations. In comparison, the trace obtained on the axis in the prototype system with a pair of the refractive singlets made from E-FDS3, is shown in Fig. 8(e). In this measurement, the pulse duration was minimized by providing the pulse with a pre-chirp of ‒3400 fs² because E-FDS3 was highly dispersive with a GDD of 394 fs²/mm. This trace was well fit by choosing a = 1.4 and t_p = 35 fs. In Fig. 8(e), a few wings are observed in the trace for 50-fs and 100-fs delays. These wings, which appeared in the traces from the off-axis beams too [17], were due to the residual pulse front distortions in the prototype. No wings were observed when propagating the 20-fs pulse through a parallel plate of E-FDS3.

From the results above, we state that the inclusion of a pair of hybrid lenses in the afocal subsystem removed not only chromatic aberrations but also pulse front distortions, through an effective management of dispersions in the pulse delivery system, and as a result, its spatio-temporal resolutions were improved to be close to their theoretical limits by tuning the pre-chirp at individual diffraction angles. For arrayed pulses, however, no single condition of pre-chirping can remove all pulse stretches at different diffraction angles. We will later discuss how to optimize the amount of pre-chirp, hence minimizing the remaining pulse stretches in the beam array.

5.3 Beam array length

Thus far, to evaluate the system, we characterized the +1-st-order diffraction beams from the blazed gratings and observed that the proposed multifocusing technique could remove pulse distortions at diffraction angles of 0.0°, 1.7°, and 2.9°, although with increasing the diffraction angle, the beam focus was affected by astigmatism, not by chromatic aberrations. Now, we use a DBS to generate a pulse beam array to investigate how far the beam array is extended with circular spots in an equal diameter.

Figure 9 shows an array of focused pulse beams generated in the best image plane set in the vicinity of F₂ using a one-dimensional (1D) splitter with 13 split counts. A total of 52 focused beam spots were aligned with a pitch of 0.12 mm, and among them, 27 circular spots with 18 µm in diameter, from the ‒13-th-order to the +13-th-order, were aligned 3.23 mm in length. The diffraction angle for ±13-th-orders was 1.9°. From these findings, we affirm that using the current optical setup, one can produce 1D pulse beam arrays 3.2 mm or larger in length and 2D pulse beam arrays in a circle of 3.2 mm or larger, depending on tolerances in the beam spot circularity and temporal pulse broadening. Outside ±14-th-order diffractions, the beam spots were found to be gradually elongated in the horizontal direction with increasing the diffraction angle, because of the remaining astigmatism in the hybrid lenses. Note that in Fig. 9, the 0-th-order spot is brighter than others because the binary phase splitter was originally designed at 532 nm, not the pulse central wavelength of 780 nm.

 

Fig. 9. Array of the focused pulse beams generated by propagating a 20-fs pulse beam through the proposed pulse delivery system. A one-dimensional diffractive beam splitter was used to split the beam.

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6. Discussion

The amount of pre-chirp can be readjusted to extend the beam array to its allowable upper bounds by considering diffraction-angle-dependent GDDs. Those GDDs through angular dispersions introduced by the DBS at F₁ are transferred to F₂ [35], and thus angle-dependent pulse stretches remain on the arrayed pulses. With the GDD given by Eq. (7), the pulse duration t at F₂ is computed as $t = \sqrt {t_\textrm{p}^2 + {{({GDD \times \Delta \omega } )}^2}} $, where t_p is the initial pulse duration. Figure 10 shows the GDDs plotted as a function of the diffraction angle u₀. If we accept a pulse stretch of 1.0 fs with t_p = 22 fs, t comes under 23 fs between u₀ = 0.0° and 1.5° (solid line). Giving the GDD an offset of ‒47 fs², t is obtained under 23 fs between u₀ = 0.0° and 2.1° (broken line). Moreover, if we offset the GDD by ‒67 fs², for a stretch of 2.0 fs, then t is obtained under 24 fs between u₀ = 0.0° and 2.5° (dotted line). In most of industrial processing applications, the amount of a pulse chirp matters and its sign hardly does. The beam array length L is given by the product of the magnification M, the DFL focal length f₀, and the outer-most diffraction angle u_max in the beam array; L = 2Mf₀u_max. With M = 1 and f₀ = 50 mm, diffraction angles u_max of 1.5°, 2.1°, and 2.5° correspond to beam array lengths of 2.6, 3.7, and 4.4 mm, respectively. The beam array presented in Fig. 9 was obtained by minimizing the pulse duration at u₀ = 1.7°: due to the limited pulse energy we were not allowed to adjust the pre-chirp by characterizing split pulses. Through this discussion, we state that the pulse stretch tolerance determines the beam array length or the working area in individual applications.

 

Fig. 10. Remaining GDDs through the angular dispersions introduced by the DBS: Solid line, broken line, and dotted lines represent the GDDs with offsets of 0, ‒47, and ‒67 fs², respectively.

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We can make the beam array longer by setting the magnification to M > 1 at the expense of an increased footprint of the optical setup. For example, if we set M = 2, then the beam array length would be doubled in theory from 3.2 mm to 6.4 mm for u₀ = 1.9°. The pulse front distortions would be reduced by a factor of (1 + 1/M) in Eq. (4). A simple way to obtain a large beam array is employing a longer pulse, e.g., 100 fs.

The compensation of pulse front distortions is absolutely important for making use of optical nonlinearity of ultrashort pulses for laser-processing materials without heat-induced damages. To ensure the thermal-free nature of the ultrashort pulses, the pulse duration is desired to be < 50 fs [36] and thus the distortion compensation should be considered in the sub-50-fs domain. In addition, the nonlinearity emerges strongly when materials are irradiated with a tightly focused pulse. As indicated in Eq. (4), the pulse front distortion Δτ is in proportion to the pulse spectral bandwidth Δλ (or reciprocal of the pulse duration t_p) and the square of pulse beam radius R, i.e., the pulse front distortions increase with shorter pulses and with thicker pulse beams.

Finally we note that pulse front tilts through angular dispersions at F₁ are also transferred to F₂. The relationship between the tilt angle α and the angular dispersion du/dλ is given by $\textrm{tan}\alpha = \lambda |{du/d\lambda } |\; $[37], and using this formula, the tilt angle at F₂ is found approximately as α = u₀. The accurate determination of spatio-temporal behaviors of diffracted and focused ultrashort pulses requires numerical simulations based on wave optics [38].

7. Conclusion

We developed an achromatic cascade optical system including diffractive-refractive hybrid lenses for multifocusing ultrashort pulse beams. Using the ray-matrix analysis, we derived simple design formulas for the system and tested the designed system using 20-fs pulses. We verified an effective dispersion management by the hybrid lenses through the improved spatio-temporal resolutions for the transmitted pulses. Beam-radius-dependent pulse broadening or pulse front distortions, which compromised the performance of the prototype, were successfully compensated. The proposed pulse delivery technique greatly enhances the practicality of materials processing technology with ultrashort pulse lasers.