## Abstract

We report an achromatic cascade optical system for multifocusing ultrashort pulse beams with a diffractive beam splitter. Distortion compensation requires the removal of pulse front distortions from arrayed pulses, which originate from beam-radius-dependent group delay dispersions. The inclusion of hybrid diffractive-refractive lenses can effectively manage system dispersions. Simple design formulas are derived using the ray-matrix analysis and the designed system is evaluated using 20-fs pulses. We confirm that the hybridized system can remove not only chromatic aberrations but also pulse front distortions, hence improving the system spatio-temporal focusing resolutions. The proposed pulse delivery technique enhances the practicality of materials processing with ultrashort pulses.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 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 (HL_{1}, HL_{2}), 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, F_{1} is an intermediate focal plane, and F_{2} 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.

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 Δ*y* is defined as $\Delta y = ({\partial y/\partial \lambda } )\Delta \lambda $, and, similarly, the longitudinal chromatic aberration Δ*z* is expressed as Δ$z = ({\partial z/\partial \lambda } )\Delta \lambda $, where Δ*λ *= *λ* ‒ *λ*_{0} and *λ* and *λ*_{0} 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 HL_{1} and HL_{2}, respectively, and *M* is the image magnification of the afocal subsystem. The corresponding focal lengths at *λ*_{0} are *f*_{0}, *F*_{0}, and *MF*_{0}, respectively.

In the diffractive subsystem, the DFL is located at a distance *s* from the DBS, and F_{1} is located at a distance *f*_{0} 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 Δ*y* at F_{1} is formulated as

*u*

_{0}is an arbitrary diffraction angle at

*λ*

_{0}. Equation (1) indicates that Δ

*y*= 0 for

*s*=

*f*

_{0}, regardless of the diffraction angle. Once the lateral chromatic aberrations are removed, the longitudinal chromatic aberrations at F

_{1}become independent of the diffraction angles and are given by${\; }{-}{f_0}{\; }({\Delta \lambda /{\lambda_0}} ),{\; }$and the diffracted pulse beams propagate parallel to each other. Because of this telecentricity, the chromatic aberrations in the arrayed beams at F

_{1}can be corrected at F

_{2}, as explained below. Note that coma, astigmatism, and curvature of the field are zero for telecentric diffractive optics [25].

To connect the diffractive and afocal subsystems in cascade, the distance between the DFL and HL_{1} is set to *f*_{0} + *F*_{0}, and that between HL_{1} and HL_{2} is set to *F*_{0} + *MF*_{0}. From the ray matrix of the afocal subsystem (See Appendix A), we find the following expression for the longitudinal chromatic aberration Δ*z* at F_{2},

*F*(λ) at

*λ*

_{0}. The first and second terms in Eq. (2) represent the contributions from the diffractive subsystem and the afocal subsystem, respectively.

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 *λ*_{0} as (see Appendix B)

*c*is the speed of light,

*R*

_{0}and

*R*are the radii of the marginal and an arbitrary ray, respectively, across HL

_{1}and HL

_{2}. From Eqs. (2) and (3), it follows that after all the chromatic aberrations are depleted,

*τ*approaches to zero, allowing the pulse and phase fronts to coincide after HL

_{2}[19]. Accordingly, we derive an expression for the radius-dependent pulse broadening Δ

*τ*at

*λ*

_{0}(see Appendix B):

*z*= 0, manifesting that space and time are entangled in ultrashort pulses. Assuming a Gaussian input pulse with a pulse duration of

*t*

_{p}(FWHM), the pulse width at F

_{2}can be expressed as $t = \sqrt {t_\textrm{p}^2 + \Delta {\tau ^2}} $. The pulse front distortion Δ

*τ*would be more problematic for shorter pulses with larger Δ

*λ*.

These formulas can be applied to off-axis beams as well as on-axis beams under the first-order approximation. The principal rays at *λ*_{0} impinge upon plane F_{1} 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, Δ*z* = 0 and Δ*τ* = 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 *f*_{1}(λ) and *f*_{2}(λ), respectively, the focal length of the hybrid lens *F*(λ) can be expressed as *F* = *f*_{1}*f*_{2}/(*f*_{1} + *f*_{2}), where the two lenses are assumed to be sufficiently thin, and *f*_{1}(λ) and *f*_{2}(λ) are given by

*λ*

_{0}. Although we use the thin-lens approximation at this point, we will later fix a real lens configuration by tuning the distance between the diffractive and refractive surfaces. To design the hybrid lens, we have to find a wavelength-dependent focal length

*F*(λ) that meets Δ

*z*= 0 as well as Δ

*τ*= 0. If we choose

*f*

_{10}as the primary design parameter, the longitudinal chromatic aberration can be corrected (Δ

*z*= 0) at one focal length, and the pulse front distortion can vanish (Δ

*τ*= 0) at another focal length: a small gap exists between these two focal lengths. This gap can be filled by selecting a weakly dispersive glass with a high Abbe number. Because system achromaticity is essential for dispersion control, we pick focal length

*f*

_{10}for Δ

*z*= 0 to fix

*f*

_{20}for a given condition of

*F*

_{0}. Using the obtained focal lengths,

*f*

_{10}and

*f*

_{20}, the effective Abbe number of a hybrid lens,

*V*

_{eff}, can be calculated as [28]

*V*

_{1}is the Abbe number of the refractive lens and

*V*

_{2}is the Abbe number of the diffractive lens. By including the hybrid lenses in the proposed optical system, the dispersion control can be substantially facilitated.

## 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 *λ*_{0} = 780 nm, beam diameter of 5.0 mm, and spectral bandwidth of Δ*λ* = 45 nm (FWHM). For a transform-limited pulse, Δ*λ* is given by $\Delta \lambda = 0.441{\lambda _0}^2/c{t_\textrm{p}}$.

Assuming *f*_{0} = 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 *F*_{0} = 150.0 mm with *M* = 1, the longitudinal chromatic aberrations were removed in theory for *f*_{10} = 132.6 mm and *f*_{20} = ‒1143.5 mm by setting Δ*z* = 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 Δ*τ* = 5.0 fs for 2*R* = 15 mm, and thus the pulse width at F_{2} was estimated as 20.6 fs, suggesting that the transmitted pulse at F_{2} 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.

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 *F*_{0} = 150.3 mm, *f*_{10} = 132.2 mm, and *f*_{20} = ‒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 *λ*_{0 }= 780 nm is 2.505 mm. The broken line shows the ray heights at F_{1}, and the solid line shows those at F_{2}. As observed, the spectral components walk off from the center of the beam spot in F_{1}, 2.2 µm at the edges of the pulse spectral range, 780 nm ± 23 nm, whereas they stay at the center in F_{2}. Walk offs of different spectral components in F_{1} are corrected by a pair of the designed hybrid lenses. (In the first-order approximation, the chief rays are concentrated at the beam center *y*_{0} = *f*_{0}*u*_{0} in F_{1}. If we incorporate the second term in the Taylor series expansion of ray height *y*, the height *f*_{0}*u*_{0} 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 F_{1}, and the solid line shows those at F_{2}. A large amount of aberrations at F_{1} is corrected at F_{2} 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 *R*_{0} for *t*_{p} = 20 fs (Δ*λ* = 45 nm) and *t*_{p} = 100 fs (Δ*λ* = 9.0 nm), which were computed by using Eq. (4). For *t*_{p} = 20 fs, the distortion at *R*_{0} = 7.5 mm was 5.0 fs. The distortion would be significantly smaller for 100-fs pulses.

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.

For pre-chirping the input pulse, we used a pair of commercially available chirped mirrors. The pre-chirp was initially set to ‒840 fs^{2} 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 fs^{2}/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* =* f*_{0}, we find an expression for angle-dependent-GDDs in the paraxial approximation

*du/dω*= ‒

*u*

_{0}

*/ω*

_{0}and

*ω*

_{0}is the central angular frequency,

*ω*

_{0}= 2.42×10

^{15}Hz. Using Eq. (7) the GDD was computed as ‒61 fs

^{2}at

*u*

_{0}= 1.7° and ‒176 fs

^{2}at

*u*

_{0}= 2.9°, respectively. The pre-chirp would be ultimately optimized by giving an offset to the GDD of the on-axis beam, hence widening the range of the diffraction orders to form a beam array.

## 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 *λ*_{0 }= 780 nm and Δ*λ *= 44 nm (FWHM). The pulsed beam from the oscillator was TEM_{00}, linearly polarized, and 4.5 mm (1/e^{2}) in diameter at the chirped mirrors, that is, the beam diameter *R*_{0}* *= 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 F_{2}. 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 fs^{2} 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/e^{2}) plotted against the detection position Δ*x* for ±2.0 mm range around F_{2}, where *u*_{0} is the diffraction angle at *λ*_{0} and hybrid lens HL_{2} lies in the negative region of Δ*x*. 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 (*u*_{0} = 1.7° and 2.9°), and no grating was in place for the measurements on the axis (*u*_{0} = 0.0°).

As shown in Fig. 7, for *u*_{0} = 0.0°, the vertical and horizontal beam widths were both minimized at F_{2}, where the minimum spot size was found to be 13 µm, comparable to the diffraction limit at *λ*_{0} = 780 nm, which was estimated as 2*λ*_{0}*F*_{0}/π*R*_{0} = 11 µm. For *u*_{0} = 1.7°, the vertical beam width was minimized at F_{2}, whereas the horizontal beam width was minimized ∼0.1 mm before F_{2} owing to the astigmatism introduced by the convex surface of hybrid lens HL_{2}, 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 *λ*_{0}*F*_{0}^{2}/π*R*_{0}^{2} = 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 *u*_{0} = 0.0° and 1.7° become circular with an equal diameter. For *u*_{0} = 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 *F*_{0}: astigmatic difference is proportional to ∼*y*^{2}/*F*_{0} [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 *u*_{0} = 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 fs^{2} for *u*_{0 }= 0.0°, ‒61 fs^{2} for *u*_{0} = 1.7°, and ‒177 fs^{2} for *u*_{0} = 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 Δ*λ* was 23 nm and *λ*_{0} was 780 nm. For the detection of GDD and/or TOD, a useful pulse characterization tool is presented in Ref. [34].

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 fs^{2} because E-FDS3 was highly dispersive with a GDD of 394 fs^{2}/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 F_{2} 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.

## 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 F_{1} are transferred to F_{2} [35], and thus angle-dependent pulse stretches remain on the arrayed pulses. With the GDD given by Eq. (7), the pulse duration *t* at F_{2} 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 *u*_{0}. If we accept a pulse stretch of 1.0 fs with *t*_{p} = 22 fs, *t* comes under 23 fs between *u*_{0} = 0.0° and 1.5° (solid line). Giving the GDD an offset of ‒47 fs^{2}, *t* is obtained under 23 fs between *u*_{0} = 0.0° and 2.1° (broken line). Moreover, if we offset the GDD by ‒67 fs^{2}, for a stretch of 2.0 fs, then *t* is obtained under 24 fs between *u*_{0} = 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 *f*_{0}, and the outer-most diffraction angle *u _{max}* in the beam array;

*L*= 2

*Mf*

_{0}

*u*

_{max}_{.}With

*M*= 1 and

*f*

_{0}= 50 mm, diffraction angles

*u*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

_{max}*u*

_{0}= 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.

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 *u*_{0} = 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 Δ*τ* is in proportion to the pulse spectral bandwidth Δ*λ* (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 F_{1} are also transferred to F_{2}. 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 F_{2} is found approximately as *α* = *u*_{0}. 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.

## Appendix A

By tracing back the beam paths from F_{2} to F_{1}, we write the ray matrix of the afocal subsystem with a pair of hybrid lenses as

*F*

_{0}is the focal length of a hybrid lens at

*λ*

_{0}and

*M*is the subsystem magnification. Thus, the matrix elements are obtained as

*z*= ‒

*B*/

*D*, we derive a formula for the longitudinal chromatic aberration

## Appendix B

Here, we formulate the propagation time difference between the pulse front and the phase front, along with the pulse broadening due to the dispersions in the achromatic cascade system. Using the analytical model described by Bor in Ref. [18], we write the time difference *t*_{d} in the diffractive subsystem as

*df/dλ*= ‒

*f/λ*and

*r*

_{0}and

*r*are the radii of the marginal and arbitrary ray, respectively, across the diffractive lens. Likewise, we write the time difference

*t*

_{h}in the hybridized afocal subsystem as

*R*and

*R*

_{0}are an arbitrary radius and the outermost radius across the refractive lens, respectively. By summing up these time differences, we obtain a formula for the total time difference

*τ*at F

_{2}

*λ*

_{0}, Eq. (B3) becomes Eq. (3) with

*f*=

*f*

_{0}and

*F*=

*F*

_{0}. By truncating the Taylor expansion of Eq. (B3) to the first order, we derive a convenient expression for the radius-dependent pulse broadening Δ

*τ*as

## Appendix C

A real hybrid lens is composed of a diffractive lens and a refractive lens with an air gap between them. Starting from the rear surface of the diffractive lens, the ray matrix of the hybrid lens is given by

*f*

_{1}and

*f*

_{2}are the focal lengths of the refractive surface and diffractive surface, respectively;

*e*

_{1}=

*t*

_{1}/

*n*(

*λ*) and

*e*

_{2}=

*t*

_{2}/

*n*(

*λ*):

*t*

_{1}and

*t*

_{2}denote the center thickness of the refractive convex lens and the substrate thickness of the diffractive lens, respectively, and

*n*(

*λ*) is the refractive index of a material, common to both the lens and substrate; and

*g*is the air gap. From Eq. (C1), we find the elements as

*F*

_{b}is formulated as

*F*

_{f}is formulated as

*F*

_{b}and

*F*

_{f}are computed as 150.9 mm and 142.1 mm at

*λ*

_{0}, respectively, with the following conditions:

*f*

_{1}= 132.2 mm,

*f*

_{2}= ‒1062.2 mm,

*n*(

*λ*

_{0}) = 1.454,

*e*

_{1}= 4.08 mm,

*e*

_{2}= 3.44 mm, and

*g*= 0.10 mm. Under the thin-lens approximation, the matrix elements reduce to

*A*= 1,

*B*= 0,

*C*= ‒1/

*F*, and

*D*= 1, where

*F*denotes the focal length of a hybrid lens, defined as $F = {f_1}{f_2}/({{f_1} + {f_2}\; } )$ by assuming that the diffractive and refractive surfaces overlap in a principal plane.

## Funding

Japan Society for the Promotion of Science (JP19K04112); Amada Foundation (AF-2017215).

## Acknowledgments

The authors thank Sumitomo Electric Hardmetal Corp. for their cooperation in the fabrication of the hybrid lenses.

## Disclosures

The authors declare no conflicts of interest.

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