## Abstract

We present simple formulas for the diffraction efficiencies of a binary phase grating that performs array illumination with ultrashort pulse beams. Using scalar diffraction theory, we formulated the efficiencies as a function of pulse spectral width by Fourier-transforming the complex-modulated frequency spectra of diffracted pulses in the far-field region. From the analytical simulations, we found that pulse array uniformity departs from unity as the spectral width increases, or the pulse duration decreases, thereby limiting the attainable split counts. This finding can be considered in the design of gratings for delivering controlled amounts of pulse energies to diffraction orders of interest.

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

## 1. INTRODUCTION

Splitting an ultrashort pulsed laser beam with a diffractive phase grating enables high-throughput parallel processing of materials. The spatiotemporal profiles of the focused pulses and diffraction efficiencies of the grating determine the quality and speed of the process. Space-temporal distortions, including chromatic aberrations and pulse broadening owing to optical system dispersions (material dispersion and angular dispersion), can be well compensated for using dispersion management techniques, which have been studied intensively for generating an array of ultrashort pulse beams [1–3]. Meanwhile, to the best of our knowledge, the diffraction efficiencies of a phase grating under ultrashort pulsed laser illumination have not necessarily been deeply discussed in the community. With continuous wave (CW) lasers and long-pulse lasers, gratings can be designed at a single wavelength, mostly the central wavelength of a narrow emission spectrum, and their diffraction properties can be accurately predicted by considering fabrication errors [4,5]. The same cannot be true, however, when the grating is used with an ultrashort pulse beam because of its broad emission spectrum. Although phase gratings have been extensively used in a variety of ultrafast material processing [6–8], these gratings are designed at the carrier frequency of a pulse, and no particular attention is paid to the potential negative effects of neglecting the wide range of spectral components in a grating design. This is seemingly because of the absence of an easy-to-use analytical model to predict the pulse energies of diffracted ultrashort laser pulses.

Among the few studies on diffraction efficiencies of gratings illuminated
with ultrashort pulse beams, Ichikawa analyzed the diffractive
characteristics of ultrashort pulses using a finite-difference time-domain
(FDTD) method for a lamellar grating in the resonance domain [9]. As manifested by the computed
efficiencies in the report, the following author’s statement is
insightful: the longer the pulse, the closer its behavior becomes to that
of a CW wave. However, performing FDTD on the complex phase profiles of
gratings designed for array illumination is not feasible. Veetil *et al*. performed elegant numerical simulations based
on wave optics to visualize the propagation of ultrashort pulses split by
a phase grating into a ${5} \times {5}$ pulse array [10]. In theory, their tool enables the computation of the
pulse energies of diffracted pulses, yet no examples have been presented.
Some research groups have implemented the spatiotemporal characterization
of diffracted ultrashort pulses passing through circularly symmetric
binary gratings, revealing interesting results on diffractive fields
behind the element. However, these studies did not address the diffraction
efficiency issues [11,12].

Through the development of an optical system for multifocusing ultrashort pulses [3], we realized the necessity of building a convenient analytical model with a few key variables to compute the pulse energies of diffracted pulses. In this study, we picked binary phase gratings [13] and formulated their diffraction efficiencies under ultrashort pulsed laser illumination. Binarized surface-relief patterns are easy to fabricate with small profile errors through mask-based photolithography, allowing the grating to perform as expected [14]. Using these formulas, we analyzed the energies of diffracted ultrashort pulses, discussed the uniformities of arrayed pulses, and proposed a grating design scheme for high uniformities, all with laser processing applications in mind.

## 2. FORMULATION

Here we address the following problem: When a binary phase grating is normally illuminated with an ultrashort pulse beam, how can the input pulse energy be distributed into diffraction orders of interest? We assume that the transmissive grating comprising 0 and $ \pi $ phases is made of fused silica, low dispersive and optically linear, and operates in the Raman–Nath regime. Using the scalar diffraction theory [15], we derived a set of simple formulas to compute the pulse energies of diffracted pulse beams. The essential idea of the formulation to be presented contains complex modulations of the frequency spectra by diffraction.

For an input ultrashort pulse, we used a Gaussian temporal field defined as [16]

The spectral width (FWHM) of ${| {{E_{\rm in}}(\omega)} |^2}$, $\Delta \omega$, is related to $ t_g $ as $\Delta \omega = 2\sqrt {2{\ln}(2)} /{t_g}$.

The grating works as a kind of spectral modifier, and thus in the far-field region, the diffracted pulses have their frequency spectra amplitude-modulated with phase delays. These modulations can be formulated as a function of $\omega$ (refer to Appendix A),

Here the subscripts $(m,n)$ and (0, 0) denote diffraction orders and ${a_{m,n}}({{\omega _0}})$ and ${a_{0,0}}({{\omega _0}})$ are the complex amplitudes of the diffracted waves at ${\omega _0}$, respectively. The moduli of the amplitudes, $| {{a_{m,n}}({{\omega _0}})} |$ and $| {{a_{0,0}}({{\omega _0}})} |$, are arbitrarily given as preconditions in the design of binary phase gratings. Using Eqs. (3a) and (3b), we construct the frequency spectra of the diffracted pulses in the far-field region as

Here we note that the spectral modulations can influence the pulse durations, which will be discussed later. And we emphasize the premise for the formulation that we neglect spatiotemporal distortions, including pulse chirps, due to dispersions while focusing on spectral modulations due to diffractions. Such distortions on the diffracted pulses are to be removed by employing dispersion control techniques [1,3]. By inversely Fourier-transforming the complex-modulated frequency spectra, we obtain the expressions for the temporal fields of the $(m,n)$th-order and (0, 0)th-order diffracted pulses in the far-field region,

Then, by integrating these temporal fields with respect to time $ t $, we derive the formulas for the energies of the $(m,n)$th-order and (0, 0)th-order diffracted pulses as

With the relation $\Delta \omega = 2\sqrt {2{\ln}(2)} /{t_{g}}$, these formulas can be rewritten as follows:

With the input pulse energy set to unity, Eqs. (7a) and (7b) present the diffraction efficiencies of a binary phase grating illuminated by an ultrashort pulse beam. Setting $\Delta \omega = 0$ in the formulas, we have ${P_{m,n}}(0) = {| {{a_{m,n}}({{\omega _0}})} |^2}$ and ${P_{0,0}}(0) = {| {{a_{0,0}}({{\omega _0}})} |^2}$, which are the efficiencies under monochromatic illumination at ${\omega _0}$. As indicated by Eqs. (7a) and (7b), the input pulse energy is preserved after diffraction and the following relation holds:

The first term in Eq. (8) is the sum of all the diffracted pulse energies, except for the (0, 0)th order. Note that $\sum \sum {| {{a_{m,n}}({{\omega _0}})} |^2} + {| {{a_{0,0}}({{\omega _0}})} |^2} = 1$.

The derived formulas indicate that under ultrashort pulse illumination, the diffraction efficiencies at individual orders depend on the spectral width, that is, the pulse duration; therefore, unless the grating is designed with consideration of such a spectral dependence of efficiencies, it may not perform as designed. This finding is supported by the example analyses in the next section. For the computations of the pulse energies, we use Eqs. (7a) and (7b), as opposed to Eqs. (6a) and (6b), making it easy to plot the computed data. These equations are applied to both 2D and 1D gratings.

## 3. EXAMPLE ANALYSES

Using Eqs. (7a) and (7b), we investigated how the input pulse energy could be distributed into a pulse beam array with odd splits and even splits separately. The choice of odd split or even split, as well as split counts, is important in laser processing applications, because the (0, 0)th-order pulse, which is included in odd splits and is excluded from even splits, has a decisive impact on the processing quality. We computed the pulse energies ${P_{0,0}}$ and ${P_{m,n}}$ and then the pulse energy ratio ${P_{0,0}}/{P_{m,n}}$ against the spectral width; this ratio represents the pulse array uniformity for odd splits and does not for even splits. Here we take notice of the uniformity, not of the pulse array efficiency, because the efficiencies show no significant variations against the spectral width, which is defined as the sum of the pulse energies in the array over the input pulse energy. For a plain explanation, we assumed in analyses that the input beam would be evenly split into the diffraction orders in the array, except for the (0, 0)th order. Note that diffraction patterns produced by a binary phase grating are of center symmetry around the (0, 0)th-order beam [13]; therefore, only odd orders are taken for even splits. Gratings for even splits can be designed by configuring the 0-phase pixels and $ \pi $-phase pixels in equal numbers within a period [17]. In the following computations, the carrier frequency ${\omega _0}$ was set at ${2.42} \times {{10}^{15}}\;{\rm rad}/{\rm s}$, corresponding to the central wavelength ${\lambda _0} = {780}\;{\rm nm}$; ${\omega _0}{\lambda _0} = {2}\pi {c}$ where $c$ is the light speed in vacuum.

#### A. Odd Split

In odd splits, the (0, 0)th-order beam is included in the array, and thus its behavior influences the pulse array uniformity. The energies of the $(m,n)$th-order and (0, 0)th-order diffracted pulses, ${P_{m,n}}$ and ${P_{0,0}}$, were computed as a function of $\Delta \omega /{\omega _0}$ for split counts of $N = {25}$ (${5} \times {5}$), 169 (${13} \times {13}$), and 1225 (${35} \times {35}$). The array efficiency at ${\omega _0}$ was set to 0.80, as an example, by referring to the potential efficiencies of binary phase gratings reported in Ref. [18], which range from 0.70 to 0.85, depending on the split counts, and thus ${| {{a_{m,n}}({{\omega _0}})} |^2} = {| {{a_{0,0}}({{\omega _0}})} |^2} = 0.80/N$ and ${P_{0,0}}(0)/{P_{m,n}}(0) = 1$. From the results shown in Figs. 1(a), 1(c), and 1(e), we find that ${P_{0,0}}$ increases and ${P_{m,n}}$ decreases against the spectral width, and these behaviors become more remarkable for large split counts. As a result, the pulse array uniformity, ${P_{0,0}}/{P_{m,n}}$, departs from unity as the spectral width increases, as shown in Figs. 1(b), 1(d), and 1(f). The (0, 0)th-order pulse responds more sensitively to the spectral width than higher-order diffracted pulses, owing to the strong coupling between the (0, 0)th-order pulse and all others, as detailed in Appendix A.

Because the nonuniformity tolerance for split pulse energies depends on the application, if we set ${P_{0,0}}({\Delta \omega})/{P_{m,n}}({\Delta \omega}) \lt 1 + \alpha$ with ${| {{a_{m,n}}({{\omega _0}})} |^2} = {| {{a_{0,0}}({{\omega _0}})} |^2}$, where $\Delta \omega \gt {0}$ and ${\alpha} \gt 0$, the split count, $ N $, is limited as

#### B. Even Split

Although even splits do not include the (0, 0)th-order pulse beam in the array, thus not affecting the pulse array uniformity, it may damage the workpiece in laser processing applications if its intensity exceeds the ablation threshold in that process. The energies of the $(m,n)$th-order and (0, 0)th-order diffracted pulses, ${P_{m,n}}$, and ${P_{0,0}}$, were computed as a function of $\Delta \omega /{\omega _0}$ for split counts of ${N} = {36}$ (${6} \times {6}$), 196 (${14} \times {14}$), and 1296 (${36} \times {36}$). The array efficiency at ${\omega _0}$ was set to 0.80 and ${| {{a_{m,n}}({{\omega _0}})} |^2} = 0.80/N$, whereas ${| {{a_{0,0}}({{\omega _0}})} |^2} = 0.0$. The results shown in Figs. 2(a), 2(c), and 2(e) indicate that ${P_{0,0}}$ increases and ${P_{m,n}}$ decreases with increasing the spectral width, similar to the behaviors in odd splits, shown in Fig. 1.

The spectral width range that meets a given condition of ${P_{0,0}}/{P_{m,n}}$ depends on the split count, as shown in Figs. 2(b), 2(d), and 2(f). For example, the spectral ranges in which ${P_{0,0}}/{P_{m,n}}$ is below 0.050 were found to be $\Delta \omega /{\omega _0} = {0.050}$ ($\sim{10}\;{\rm fs}$) for $N = {36}$, $\Delta \omega /{\omega _0} = {0.021}$ ($\sim{30}\;{\rm fs}$) for $N = {196}$, and $\Delta \omega /{\omega _0} = {0.0082}$ ($\sim{70}\;{\rm fs}$) for $N = {1296}$. The larger the split count, the narrower the acceptable spectral range. For large split counts, for example, ${\gt}1000$, the (0, 0)th-order pulse may be problematic even for a 100-fs pulse because the (0, 0)th-order pulse energy approaches the arrayed pulse energies, which makes it difficult to find a pulse irradiation condition for laser ablations. Note that in even splits, the pulse array uniformity remains unaffected because the couplings among the arrayed pulses, except for the (0, 0)th-order pulse, are negligible, as explained in Appendix A.

## 4. DISCUSSIONS

How can the pulse spectral width be considered in the grating design for perfect uniformity in odd splits, that is, ${P_{0,0}}({\Delta \omega})/{P_{m,n}}({\Delta \omega}) = 1$ at a given $\Delta \omega$? Equations (7a) and (7b) allow us to derive the following expressions for the diffraction efficiencies at $\Delta \omega = {0}$ that enable perfect uniformity at $\Delta \omega \ne {0}$:

Figure 3 compares the computed uniformities with split counts of $N = {25}$ and 196, assuming $S = {0.80}$ in common. For $N = {25}$, ${| {{a_{m,n}}({{\omega _0}})} |^2}$ and ${| {{a_{0,0}}({{\omega _0}})} |^2}$ should be set to 0.0321 and 0.0306, respectively, and thus the pulse array uniformity, ${P_{0,0}}/{P_{m,n}}$, is 0.95 at $\Delta \omega /{\omega _0} = {0.0}$ (CW) and 1.0 at $\Delta \omega /{\omega _0} = {0.0574}$ (10 fs), whereas for $N = {169}$, ${| {{a_{m,n}}({{\omega _0}})} |^2}$ and ${| {{a_{0,0}}({{\omega _0}})} |^2}$ should be tuned to 0.00474 and 0.00328, respectively, and thus the pulse array uniformity, ${P_{0,0}}/{P_{m,n}}$, is 0.69, at $\Delta \omega /{\omega _0} = {0.0}$ (CW) and turns up to 1.0 at $\Delta \omega /{\omega _0} = {0.0574}$ (10 fs), as shown in Fig. 3(a). However, if the gratings have 2% depth errors, ${P_{0,0}}/{P_{m,n}}$ at $\Delta \omega /{\omega _0} = {0.0574}$ increases from 1 to 1.03 for $N = {25}$ and from 1 to 1.21 for $N = {169}$, as shown in Fig. 3(b). With grating depth errors, ${| {{a_{m,n}}({{\omega _0}})} |^2}$ moves down, whereas ${| {{a_{0,0}}({{\omega _0}})} |^2}$ moves up, and as a result, the spectral width that satisfies ${P_{0,0}}/{P_{m,n}} = 1$ becomes smaller than the chosen one; in other words, the pulse duration that satisfies ${P_{0,0}}/{P_{m,n}} = 1$ becomes longer.

Given these findings, one can make a compromise in the split counts to choose, thereby decreasing the gradient of plot line ${P_{0,0}}/{P_{m,n}}$ to minimize the deviation from the unity of the array uniformity. There are two reasons why two plot lines for $N = {25}$ look similar and those for $N = {169}$ do not in Fig. 3: 1) From Eqs. (7a) and (7b), ${P_{0,0}}/{P_{m,n}}$ increases less against $\Delta \omega$ for smaller split counts $ N $. 2) From Eqs. (A6a) and (A6b), ${P_{0,0}}/{P_{m,n}}$ at $\Delta \omega = {0}$ moves up less with grating depth errors for smaller split counts $ N $.

From Eq. (10b), the split count $ N $ has an upper limit, ${N_{\max }}$, for a given spectral width and a given efficiency, and ${N_{\max }}$ is determined by

Figure 4 shows ${N_{\max }}$ plotted against $\Delta \omega /{\omega _0}$ with $S = {0.80}$, where ${N_{\max }}$ is found to be 54,569 at $\Delta \omega /{\omega _{0}}={0.00574}$ (100 fs), 4911 at $\Delta \omega /{\omega _{0}} = {0.0287}$ (30 fs), and 546 at $\Delta \omega /{\omega _{0}} = {0.0574}$ (10 fs). From the comparison of Eq. (11) with Eq. (9), it is evident that attainable split counts can be substantially increased by incorporating the spectral width in the design of gratings. As indicated by Eq. (11), ${N_{\max }}$ is further limited with lower efficiencies at ${\omega _0}$. For split counts above ${N_{\max }}$, it is impossible to achieve the perfect uniformity ${P_{0,0}}/{P_{m,n}} = 1$.

Thus far, we have focused on the perfect uniformity in pulse energy, which is not enough to guarantee uniformity in material processing with arrayed ultrashort pulses. The pulse peak intensity is as crucial for light–matter interactions as the pulse energy [22]. As the peak intensity is given by the ratio of the pulse energy over the pulse duration, we discuss the pulse durations of the diffracted pulse beams. As indicated by Eqs. (4a) and (4b), the spectral modulations are caused by the grating, thereby compressing or stretching the transmitted pulses. As an example, the far-field intensity profiles ${| {{E_{m,n}}(t)} |^2}$ and ${| {{E_{0,0}}(t)} |^2}$ computed for a 10 fs pulse are illustrated in Fig. 5, for $N = {169}$ (${13} \times {13}$) in Figs. 5(a) and 5(b), and for $N = {196}$ (${14} \times {14}$) in Figs. 5(c) and 5(d). The array efficiency at ${\omega _0}$ is $S = {0.80}$ in common. The profiles were normalized to 1.0 for comparison. On the time axis, $t = {0}$ is where the input pulse lies, unless diffracted by the grating. As illustrated, the $(m,n)$th-order diffracted pulse delays by $\sim{0.7}\;{\rm fs}$, proportional to the grating depth $\sim\pi /{\omega _0}$, and is slightly stretched by $\sim{0.1}\;{\rm fs}$, independent of diffraction orders or split counts, whether the split count is odd or even [Figs. 5(a) and 5(c)]. On the other hand, the (0, 0)th-order pulse is compressed by $\sim{2}\;{\rm fs}$ [Fig. 5(b)] and further compressed with a growing secondary peak in odd splits as the split count increases. The pulse compression was found to be $\sim{0.8}\;{\rm fs}$ for a 20 fs pulse and $\sim{1.2}\;{\rm fs}$ for a 5 fs pulse. The intensity profile of the (0, 0)th-order pulse has double peaks with zero intensity between them in even splits [Fig. 5(d)]. These peculiar behaviors of the (0, 0)th-order pulse are because of the spectral filtering by the grating that filters out the spectral components at and around ${\omega _0}$ [23]. To summarize, diffractions can significantly compress the (0, 0)th-order pulse and hardly stretch higher-order pulses. Therefore, even if the pulse array uniformity looks perfect in pulse energy, the (0, 0)th-order pulse can protrude among the arrayed pulses in pulse peak intensity.

These results make it clear that the (0, 0)th-order pulse can be significantly affected by diffractions in terms of its temporal intensity profile both in odd splits and even splits. Regarding odd splits, if the compression of the (0, 0)th-order pulse by the grating cannot be ignored from the application perspective, one should choose a small split count to hold the pulse compression at an acceptable level. Otherwise, even split counts can be selected; however, the processing outcome suffers from the (0, 0)th-order pulse if its peak intensity exceeds the damage threshold.

Finally, we comment on the experimental validation of the proposed grating design scheme. This can be implemented by comparing two sample phase gratings with an odd split count in common fabricated on two distinct design conditions of ${P_{0,0}}({\Delta \omega})/{P_{m,n}}({\Delta \omega})$: Condition 1) ${P_{0,0}}/{P_{m,n}} = 1$ at $\Delta \omega = {0}$ and $\gt1$ at $\Delta \omega \ne \;{0}$; condition 2) ${P_{0,0}}/{P_{m,n}} \lt 1$ at $\Delta \omega = {0}$ and ${P_{0,0}}/{P_{m,n}} = 1$ at $\Delta \omega \ne {0}$. In condition 2, the uniformity difference between $\Delta \omega = {0}$ and $\Delta \omega \ne {0}$ should be sufficiently large such that the plot line ${P_{0,0}}/{P_{m,n}}$ can surely cross the line of unity near a chosen $\Delta \omega$ if fabrication errors are under strict control.

## 5. CONCLUSION

Simple formulas were derived to predict the diffraction efficiencies of binary phase gratings for array illumination with ultrashort pulse beams. Through example analyses with the formulas, we found that the distribution of pulse energies in the array is strongly influenced by the pulse spectral width, particularly when large split counts and/or short pulse durations are employed. Thus, we conclude that the pulse spectral width, that is, the pulse duration, should be properly considered, enabling the designed gratings to perform as predicted.

## APPENDIX A

The phase modulation depth of a transmissive binary phase grating $\phi$ is given by $\phi (\omega) = \omega [{n(\omega) - 1}]h/c$, where $\omega$ denotes the frequency, $c$ is the speed of light in vacuum, $n(\omega)$ is the refractive index of the grating material, and $ h $ is the grating depth. By setting $\phi = \pi$ at ${\omega _0}$, the phase modulation depth can be represented as

For a binary phase grating comprising ${\rm K} \times {\rm L}$ pixels in one period in the $ x $ and $ y $ directions, the scalar diffraction theory [15] enables the complex amplitude of the $(m,n)$th-order diffracted wave in the far-field region to be computed by

Here the periods of the grating are normalized to unity in both the $ x $ and $ y $ directions, and ${\phi _{k,l}}$ is the modulation phase depth at the $({k,l})$th pixel. This equation can be rewritten as

where $ A $ is the contribution from the pixels that have a 0 phase and $ B $ is the contribution from the pixels that have a $ \pi $ phase. For the $(m,n)$th-order diffracted wave, ${a_{m,n}}(0) = A + B = 0$ and ${a_{m,n}}(\pi) = A + \exp ({j\pi})B = A - B$, and thus we obtain $A = {a_{m,n}}({{\omega _0}})/2$ and $B = - {a_{m,n}}({{\omega _0}})/2$. Using relations $ A $ and $ B $ with Eq. (A2), we find the following expression for the $(m,n)$th-order wave:Similarly, the expression for the (0, 0)th-order wave is obtained as

These expressions, ${a_{m,n}}({{\omega _0}})$ and ${a_{0,0}}({{\omega _0}})$, work as the modulation functions for the frequency spectra of diffracted pulses in the far-field region.

We emphasize that in binary phase gratings, the (0, 0)th-order beam is unique as compared with other orders in terms of power exchanges between orders. From Eq. (A4), the formulas for the diffraction efficiencies at ${\omega _0}$ are derived as

Therefore, $\sum \sum {| {{a_{m,n}}(\phi)} |^2} + {| {{a_{0,0}}(\phi)} |^2} = {{\sin}^2}({\phi /2}) + {{\cos}^2}({\phi /2}) = 1$. From Eqs. (A6a) and (A6b), we find that the efficiencies of higher orders decrease by a factor of ${{\sin}^2}(\phi /2)$, and adding up these decreases make up for the increase in the (0, 0)th-order efficiency. Otherwise stated, the (0, 0)th-order beam is much more sensitive to phase changes than higher-order beams. The coupling between the (0, 0)th-order and others is strong, whereas the couplings among higher orders are negligible. We set $\phi = \pi$ at ${\omega _0}$ because ${| {{a_{m,n}}(\phi)} |^2}$ peaks at $\phi = \pi$, as indicated by Eq. (A6a). The approximate relation $\phi (\omega) = \pi \omega /{\omega _0}$ and other formulas based on it can be applied to reflective binary gratings.

## Funding

Japan Society for the Promotion of Science (JP19K04112).

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

All data plotted in the figures were computed using the formulas presented in the text.

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