Fabrication of double-grooved gratings using coplanar multibeam holographic lithography with a phase mask

Double-grooved gratings,^1,2 with two grooves and subridges, feature high efficiency within a wide spectral bandwidth^3,4 or across a broad incidence angular range.^5,6 Such gratings are promising for various fields such as spectrometers,⁷ achromatic displays,^3–6,8–10 optical computing,² and anomalous scattering.¹¹ The research of double-grooved gratings in early times aimed to overcome the problems of wavelength dependence of grating response in spectrometers.^12–14 For example, double-grooved fused silica gratings demonstrate nearly wavelength-independent efficiency curves for either polarized or unpolarized light ranging from 300 to 800 nm at the incident angle of −5°.³ Recently, the main potential applications of double-grooved gratings include light deflectors^15–19 and beam splitters.^20–23 First, with the development of miniaturization and integration of optical systems, diffraction gratings face up to the challenges of low efficiency at large diffraction angles. In contrast, double-grooved gratings show high efficiency at large diffraction angles. For instance, double-grooved TiO₂ gratings with slanted¹⁹ profiles can deflect incident green light to 80° with a transmission efficiency as high as 89.5%. This research represents an effective approach for light manipulation and may promote the practical applications of metadevices based on double-grooved gratings.^24–26 Second, double-grooved grating beam splitters from two-port to even seven-port have been researched by different groups.^20–23 Compared with the conventional single-groove grating beam splitters, the double-grooved grating splitters may have a wide incident bandwidth and a large tolerance of grating period. For instance, a double-groove dual-port reflection beam splitter grating with zeroth-order suppression is designed by using rigorous coupled-wave analysis (RCWA) and optimized by using simulated annealing algorithms.²³ In particular, the efficiencies of the ±1st orders of the optimized double-grooved gratings can reach 48.15% and 46.26% under TE and TM polarization, respectively. Until now, most double-grooved gratings have been fabricated by electron beam lithography.^3,5,7 One of the major challenges that handicap common applications of such metagratings is the lack of a high-throughput and cost-effective fabrication technology. Hence, it is of urgent need to develop such a low-cost and high-throughput fabrication technology for double-grooved gratings.

Electron beam lithography is superior to traditional holographic lithography in terms of resolution and flexibility of the patterns. However, holographic lithography is cost-effective for the formation of submicrometer patterns with high throughput. Compared with conventional holographic lithography, near-field holography (NFH),²⁷ i.e., holographic lithography based on a phase mask, features high throughput, low cost, and a compact setup with a short optical path. During NFH, a phase mask is used as a beam splitter and the fields from two orders, as diffracted by the phase mask, interfere.

NFH based on double-beam interference is typically used to generate periodic gratings^27–29 or varied line spacing gratings.^30,31 Under normal incidence exposure conditions, grating patterns with half of the period of the phase mask can be produced by using the interference of the light as diffracted by the phase mask.^32–37 For this exposure configuration, the reported investigation mainly emphasizes the importance of suppressing the transmitted light in the zeroth order to improve the quality of the gratings. There has been only limited work that studies the effect of the unsuppressed light in the zeroth order on double-grooved gratings.^27,33,37

In order to explore alternative methods for double-grooved gratings, it is important to develop the fabrication of double-grooved gratings based on the holographic lithography technology, which may also enrich the diversity of grating profiles by NFH and promotes the application of double-grooved metagratings. Hence, in this study, we propose and demonstrate the capability of NFH to create double-grooved gratings based on multibeam interference. The formation mechanism and evolution of double-grooved gratings will be elucidated.

Section II introduces the basic principle of multibeam interference and the simulation of NFH based on three typical coplanar three-beam and four-beam interference schemes for symmetric and asymmetric double-grooved gratings. In Sec. III, the experiment and characterization are introduced. Section IV is dedicated to the experimental results and a discussion of the double-grooved gratings produced by NFH based on multibeam interference. Section V summarizes this study and gives an outlook to future steps.

Here, A_m is the amplitude of the diffracted light in the mth order and $A m 2$ corresponds to the efficiency in the mth order of the phase mask. N is the number of the maximum diffraction order, which depends on the wavelength, the incidence angle, and the period of the phase mask. m and n are integers less than or equal to N, corresponding to the diffraction order. The total intensity behind the mask involves the background intensity (the first term) and the intensity due to mutual interference of the diffracted beams (the second term). According to Eq. (2), when the number of diffracted beams in the interference pattern is larger than 2, the light field of the interference behind the phase mask varies as a function of both x and z. The depth of focus³⁹ of the interference fringes behind the mask is reduced from infinity to limited length related to the Talbot distance.⁴⁰ To achieve the exact self-images of the phase mask, it is necessary that all plane waves are in phase at a certain distance between the mask and the substrate.⁴¹ 

According to Eq. (2), the intensity curves for the formation of double-grooved gratings on the photoresist layer can be understood as the superimposition of different portions of mutual interference of the diffracted beams (the second term). Within this multibeam interference, every two of them interfere and form their latent patterns with a certain period, which is called subgrating. The period d_mn of the subgrating can be written as d_mn = λ⁄(sinθ_m + sinθ_n). The amplitude of the subgrating depends on the diffraction efficiencies of the diffracted beams in the mth and nth orders (⁠ $A m 2$ and $A n 2$⁠).

To demonstrate the idea of fabricating double-grooved gratings by optimizing the conditions of multibeam interference, Fig. 2 shows the light field intensity distribution of coplanar three-beam and four-beam interference that can generate double-grooved gratings under the three typical incident conditions as shown in Fig. 1. The optional parameters for the multibeam interference conditions include the number of involved coplanar interference beams N, the intensity of each beam (corresponding to the diffraction efficiency of the mask in each order) I, and the incident angle α, which can be used to form symmetric and asymmetric double-grooved structures on the photoresist layer by tailoring the periodicity and symmetry of the light field distribution along the x and z directions. The simulation shown in Fig. 2 may provide a criterion for the optimization of the structural parameters of the phase mask (duty cycle and depth of a rectangular grating) for the double-grooved gratings.

As shown in Eq. (3), the total light field I_Tot of the interference field includes three terms. The first term on the right-hand side in Eq. (3) describes a constant intensity I_Background, caused by the transmission in the zeroth order of the mask, which reduces the contrast of the periodic linear pattern generated by the second and third terms. The second term represents the interference of the diffracted beams in the −1st and +1st orders, corresponding to patterns with a period of d/2 called “2ω patterns.” The third term results from the interference of the beams between the 0th and +1st, and the 0th and −1st orders, which generates 1ω grating patterns with a period of d identical to that of the phase mask. In addition, this third term shows, due to η₀, that the intensity distribution is no longer uniform along the z-direction, as also depicted in Figs. 2(a1) and 2(a3). In particular, η₀ plays an important role in adjusting the diversity of grating profiles, as illustrated in Figs. 2(a2) and 2(a4). In this case, the Talbot distance of the self-images can be written as $z T 1=nλ/[1− 1 − ( λ / d ) 2],$ $n=1,2,…$⁠, and the corresponding half-Talbot distance^41,42 can be written as $z t 1=(n−1/2)λ/[1− 1 − ( λ / d ) 2], n=1,2,…$⁠. Figures 2(a1) and 2(a3) show the intensity distribution behind phase masks with η₀ = 0.1% and η₀ = 5%, respectively. The wavelength used in the calculation is 397.5 nm, and the phase mask period is 666.7 nm, to ensure only three transmitted beams behind the phase mask. Figures 2(a2) and 2(a4) correspond to the cross sections of the intensity shown in Figs. 2(a1) and (a3) at a distance of the half-Talbot distance (z₁ = z₂ = z_t1). Both I_Tot (x = d/2) and I_Tot (x = d) can be tuned by varying the value of η₀, which may also lead to two different intensity peaks within one period in the x-direction.

Equation (4) shows that the total intensity distribution I_Tot of the interference pattern originates from four terms. The first term on the right-hand side in Eq. (4) describes the constant intensity I_Background, caused by all beams transmitted through the phase mask. The second and third terms represent the interference of the diffracted beams in the zeroth and third orders, corresponding to patterns with a period of d/(2sinθ₀). The third term is due to the interference of the diffracted light in the first and second orders with a period of d/(2sinθ₁). The fourth term results from the interference of the four transmitted beams in the zeroth, first, second, and third orders, which makes I_Tot a function along the z-direction, as shown in Figs. 2(b1) and (b3). In this case, the Talbot distance z_T2 of the self-images can be written as $z T 2=nλ/[ 1 − ( λ / 2 d ) 2− 1 − ( 3 λ / 2 d ) 2], n=1,2,…$⁠, and the corresponding half-Talbot distance can be written as $z t 2=(n−1/2)λ/[ 1 − ( λ / 2 d ) 2− 1 − ( 3 λ / 2 d ) 2], n=1,2,…$⁠. Figure 2(b1) shows the simulated light intensity distribution I_Tot behind a phase mask with a period of 950 nm. The phase mask is illuminated by a laser source with a wavelength of 397.5 nm at an incidence angle of −38.9°. The partial intensities (η₀, η₁, η₂, and η₃) of the phase mask are assumed as (40%, 10%, 10%, and 40%), omitting the material and structural parameters of the phase mask. The distance z₃ between the mask and the substrate is equal to z_t2 [dashed lines in Fig. 2(b1)]. To see the grating profile clearly, the light intensity at the distance z₃ is shown in Fig. 2(b2).

As shown in Fig. 2(b2), at the position z₃ = z_t2, the intensity I_(0/1/2/3) [the fourth term in Eq. (4)] has the phase shift of π,^43,44 and the corresponding intensity distribution curve is shown in Fig. 2(b2). Hence, the destructive interference of these intensities occurs at x = n⋅d/2, n = 1,2, … , which corresponds to two crests and valleys, which are visible within one period along the x-direction, at a lateral distance of d/2. This indicates that the superimposition of these intensities results in a double-grooved grating with a distance of d/2 between the two ridges. Moreover, $I(x=d)=4( A 0 2+ A 1 2−2 A 0 A 1)$ and $I(x=d/2)=8 A 0 A 1$ for x = d/2. Lower intensities I(x = d/2) and I(x = d) are beneficial to destructive interference and the formation of double-grooved gratings. Therefore, we optimized the parameters of the phase mask with the goal of minimizing I(x = d/2) and I(x = d).

With the identical wavelength and incidence angle, Fig. 2(b3) shows the simulated light intensity distribution I_Tot behind a phase mask with a period of 950 nm. Likewise in Figs. 2(b3) and 2(b4), (η₀, η₁, η₂, and η₃) of the phase mask are assumed as (25%, 16%, 34%, and 25%), respectively. For this asymmetric efficiency configuration, the light intensity behind the mask is shown in Eq. (A5) in the  Appendix. The distance z₄ between the mask and the substrate is z_t2 [dashed lines in Fig. 2(b3)]. The light intensity at the distance z₄ is shown in Fig. 2(b4). Destructive interference can be seen in Fig. 2(b4) due to the phase shift π of the interference intensity curves of I_0/1/2/3. Moreover, I(x = d) = I_background + 2(A₀A₃ + A₁A₂ − A₀A₁ − A₀A₂ − A₁A₃ − A₂A₃) and I(x = d/2) = I_background + 2(A₀A₁ + A₂A₃ − A₀A₂ − A₀A₃ − A₁A₂ − A₁A₃). Therefore, at the position z₄ (half-Talbot distance), two crests and valleys are seen in intensity curves along the x-direction as shown in Fig. 2(b4). Note that, according to the data shown in Figs. 2(b3) and 2(b4), it can be deduced that an asymmetric distribution of efficiencies is unnecessary for the formation of double-grooved gratings.

To demonstrate the effect of the transmitted efficiency of the zeroth order of the phase mask (η₀) on the groove profile of photoresist gratings, we first simulated the intensities of the light field of every term in Eq. (3), i.e., I_Tot, I_Background, I_±1, and I_{0, ±1} as a function of distance along the x-direction, as shown in Figs. 3(a1)–3(a6). The period of the phase mask is d = 666.7 nm. The wavelength of the incident laser beam is λ = 397.5 nm. To demonstrate typical situations, the gap or distance between the mask and the substrate is z_t1. The efficiency of the phase mask used in this and following sections were simulated as functions of structural parameters (duty cycle and depth) using RCWA.⁴⁵ In particular, the efficiency (η₀, η₁) of six representative phase masks for the three-beam interference as shown in Figs. 3(a1)–3(a6) is (90%, 2%), (42%, 21%), (16%, 32%), (10%, 36%), (4.4%, 43%), and (0%, 45%), respectively. Correspondingly, the efficiency ratio R = η₀/η₁ is 45, 2, 0.5, 0.28, 0.10, and 0. Clearly, the 1ω and 2ω grating patterns result from the interference terms of I_{0, ±1} and I_±1, respectively. The duty cycle of a conventional rectangular grating is defined as the ratio of the width of ridge w to the period d of the grating [Fig. 3(b2)]. The corresponding structure parameters of these gratings are shown in Fig. 3(b).

The nonlinear effect of photoresist is utilized for the formation of the rectangular groove profile of diffraction gratings.^46–48 Moreover, a double-grooved grating is a special rectangular grating. Hence, for a better understanding of the influence of η₀ on the morphology evolution of double-grooved gratings, the photoresist grating profiles h(x), corresponding to the light intensity as shown in Figs. 3(a1)–3(a6), was further simulated considering the nonlinear effect of photoresist, as shown in Figs. 3(b1)–3(b6). As η₀ decreases, three typical profiles are generated in the sequence: 1ω gratings with rectangular profiles, i.e., conventional rectangular gratings [Figs. 3(b1) and 3(b2)], double-grooved gratings [Figs. 3(b3)–3(b5)], and 2ω grating [Fig. 3(b6)].

When the value of η₀ is relatively high, the diffraction efficiency ratio R of the corresponding mask R = η₀/η₁> 0.5, the effect of I_±1 on the resulting grating patterns can be neglected. Therefore, the dominant patterns on the photoresist layer are 1ω grating Figs. 3(b1) and 3(b2), which is induced from I_{0, ±1}. If η₀ = 0 (corresponding to R = 0), only 2ω grating can be generated [Fig. 3(b6)]. In this case, the three-beam interference is reduced to normal double-beam interference. Therefore, the spatial distribution of the field behind the mask, I_±1, is independent of the propagation distance z.

As η₀ decreases from 16% to approximately 0, corresponding to 0 < R < 0.5, the groove profiles of double-grooved gratings evolve from initial formation [Fig. 3(b3)] and well-growth [Figs. 3(b4) and 3(b5)]. The corresponding subridge spacing (w1) and subridge width (w2), and groove height (H) and residual groove height (h) are all shown in Fig. 3(b4). According to Eq. (3), the light intensity I_Tot of three-beam interference recorded on photoresist is the sum of 1ω and 2ω intensities with different amplitudes. Correspondingly, the formation of the groove profile of a double-grooved grating is attributed to the superimposition of those of different proportions of 1ω and 2ω gratings. As η₀ decreases and η₁ increases, the percentage of 2ω gratings in I_Tot increases, which leads to the split of the ridge of the 1ω grating. The higher η₁ is, the more pronounced the splitting of the ridges of the 1ω grating is. Thus, w1 increases and h tends to 0 gradually. For such double-grooved gratings, since the subridges are split from the ridges of the 1ω gratings, the width of these subridges is narrower than that of the original 1ω grating. It demonstrates a method to form subridge gratings based on interference lithography.

Since it is not convenient to generate double-grooved gratings by conventional interference lithography, this NFH may be useful for the fabrication of double-grooved gratings with low cost and high throughput, which may be beneficial to the application of such structures further.

To clarify the formation of asymmetric double-grooved gratings at four-beam interference, we simulated the groove profile of double-grooved gratings as a function of exposure time generated by four-beam interference NFH at oblique incidence and a distance of z₃= z_t2 from the phase mask [Figs. 4(a)–4(e)], according to Eq. (4) and considering the nonlinear effect of photoresist. The period of the phase mask is 950 nm, and the incidence angle is −38.9°, which is the same as those in Sec. II B. The employed efficiency (η₀, η₁, η₂, and η₃) of the phase mask is (26.2%, 12.6%, 12.6%, and 26.2%), as shown in Table I. Similar to those in Figs. 2(b1)–2(b4), the grating ridges remain due to the destructive interference of I_0/3, I_1/2, and I_0/1/2/3 at x = d and x = d/2. In contrast, constructive interference, corresponding to peaks of intensity curves, leads to grooves formed in the photoresist layer. Thus, two grooves and subridges are formed. It is also shown in Figs. 4(a1)–4(a5) that the distance w21 between the two subridges is d/2, which does not depend on exposure time. However, the widths (w22, w23) of subridges of the double-grooved gratings decrease with the increasing exposure time. All these simulation results are identical to those in Sec. II B.

Figures 4(b1)–4(b5) show the simulated groove profiles of asymmetric double-grooved gratings as a function of distance between the phase mask and substrate at incidence angle of 2°, corresponding to the four-beam interference at asymmetric incidence angles (Table I). The period of the phase mask is 1000 nm. The phase and the diffraction efficiencies in the −2nd and 2nd, and −1st and 1st orders of the phase mask are designed to be as similar as possible, where η₋₂ ≈ η₂ and η₋₁ ≈ η₁. Moreover, the efficiency in the zeroth order is suppressed to approach 0. The optimized η₋₁ and η₁ are approximately 38%, and η₋₂ and η₂ are about 9% (Table I). At near normal incidence, the light intensity distribution behind the phase mask is no longer a conventional self-imaging, which becomes more complex than that at normal incidence. The distance between the mask and substrate of Figs. 4(b1)–4(b5) varies from 1.3 to 2.6 μm, 3.9, 5.1, and 6.4 μm, respectively. The exposure time is 55 s. As shown in Figs. 4(b1)–4(b5), with an increasing distance of 1.3–6.4 μm, the depth and width of the two grooves within each period tend to be identical to each other. This can be attributed to the fact that the second peak of the intensity P₂ is gradually up to that of the first peak P₁ [Figs. 2(c2) and 2(c3)].

As mentioned in Sec. II, different fused silica phase mask with rectangular profiles was used as the beam splitter for NFH in this study.³⁰ The resulting multibeam interference behind the mask formed double-grooved patterns that were recorded by the AZ^® MiRTM 703 photoresist (coplanar three-beam NFH) and AR-P 3740 (coplanar four-beam NFH) photoresist on the grating substrates.

The phase mask is illuminated at a wavelength of 397.5 nm. During NFH, the optical power of the exposure field measured by a detector with an area of 10 × 10 mm² is approximately 850–900 nW. The groove profile of the fabricated photoresist gratings was characterized using atomic force microscopy (AFM) in the tapping mode and scanning electron microscopy (SEM).

In the case of coplanar three-beam NFH at normal incidence, the effect of the η₀ of a phase mask on the formation of symmetric double-grooved gratings is shown in Fig. 5, to demonstrate the important role of η₀ in tailoring groove profiles. Figure 5 shows the AFM images of the fabricated double-grooved gratings using six phase masks with different η₀ (Table II). The period of all these phase masks is 666.7 nm. The nominal distance between the mask and each substrate is set at 5 mm, close to the Talbot distance. The initial thickness of the photoresist layer is ∼260 nm. All these experimental parameters were selected as consistent as possible to those in the corresponding simulation [Fig. 3]. Note that, to demonstrate more double-grooved gratings using masks with different η₀, the η₀ of the phase masks in experiments is not the same as those shown in Fig. 3.

As shown in Fig. 5, the evolution of the groove profile of the fabricated double-grooved gratings is similar to that shown in Fig. 3. The results of the evolution of the grating profile with η₀ [Fig. 5] are in relatively good agreement with the simulation [Fig. 3]. The efficiency in the zeroth order, η₀, of a phase mask plays a key role in the type of groove profile of a grating. This means that once η₀ of a phase mask is fixed, it is almost determined that whether a conventional rectangular grating or a double-grooved grating can be formed on photoresist. The lower the η₀ of the phase mask is, the more pronounced a double-grooved grating is. Other conditions, such as exposure and development times, as well as the gap between the mask and the sample, can only tune the groove profile of gratings within a certain range.

To demonstrate the formation of asymmetric double-grooved grating with NFH, we further conducted the NFH based on coplanar four-beam interference. If not specified, the experimental conditions in the coplanar four-beam NFH are identical to those in the coplanar three-beam NFH. The efficiencies (η₀, η₁, η₂, and η₃) of the phase mask used for the coplanar four-beam NFH at oblique incidence angle of −38.9° is (23.8%, 18%, 11.1%, and 30.5%) at the diffracted angle of (−38.9°, −12.1°, 12.1°, and 38.9°), respectively (Table II). The initial thickness of the photoresist is ∼200 nm. Figure 6 shows the SEM and AFM images of the groove profiles and the corresponding feature sizes of the fabricated asymmetric double-grooved gratings with exposure dose from 2.5 to 5.5 mJ/cm². The evolution of the groove profiles of the asymmetric double-grooved gratings in experiments [Fig. 6] resembles that in simulation [Fig. 4(a)]. In particular, the distance of the two subridges of double-grooved gratings does not depend on exposure dose. As shown in Fig. 6, the structural parameters of the fabrication asymmetric double-grooved gratings can be tuned by varying the exposure dose. With increasing exposure dose, the widths of the subridges decrease, while the widths of the double grooves increase. Furthermore, the width of the finer subridges decreases from 220 to 95 nm, corresponding to the duty cycle of 0.23–0.10. The width of the coarser subridges decreases from 420 to 290 nm, corresponding to the duty cycle of 0.44–0.30.

Finally, the NFH of coplanar four-beam asymmetric exposure was carried out using the phase mask with a period of 1000 nm, whose efficiencies (η₋₂, η₋₁, η₁, and η₂) are (7.7%, 37.3%, 36.3%, and 7.4%) at near normal incidence (Table II). We observed variable groove profiles of asymmetric double-grooved gratings (insets in Fig. 7) along the direction of the grating vector, which is due to the fact that the phase mask and the substrate are not strictly parallel to each other, i.e., the different distances between the phase mask and the substrates at the observed positions. This indicates that it is necessary to precisely control the distance between the mask and the substrate for uniform double-grooved gratings, which is common to other multibeam interference configurations.^49–51 On the other hand, it provides a cost-effective fabrication method of variable double-grooved gratings.

Similar to conventional interference lithography methods, interference lithography based on a phase mask can generate complex periodic micro- and nanostructures including double-grooved gratings from two-beam to multibeam exposure. On the other hand, the depth of focus of the formed patterns behind the phase mask will decrease from infinite to finite value. The limited depth of focus of grating patterns in multibeam holographic lithography may be a challenge for practical applications of this method. This means that for this multibeam holographic lithography, the larger the area of uniform grating is, the higher the requirement for the precision control of the mask and substrate is.

To overcome this challenge, the precise control of the phase mask and substrate is necessary. Several factors and the corresponding resolutions to solve the issue are summarized as follows:

1. Nanopositioning stage is needed to precisely adjust the distance between the mask and substrate, which has been used in this study.

2. The photoresist-coated substrate needs to be aligned in the direction parallel to the surface of the mask. In particular, the techniques of Moiré⁵² or alignment^53,54 have demonstrated to provide a simple way for the alignment during multibeam interference lithography, which can be considered in our future work.

3. With the increasing grating area, the flatness or wavefront of the phase mask and the grating substrate may also affect the distance between them, which may play a key role in the uniformity of the fabricated gratings.

4. In addition, similar to conventional double-beam interference lithography based on a phase mask, due to the coherence of the laser source, it is also important to suppress the interface reflections from each optical element of the exposure setup, which may lead to unwanted long-period modulation of the fabricated gratings. Such long-period modulation patterns can be suppressed by using refractive index liquids between the mask and the substrate³⁴ or integrating antireflection coatings on the mask and the substrates shown in our previous publication.⁵⁵ 

We presented an alternative method based on multibeam interference lithography with a phase mask for the fabrication of double-grooved gratings with unique properties. Submicrometer double-grooved gratings with symmetric and asymmetric groove profiles have been fabricated in a relatively simple way. Symmetric double-grooved gratings with a period of 666.7 nm, along with asymmetric double-grooved gratings with periods of 950 and 1000 nm, have been fabricated using the proposed coplanar multiple beam NFH method. For symmetrical double-groove gratings, the ratio of the subridge widths of the period ranges from 0.11 to 0.23, the aspect ratio varies from 1.3 to 2.7, and the groove depth is between 180 and 250 nm. Additionally, for asymmetric double-groove gratings, the ratio of the thinner subridge widths of the period ranges from 0.1 to 0.23, while the ratio of the thicker subridge widths of the period ranges from 0.3 to 0.44. The aspect ratio of the grating ridges spans from 0.47 to 2.1, and the groove depth is between 195 and 210 nm. The simulated and experimental results of the evolution of the grating profiles of double-grooved gratings produced by this method agree well. The demonstrated holographic interference lithography technology may overcome the limitations in the fabrication of double-grooved grating using a serial direct-writing method. Furthermore, this study enriches the diversity of grating profiles from conventional rectangular structures to periodic metastructures, which may provide insights into the development of holographic interference lithography with a phase mask.

In future, further investigations, e.g., the improvement of the pattern uniformity and the transfer of the double-grooved patterns from the photoresist to underlying materials, will be carried out to widen the applications of double-grooved gratings in spectrometers,^3,7 displays,^5–10 and optical computing.².

This work was supported by the Sino-German Centre for Research Promotion (Grant No. GZ 983) and the National Natural Science Foundation of China (NSFC) (Grant No. 11675169). Y. L. would like to thank Christoph Braig for the fruitful discussions with him and for his careful proofreading of the manuscript.

The authors have no conflicts to disclose.

Shiyang Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Shuhu Huan: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Tao Ren: Investigation (equal); Writing – review & editing (equal). Ying Liu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yilin Hong: Resources (equal); Supervision (equal); Writing – review & editing (equal). Shaojun Fu: Conceptualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

At position z_t2, 2πzλ/Δφ = π means that I_0/1(2/3) and I_0/2(1/3) have the phase shift of π. Moreover, $I(x=d)= I background$ $+2( A 0 A 3+ A 1 A 2− A 0 A 1− A 0 A 2− A 1 A 3− A 2 A 3)$ and $I(x=d/2)$ $= I background+2( A 0 A 1+ A 2 A 3− A 0 A 2− A 0 A 3− A 1 A 2− A 1 A 3)$⁠. At position z_T2, I_0/1(2/3) and I_0/2(1/3) have the phase shift of 2π. Moreover, $I(x=d)= I b a c k g r o u n d+2( A 0 A 3+ A 1 A 2+ A 0 A 1+ A 0 A 2$ $+ A 1 A 3+ A 2 A 3)$⁠.