## Abstract

Global research on the solution-processable colloidal quantum dots (CQDs) constitutes outstanding model systems in nanoscience, micro-lasers, and optoelectronic devices due to tunable color, low cost, and wet chemical processing. The two-dimensional (2D) CQDs quasicrystal lasers are more efficient in providing coherent lasing due to radiation feedback, high-quality-factor optical mode, and long-range rotational symmetry. Here, we have fabricated a 2D quasicrystal exhibiting 10-fold rotational symmetry by using a specially design pentagonal prism in the optical setup of a simple and low-cost holographic lithography. We developed a general analytical model based on the cavity coupling effect, which can be used to explain the underlying mechanism responsible for the multi-wavelength lasing in the fabricated 2D CQDs holographic photonic quasicrystal. The multi-wavelength surface-emitting lasers such as λ_{0} = 629.27 nm, λ_{1} = 629.85 nm, λ_{−1} = 629.06 nm, λ_{2} = 630.17 nm, and λ_{−2} = 628.76 with a coupling constant *κ* = 0.38 achieved from the 2D holographic photonic quasicrystal are approximately similar with the developed analytical model based on cavity coupling effect. Moreover, the lasing patterns of the 2D CQDs photonic quasicrystal laser exhibit a symmetrical polarization effect by rotating the axis of polarization with a difference of 120^{0} angle in a round trip. We expect that our findings will provide a new approach to customize the 2D CQDs holographic photonic quasicrystal lasers in the field of optoelectronic devices and miniature lasing systems.

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

## 1. Introduction

In the field of micro-cavity lasers, periodic structure plays a dominant role and the different behavior of light lead to various effects such as Anderson localization, extended Bloch states, photonic bandgap, and coherent backscattering enhancement [1–6]. Although, periodic structures have numerous applications but the deviation from the periodicity can lead to high complexity and give rise to a variety of astonishing results. Photonic quasicrystals contain such type of deviation in which building blocks are arranged in the well-defined long-range pattern but lack of translational symmetry [7–9]. The sharp diffraction pattern and self-similarity properties make quasicrystal different from the periodic and disorder structures, which can add intriguing optical design opportunities and richness in engineering the specific optical design properties of the micro-cavity lasers, which need to be addressed.

The remarkable finding of the quasicrystal provides new research opportunities in the field of crystallography when Schehtmen et al. [10] in 1984 grown a solid crystal showed a ten-fold rotational symmetry and a sharp diffraction pattern, received a Nobel prize in 2011. Interestingly, the complex structure of the photonic quasicrystals provides an instructive framework for investigating the optical behavior of the one-dimensional (1D) [11–13] and higher dimensions (2D and 3D) dielectric media [14–16]. In these structures, 1D dielectric media composed of multilayer stacks having two different dielectrics of permittivity ɛ_{1} and ɛ_{2} arranged in Fibonacci and Octonacci sequence commonly named as a non-resonant and resonant case [12,13]. Photonic quasicrystals with higher dimensions offer a long-range deterministic structure and rotational symmetry (more than 14-fold). Owing to these properties, higher dimension photonic quasicrystals have significant flexibility over the 1D quasi-periodic structures in the geometrical design, which shows potential applications in the optoelectronic and micro-cavity lasing devices [17–21]. The complex geometry of the higher dimensions photonic quasicrystal has several scientific and technical impacts including efficient feedback mechanism in different directions, small beam divergence angle for every lasing pattern, low lasing thresholds, strong localization of the optical modes, and independent surface emissions can be attained at different angles. Several 2D photonic quasicrystals can be derived from the1D Fibonacci structure [22,23]. Consequently, Roger Penrose established a tilling rule for the 2D quasiperiodic structures but a similar design rule would be increasingly complex for the broad range aperiodic structure having a rotational symmetry more than 14-fold [24]. Photonic quasicrystal, however, cannot support strict Brillouin zone but having ‘pseudo-jones’ zone (extended band-like states) at the edge of the bandgap similar to periodic structures and also support localized modes analogous to Anderson modes in the randomly arranged structures [13]. Extensive recent research projects examine the behavior of light and lasing phenomena in the 2D photonic quasicrystals because of their long-range quasiperiodic order, which exhibits unique and rich rotational symmetries in the Fourier space (such as octagonal, decagonal, or dodecagonal) and point-group symmetry [25,26]. In order to observe lasing phenomena in 2D quasiperiodic structures gain medium plays a vital role in the stimulated emission process. Up to date, some gain materials were suitable for coherent lasing action in higher dimension quasicrystals such as organic polymers, holographic polymer dispersed liquid crystals (H-PDLCs), and dyes dye-doped 2D H-PDLC [25,27,28]. Recently, colloidal quantum dots (CQDs) are regarded as the most favorable gain material for nanoscience, micro-lasers, and optoelectronic devices. Similarly, CQDs have attracted a lot of research interest globally due to various intriguing properties such as high quantum yield (100%), chemical composition, morphology, and tuneable emission wavelength [29–32]. Owing to their unique and admirable optical abilities, CQDs can be used as gain material to improve the lasing performance of the 2D photonic quasicrystals. Nanostructures provide an essential feedback mechanism for attaining the coherent beam of photons in the micro-cavity lasers. Since, Notomi et.al, fabricated 2D quasicrystal with Penrose lattice (10-fold rotational symmetry) showing complicated features as compared to distributed feedback lasers/ random laser, in which coherent lasing action was well explained by considering their reciprocal lattice constant [33]. D. Luo et al. reported the multi-mode lasing from the organic quasicrystals fabricated via holographic lithography, where the lasing mechanism was explained on the basis of photonic bandgap obtained from the nanostructures [25]. Recent literature reported some appealing results regarding the lasing mechanism in higher dimensions by considering the transmission spectra from the finite difference time domain (FDTD) simulations satisfying photonic bandgap and desecrate sets of reciprocal lattice space [34,35]. However, the feedback mechanism in the 2D photonic quasicrystals has not been explained based on the cavity mode coupling effect [36]. On the other hand, it is difficult to fabricate higher dimension photonic quasicrystals with the conventional method used for photonic crystals such as semiconductor lithography, multiphoton absorption, and template technique. The essential nanostructures needed for the feedback mechanism can be introduced into 2D and 3D photonic quasicrystals by various lithographic techniques such as direct laser writing, interference between multiple laser beams, stereo lithography, and holographic lithography [35–37]. Among them, holographic lithography is the most convenient, cheap, rapid, and effective technique for fabricating 2D and 3D quasicrystal [38–46]. To our knowledge, the analytical solution based on the cavity mode coupling effect that can be used to evaluate the output emission wavelengths of the colloidal quantum dot laser have not been reported in holographic photonic quasicrystals.

In this article, we fabricated 2D quasiperiodic structures experimentally and developed an analytical solution based on the cavity mode coupling effect which would allow us to engineer the lasing performance of the colloidal quantum dots 2D photonic quasicrystal lasers. 2D photonic quasicrystals were fabricated by using a simple and low-cost holographic lithography technique, in which an especially design pentagonal prism divides a single beam of the incident light source into five beams. The center of the pentagonal prism was covered with a black paper to avoid the generation of the 3D holographic quasicrystal. The gain medium CQDs have been covered on the 2D quasiperiodic structures via spin coating technique. The developed analytical solution based on the cavity mode coupling effect has been utilized to demonstrate multi-wavelength lasing emission obtained from the fabricated CQDs 2D holographic photonic quasicrystal laser. We also examined the polarization effect of the output lasing patterns.

## 2. Design and fabrication of 2D holographic photonic quasicrystals

#### 2.1 Analytical model based on the cavity mode coupling effect for the 2D holographic photonic quasicrystals laser

The assessment of the feedback mechanism in the 2D periodic and quasiperiodic structures is an interesting topic for the research. Generally, the emission wavelength in 1D periodic micro-cavities is related to the Brag condition along with other parameters of grating, e.g., shape, its aspects ratio, and height. Nowadays, 2D nanostructures are receiving much interest due to good confinement, different lasing spot symmetries, and the output beam can be engineered by controlling the shape and size of the respective cavities.

In this section, we have developed our analytical solution based on the cavity mode coupling effect for the 2D quasicrystal. Figure 1(a) represents Matlab simulated interference patterns arranged for five beam interference lithography. The yellow and blue colors indicate bright and dark fringes respectively. The dark red circles describe the effective area considered during the cavity mode coupling effect, the dark red arrow is pointed towards the same effective area of the microcavities obtained by considering five grating structures (same period for each), see Figs. 1(a) and 1(b). Similarly, the schematic diagram of the analytical model based on the cavity mode coupling effect for five grating structures arranged symmetrically in 2D space (*x* and *y* coordinates) is depicted in the Fig. 2(b). The nanostructures in the diagram showing ten-fold rotational symmetry are created when the grating cross each other (the effective area for cavity mode coupling, dark red dotted circle). We aim to develop a general analytical solution based on the cavity mode coupling effect to explain the mechanism of the multi-wavelength lasing emission in the 2D photonic quasicrystal lasers. According to the Fig. 1(b), five 1D grating structures having the same grating periods are symmetrically distributed in the *x* and *y* coordinate system and uniform in the *z*-direction, forming five independent 2^{nd} order distributed feedback (DFB) cavities. Similarly, the center of the assumed grating structure exhibits similar 10-fold rotational symmetry, as the Matlab simulated. If we extend a similar idea to other points in the schematic of the developed model, similar 2D quasiperiodic geometry can be obtained. Figure 1(c) shows the enlarged view of the schematic 2D photonic quasicrystals in which nanostructures are symmetrically arranged in the two-dimensional space with 10-fold rotational symmetry. When we consider each of the grating structures as a source of single 2^{nd} order DFB cavity coupling between the cavities is ignored. There will be one emission wavelength and the lasing spots would be symmetrically distributed along with the symmetry of the 2D photonic quasicrystals. Therefore, the scalar equation for the dynamic of resonance amplitude can be written as:$[da/dt] ={-} i\omega t$, where *a* is the field amplitude in the cavity and $\omega $is the resonant frequency. Let us assume the cavity mode coupling effect between the cavities, as shown in the Fig. 1(a). The resonant amplitude of each cavity can enter into the neighbouring cavity with a coupling constant. The coupling constant for each cavity is represented by${\kappa _1},{\kappa _2},{\kappa _3},{\kappa _4}\textrm{, and }{\kappa _5},$ respectively, described in the Fig. 1(b). We ignore the coupling between the non-adjacent cavities. Supposing no gain and losses in the relevant cavities forming 2D quasicrystal. Therefore, the overlap integral of the cavity modes will give us the amount of coupling.

From the equation of the coupled wave theory, the cavity mode coupling amplitude for the five cavities arranged symmetrically in the Fig. 1(a) and their expansion to the *N* coupling cavities can be illustrated as follows:

^{nd}orders DFB cavities (having the same grating periods for each cavity) arranged symmetrically in 2D space forming a 2D quasiperiodic structure. Suppose ${\kappa _{ij}}(i,j = 1,2,3,4,\textrm{ and }5)$ is the coupling coefficient when the mode of i cavity coupled to the mode of j cavity and${\kappa _{ij}} = {\kappa _{ji}} = {\kappa _i}$. The Fourier transform of the field amplitude a can be written as: With all these assumptions inserting the value of Eq. (2) in Eq. (1), after solving, the coupling wave equations in the matrix form for the five 2

^{nd}order DFB cavities arrange symmetrically in 2D space forming a quasiperiodic structure can be expressed as:

^{nd}order DFB cavity forming a 2D quasiperiodic is the same. So, we assumed that the coupling constant${\kappa _1} = {\kappa _2}\ldots . = {\kappa _5} = \kappa $. Therefore Eq. (6) can be written as:

The solution of the Eq. (9) will give us the resonant frequencies when we put every root of the${D_5}$ equal to zero:

Therefore, Eq. (10) gives us the resonant frequencies due to the cavity coupling effect of the five identical 2^{nd}order DFB cavities symmetrically distributed in the 2D space forming quasiperiodic structures. The developed analytical model based on the cavity coupling effect can be applied to less than identical five DFB cavities and also for N identical cavities arranged symmetrically in the 2D space.

#### 2.2 Experimental arrangement for the holographic lithography

The optical layout used for the fabrication of 2D quasi-periodic structures is schematically illustrated in the Fig. 2(a). In our experiment, a specially designed pentagonal prism was used to avoid the alignment complexity and vibrational uncertainties in the optical layout. The pentagonal prism organizes five beam interference at the bottom surface for generating 2D quasicrystal with 10-fold rotational symmetry, as shown in the Fig. 3(a).

A specially designed prism was fabricated from the H-K10 glass with a refractive index of 1.54. From the Fig. 2(b), it can be seen that some essential parameters should be kept in mind to design a prism for the interference holography. For example, both the top and bottom surface of the prism contain equilateral pentagon with side length *L*_{o} (12.15 mm) and *L*_{1} (22.91 mm), height h (30.00 mm), and the prism angle φ (76.14^{o}) respectively. As described in Fig. 2(a), a linearly polarized beam from the short-pulsed diode-pumped solid-state laser (343 nm, 1 ns, 550-Hz repetition rate, Coherent Inc., Santa Clara, CA, USA) after passing through the optical layout falls on the ¼ wave plate [putting fast axis at 45^{0} to the linearly polarized light) changes the beam into a circularly polarized beam. A single circularly polarized collimated beam impinges perpendicularly on the top surface of the pentagonal prism shielded with a black paper at the center of the top surface to avoid the beam along the z-axis and ensure the 2D quasiperiodic structure. Therefore, a single circularly polarized beam enters from the remaining top surface of the prism and falls on each side of the pentagonal prism. The internally reflected beam from each side of the pentagonal prism will emerge from the bottom surface to encode five-beam interfere patterns. A positive photoresist resist (PR, AR-P 3170) developer, Allresist)] thin film on the glass substrate was fixed in a holder at the bottom surface for recording the interference patterns.

The schematic of the five-beam interference pattern recorded at the bottom of the pentagonal prism is represented in the Fig. 2(c). As we discuss earlier that black paper was stuck at the top surface of the prism to avoid the beam from the center for generating a 2D quasiperiodic structure. In our experiment, the electric field distribution vectors at the bottom surface of the pentagonal prism to record inference patterns on the recording material can be written as:

*x-y*plane, $\theta = {46.84^0}$ is the angle between the refracted laser beams at the bottom of pentagonal prism and z-axis in free space. Similarly, the intensity distribution for five-beam interference patterns can be written as:

*n*is the refractive index of the recording materials.

In this experiment, we spin-coated the positive photoresist solution on a glass substrate (20 × 20 × 1 mm) at a speed of 2000 rpm for 30 s. The photoresist (PR) samples were heated on a hot plate at 110 ^{o}C for 60 s to evaporate the solvent. Similarly, the prepared samples with PR film were exposed for 4 mints to the nanosecond laser with 343 nm wavelength by using the optical layout arranged for the holographic lithography, as shown in the Fig. 2(a). The samples with recorded inference patterns on the PR film were developed in a developer (AR 300-47) for two seconds to create 2D quasi-periodic structures with long-range rotational symmetry. The gain material oil-soluble CdSe CQDs (Beijing Beida Junbang science and technology Co. Ltd) dissolved in toluene at a concentration of 40 mg/ml. After the developing process, CQDs spin-coated on the developed 2D quasicrystal at 1000 rpm for 30 s (heated at 90 °C for 60 s to evaporate the solvent) forming lasing device and used for further detailed study.

#### 2.3 Morphology of the 2D photonic quasicrystals

We have simulated the maximum and minimum five beam interference patterns similar to our experimental condition by a commercial Matlab software. Figure 3(a) illustrates the simulated 2D quasicrystal geometry found from the bright (red colors) and dark (blue colors) fringes of the respective five beam interference patterns. A small area is also simulated, as shown in the Fig. 3(b). The white dotted circles in both the simulated patterns indicate a 10-fold rotational symmetry, which is uniformly distributed over the whole 2D quasiperiodic geometry.

The top view scanning electron microscope (SEM) image of the 2D holographic photonic quasicrystal is depicted in the Fig. 3(c). It can be seen that a red dotted circle at the center of the 2D quasiperiodic geometry indicates that the similar 10-fold microstructures fill the whole 2D space with a long-range rotational symmetry. The enlarged view of the 2D quasicrystal is shown in the Fig. 3(d). The red dotted circle along with the collection of lines are drawn to display the clear image of the 10-fold rotational symmetry obtained in our experiment using a specially designed pentagonal prism for holographic lithography. Clearly, the simulated interference pattern displays a 10-fold rotational symmetry that agrees well with the SEM topography of the 2D photonic quasiperiodic nanostructures. The top surface SEM image of the CQDs spin-coated on the respective 2D quasiperiodic geometry is described in the inset of Fig. 3(c). The surface of the device almost becomes rough due to the quasiperiodic microstructure after spin coating CQDs. The cross-sectional SEM image of the whole device indicates that CQDs are filled in the gaps of the microstructures, see inset of Fig. 3(d).

## 3. Results and discussions

Figure 4(a) schematically depicts the optical setup used for the measurement of lasing spectra of the fabricated CQDs 2D holographic photonic quasicrystal lasers. In this experiment, a short-pulse laser (FLARE NX, Coherent Santa Clara, CA, USA) with 343 nm, 1 ns pulse, and 800 Hz was employed as a pump source. The power of the linearly polarized beam from the pump source about 2 mm in diameter was adjusted by the optical attenuator. We put a quarter-wave plate (fast axis at 45^{o}) in the optical layout to change the linear polarization into circular polarization. The circularly polarized beam will pump the 2D quasiperiodic structure from different orientations according to their symmetry, which will improve the emission properties of the fabricated CQDs 2D holographic photonic quasicrystal laser. The sample was excited with the CQDs on the 2D quasiperiodic side facing the pump beam normally and the emission spectra were measured from the other side by a spectrometer (Maya 2000 Pro, Ocean Optics). The emission spectra of the CQDs 2D holographic photonic quasicrystal laser under different pump energies, as shown in the Fig. 4(b). As we can observe from the Fig. 4(b), lasing started to appear at a pump energy density of about 26.5 µJ/cm^{2}. As with the small increase in the pump energy density, the first two lasing peaks appear at about 33.1 µJ/cm^{2}. Similarly, following the increase in the pump energy density more peaks, i.e., three, four, and five emerge at 42.5 µJ/cm^{2}, 62.2 µJ/cm^{2}, and 76.4 µJ/cm^{2}, respectively. Figure 4(c) depicts the five lasing peaks measured at 76.4 µJ/cm^{2}. From the Fig. 4(c), it can be clearly seen that four lasing peaks (λ_{1} = 629.85 nm, λ_{-1} = 629.06 nm, λ_{2} = 630.17 nm, and λ_{-2} = 628.76 nm) are symmetrically distributed around the λ_{0} = 629.47 nm due to cavity coupling effect. In our experiment, we consider the case where the optical cavity is coupled to the adjacent cavity. The coupling constant can be determined from the difference between the two nearest emission wavelengths$({\kappa = {\lambda_1} - {\lambda_0} = {\omega_1} - {\omega_0}} )$. The value of the coupling constant in our experiment is equal to −1.80 s^{-1}. We have used COMSOL software to evaluate the eigenmode distribution of the 2D quasicrystals at 629.47 nm, as shown in the Fig. 4(d). The red and blue colors in the simulated image indicate the lasing modes and 2D quasiperiodic geometry.

To confirm our interpretation, a comparison has been made for the resonant frequencies found from both the experimental results and the developed analytical model based on the cavity mode coupling effect, as shown in the Fig. 4(e). Here, we consider the coupling between the five cavities, which are symmetrically distributed in the two- dimensional space forming 2D photonic quasicrystals. (Figures 1 and 2). Figure 4(e) plots the comparison of the results observed in the experimental and developed analytical model, which displays the relationship between resonant frequencies and the wavenumber for *N* = 5 (the case of the experiment). The plotted results almost identical with the developed analytical model based on the cavity coupling effect for the 2D quasiperiodic structures. The detailed information about the important parameters used in both the experimental process and the developed analytical model based on cavity coupling effect, i.e., coupling constant, emission wavelengths, resonant frequencies, and wavenumber can be found in the Table 1.

The feedback back mechanism in 1D and 2D 2^{nd} order DFB is limited to one or two dimensions. This problem can be resolved, by using higher dimensions quasicrystals in which cavity periodicity is carefully perturbed by implementing hybrid DFB patterns that can maximize lasing output, multi-wavelength emissions, and lower the lasing threshold. Similarly, our work includes such a type of 2D photonic quasicrystal in which periodicity is perturbed by arranging five 2^{nd} order DFB cavities in 2D space forming quasicrystals. Figure 4(f) illustrates the multi-wavelength output intensity of the fabricated 2D photonic quasicrystal as a function of the pump fluence. The threshold point of the lasing device is at about 26.5 µJ/cm^{2}, indicated by a black arrow, see Fig. 4(f). In this experiment, the lasing threshold of a 2D photonic quasicrystal is too much low as compared to our previously reported 1D and 2D DFB laser using the same gain material (CQDs) [32]. Further, we have got multi-wavelength lasing and nice lasing patterns that are symmetrically distributed in 2D space [see Figs. 4(c) and 5(a)].

The multi-directional surface-emitting DFB lasing patterns realized in the CQDs 2D holographic quasicrystal laser exhibit excellent polarization properties in a different direction by inserting an optical polarizer. The photograph of the multi-wavelength lasing patterns obtained during the operation of the 2D holographic quasicrystal was captured by using a camera at a pump energy density of about 76.4 µJ/cm^{2}, which is depicted in the Fig. 5(a). The lasing pattern consists of five DFB lasing patterns [demonstrating by the numbers 1, 2, 3, 4, and 5 along with blue circles in the Fig. 5(a)], which are symmetrically divided into two-dimensional space. The middle of the lasing pattern is similar to the shape of the star, showing excellent lasing properties.

The polarization response of the fabricated lasing device was examined by introducing a polarizer between the lasing samples facing CQDs film on the quasiperiodic structure and the recording white paper screen at the pump energy density of 76.4$\; \mathrm{\mu }\textrm{J}/\textrm{c}{\textrm{m}^2}$. Figure 5(b) represents the schematic optical layout used for the measurement of the polarization at different polarization axis [shown by an angle$\; \alpha $ in the Fig. 5(b)]. A circularly polarized beam from the 343 nm lasers at a pump energy density of 70.6$\; \mathrm{\mu }\textrm{J}/\textrm{c}{\textrm{m}^2}$ impinges at a certain angle on the sample and the surface-emission lasing patterns pass through the polarizer. The axis of the polarizer [black dotted lines in Fig. 5(b)] rotates at a certain angle $\mathrm{\alpha }$ with respect to the lasing patterns from the 2D CQDs holographic photonic quasicrystal laser. Therefore, the lasing patterns parallel to the polarizer axis will be observed on the white screen. The photographs of the lasing patterns at different polarizations axis are shown in the Figs. 5(c)–5(e). The blue arrows indicate the directions of the polarizer axis with respect to lasing patterns. Only one lasing pattern can be seen in the Fig. 5(c) when the polarization angle $\mathrm{\alpha } = {0^\textrm{o}}$ (the blue arrow axis of polarization) is parallel to the polarization of the lasing pattern. Similarly, two more lasing patterns appear on the white screen by rotating the axis of polarization at$\; \mathrm{\alpha } = {120^\circ }$, represented in the Fig. 5(d). Figure 5(e) shows the photograph of the remaining two lasing patterns on the white screen by rotating the axis of the polarization at angle of$\; \mathrm{\alpha } = {240^\circ }$. Therefore, in our experiment, we observe three polarization directions of the lasing patterns with a difference of 120^{°} angle in a complete round trip.

## 4. Conclusion

In conclusion, we use a specially design pentagonal prism in holographic lithography to fabricate 2D CQDs multi-wavelength surface-emitting micro-cavity laser operating in the visible range, exploiting the feedback mechanism in the quasi-crystalline photonic patterns. We have developed a general analytical solution based on the cavity coupling effect to engineer the mechanism of the multi-wavelength lasing emission in the 2D photonic quasicrystals. In the proposed analytical model five independent grating structures are symmetrically distributed in the two-dimensional space behave as single DFB lasers and the cavity coupling effect between the nearest cavities is taken into account. The fabricated 2D CQDs holographic photonic quasicrystals exhibit long-range 10-fold rotational symmetry. The short-pulse semiconductor laser operating at 343 nm wavelength was employed as a pump source to study the lasing characteristic of the 2D quasicrystal laser. Hence, the respective 2D quasicrystal samples displayed beautifully five lasing patterns with a shape similar to the star in the middle. We have calculated the resonant wavelength by considering the cavity coupling effect. The results obtained from the developed analytical model provide a quantitative agreement with the experimental results. Further, the polarization response of the 2D CQDs quasicrystal laser was studied by rotating the polarizer axis at 0°, 120°, and 240°. Therefore, the desired approach of the developed analytical solution and simple holographic lithography provides a new route to design laser devices exhibiting 2D quasiperiodic structures.

## Funding

National Natural Science Foundation of China (61822501); Beijing Municipal Natural Science Foundation (Z180015).

## Disclosures

The authors declare no conflicts of interest.

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