1. Introduction

Spectral measurement plays a central role in nearly all aspects of modern optical science and engineering. Optical grating which diffracts light into different directions is the key dispersive component of the frequency domain spectrometers. Usually, spectrum unscrambling only needs the 1st order diffraction of the grating to realize the light dispersion. However, a major problem of the traditionally used binary relief gratings is that the undesired high order diffractions always overlaps the 1st one, and thus greatly degrades the resolving power and precision of analysis [1–4]. To overcome overlapping of the 1st and the neighboring diffractions, considerable efforts have been made to design and fabricate gratings which can suppress the high order diffractions. For the visible range, a solution to overcome this difficulty is to utilize one dimensional sinusoidal amplitude transmission grating instead of ordinary black white grating [5]. Unfortunately, for the short wavelengths such as soft x-ray and extreme ultraviolet, sinusoidal amplitude transmission grating introduces an additional associated phase shift and thus reproduces the high order diffractions [6]. Another solution relies on the fact that the high order diffraction modes become evanescent waves when the grating period is smaller than the wavelength in free space. As a result, metasurfaces or sub-wavelength gratings can be used for the spectral dispersion of the visible and infrared light [7–13]. However, the fabrication of sub-wavelength structures at optical frequencies is very difficult, expensive and time consuming by current nanofabrication technologies such as e-beam lithography [14]. Furthermore, it is now impossible to fabricate sub-wavelength structures for soft x-ray and extreme ultraviolet range by the current technologies. Therefore, it is a goal to design the binary relief structure much larger than the wavelength with the suppression of high order diffractions.

In recent years, several structures have been reported to realize single order diffraction gratings [15–19]. Torcal-Milla et al. pointed out that the intensity of the high order diffractions decreases with the roughness of the gratings [15]. Cao et al. proposed a grating of sinusoidal-shaped apertures, which realizes sinusoidal amplitude transmission at one direction [16,17]. In principle, this grating can completely suppress the high order diffractions. However, the fabrication of sinusoidal-shaped apertures is a big challenge for the current nanofabrication technology. At the same time, Gao et al. also reported a quasi-periodic grating modulating the positions of slits, which suppressed the high order diffractions by the average transmission of the large amount of modulated slits [18,19]. For this quasi-periodic grating, the smallest gap between the two adjacent slits or holes is indefinitely small and the fabrication of quasi-periodic gratings is also a challenge task. All the above mentioned approaches are based on the one dimensional grating and need supporting membrane as the substrate, which is especially undesired for the x-ray region since it often absorbs above 70% incident energy. Therefore, we are in great need of a free-standing grating structure whose lattice spacing and characteristic size are large enough to be fabricated by the current planar silicon technology.

Recently, photon sieves with aperiodic distributed holes have drawn great attention owing to their novel properties such as super-resolution focusing and imaging beyond the evanescent region [21–25]. In the aperiodic structures, the locations of the holes can be designed to create constructive interference, leading to a subwavelength focus of prescribed size and shape. In this prospective, photon sieves are similar to the single-order diffraction grating with modulated slits. Photon sieves with aperiodic distributed holes can acquire rich degrees of freedom (spatial position and geometric shape of holes) to realize complex functionalities, which are not achievable through periodic features with limited control in geometry [24,25]. It should be noted that photon sieves are of the membrane structures milled with holes, which can be free-standing. By analogy with the photon sieves, we can interpret the suppression of high-order diffractions from the viewpoint of destructive interference of light from two dimensional lattices with specific distribution of the holes. At the same time, the 1st order diffraction results from the constructive interference of lights from the different holes. Because the interference of lights from different circular holes is controlled by the hole position, the desired diffraction pattern only containing the 0th order and ± 1st order diffractions can be achieved by tailoring the distribution of holes according to some statistical law. Consequently, we can design quasi-periodic two-dimensional grating comprised of holes to realize suppression of the high order diffractions.

In this paper, we propose a quasi-periodic two-dimensional grating comprised of a large number circular holes, which can completely suppress the 3rd and even order diffractions. The circular shape of holes, the enough deviation tolerance of hole size and the relatively large spacing of adjacent holes make our grating much easy to be fabricated than the reported structures [16–19]. Moreover, the proposed two-dimensional grating comprised of circular holes can be free standing and thus avoid the potential absorption from the substrate. The ability of free-standing is very important when the grating scales down to the ultraviolet and soft x-ray regions where the strong absorption will severely decrease the performance of the devices. This paper is organized as follows. In Sec. 2, by using Kirchhoff’s diffraction theory, we sets up our general framework for the analysis of the dependence of the diffraction property on the structure parameters of the quasi-periodic two-dimensional grating of circular holes. The special statistical law of the location distribution of holes leads to the diffraction intensity pattern with the sinc function, which has periodic zero crossings and thus can eliminate even diffractions. Then, we theoretically and experimentally demonstrate the diffraction pattern of the quasi-periodic two-dimensional grating. It is found that the 3rd and even order diffraction are completely suppressed. And the 5th order diffraction is as low as 0.02% of the 1st order diffraction. These results will find an application in the wide spectrum unscrambling from the infrared to the x-ray region.

2. Theory

As mentioned above, periodic structures usually cannot realize the single order diffraction. In order to suppress the high order diffractions, we introduce the location modulation of holes of the triangle array. The circular hole is selected since it can be easily fabricated. Schematic illustrations of the proposed the quasi-triangle array of circle holes are shown in Fig. 1(a). The dashed lines denote the triangle array and the crossings of dashed lines are lattice points. The holes are shifted by $s$ from the lattice points along the $\xi$ axis according to the probability distribution $\rho (s)=1/(2a)$, $\left|s\right|\le a$, where $a$is the shift range of circle holes along the $\xi$ axis. Here, we only consider the shift $s$ along the $\xi$ axis since spectral measurement is usually performed at this direction. Moreover, we choose the triangle array rather than the square array, this is because the spacing between any two adjacent holes of the triangle array is larger than that of the square array for the same period and hole size. Thus, it leads to larger characteristic size and more stable free standing structure.

 

Fig. 1 (a) The quasi-triangle array of circular holes: each hole shifts $s$from the lattice points along the $\xi$ axis according to the probability distribution $\rho (s)$. (b) The coordinate systems of the grating plane and the diffraction plane.

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As shown in Fig. 1(b), we denote the coordinate systems of the grating plane and the diffraction plane by $\left(\eta ,\xi \right)$ and $(x,y)$, respectively. For the quasi-triangle array of $N_{\xi }{N}_{\eta }$circular holes with the radius $r$ as shown in Fig. 1(a), the diffraction intensity pattern in the Fraunhofer diffraction is given by [4]:

Since the spectral measurement in real application is usually performed at one direction, now we focus on the suppression of high order diffractions along the $x$ axis. According to Eq. (1), the intensity along the $x$ axis is:

Now we investigate the dependence of all order diffractions on the radius $r$and random range $a$. According to Eq. (2), the m-th order diffraction intensity along the $x$ axis is:

In the real spectral measurement, only the adjacent diffractions (such as the 2nd and 3rd order diffractions) will overlap the 1st order diffraction. The higher order diffractions are usually very small and have little effects on the 1st order diffraction. Thus we pay the utmost attention to the structure parameters which lead to the vanishing of the 2nd and 3rd order diffractions. According to Eq. (3), we calculate the m-order diffraction intensity versus $r/P_{\xi }$ and $a/P_{\xi }$. Figure 2(a) and 2(b) present the dependence of the 2nd and 3rd order diffractions on $r/P_{\xi }$ and $a/P_{\xi }$. The diffractions disappear as $r$, $a$and $P_{\xi }$ satisfy some proportional relations (dash lines in Fig. 2). The vertical and horizontal lines are respectively the zero crossings of ${\left[2J_1(2\pi mr/P_{\xi })/(2\pi mr/P_{\xi })\right]}^2$ and $\sin c^2(2ma/P_{\xi })$. Thus, the 2nd and 3rd order diffractions simultaneously disappear as $(r/P_{\xi },a/P_{\xi })$ take some special values: (0.203, 1/4), (0.305, 1/6), (0.305, 1/3), (0.372, 1/4) (the cross points of white and blue lines). Considering the fabrication tolerance, we select $r/P_{\xi }=0.203$ and $a/P_{\xi }=0.25$ since the smallest spacing $\sqrt{{(P_{\xi }-2a)}^2+{(P_{\eta }/2)}^2}-2r=0.3011P_{\xi }$ (for $P_{\xi }=P_{\eta }$) of arbitrary adjacent holes is the largest one in the four cases of $(r/P_{\xi },a/P_{\xi })$. Here, it should be noted that the sinc function with $a/P_{\xi }=0.25$ eliminates not only the 2nd order diffraction but also all the even order diffractions since it has periodic zero crossings.

 

Fig. 2 (a) The dependence of the 2nd order diffraction intensity on $r/P_{\xi }$ and $a/P_{\xi }$. The dash lines denote the 2nd order diffraction disappears. (b) Same as (a), except for the 3rd order diffraction.

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Figure 3 presents the diffraction intensity pattern of $r/P_{\xi }=0.203$ and $a/P_{\xi }=0.25$ according to Eqs. (1) and (2). As expected, Fig. 3(a) shows that the 0th and 1st order diffractions are kept along $x$ axis, and high order diffractions disappear. Insets in Figs. 3(b) show clearly intensity distributions of the 0th and 1st order diffractions. The diffraction intensity along $x$ axis in Fig. 3(b) presents clearly the complete suppression of the 2nd, 3rd 4th and 6th order diffractions. The 5th order diffraction of $5.370\times {10}^{-5}$ is as low as 0.02% of the 1st order diffraction of 0.2637. As a result, it will be submerged in the background noise, i.e., it decay to a neglectable value in real experimental measurements.

 

Fig. 3 (a) The far-field diffraction intensity pattern of the quasi-triangle array of circular holes. (b) The diffraction intensity along the x axis. Insets: the 0th and 1st order diffractions.

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In order to verify the validity of the theoretical analysis, we simulated the diffraction intensity pattern of the quasi-triangle array of 301 × 301 circular holes according to Eq. (1) from Fraunhofer approximation. The locations of holes are determined by generating uniformly distributed pseudorandom numbers. The two-dimensional grating has the period $P_{\xi }=P_{\eta }=10\mu m$ and the area$3.01mm\times 3.01mm$. Figure 4 shows that there exists the 0th and 1st order diffractions along $x$ axis, and the 2nd, 3rd, 4th and 6th order diffractions disappear. Insets in Figs. 4(b) show clearly intensity distributions of the 0th and 1st order diffractions. The 5th order diffraction of $5.273\times {10}^{-5}$ is as low as 0.02% of the 1st order diffraction of 0.2638. These agree very well with the theoretical results. Different from the theoretical results, the noise is introduced between any adjacent diffractions. Fortunately, the noise is much smaller than the 5th order diffraction and can be submerged in the background noise.

 

Fig. 4 (a) The far-field diffraction intensity pattern of the quasi-triangle array of 90601 circular holes. (b) The diffraction intensity along the x axis. Insets: the 0th and 1st order diffractions.

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Up to now, the quasi-periodic two dimensional gratings discussed above are the binary amplitude type, the grating thickness is neglected, and the incident light can transmit through the circular holes and be completely absorbed outside holes. At the visible range, a thin metal layer such as 110 nm thick chromium can almost completely absorb the incident light. Therefore, the practical grating thickness is much smaller than the wavelength, and the grating period is much larger than the wavelength. Thus the effect of the grating thickness can be neglected and the above results according to the scalar theory (Fraunhofer approximation) are sufficiently accurate [3]. However, at x-ray range, the incident light may not be completely absorbed outside holes, and the grating area outside holes are partially transparent. At x-ray range, the grating material usually has the refractive index $n=1-{\delta }_1+i{\delta }_2$, here ${\delta }_1$ and ${\delta }_2$ are much smaller than one. For the grating with the thickness $d$, the transmittance coefficient outside holes is not zero but a complex value $\alpha e^{-i\beta }$ ($\alpha =\exp (-kd{\delta }_2)$ and $\beta =kd{\delta }_1$, $k$ is the wave vector) including the phase shift effect. Similarly, according to the Fraunhofer approximation, the m-th order diffraction intensity $I_{\alpha \beta }(m)$ along the $x$ axis is:

3. Experimental results and discussions

We performed a proof-of-principle experiment to confirm our theoretical predictions and numerical results. We fabricated a quasi-periodic two-dimensional chromium grating comprised of a triangle array of circular holes on a soda glass substrate by direct writing laser lithography system and the wet etching technique. The blank sample comprises a soda glass with 2.286 mm thickness, a 110 nm thick chromium layer and 500 nm thick AZ1500 photoresist from the bottom to the top. The radius and positional data of the quasi-periodic hole array is calculated and saved in CIF format by Matlab. A direct writing laser lithography system (DESIGN WRITE LAZER 2000) reads the data in CIF format, and generate the pattern of the quasi-periodic triangle array of circular holes on AZ1500 photoresist. The exposure wavelenght is 413 nm and the exposure power is 80 milliwatt. Then, the pattern of the quasi-periodic triangle array of circular holes is transferred to the chromium layer by wet etching technique. The chromium grating includes about 4000 × 4000 holes over $4cm\times 4cm$ area. The microphotograph of the fabricated structure is illustrated in inset of Fig. 5. Periods $2P_{\xi }$ and $P_{\eta }$of the quasi-triangle array along the $\xi$ and $\eta$ axes are respectively $20\mu m$ and $10\mu m$. The hole diameter is$0.406P_{\xi }\approx 4\mu m$. It is shown clearly that the spacing between any two adjacent holes is larger than $0.3011P_{\xi }\approx 3\mu m$. The experimental setup for optical demonstration is shown in Fig. 5. A collimated laser beams from Sprout (Lighthouse Photonics) with the wavelength of 532 nm was used to illuminate the two-dimensional grating, and the far-field diffraction pattern from the grating is focused by a lens and then recorded on a charge coupled device (CCD) camera (ANDOR DU920P-BU2) with 1024 × 256 pixels.

 

Fig. 5 Experimental setup for the optical measurement. Inset: Microphotograph of the fabricated quasi-triangle array of circular holes.

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The measurement results at −85 centidegree are shown in Fig. 6. It was shown that only 0th and the 1st orders exist along the $x$ axis, which agrees well with the theoretical and simulation results. The 5th order diffraction (theoretical value 5.37 × 10⁻⁵) cannot be observed, which is submerged in the background noise of 5 × 10⁻⁴. The 1st order diffraction efficiencies is 23.99%, which is a little smaller than the theoretical value 26.37%. The difference between experimental and theoretical values is attributed to the fabrication and measurement errors. In addition, the red vertical lines in Fig. 6 are crosstalk along $y$ direction due to our one-dimension CCD.

 

Fig. 6 (a) The far-field diffraction intensity pattern of the quasi-triangle array of circular holes. (b) The diffraction intensity along the $\xi$.

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In the above discussions, we have assumed that the size and shape of holes are perfect. In practice, the size of fabricated holes can be a little larger or smaller than the designed target and the shape cannot be perfectly round. Thus the diffraction pattern will not be the same as the designed one. Numerical simulation based Eq. (3) is carried out and we obtain the 2nd, 3rd, 4th and 5th diffraction intensities versus the hole size in Fig. 7. The vertical grey dot line denotes the optimized diameter $2r=4.0656\mu m$. Figure 7 shows that the 2nd and 4th order diffraction intensities are always zeros regardless of whether the hole diameter deviate the optimized value or not. This is because the disappearance of the even order diffractions result from zero crossings of the normalized sinc function, which is from the location randomness of holes. As $2r\in (3.6,4.8)\mu m$, the 3rd order diffraction intensity will not be larger than $5\times {10}^{-4}$, even though it increases with the deviation of hole diameter from the optimized value. Similarly, the 5th order diffraction intensity is smaller than $3\times {10}^{-4}$ as $2r\in (3.25,4.9)\mu m$. Therefore, the quasi-periodic two-dimensional grating comprised of circular holes can, at least, tolerate ± 10% deviation of hole size. This large tolerance makes our structure can be easily fabricated by the current planar silicon technology.

 

Fig. 7 The 2nd, 3rd, 4th and 5th order diffraction intensities versus the hole diameter.

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4. Summary

In summary, we have proposed to use quasi-periodic two-dimensional gratings comprised of a large number circular holes to considerably suppress the high-order diffractions. The grating can avoid the overlapping of the 1st and the neighboring diffractions, thus it can greatly increase the resolving power and precision of analysis in spectral measurement. It is worth mentioning that the smallest spacing between any adjacent holes of the quasi-triangle array is $0.3011P_{\xi }\approx 3\mu m$, which is larger than the previous reports [17–19]. Both the large characteristic size and the enough deviation tolerance up to ± 10% of hole size benefit the fabrication. Especially, our grating has the ability to form free-standing structures, which is highly desired for the x-ray region. In addition, the binary relief structure can be scalable from far infrared to x-ray wavelengths. The theoretical value of the relative diffraction efficiency of the 1st order diffraction is 26.37%. The experimental one is 23.99%, which agrees with the theoretical prediction. Both theoretical and experimental results demonstrate the 3rd and even order diffractions are completely suppressed. And the 5th order diffraction is as low as 0.02% of the 1st order diffraction and can be submerged in the noise for the real applications. We anticipate that such quasi-periodic two-dimensional grating of circular holes could be widely used in high-accuracy spectral measurement.