Conventional diffraction gratings comprising regular grooves are used to disperse incident light and provide spectral information for many applications over wavelengths ranging from the infrared to SXRs^[1–4]. Generally, spectral unscrambling and monochromatization only need one diffraction order of a grating. However, from the grating equation, at the same incident angle, the nth order (n＝2, 3, 4, …) diffraction of the nth order harmonics (λ/n) of the fundamental wavelength λ is diffracted at the same angle as the first diffraction order of λ. This coincidence introduces harmonics overlapping or contamination^[5–6], which is particularly strong for extreme ultraviolet (EUV) and soft X-ray (SXR) range ^[7–9] light.

The ideal sinusoidal grating may be a radical solution, which gives only 0^th and ±1^st orders of diffraction but has not been realized for EUV and SXR gratings because of limitations in the fabrication technology^[10]. Nevertheless, several single-order or quasi-single-order binary diffraction structures have been developed in recent years by a few novel methods. One method is to use periodic arrangements of nanoscale holes array with complex profiles, making structures as the binary sinusoidal grating^[10], the zigzag grating^[11], the inclined rectangular aperture grating^[12], the trapezoidal transmission grating^[13] and the hexagonal holes gratings^[14]. All of these can suppress the higher order diffractions in some directions, but the size bias of the holes will reintroduce higher order diffractions, especially the 2^nd order^[12]. Another method introduces quasi-periodical hole arrays rather than grooves to form a grating. It can achieve an approximate integral sinusoidal transmissivity by setting an appropriate probability density distribution function that describes the probability of occurrence of a hole at some coordinate. For example, the quantum-dot-array grating^[15] employs a number of holes, which obeys the sinusoidal distribution function in each period and can theoretically suppress all the higher order diffractions. However, the size of the processed holes is close to 1/50 of the period to avoid overlapping for this distribution function is continuous, which greatly increase data volume and processing time with electron beam lithography tools. To decrease the data volume, the spectrum photon sieve^[16-18] uses holes with the size close to half of the grating period and a non-continuous rectangular distribution function to obtain an approximate integral sinusoidal transmissivity. The size bias of the holes restrains the inhibitive effects on the ±3^rd order diffractions^[19–20].

From the above analysis, the key to a single-order or quasi-single-order grating is selecting appropriate holes and their probability density distribution function. The machined holes are mainly affected by the data volume^[20–21] or size deviation^[12], making it difficult to meet the design requirements. It is shown that any fine optimization of the hole design is futile. Therefore, the probability density distribution function will be the only optimization parameter. In this paper, we propose a non-continuous trapezoidal probability density distribution function, which is a higher order approximation of a sinusoidal function and theoretically suppresses the ±2^nd, ±3^rd, and ±4^th orders by the statistical effect of the distribution, no matter what size of the holes. Besides, its non-continuity makes the maximum size of the holes at the period direction reach 1/6 of the period. Therefore, it yields both larger process tolerance in the size and lower data volume that greatly benefit the processing and application in spectrum measuring and unscrambling in the waveband distributed across the infrared to the X-ray regions.

For a metal membrane containing an array with a number of identical holes, the Fraunhofer diffraction intensity distribution is

where $ \otimes $ denotes convolution, p＝x/z, q＝y/z, z denotes the distance between the plane of the array and the plane of the diffraction pattern, which are denoted by (x, y), $g(\xi ',\eta ')$ represents the binary amplitude transmission of a single hole, ${L_n}$ the distribution function of the coordinates (ξ_n, η_n) of the centers of the holes, and FT(·) the Fourier transform. Normally, ${L_n}$ is an exact two-dimensional periodic array of δ-functions corresponding to a rectangular lattice: $\left( {{\xi _n},{\eta _n}} \right){\rm{ = }}\left( {h{d_1},l{d_2}} \right)$, where h denotes the ordinal value of the rows of the lattices ( $h = 0, \pm 1, \pm 2, \cdot \cdot \cdot , \pm H$), l the ordinal value of the columns ( $l = 0, \pm 1, \pm 2, \cdot \cdot \cdot , \pm L$); d₁ and d₂ denote the periods of the array in the ξ and η directions, respectively. The total number of holes is N＝(2H＋1)(2L＋1). A given high order of diffraction cannot be suppressed completely by optimizing only $g(\xi ',\eta ')$ because fabrication errors are unavoidable.

However, ${L_n}$ may be generalized to a two-dimensional periodic array of arbitrary identical positive functions.

where ${\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right)$ is the distribution function over one period and $\delta \left( {{\xi _n},{\eta _n}} \right)$ is a δ-function (Fig. 1). The probability density distribution ${L_n}$ for one hole satisfies

The square of its Fourier transform is

where ${C_n} = \exp \left[ {{\rm{i}}k\left( {{\xi _n}p + {\eta _n}q} \right)} \right]$, k denotes the wave number, and n≠m.C_n and $C_m^*$ are not independent variables, hence we must use the joint distribution function L_nm of (ξ_n, η_n) and (ξ_m, η_m), which reads

where the coordinates of (h, l) and (h′, l′) are different in each term. The mean diffracted intensity distribution yields^[19]

The first term in the bracket of Eq. (6) represents a linear superposition of the diffraction intensity from all holes; the second involves an interference factor between all pairs of holes. Note that, unlike a traditional grating, there is a modulation factor B that gives another degree of freedom to suppress the higher orders of diffraction rather than ${I_0}(p,q)$. Clearly, if ${\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right)$ is a sinusoidal function, the diffraction pattern retains only the 0^th and ±1^st orders. Because the sinusoidal function is a continuous distribution function, $g(\xi ',\eta ')$ must be a δ-function to avoid holes overlapping. However, a large number of small holes (corresponding to δ-functions) which are reduced considerably in dimension from that of one period, are very difficult to fabricate, especially for the EUV and SXR gratings. The next best solution is to suppress the higher orders adjacent to the ±1^st orders, which have a real impact on the performance of grating spectrometers and monochromators.

It is noted that an amplitude grating with a duty cycle of 1/2 produces only 0^th and odd orders of diffraction, while an amplitude grating with a 1/3 duty cycle eliminates ±3t^th orders of diffraction (t＝1, 2, 3,…). Setting the normalized distribution function for one period as

and substituting it into Eq. (7), we obtain the modulation factor

From Eq. (6), this design clearly suppresses the ±2^nd, ±3^rd, and ±4^th orders of diffraction along the p-axis of the diffraction plane, regardless of the hole shape and size. In this instance, reducing the requirements of processing precision is very beneficial.

Indeed, the distribution function of Eq. (8) is a trapezoidal function. Fig. 2 gives one example, the design consists of a number of rectangular holes with side lengths d/6 and d (d₁＝d₂＝d). Each hole is randomly positioned in a square lattice (d×d square), and the central location (ξ_n, η_n) of the holes obey the probability density distribution function ${\mathit{\Gamma}} \left( {{\xi _n}} \right)$ in Eq. (8). In every lattice, there is only one hole.

From Eqs. (6), (7), and (9), the mean diffracted intensity peak for each order obtains

where m denotes the diffraction order. The 0^th order has the same intensity as that of a periodic grating with a duty cycle of 1∶5; the diffraction intensity of the 1^st order is about 52.7% of that of a periodic grating with a duty cycle of 1∶5 when $ N \gg 1 $, and there are no diffraction peaks for the ±2t^th and ±3t^th orders. The ratio of the diffraction efficiency of the nearest higher order (5^th order) to that of the 1^st order is found to be

which represents a decrease by 625 times than that of a periodic grating with a duty cycle of 1∶5 when N$\gg $1.

The intensity distributions based on the calculated results of Eq. (6) and the numerical results from Eq. (1) (Fig. 3(a) and (b), respectively) indicate that our model, Eqs. (6)–(10), agrees well with the numerical results. Compared with a periodic grating with a duty cycle of 1∶5 under identical conditions (Fig. 3(c) and (d)), the ±2^nd to ±4^th orders of diffraction seem to vanish and the ±1^st orders attenuate significantly, meaning that part of the energy of incident light is diffracted elsewhere. The distinctive background intensity distributions (the two “wings” seen in Fig. 3(a) and (b)) confirm this well. Its hollow structure is a typical characteristic of the diffraction pattern from a quasi-periodical array.

For a quasi-periodical structure, intensity fluctuations in its diffraction pattern are inevitable. The numerical results (Fig. 3(b)) obtained using Eq. (1) show the fluctuations appearing as speckles in the two “wings”. In addition, there are also many background fluctuations between different diffraction peaks in the profile (Fig. 3(d)) across the p-axis in the numerical results obtained using Eq. (1). Their mean value accords with the theoretical value of Eq. (6), which is comparable to the background level of a periodic grating.

Fig. 4(a) shows the intensity profiles of Fig. 3(a)–(c) across the line $q=\lambda /\left( {2}d \right)$(indicated by a yellow dashed line in Fig. 3). The background intensity of a quasi-periodical array is much higher than that of a periodic grating, verifying that the diffracted energy from the higher orders is transferred to the background and increases the mean background level. However, the p-axis is the direction that is applied in the spectroscopic measurements, where there is no significant increase in the background intensity.

The size bias of a microstructure has a large impact on the suppression of higher orders. The process bias of an inclined rectangular aperture grating needs to be controlled within ±9% of the target value to ensure the suppression ratio is less than 1/20. This precision is very difficult for electron beam lithography technology^[12]. To verify the process tolerance of our design on the suppression of higher order diffractions, Fig. 4(b) shows the effects of process errors on this design, where the side lengths in the direction of the periods of the rectangular pinholes are 120% and 80% of the design values in Fig. 2. This quasi-periodical array still restrains the 2^nd to 4^th orders, and the ratio of the diffraction efficiency of the 5^th order diffraction to that of the 1^st order is always below 0.01%. The elimination of the ±2t^th and ±3t^th (t＝1, 2, 3 …) order diffractions produced by the design is a statistical effect of distribution of a number of holes, being independent of the process accuracy of the individual hole size. Accordingly, the design gains a higher tolerance to process errors.

For higher-order diffraction suppression, a quasi-periodical array of rectangular holes has been proposed displaying a larger process tolerance than that previously reported. A general analytical description has been established to describe the diffraction properties of the quasi-periodical holes array using an arbitrary periodic probability density distribution function, as well as to reveal the physical mechanism behind the effective suppression of the higher orders and large process tolerance of the design. Its binary structure easily accommodates infrared to X-ray light fields. The quasi-periodical array grating has potential applications in devices performing high-accuracy spectral measurements and monochromators as well as a standard for other devices that require high process accuracy.