Two-dimensional gratings of hexagonal holes for high order diffraction suppression

Ziwei Liu; Lina Shi; Tanchao Pu; Hailiang Li; Jiebin Niu; Guanya Wang; Changqing Xie

doi:10.1364/OE.25.001339

1. Introduction

Grating is a fundamental component for spectral measurement equipment. Generally, spectrum unscrambling only needs the ± 1st order diffraction of the grating. Unfortunately, the high order diffractions of the traditional black-white grating always overlap the 1st one, and thus obstruct spectroscopic analysis [1,2]. For the visible range, there are two solutions to deal with the overlapping: One is the sinusoidal transmission grating with only 0th and + 1st/-1st order diffractions [3–5], and the other is the grating or metasurface with the period $d < 2 λ$ , $λ$ is the wavelength [6–12]. However, the sinusoidal transmission gratings are much more difficult to fabricate than the black-white ones [4,5]. Furthermore, it is difficult to scale the grating period down to the wavelength size of 100 nm by the current nanofabrication technology [13]. Therefore, it has been a goal to design novel black-white gratings, which can suppress high order diffraction and have much larger period than the wavelength.

Several single-order diffraction structures with $d > > λ$ have been developed, and the key is to modulate the groove position or to introduce structures with complicated shapes [14–19]. However, the modulation of the groove position leads to the background noise and the very small gaps between the two adjacent grooves. Thus the reported works cannot obtain complete suppression of high order diffractions, since it’s difficult to realize the complex shapes or the very small gaps. Moreover, the background noise arising from the modulation of the groove position may interfere with the 1st order diffraction. In addition, the reported structures based on the one dimensional grating need the support mesh, which not only produce diffraction spikes in the diffraction plane, but also decrease the diffraction efficiency [20]. Thus, we stand in need of a free-standing grating structure without the modulation of the groove position, whose period and characteristic size are large enough to be fabricated by the current planar silicon technology.

Recently, photon sieves have drawn great attention owing to their novel properties, such as super-resolution focusing and imaging beyond the evanescent region [21–26]. The key idea of photon sieves is to create constructive or destructive interference by designing the hole position. In this point, photon sieves are similar to the single-order diffraction grating with the modulation of the groove position. It should be noted that photon sieves are the membrane structures with holes, which can be free-standing. Triggered by photon sieves, we can design two-dimensional grating comprised of holes to realize the free-standing structure. And at the same time, in order to avoid the background noise, we optimize the shape and size of holes other than the hole position to suppress the high order diffractions.

In this paper, we propose a two-dimensional grating comprised of hexagonal holes, which can completely suppress the 2nd, 3rd, and 4th order diffractions. The hexagonal holes of our grating are much easier to fabricate than sinusoidal holes [18,19]. Moreover, the two-dimensional grating avoids the background noise, which is introduced by the modulation of the groove position [15,16]. In addition, the two-dimensional grating comprised of hexagonal holes can be free standing and thus the absorption of the supporting is avoided. And our black-white structure can be scalable from soft x-ray to far infrared wavelengths. First, we theoretically analyze the dependence of the diffraction property on the structure parameters of the two-dimensional grating. The special size of hexagonal holes results in our desired diffraction pattern. Then, we theoretically and experimentally demonstrate the diffraction pattern of the two-dimensional grating with the effective suppression of high order diffractions. The grating of the triangle array of hexagonal holes will benefit the fabrication due to the larger spacing between any two adjacent holes for the same period and the same hole size. These results will find an application in the wide spectrum unscrambling from the infrared to the x-ray region.

The work is organized as follows. In Section 2 we present the theoretical analysis and structure design of the two-dimensional grating of the hexagonal holes. Section 3 is devoted to experimental results and discussions. A conclusion is drawn in Section 4.

2. Theory

As mentioned above, we optimize the hole shape and size to suppress the high order diffractions of two-dimensional grating. The hexagonal hole is selected since it’s easily fabricated and at the same time has more degrees of design freedom than the circular one. Schematic illustrations of the proposed structures are shown in Figs. 1(a) and 1(b). As shown in Fig. 1(c), we denote the coordinate systems of the grating plane and diffraction plane by $(x, y)$ and $(η, ξ)$ respectively.

Fig. 1 The two-dimensional gratings comprised of hexagonal holes. The side of the hexagonal hole along the $x$ axis is $2 a_{1}$ , the diagonal along the $x$ axis is $2 a$ and the height along the $y$ axis is $2 b$ . (a) The square array with periods $P_{x}$ and $P_{y}$ . (b) The triangle array with periods $2 P_{x}$ and $P_{y}$ . (c) The coordinate systems of the grating plane and the diffraction plane.

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For a membrane that contains a large number of identical and similarly oriented holes, the light distribution in the Fraunhofer diffraction pattern is given by [3]

U (p, q) = C \sum_{N} e^{- i k (p x_{n} + q y_{n})} \iint_{Α} e^{- i k (p x' + q y')} d x' d y' .

Here

C = \sqrt{P} / (λ R)

,

P

is the power density incident on the hole array,

λ

is the incident light wavelength,

R

is the distance between the hole array plane and the diffraction plane. The coordinates of the hole center are

(x_{1}, y_{1})

,

(x_{2}, y_{2})

, …

(x_{N}, y_{N})

,

k

is the wave vector in free space,

p = ξ / R

, and

q = η / R

. The integration extends over the hole area and the integral expresses the effect of a single hole. The sum represents the superposition of the coherent diffraction patterns.

For the square array of $N_{x} \times N_{y}$ hexagonal holes with side $2 a_{1}$ along the $x$ axis, the diagonal $2 a$ along the $x$ axis, and the height $2 b$ along $y$ axis as shown in Fig. 1(a), the diffraction intensity pattern is

\begin{array}{l} I (p, q) = U (p, q) \cdot U^{*} (p, q) \\ = I_{0} \cdot \frac{\sin^{2} (N_{x} k p P_{x} / 2)}{N_{x}^{2} \cdot \sin^{2} (k p P_{x} / 2)} \cdot \frac{\sin^{2} (N_{y} k q P_{y} / 2)}{N_{y}^{2} \cdot \sin^{2} (k q P_{y} / 2)} \\ \cdot {| \frac{\cos (k p a_{1} - k q b) - \cos k p a}{k p (a + a_{1}) \cdot (k p (a - a_{1}) + k q b)} + \frac{\cos k p a - \cos (k p a_{1} + k q b)}{k p (a + a_{1}) \cdot (- k p (a - a_{1}) + k q b)} |}^{2} . \end{array}

Here

I_{0} = C^{2} \cdot {(N_{x} N_{y} \cdot 2 (a + a_{1}) b)}^{2}

is the peak irradiance of the diffraction pattern. And the light intensity without the hole array is

I_{00} = C^{2} \cdot S^{2}

, where

S

is the total area of the hole array [3].

P_{x}

and

P_{y}

as shown in Fig. 1(a) are respectively the periods along the

x

and

y

axes.

Similarly, for the triangle array as shown in Fig. 1(b), the diffraction intensity pattern is given by:

\begin{array}{l} I (p, q) = U (p, q) \cdot U^{*} (p, q) \\ = I_{0} \cdot \frac{\sin^{2} (N_{x} / 2 \cdot k p 2 P_{x} / 2)}{{(N_{x} / 2)}^{2} \cdot \sin^{2} (k p 2 P_{x} / 2)} \cdot \frac{\sin^{2} (N_{y} k q P_{y} / 2)}{N_{y}^{2} \cdot \sin^{2} (k q P_{y} / 2)} \cdot \cos^{2} (\frac{k p P_{x}}{2} + \frac{k q P_{y}}{4}) \\ \cdot {| \frac{\cos (k p a_{1} - k q b) - \cos k p a}{k p (a + a_{1}) \cdot (k p (a - a_{1}) + k q b)} + \frac{\cos k p a - \cos (k p a_{1} + k q b)}{k p (a + a_{1}) \cdot (- k p (a - a_{1}) + k q b)} |}^{2} . \end{array}

The parameters are same as the square array except the period

2 P_{x}

along the

x

axis in Fig. 1(b).

Here, we focus on the diffraction intensity along the $ξ$ axis since spectral measurement is usually at this direction. For both the square and triangle arrays, the diffraction intensity along the $ξ$ axis according to Eq. (2) and Eq. (3) is given by:

I (p) = I_{0} \cdot \frac{\sin^{2} (N_{x} k p P_{x} / 2)}{N_{x}^{2} \cdot \sin^{2} (k p P_{x} / 2)} \cdot {(\frac{\sin (k p (a + a_{1}) / 2) \cdot \sin (k p (a - a_{1}) / 2)}{k p (a + a_{1}) / 2 \cdot k p (a - a_{1}) / 2})}^{2} .

The same diffraction intensity along the

ξ

axis of both the square and triangle arrays are due to the same average transmission function

t (x)

along the

x

axis, which can be obtained by integrating the transmission function along the

y

axis:

t (x) = {\begin{cases} 2 b,_{}_{}_{}_{} | x | \leq a_{1}, \\ \frac{2 b (a - x)}{a - a_{1}},_{} a_{1} < | x | \leq a, \\ 0,_{}_{}_{}_{} a \leq | x | \leq P_{x} / 2. \end{cases}

And according to Fraunhofer diffraction formula, we can also obtain the intensity along the

ξ

axis directly from Eq. (5):

\begin{array}{l} I (p) = {| U (P) |}^{2} \\ {_{}}^{}_{^{}} = \frac{1 - \cos N_{x} k p P_{x}}{1 - \cos k p P_{x}} {| C \int_{- a}^{a} t (x) \cdot e^{- i k p x} d x |}^{2} \\ _{} = C^{2} (^{N_{x} 2 (a + a_{1}) b) 2} \frac{\sin^{2} (N_{x} k p P_{x} / 2)}{N_{x}^{2} \cdot \sin^{2} (k p P_{x} / 2)} \cdot {(\frac{\sin (k p (a + a_{1}) / 2)}{k p (a + a_{1}) / 2} \cdot \frac{\sin (k p (a - a_{1}) / 2)}{k p (a - a_{1}) / 2})}^{2} . \end{array}

Equation (4) shows that the relative diffraction efficiency along the $ξ$ axis has no relation to the parameters of $b$ and $P_{y}$ . The parameters $b$ and $P_{y}$ will affect the absolute diffraction efficiency and structure firmness. For the case of $b = P_{y} / 2$ , the hexagonal holes along $y$ axis connect with each other and become a groove, the absolute diffraction efficiency reaches the largest value. This groove structure cannot be free-standing. With $b$ decreasing, the absolute diffraction efficiency decreases, and the membrane structure with holes becomes more stable. In real applications such as the spectral measurement, the stable structure is more important since the absolute diffraction efficiency will not change the diffraction pattern. Fortunately, the triangle array will benefit the fabrication since the spacing between any two adjacent hexagonal holes of the triangle array is larger than that of the square array for the same period and hole size.

From Eq. (4), the m-th order diffraction along the $ξ$ axis can be expressed as:

I (m) = I_{0} \cdot {(\frac{\sin (m (a + a_{1}) π / P_{x}) \cdot \sin (m (a - a_{1}) π / P_{x})}{m (a + a_{1}) π / P_{x} \cdot m (a - a_{1}) π / P_{x}})}^{2} .

In the real spectral measurement, only the adjacent diffractions (such as the 2nd, 3rd, and 4th order diffractions) will overlap the 1st order diffraction. The far field diffractions are usually very small and have little effects on the 1st order diffraction. Thus we should pay the utmost attention to the structure which leads to the vanishing of the 2nd, 3rd, and 4th order diffractions. According to Eq. (7), we calculate the m-order diffraction intensity for different values of $a$ , $a_{1}$ and $P_{x}$ (shown in Fig. 2).

Fig. 2 The dependence of the m-th order diffraction intensity on $a$ , $a_{1}$ and $P_{x}$ . The white dash line denotes the m-th order diffraction intensity vanishes: the 2nd order diffraction (a), the 3rd order diffraction (b), and the 4th order diffraction (c).

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Figure 2 shows that the m-th order diffractions disappears once the parameters $a$ , $a_{1}$ and $P_{x}$ satisfy some proportional relations (white dash lines in Fig. 2)

a + a_{1} = \frac{n}{m} P_{x} or | a - a_{1} | = \frac{n}{m} P_{x}, n = 1, 2, 3, ...

According to Eq. (8), the 2nd, 3rd and 4th order diffractions simultaneously disappear as $a_{1} = P_{x} / 12$ and $a = 5 P_{x} / 12$ . And with these parameters, the m-order diffraction intensity is $I (m) = I_{0} \cdot {(\frac{\sin (m π / 2) \cdot \sin (m π / 3)}{m π / 2 \cdot m π / 3})}^{2} .$ For the case of $b = P_{y} / 2$ , we can obtain the 0th order diffraction $I (0) = I_{0} = C^{2} \cdot {(N_{x} N_{y} \cdot 2 (a + a_{1}) b)}^{2} = 25 % C^{2} S^{2} = 25 % I_{00}$ , the 1st order diffraction $I (1) = 0.2772 I_{0} = 6.93 % I_{00}$ , which is higher than $I (1) = 0.25 I_{0} = 6.25 % I_{00}$ of the ideal sinusoidal transmission grating. In addition, we can also obtain the 5th order diffraction $I (5) = 0.0004435 I_{0} = 0.0016 I (1) = 0.01 % I_{00}$ is small enough to recognize the 1st order diffraction. Similarly, for the case of $b = P_{y} / 3$ , the 0th, 1st, and 5th order diffraction intensities are respectively $I (0) = I_{0} = C^{2} \cdot {(N_{x} N_{y} \cdot 2 (a + a_{1}) b)}^{2} = 11.11 % I_{00}$ , $I (1) = 0.2772 I_{0} = 3.08 % I_{00}$ and $I (5) = 0.0004435 I_{0} = 0.0016 I (1) = 0.0049 % I_{00}$ .

In order to evaluate the loss for spectral analysis, we discuss 2D intensity distribution along the $η$ axis. According to Eq. (2), we can obtain the n-order diffraction intensity along the $η$ axis for the square array of hexagonal holes with $a_{1} = P_{x} / 12$ and $a = 5 P_{x} / 12$ :

\begin{array}{l} I (n) = I_{0} \cdot {(\frac{2 a_{1}}{a + a_{1}} \cdot \frac{\sin (2 n π b / P_{y})}{2 n π b / P_{y}} + \frac{a - a_{1}}{a + a_{1}} \cdot {(\frac{\sin (2 n π b / 2 / P_{y})}{2 n π b / 2 / P_{y}})}^{2})}^{2} \\ \begin{matrix}  \end{matrix} = I_{0} \cdot {(\frac{1}{3} \cdot \frac{\sin (2 n π b / P_{y})}{2 n π b / P_{y}} + \frac{2}{3} \cdot {(\frac{\sin (n π b / P_{y})}{n π b / P_{y}})}^{2})}^{2} . \end{array}

As

b = P_{y} / 2

, the absolute diffraction efficiencies of the 0th, 1st, 2nd, 3rd, 4th, and 5th order diffractions are respectively

25 % I_{00}

,

1.83 % I_{00}

,

0.00 % I_{00}

,

0.02 % I_{00}

,

0.00 % I_{00}

, and

0.00 % I_{00}

. The absolute diffraction efficiency of the 1st order diffraction along the

η

axis is much less than the 1st order diffraction intensity

6.93 % I_{00}

along

ξ

axis. As

b = P_{y} / 3

, the absolute diffraction efficiencies of the 0th, 1st, 2nd, 3rd, 4th, and 5th order diffractions are respectively

11.11 % I_{00}

,

3.92 % I_{00}

,

0.02 % I_{00}

,

0.00 % I_{00}

,

0.04 % I_{00}

,

0.00 % I_{00}

.

For the triangle array of hexagonal holes, the 1st order diffraction intensity along $η$ axis ( $n = 1, m = 0$ ) are almost zeros. We can observe this point in Fig. 3(b). For the triangle array, the largest diffraction spot along $η$ axis locate at $m = 1 / 2$ .Similarly, according to Eq. (3), the n-order diffraction intensity at $m = 1 / 2$ is:

\begin{array}{l} I (\frac{1}{2}, n) = I_{0} \cdot \cos^{2} (n π / 2) \cdot | \frac{\cos (π a_{1} / P_{x} - 2 n π b / P_{y}) - \cos (π a / P_{x})}{π (a + a_{1}) / P_{x} \cdot (π (a - a_{1}) / P_{x} + 2 n π b / P_{y})} \\ {^{} {_{}}^{} {_{}}_{} + \frac{\cos (π a / P_{x}) - \cos (π a_{1} / P_{x} + 2 n π b / P_{y})}{π (a + a_{1}) / P_{x} \cdot (- π (a - a_{1}) / P_{x} + 2 n π b / P_{y})} |}^{2} . \end{array}

We can obtain the 1st order diffraction intensity efficiencies are respectively

0.87 % I_{00}

and

2.58 % I_{00}

for

b = P_{y} / 2

and

b = P_{y} / 3

. The 1st order diffraction intensity efficiency along

η

axis of the triangle array is smaller than that of the square one.

Fig. 3 (a) The far-field diffraction intensity pattern of the square array of hexagonal holes. (b) The same as (a) except for the triangle array. (c) The intensity distribution of the 0th and 1st order diffractions of the square array of hexagonal holes. (d) The same as (c) except for the triangle array. (e) Blue curve: the diffraction intensity along the $ξ$ axis of the square array of hexagonal holes. Red curve: the diffraction patterns of 1:1 transmission gratings. (f) The same as (c) except for the triangle array.

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Figure 3 presents the normalized diffraction intensity pattern of the grating of hexagonal holes with $a_{1} = P_{x} / 12$ , $a = 5 P_{x} / 12$ and $b = P_{y} / 3$ according to Eq. (2) and Eq. (3). The intensity is normalized by the 0th order diffraction intensity $I (0) = I_{0} = 11.11 % I_{00}$ . As expected, the 0th and 1st order diffractions in Figs. 3(a) and 3(b) are kept along $ξ$ axis, and the 2nd, 3rd and 4th order diffractions disappear. Figures 3(c) and 3(d) show clearly intensity distributions of the 0th and 1st order diffractions. The diffraction intensity along $ξ$ axis in Figs. 3(e) and 3(f) present clearly the complete suppression of the 2nd, 3rd and 4th order diffractions. The 5th order diffraction is smaller than the noise between the 0th and 1st order diffractions. Comparing with the diffraction pattern of conventional 1:1 transmission gratings (red curve in Figs. 3(e) and 3(f)), the 5th order diffraction components of the hexagonal hole array are strongly suppressed, which are weak enough for real measurement. Figure 3 shows that the high order diffractions along $ξ$ axis are effectively suppressed by the two-dimensional grating of hexagonal holes.

Considering the parameters of fabricated hexagonal holes can be a little larger or smaller than the designed target in real experiment of measurement, the diffraction pattern will not be the same as the designed one. In the following, we analyze and evaluate the influence of the size deviation of the hexagonal holes on the practical performance. We define a deviation $D_{1} = a - a_{f} = a_{1} - a_{1 f}$ , where $a_{f}$ and $a_{1 f}$ are the actual parameters of the fabricated hexagonal holes. Numerical simulation based Eq. (7) is carried out and we obtain the 2nd, 3rd, 4th diffraction intensities versus $D_{1}$ as shown in Fig. 4.

Fig. 4 The absolute diffraction efficiencies of the 2nd, 3rd, and 4th order diffractions versus the deviation $D_{1}$ .

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From Fig. 4 one can see that the 3rd order diffraction intensity is always zero regardless of whether the size of hexagonal holes is changed. This is because the suppression of 3rd order diffraction depends on $a - a_{1}$ , which has discussed in the theory. As $D_{1} \in (- 0.359, 0.32) μ m$ , both of the 2nd and 4^rd order diffractions intensity will not larger than $6 \times 10^{- 5}$ . The hexagonal holes can tolerate $\pm 7.5 %$ deviation of $2 a_{1}$ . This tolerance makes our structure can be fabricated by the current technology.

It should be noted that the sharp corner can’t be achieved in real fabrication process. Therefore, we also discussed effects of fabrication error of the corners on the diffraction pattern. We calculated the 2nd, 3rd and 4th order diffraction intensity pattern of the gratings of hexagonal holes with round corner (as shown in Fig. 5(a)). We define a deviation $D_{2}$ , which is the distance between the round corner and the vertex of the sharp corner as shown in Figs. 5(c) and 5(d). Figure 5(e) presents that the 2nd, 3rd and 4th order diffraction intensities increase with $D_{2}$ . As $D_{2} = 0.4 μ m$ , the 2nd, 3rd and 4th order diffraction intensities are less than $3 \times 10^{- 6}$ . The tolerance of deviation of sharp corner is at least $\pm 10 %$ of $2 a_{1}$ . This large tolerance greatly relaxes the fabrication requirement.

Fig. 5 (a) The fabricated hexagonal hole. (b) The simulated hexagonal hole. (c) The round corner 1 of hexagonal hole. (d) The round corner 2 of hexagonal hole. (e) The absolute diffraction efficiencies of the 2nd, 3rd, and 4th order diffractions versus the deviation $D_{2}$ .

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3. Experimental results and discussions

A proof-of-principle experiment was performed to confirm our theoretical predictions. We fabricated two-dimensional gratings with $4 c m \times 4 c m$ area on a glass substrate by laser direct writing system and the wet etching technique. The gratings comprise a square or triangle array of hexagonal holes in a chromium film with 110 nm thickness. The microphotographs of the fabricated structures are illustrated in Figs. 6(a) and 6(b) Periods $P_{x}$ and $P_{y}$ along the $x$ and $y$ axes are respectively $24 μ m$ and $28 μ m$ . The structure consists of 1667 × 1429 holes. It is shown clearly that the spacing between any two adjacent hexagonal holes of the triangle array is larger than that of the square array. Thus the triangle array will benefit the fabrication. The experimental setup for optical demonstration is shown in Fig. 6(c). A collimated laser beams 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 CCD camera.

Fig. 6 (a) Microphotograph of the fabricated two-dimensional grating with the square array of hexagonal holes. (b) The same as (a) except for the triangle array. (c) Experimental setup for the optical measurement.

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To decrease the background noise, CCD was cooled to minus 85 degree Celsius. The measurement results are shown in Fig. 7. Both the square and triangle array have the similar diffraction property along the $ξ$ axis. Only 0th and the 1st orders exist along the $ξ$ axis, which agrees well with the theoretical results. Figures 7(c) and 7(d) show clearly intensity distributions of the 0th and 1st order diffractions. Each diffraction spot on the camera includes about 12 pixels. The 5th order diffraction cannot be observed, which is submerged in the background noise. The counts of the 0th and 1st order diffractions are $6.462 \times 10^{4}$ and $1.825 \times 10^{4}$ for the square array. We measured the absolute diffraction efficiency 10.89% of the 0th order diffraction. Thus the corresponding absolute diffraction efficiency of 1st order diffraction is 2.98%, which is less than the theoretical value 3.08%. Similarly, for the triangle array, the counts of 0th and 1st order diffractions are $6.519 \times 10^{4}$ and $1.899 \times 10^{4}$ . The absolute diffraction efficiency 10.24% of the 0th order diffraction is obtained. The corresponding absolute diffraction efficiency of 1st order diffraction is 2.89%, which is also less than the theoretical value 3.08%. The difference between experimental and theoretical values is attributed to the fabrication and measurement errors. In addition, the red vertical lines in Figs. 7(a) and 7(b) are crosstalk along $y$ direction due to our one-dimension CCD.

Fig. 7 (a) The far-field diffraction intensity pattern of the square array of hexagonal holes. (b) The same as (a) except for the triangle array. (c) The intensity distribution of the 0th and 1st order diffractions of the square array of hexagonal holes. (d) The same as (c) except for the triangle array. (e) The diffraction intensity along the $ξ$ axis for the square array. (f) The same as (c) except for the triangle array.

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4. Summary

In summary, the two-dimensional grating comprised of hexagonal holes has been proposed to suppress the high-order diffractions which may lead to wavelength overlapping in spectral measurement. And the smallest characteristic $2 a_{1}$ is one sixth of period $P_{x}$ , which is larger than the previous reports [15,16]. For the case of $b = P_{y} / 2 a$ , we obtain the 1st order diffraction intensity efficiency of 6.93%, which is a little higher than that of the ideal sinusoidal transmission grating. For the case of $b = P_{y} / 3$ , the 1st order diffraction intensity efficiency is 3.08%. For the case of $b < P_{y} / 2$ , the membrane with holes can be free-standing and thus avoid diffraction spikes [20]. The black-white structure can be scalable from far infrared to x-ray wavelengths. Both theoretical and experimental results demonstrate the 2nd, 3rd and 4th order diffractions are completely suppressed. And the 5th order diffraction is as low as 0.01% and which can be submerged in the noise for the real application. The two-dimensional grating of the triangle array of hexagonal holes is easier to fabricate than that of the square array due to the larger spacing between any two adjacent hexagonal holes. The two-dimensional grating with hexagonal holes offers an opportunity for high-accuracy spectral measurement and will possess broad potential applications in optical science and engineering fields.

Funding

National Natural Science Foundation of China (NSFC) (61107032, 61275170); Major National Scientific Instruments Developed Special Project (2013YQ1508290602, 2012YQ13012504); Research and Development of Major Research Equipment of Chinese Academy of Sciences (YZ201446); the Opening Project of Key Laboratory of Microelectronics Devices and Integrated Technology, Institute of Microelectronics of Chinese Academy of Sciences.

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Two-dimensional gratings of hexagonal holes for high order diffraction suppression

Abstract

1. Introduction

2. Theory

3. Experimental results and discussions

4. Summary

Funding

References and links

Cited By

Figures (7)

Equations (10)

Optics Express