The Quasi-Triangle Array of Rectangular Holes with the Completely

Suppression of High Order Diffractions

Lina Shi, Hailiang Li, Ziwei Liu, Tanchao Pu, Nan Gao and Changqing Xie

Key Laboratory of Microelectronic Device & Integrated Technology,

Institute of Microelectronics of Chinese Academy of Sciences, Beijing 100029, China

Jiangsu National Synergetic Innovation Center for Advanced Materials, Nanjing 210009, China

Keywords: Diffraction Gratings, Binary Optics, Optical Design and Fabrication, High-Resolution Spectroscopy.

Abstract: We propose the quasi-triangle array of rectangular holes with the completely suppression of high order

diffractions. The membrane with holes can be free-standing and scalable from X-rays to far infrared

wavelengths. Both numerical and experimental results demonstrate the completely suppression of high order

diffractions. The desired diffraction pattern only containing the 0

and +1

/-1

order diffractions results

from the constructive interference of lights from different holes according to some statistical law

distribution. The suppression effect depends on the number of holes. Our results should be of great interest

in a wide spectrum unscrambling for any wavelength range.

1 INTRODUCTION

Gratings are the key component of the spectrometers.

Spectrum unscrambling only needs the first order

diffraction of the traditional black-white grating.

However, unwanted higher order diffraction always

overlaps the first diffraction, and thus greatly

degrade precision of analysis (Palmer, 2005). The

sinusoidal transmission gratings only have 0

and

/-1

order diffractions (Born, and Wolf, 1980),

but they are much more difficult to fabricate than the

black-white ones (Jin, Gao, Liu, Li, and Tan, 2010,

Vincent, Haidar, Collin, Guérineau, Primot,

Cambril, and Pelouard, 2008). The high order

diffractions can become evanescent waves with a

grating period in the range of the wavelength

(Clausnitzer, Kämpfe, Kley, Tünnermann,

Tishchenko, and Parriaux, 2008, Zhou, Seki,

Kitamura, Kuramoto, Sukegawa, Ishii, Kanai,

Itatani, Kobayashi, and Watanabe, 2014, Warren,

Smith, Vawter, and Wendt, 1995). Unfortunately,

for short wavelengths less than 100 nm, it’s difficult

to scale the grating period down to the wavelength

size by the current nanofabrication technology

(Pease, Deshpande, Wang, Russe, and Chou, 2007).

Therefore, it has been a goal to design the black-

white structure much larger than the wavelength

with the suppression of high order diffractions.

Several structures with the suppression of high

order diffractions have been developed, and the

points are to modulate the groove position or to

introduce structures with complicated shapes (Gao,

and Xie, 2011, Fan, et al, 2015, Cao, Förster,

Fuhrmann, Wang, Kuang, Liu, and Ding, 2007).

Unfortunately, the reported works can’t obtain

complete suppression of high order diffractions

since it’s difficult to realize the complex shapes or

the very small gaps between the two adjacent

grooves. Moreover, these structures based on the one

dimensional grating can’t be free-standing.

Unfortunately, the supporting membrane will absorb

80% energy of the soft X-ray.

Recently, photon sieves with aperiodic

distributed holes have drawn great attention owing

to their novel properties, super-resolution focusing

and imaging beyond the evanescent region (Kipp,

Skibowski, Johnson, Berndt, Adelung, Harm, and

Seemann, 2001, Huang, Liu, Garcia-Vidal, Hong,

Luk’yanchuk, Teng, and Qiu, 2015, Huang, Kao,

Fedotov, Chen, and Zheludev, 2008, Huang,

Zheludev, Chen, and Abajo, 2007). The numerous

holes can be designed that creates constructive

interference, leading to a subwavelength focus of

prescribed size and shape. 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.

Shi L., Li H., Liu Z., Pu T., Gao N. and Xie C.

The Quasi-Triangle Array of Rectangular Holes with the Completely Suppression of High Order Diffractions.

DOI: 10.5220/0006100700540058

In Proceedings of the 5th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2017), pages 54-58

ISBN: 978-989-758-223-3

Triggered by photon sieves, we propose the

quasi-triangle array of rectangular holes with the

completely suppression of high order diffractions.

The membrane with holes can be free-standing and

scalable from X-rays to far infrared wavelengths.

The location distribution of holes according to some

statistical law results in a desired diffraction pattern.

We numerically and experimentally demonstrate the

diffraction pattern of the quasi-periodic hole array

with only 0

and +1

/-1

orders. Furthermore, we

investigate the effect of the hole number on the

diffraction property.

2 DESIGN AND SIMULATIONS

We start the design from the triangle array of

rectangular holes, and the holes are shifted by

along the

axis (Fig. 1(a)) according to the

probability distribution

() ( / )cos(2 / )sP sP

ξξ

ρπ π

=⋅

|| /4sP

≤

, where

is the period of the triangle

array along the

axis. Here, we only consider the

shift

along the

axis since spectral measurement

is usually at one direction. Moreover, we choose the

triangle array rather than the square array since 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. This will benefit

Figure 1: (a) The quasi-triangle array of rectangular holes:

each hole shifts

s along the

axis according to the

probability distribution

()s

. (b) Microphotograph of the

fabricated quasi-triangle array with

44cm cm×

area.

the fabrication, and lead to more stable free standing

structure. In addition, rectangular hole is selected

since its diffraction intensity pattern is beneficial to

the suppression of high order diffractions.

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]

()

(' ')

(,) ' '.

ik p q

Upq C e e d d

ξη

−+





(1)

Here

/( )CPz

，

is the power density

incident on the hole array,

is the incident light

wavelength,

is the distance between the hole array

plane and the diffraction plane. The coordinates of

the hole center are

(, )

ξη

(, )

ξη

, …

(, )

ξη

, and

is the wavevector.

/pxz=

/qyz=

. 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 quasi-triangle array of

2NN

ξη

rectangular

holes of sides

2/2aP

and

2/2bP

as shown in

Fig. 1, the diffraction intensity pattern is

(,) (,) (,)

(4 ) sin sin

()()

| ( ) exp( ( ) ) |

cos ( /4) cos ( 2 /4 /4)

(1 / 2 / ) (1 / 2 / )

sin 2

sin sin

()()(

sin

Ipq Upq U pq

Pab kpa kqb

z kpa kqb

s ikp s ikq ds

kpP kp P kqP

kpP kpP

Nkp P

kpa kqb

kpa kqb N

ξξη

ξξ

ρξη

ππ

=∗

⋅

=⋅

⋅−+−

⋅+

=⋅

−+

⋅⋅





sin

)( ).

2sin

NkqP

kp P N kqP

ηη

ξη η

(2)

Here

(2 4 )IC NNab

ξη

=⋅ ⋅

is the peak irradiance

of the diffraction pattern. For simplicity, in this letter

we set

1I =

and

are respectively the periods

along the

and

axes.

Figure 2 presents the diffraction intensity pattern

according to Eq. (2). As expected, the 0

and 1

order diffractions are kept along x axis, and the

high-order diffractions disappear. The logarithm of

diffraction intensity along x axis in Fig. 2(b)

presents clearly the complete suppression of the high

order diffractions. Insets in Fig. 2 shows clearly

intensity distributions of the 0th and 1st order

diffractions. Figure 2 shows that the diffraction

pattern of the quasi-triangle array of rectangular

holes along x axis is the same as the ideal sinusoidal

transmission grating.

The Quasi-Triangle Array of Rectangular Holes with the Completely Suppression of High Order Diffractions

Figure 2: (a) The far-field diffraction intensity pattern of

the quasi-triangle array of rectangular holes. (b) The

diffraction intensity along the x axis. Insets: the 0

and 1

order diffractions.

Numerial simulation based on Eq. (2) is carried

out to evaluate the diffraction property of the quasi-

triangle array of 100000 rectangular holes. The

logarithm of diffraction intensity along x axis is

shown in Fig. 3, and high-order diffraction is much

less than the noise of 10-5 between 0th and 1st

diffraction (red dash line in Fig. 3), which agrees

well with the theoretical prediction of Eq. (2) and

Fig. 2.

Figure 3: The diffraction intensity along x axis of the

quasi-triangle array with 100000 rectangular holes.

3 EXPERIMENTAL RESULTS

AND DISCUSSIONS

A proof-of-principle experiment is performed to

confirm our theoretical and numerical results. The

experimental setup is shown in Fig. 4. A plane wave

is produced by expanding a 632 nm laser beam from

Sprout (Lighthouse Photonics) and then illuminating

the quasi-triangle array of holes. The far-field

diffraction pattern from the array is focused by a

lens and recorded by a CCD (ANDOR DU920P-

BU2). The quasi-triangle array of rectangular holes

with

44cm cm×

area is fabricated on a glass substrate

by DESIGN WRITE LAZER 2000 from Heidelberg

Instruments Mikrotechnik GmbH. The microphoto-

graph of fabricated structure is illustrated in Fig.

1(b). Periods

and

of the quasi-triangle array

along the

and

axes are respectively

20 m

and

10 m

. The side of square holes is

5 m

Figure 4: Experimental setup for the optical measurement.

Figure 5 presents the recorded diffraction pattern

and it is obvious that high order diffractions of the

quasi-triangle array of holes are effectively

suppressed. The diffraction intensity along x axis in

Fig. 5(b) is almost the same as the ideal sinusoidal

transmission gratings. The ratio of 1st order

diffraction intensity to the 0th order diffraction

intensity is 73.56% and much larger than the

theoretical predition and numerical value 25%. This

is because the CCD is saturated by the 0th order

diffraction intenstiy. In addition, the red vertical

lines in Fig. 5(a) are crosstalk along y direction due

to our one-dimension CCD.

Figure 5: (a) The far-field diffraction intensity pattern of

the quasi-triangle array of rectangular holes. (b) The

diffraction intensity along the

axis.

Now we focus on the complete suppression of

high order diffractions along the x axis. As

large enough, the intensity according to Eq. (2)

along the x axis is given by:

PHOTOPTICS 2017 - 5th International Conference on Photonics, Optics and Laser Technology

222

sinc ( 8 / )

()

(1 2 / ) (1 2 / ) cos ( 4 )

INkpa

kp a kp a kp a

ππ

⋅

−+











(3)

Equation (3) demonstrates that the quasi-triangle

array of infinite rectangular holes can generate the

same diffraction pattern as sinusoidal transmission

gratings along the x axis. Only three diffraction

peaks (the 0

order and +1

/-1

orders) appear on

the

xy−

plane. The above theoretical results are

scalable from X-ray to far infrared wavelengths.

To obtain physical insight into the diffraction

property of the quasi-triangle array of rectangular

holes, the average transmission function along

axis

is calculated by integrating the probability

distribution over

axis:

|| /4

() () (1 cos( )).

Tsds

ξρ ξ

−

==+



(4)

Equation (4) shows that the quasi-triangle array

of infinite rectangular holes has the same

transmission function along the

axis as sinusoidal

transmission gratings. It is the average diffractive

effect similar to sinusoidal grating that eliminates

high-order diffractions.

We can also understand the suppression of high-

order diffractions by the interference weakening or

strengthening. It’s known that diffraction peaks is

from the constructive interference of lights from the

different holes. The interference of lights from

different rectangular holes is controlled by the hole

position. The desired diffraction pattern only

containing the 0

order and +1

/-1

order diffra-

ctions can be tailored by the location distribution of

holes according to some statistical law.

In the above discussions, we assume the size of

the quasi-triangle array is infinite. In practice, the

number of holes is finite and thus the diffraction

pattern will not be perfect as that described by Eq.

(2) and (3). Numerical simulation based Eq. (2) is

carried out and we obtain the noise intensity at 3rd

order diffraction location for different number of

holes in Fig. 6. It reveals that the noise intensity is

approximately inversely proportional to the number

of holes. This can be attributed to the approaching of

the average transmittance function of quasi-triangle

array to the ideal sinusoidal function with the hole

number increasing. The noise intensity around the

location of the 3rd order diffraction becomes less

than 10-5 as the hole number reaches to 30000.

Figure 6: The maximum noise intensity around the

location of the 3

order diffraction versus the number of

holes.

4 CONCLUSIONS

In conclusion, the binary quasi-triangle array of

rectangular holes has been proposed to suppress the

high-order diffractions which may lead to

wavelength overlapping in spectral measurement.

The new design of the membrane with holes can be

free-standing and scalable from X-rays to far

infrared wavelengths. Both numerical and

experimental results have demonstrated the high-

order diffractions are efficiently suppressed. The

total number of holes illuminated by the light affects

the suppression of the high-order diffractions and the

noise intensity will be less than 10

-5

as the hole

number larger than 30000. The binary quasi-periodic

hole array offers an opportunity for high-accuracy

spectral measurement and will possess broad

potential applications in optical science and

engineering fields.

ACKNOWLEDGEMENTS

This work is supported by National Natural Science

Foundation of China (NSFC) (61107032, 61275170)

and the Opening Project of Key Laboratory of

Microelectronics Devices and Integrated Techno-

logy, Institute of Microelectronics of Chinese

Academy of Sciences.

REFERENCES

Palmer, C., 2005. Diffraction Grating Handbook,

Richardson Grating Laboratory, 6

edition.

Born, M., and Wolf, E., 1980. Principle of Optics.

Pergamon.

Jin, P., Gao, Y., Liu, T., Li, X., and Tan, J., 2010. Resist

shaping for replication of micro-optical elements with

The Quasi-Triangle Array of Rectangular Holes with the Completely Suppression of High Order Diffractions

continuous relief in fused silica. Opt. Lett. 35, 1169-

1171.

Vincent, G., Haidar, R., Collin, S., Guérineau, N., Primot,

J., Cambril, E., and Pelouard, J.-L., 2008. Realization

of sinusoidal transmittance with subwavelength

metallic structures. J. Opt. Soc. Am. B 25, 834-840.

Clausnitzer, T., Kämpfe, T., Kley, E.-B., Tünnermann, A.,

Tishchenko, A. V., and Parriaux, O., 2008. Highly-

dispersive dielectric transmission gratings with 100%

diffraction efficiency. Opt. Express 16, 5577-5584.

Zhou, C., Seki, T., Kitamura, T., Kuramoto, Y.,

Sukegawa, T., Ishii, N., Kanai, T., Itatani, J.,

Kobayashi, Y., and Watanabe, S., 2014. Wavefront

analysis of high-efficiency, large-scale, thin

transmission gratings. Opt. Express 22, 5995-6008.

Warren, M. E., Smith, R. E., Vawter, G. A., and Wendt, J.

R., 1995. High-efficiency subwavelength diffractive

optical element in GaAs for 975 nm. Opt. Lett. 20,

1441-1443.

Pease, L. F., Deshpande, P., Wang, Y., Russe, W. B., and

Chou, S. Y., 2007. Self-formation of sub-60-nm half-

pitch gratings with large areas through fracturing.

Nat. Nanotechnol. 2, 545-548.

Gao, N., and Xie, C., 2011. High-order diffraction

suppression using modulated groove position gratings.

Opt. Lett. 36, 4251-4253.

Fan, Q., Liu, Y., Wang, C., Yang, Z., Wei, L., Zhu, X.,

Xie, C., Zhang, Q., Qian, F., Yan, Z., Gu, Y., Zhou,

W., Jiang, G., and Cao, L., 2015. Single-order

diffraction grating designed by trapezoidal

transmission function. Opt. Lett. 40, 2657-2660.

Cao, L. F., Förster, E., Fuhrmann, A., Wang, C. K.,

Kuang, L. Y., Liu, S. Y., and Ding, Y. K., 2007.

Single order x-ray diffraction with binary sinusoidal

transmission grating. Appl. Phys. Lett. 90, 053501.

Kipp, L., Skibowski, M., Johnson, R. L., Berndt, R.,

Adelung, R., Harm, S., and Seemann, R., 2001.

Sharper images by focusing soft X-rays with photon

sieves. Nature, 414, 184-188.

Huang, K., Liu, H., Garcia-Vidal, F. J., Hong, M.,

Luk’yanchuk, B., Teng, J., and Qiu, C.-W., 2015.

Ultrahigh-capacity non-periodic photon sieves

operating in visible light. Nat. Commun., 6, 7059.

Huang, F. M., Kao, T. S., Fedotov, V. A., Chen, Y., and

Zheludev, N. I., 2008. Nanohole array as a lens. Nano

Lett. 8, 2469-2472.

Huang, F. M., Zheludev, N. I., Chen, Y., and Abajo, F J.

G., 2007. Focusing of light by a nanohole array. Appl.

Phys. Lett. 90, 091119.

PHOTOPTICS 2017 - 5th International Conference on Photonics, Optics and Laser Technology