Optical intensity far-field distribution of MEMS micro-mirror arrays by Fraunhofer diffraction

MEMS mirror arrays perform superior than MEMS mirror to assemble a Lidar, but few published papers about the optical intensity and distributions when laser diffracted by MEMS arrays are available, this letter focuses on the issues. Firstly, the complex amplitudes of laser which is diffracted by 1D (dimension) and 2D arrays are presented, respectively. Then we give the optical intensity and distributions on the observation plane. Finally, the simulation diagrams of these distributions are shown, and the correctness of the results is verified by Lumerical FDTD Solutions simulations and Young’s double-slit experiment.

M EMS (viz. micro-electromechanical systems) sensors, such as piezoresistive silicon pressure sensor, strain gauge, etc., [1][2][3][4] are the product of the development of the semi-conductor industry which was born in 1954. 5) In seriously, the sign of the MEMS sensor birth was the pressure sensor invention in the 1970s. 6,7) In subsequent, with the rapid development of micromachining, massive production of inexpensive MEMS made MEMS widely used in the automobile industry [8][9][10][11] and daily consumer electronics 12,13) possible.
MEMS micro-mirror and MEMS micro-mirror arrays were extensively applied in optics to date, 14,15) and they are indispensable in laser communication and imaging, 16) especially in laser detection. MEMS micro-mirror has unique endowment when using it to design a compact Lidar. 17) But compared with MEMS micro-mirror, MEMS micro-mirror arrays perform more excellent in the application field of large optical aperture, e.g. MEMS micro-mirror arrays can achieve a larger scanning angle in space laser communication system. 18) Hitherto, the MEMS micro-mirror arrays are extensively used in adaptive light detection. 19) The difference between MEMS mirror with MEMS mirror arrays to steer laser is the former does it by the mechanical vibration of the mirror but the latter by the combination of light interference and mirrors vibration. 20) So the laser blueprints steered by MEMS mirror arrays are much more meticulous than by single MEMS mirror on the observation plane. Although MEMS mirror Lidar performs outstanding respect to mechanical one, it still has some disadvantages, such as short detection range, insufficient scanning angle, et al. 20,21) In order to forbid the shortcomings of MEMS Lidar, we intend to design a Lidar based on MEMS mirror arrays. However, the prerequisites of devising a Lidar based on MEMS arrays are the analysis of optical intensity and distribution on the observation plane when the laser beam is diffracted by the arrays, they are seldom reported so far.
To date, most of the light intensity distribution of MEMS arrays diffracted laser on the observation plane is used by numerical analysis. 20) In this paper, we would theoretically solve the light intensity distribution on the observation plane when the laser is diffracted by MEMS arrays. We calculate the optical intensity distribution in far-field from the arrays when the laser is diffracted by 1D and 2D MEMS micromirror arrays-based on Kirchhoff and Fraunhofer diffraction theory. Kirchhoff theory is the centerpiece of wave optics, and mathematically discovers the mystery of diffraction. Meanwhile, Kirchhoff theory is a mathematical reinterpretation for Fresnel-Huygens principle, Gustav Kirchhoff established the theory employing Green theorem with special boundary conditions in 1883. 22) Kirchhoff integral formula is a special value integral of Kirchhoff theory in a specific condition. While Fraunhofer theory is the far-field approximation of Kirchhoff integral formula. 23) We calculate the intensity distribution of the laser diffracted by 1D and 2D MEMS arrays on the observation plane, and we show some simulation figures to verify the calculations.
The Fraunhofer theory is an approximation of Kirchhoff diffraction theory with the Fraunhofer diffraction condition. 23) Setting the coordinate system as Fig. 1(a) shown, the Fraunhofer diffraction condition is 25 cis the number of waves per second, w and c are the angle frequency and the speed of light, respectively, z 1 is the distance between the aperture of M with the plane of xP y. Taking the parameter of aperture M, whose sizes iś a b 2 2 as shown in Fig. 1 We take the beam as coherent light, so˜( ) E x y , 1 1 equals a constant, and let it equal A, then integrating formula (3), it becomes the equation of then taking a and b into Eq. (6), there is If the mirror is a square with side length a 2 , just substituting a for b of Eqs. (5)-(7) would be the accordingly results.
Supposing 1D arrays is composed of N pieces ofá b 2 2 mirror, and the pitch between two consecutive mirrors is d as shown in Fig. 1(d). We set the coordinate system as Fig. 1(c), then the complex amplitude of laser on xP y 0 plane is   where combining Eq. (9) with Eq. (10) one obtains Then we can analyze the distribution characteristics of optical intensity in the plane of xP y 0 based on Eqs. (11) and (12). On xP y 0 plane, when the coordinates of one point are equal to the roots of Eq. (12), the brightness of the point would be the minimum. While getting the stripes of maximum brightness by analysis methods, we should take advantage of Eq. (12) derivative, but this is rather challenging. In order to illustrate the contrast and resolution of the maximum stripes on the observation plane, we would show the simulation diagram of Eq. (12).
Supposing that 2D MEMS micro-mirror arrays are consist ofŃ N pieces of mirror, and the Cartesian coordinate system is established as Fig. 1(c). Based on Fraunhofer diffraction theory, the complex amplitude of P 1 point in xP y 0 plane is gotten as  Equations (12) and (15) contain the common factor of | | Cab 4 , 2 which is the optical intensity of the brightest spot in the observation plane when the laser is diffracted by single mirror. For the convenience of calculating, it will be taken as a unit of optical intensity in the following paragraphs. When the laser is diffracted by 1D MEMS micro-mirror arrays, the optical intensity of the points in xP y 0 plane is the minimum if their coordinates fulfill the equations of  Compared with dark spots, the maximum optical intensity locations are rather difficult to locate. Although the locations of the maximum optical intensity diffracted by 1D and 2D arrays can be derived from the derivative of Eqs. (12) and (15) theoretically, their derivatives are very complicated, and it is rather challenging to do it analytically. Furthermore, the exact coordinates of all maximum optical intensity locations are less important than the detailed contrast of optical intensity in the observation plane. In other words, the specifics of intensity contrast are more important than the locations of maximum intensity in the observation plane to design a Lidar based on MEMS arrays. Meanwhile, the simulation of the intensity contrast is easier than calculating the derivatives of Eqs. (12) and (15).
The distributions of optical intensity expressed by Eqs. (12) and (15) would be shown by simulation diagram. We also use Lumerical FDTD Solutions to simulate the diffracted results of MEMS arrays. Finally, we compare two kinds of simulation results to verify the rationality of Eqs. (12) and (15). Furthermore, we compare the simulation diagram with Young's double-slit experiment results to verify the correctness of Eqs. (12) and (15)  The abscissa and ordinate of Fig. 2(a) stand for the coordinates of the observation plane as shown in Fig. 1(c). While in Fig. 2(b), the abscissa stands for x-axis of the observation plane of Fig. 1(c), and the ordinate stands for the light intensity. Colors of Figs. 2(a) and 2(b) represent the contrast of different light intensities. In Fig. 2(c), the colors stand for the different relative light intensity of laser diffracted by the arrays on the observation plane, and the white circles stand for different zenith angles on the observation plane. And the length unit of both figures is meter. From Fig. 2(b), we figure out the result that the deflection angles of the first-level maximum and second-level maximum of light intensity to the source are 4 degrees (arctan(1.42/20) = 4.07°) and 8 degrees (arctan(2.84/20) = 8.08°), respectively. And from Fig. 2(c), we also get the deflection angles of the first and second maximum of light intensity are almost the same as Fig. 2(b) shown. On the other side, both diagrams show a wide gap in light intensity among different maximums. Furthermore, with the result of Eq. (12), we take not more than 30 s to simulate the distribution, but we spend more than 3 h to get the distributions by FDTD. With the increase in the size and number of micro-mirror, computer hardware requirements will significantly increase. Then we also show the diagram of 2D arrays simulated by MATLAB and FDTD, respectively. The legends and coordinates of Fig. 3 are the same as Fig. 2. From Fig. 3(b), we figure out the deflection angles of the first-level maximum and second-level maximum of light intensity to the source are 3.9 degrees (arctan(1.36/ 20) = 3.89°) and 8.1 degrees (arctan(2.85/20) = 8.11°), respectively. The result is nearly the same as Fig. 3(c). The contrast of light intensity in both imitation figures is almost the same, and the central maximum is much bright than other level maximum. Meanwhile, using Eq. (15) to simulate the diffraction result of 2D MEMS arrays save much more time than using FDTD. Due to the limitation of the calculating power of our computer, we can only calculate MEMS arrays of 4 × 4. If the number of micro-mirror in arrays increases, the light intensity contrast and angular resolution would significantly improve. And this is vital important for designing a scanning MEMS arrays-based Lidar.
We compare the results of Eq. (15) with Young's doubleslit interference experiment. Although formulas (12) and (15) are obtained with the Fraunhofer diffracted condition (1), to some extent, we can prove the light intensity distributions of Eqs. (12) and (15) are correct by comparing with Young's double-slit interference experiment in the plane of 800 mm. Where the size of double slits is 0.02 mm × 0.02 mm, the pitch between slits is 0.1 mm, and the wavelength l is 1064 nm. Figures. 4(a) and 4(b) are simulation diagrams of double-slit interference simulated by Eq. (12), and Figs. 4(c) and 4(d) are the diagram diffracted by 2 × 2 arrays simulated by Eq. (15). The legend of Fig. 4 are the same as Fig. 2.
In Fig. 4, the contrast of optical intensity is lower than which of Figs. 2 and 3, and the conclusion is consistent with the phenomenon of Young's double-slit experiment, in which the intensity is almost the same among different stripes. Meanwhile, Eqs. (12) and (15) also indicate the number N of mirrors is one of key determinants to optical intensity and contrast on the observation plane. Besides, according to Young's theory, 24) the period length of stripes is The results of Figs. 4(a) and 4(c) are almost consistent with Eq. (20), which proves Eqs. (12) and (15) are reasonable.
We have customized some MEMS arrays, but the parameter and technology standard cannot fulfill the academic requirements with the limitation of engineering technology (the pitch must be the same order of magnitude as wavelength), 20) so we cannot directly prove the correctness of Eqs. (12) and (15). That's why we prove the correctness of Eqs. (12) and (15) by Lumerical FDTD Solutions and the result of Young's double-slit interference experiment.
In this letter, based on the Kirchhoff diffraction and Fraunhofer diffraction theory, the distribution functions of optical intensity are deduced when the laser is diffracted by 1D and 2D MEMS arrays; the simulation results of the distributions are provided. Also, we have proven the correctness of the results by Lumerical FDTD Solutions and Young's double-split experiment. The optical intensity of the greatest maximum stripe is much stronger than others, and the optical intensity in the observation plane would increase rapidly when the number of mirrors rises, so are the angular resolution of arrays. And the theory in this paper reveals that using MEMS micro-mirror arrays to steer the laser beam can improve the resolution of Lidar comparing with using MEMS micro-mirror. The resolution is one of the critical properties of Lidar, so the results we provided will give some references to assemble a laser scanning Lidar, especially for choosing laser radiation source and receiver sensor.