## Abstract

The periodic structure on the optical surface affects the beam shape and its propagation. As the size of the optical elements becomes larger and its shape becomes complicated, the quantitative analysis of the effect of the periodic structure on the optical surface becomes indispensable given that it is very difficult to completely eliminate the microscopic periodic structures. Herein, we have experimentally investigated Bragg scattering from an optical surface with extremely small aspect ratios (~10^{−5}) and groove densities (0.5 lines/mm). We observed the period of the constructive interference formed due to the propagation of the 0th, 1st, and −1st beam modes caused by Bragg scattering. When the periodic structure has a modulation depth of ± 50 nm, the intensity increase of constructive interference between the beam modes formed by Bragg scattering was > 10 times greater than the intensity of a flat surface at the propagation distance at which constructive interference was most pronounced. This study is envisaged to open new avenues for the quantification of the effect of periodic structures based on the observation of the interference on the beam profile formed by Bragg scattering during the beam propagation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The light that is reflected (or transmitted) from various optical elements typically includes information about the surface (or internal) structure of the optical elements, in a direct or indirect forms. In the cases of the large-size optical elements that have been extensively used in high-power lasers, space/astronomical optics, and accelerators, the quality of the optical surface is crucial and has a significant influence on the performance of the entire system [1–16]. Furthermore, in recent years, the demand for advanced studies based on the application of the freeform optics, such as off-axis and aspherical optics, beyond the existing flat and spherical optics has been increasing [8–16]. However, as the size of the optical elements increases and the surface shape becomes more complicated, it becomes increasingly difficult to achieve a surface quality beyond a certain level. If the optical surface has a periodic structure, it undergoes Bragg scattering [17–20] depending on the conditions. A typical example is a diffraction grating, which has a period that varies between the submicrometer (sub-μm) and micrometer (μm) range, and a modulation depth from a few μm to a few tens of μm. Thus, it has an aspect ratio in the range of a few to a few tens of units and a groove density from a few hundreds to thousands of lines/mm [21,22].

Given that the optical surface is manufactured by a mechanical process, such as that which uses the diamond turning machine (DTM) [23], a periodic texture structure can be typically formed on the surface. One of the two prominent texture characteristics is the roughness [24] which has a period of the order of several μm attributed to the radius and the speed of the cutting tool. The other is the waviness [25], which has a period of the order of several mm and is dependent on the periodicity of environmental factors, such as temperature and humidity. As the size of the optical elements increases and the surface shape becomes complicated, it is almost impossible to completely eliminate structural periodicity, such as the roughness and waviness even if the polishing process is performed. This periodic structure can cause Bragg scattering and can affect the beam profile which propagate.

The roughness on the optical surface can be studied by observing the diffraction signal [26,27]. This is considered as one of the methods used to evaluate the optical surface. However, in the case of roughness, which has a period of the order of several μm, the higher order modes beyond the 0th order mode may not be significant because they spread rapidly at very large angles owing to Bragg diffraction. In the case of waviness, which has a period of the order of several mm, the higher order modes beyond the 0th order can affect the laser beam profile which propagates a long distance because it travels at a very small diffraction angle with respect to the beam direction of the 0th order.

In terms of the aspect ratio and groove density, the periodic structure given by the waviness on the optical surface is very different from the typical diffraction grating which generates Bragg diffraction. In the range of the optical wavelength corresponding to visible light, it is close to the plane surface owing to its extremely low-aspect ratio. For this reason, the diffraction-based approach has not been investigated until now. Usually, the information on the optical surface can be acquired with the use of laser interferometry [28,29]. However, the information of the beam profile reflected on the optical surface, and the variation of the beam profile owing to its propagation cannot be acquired by laser interferometry. Consequently, a quantitative analysis is required to evaluate how the Bragg scattering is caused following the interaction between the light and structure with a periodicity at the millimeter (mm) scale, including the waviness. For instance, in a very large optical system, such as in a high-power laser facility [1–7], the waviness on the optical surface may affect the beam profile which propagates in the entire optical system. This is because the Bragg scattering formed by the waviness becomes pronounced as the beam size increase to a few hundreds of mm and as the beam propagation increases to a few tens of meters. Moreover, the diffraction angle between the beam modes becomes much smaller as the period of the waviness increases so that the spatial overlap between the beam modes is maintained over a very long propagation distance. The fringe patterns which are formed by the constructive and destructive interferences between the beam modes can affect the propagating beam profile in the optical system.

Thanks to recent technological advances, the precision of mechanical machining has achieved surface roughness values at the nanometer (nm) scale [30]. Furthermore, the period and the modulation depth of waviness became adjustable through the precise control of temperature and humidity during the manufacturing process. In this study, several optical samples with a waviness period of 2 mm and modulation depths of ± 15 nm, ± 30 nm, and, ± 50 nm were manufactured using DTM. To the best of our knowledge, we have succeeded in observing for the first time Bragg scattering from optical surfaces with extremely small aspect ratios (~10^{−5}) and groove densities (0.5 lines/mm). One of the most remarkable results of this study is the observation of the period of the constructive interference of the center intensity formed by the overlap of three propagation lines (which involve the 1st and the −1st, the 0th and the 1st, and the 0th and the −1st beam modes) caused by Bragg scattering. In addition, the increment of the center intensity owing to the constructive interference was > 10 times larger compared to that evoked by a flat mirror at the propagation distance at which constructive interference was most pronounced. The most important aspect is the fact that even though the period is very long (> 10^{3}) compared to the laser wavelength (808 nm), and the height is extremely low ( ± 15 nm), Bragg scattering is clearly observed whenever the periodic structure is formed on the optical surface.

## 2. Experiment

Two-dimensional (2D) surface images and line profiles of Aluminum (Al) samples plated with Nickel Phosphorus (NiP), which were manufactured by DTM (Model: Nanoform L1000), were measured with a Fizeau interferometer (Model: Apre S150). As shown in Fig. 1(a), a circular waviness pattern with 25 periodic structures was observed on the Al sample with a diameter of 100 mm. The line profiles (black dotted line) on the 2D surface image exhibits a sinusoidal structure with a period of approximately 2 mm (see Fig. 1 (b)). The modulation depths were maintained at ± 50 nm (black), ± 30 nm (red), and ± 15 nm (blue) on average, and exhibited a variation from a few nm to a few tens of nm. Conversely, the height variation of the flat mirror (shown in cyan color) spans a few nm.

The experimental configuration for the observation of the laser beam profile at increasing propagating distances after the reflection on the surface of the sample is shown in Fig. 2. The laser diode (Model: 1Q1C, Power Technology) is shown at the lower left, and emits light with a wavelength of 808 nm and a power of 140 mW. Additionally, there are two partially reflective mirrors that are used to reduce the laser energy. Extended collimation is applied by the configuration of convex and concave mirrors to enlarge the beam size up to 100 mm. The incident angle of the beam on the surface of the sample is 3.15°. After the reflection, the reduced collimation is applied by the configuration of a concave and convex mirror to decrease the beam size up to 17 mm. The incident angle of the beam on the surface of the collimating mirrors is minimized to reduce the beam aberrations. To acquire the 2D beam image, a beam profiler (Model: BGS–USB3–LT665, Ophir–Spiricon) was applied. A folding type using a number of gold-coated mirrors was also applied to achieve a long propagating distance within the limited area. Overall, the reflective optical elements were used to minimize the degradation of beam quality that could be attributed to the transmitted optical elements, such as the lens and the filters.

## 3. Results

Figure 3 shows the variation of the center intensity in the 2D beam profile observed by changing the propagation distance from 10 to 250 cm after collimation reductions. In this figure, one of the most significant features was the intensity modulation with a periodic structure of ~30 cm observed along the beam propagation direction. This was mainly due to the constructive and destructive interferences among the three propagating beam modes (0th, 1st, and −1st) caused by Bragg scattering (this is described in the Discussion section in more detail). Conversely, in the case of a flat mirror which has no waviness, such as the case at which the periodicity is of the order of mm, the center intensity is almost constant at low levels.

In addition, the center intensity of the 2D beam profile increases as the modulation depth of the waviness increases. The center intensities measured for waviness values equal to ± 30 nm and ± 50 nm, respectively increase by approximately three and seven times compared to the intensity measured for a waviness of ± 15 nm. Additionally, the intensity observed at the modulation depth of ± 50 nm increases by approximately 14 times compared to that of the flat mirror. The variation of the intensity is quite substantial despite the fact that the modulation depths of ± 15 nm (<λ/20), ± 30 nm (<λ/20), and ± 50 nm (<λ/6) are very low with respect to flat surfaces. This variation can be related to the surface damage of optical elements [31–33] because they can affect the subsequent optical components placed at a distance which generates the highest intensity based on constructive interference attributed to Bragg scattering.

The degree of beam intensity formed by the interference among the beam modes depends on the diffraction efficiency of each beam mode. The diffraction efficiency of the m^{th} order for a sinusoidal period is proportional to J_{m}^{2} (2kacos*θ*) (where J = Bessel function of the 1st kind, k = 2π/λ, m = m^{th} beam mode, a = amplitude of the sine wave, and *θ* = angle of incidence) [34]. It is important to note that J_{m} depends on a, which denotes the modulation depth of waviness. The 0th (J_{0}^{2}) and 1st (J_{1}^{2}) values of J_{m}^{2} (2kacos*θ*) are respectively equal to 0.973 and 0.013 at the modulation depths of ± 15 nm (a = 15 nm), 0.896 and 0.051 at the modulation depths of ± 30 nm (a = 30 nm), and 0.731 and 0.129 at the modulation depths of ± 50 nm (a = 50 nm). Based on these, the intensity owing to the interference among the beam modes is maximized at the condition at which the efficiency difference between the 0th and 1st beam modes is the smallest (i.e., the efficiency of 1st beam mode is the highest), i.e., at the modulation depth of ± 50 nm.

The intensity decreases locally in the early periodic modulation formed by the waviness with a modulation depth of ± 50 nm (see plot in black color in Fig. 3). This can be attributed to the higher order beam modes (higher than the 2nd order), but additional studies are needed to confirm this speculation. Conversely, these phenomena do not emerge for modulation depths of waviness equal to ± 15 nm or ± 30 nm because the diffraction efficiency for generating higher order modes depends on the modulation depth of waviness, as mentioned earlier. After the 2nd ( ± 15 nm, ± 30 nm) or 3rd ( ± 50 nm) periodic modulations in Fig. 3, the overall center intensity gradually decreases at increasing propagation distances. This is because the spatial overlap between the 0th, 1st, and −1st beam modes gradually decreases owing to the differences of the diffraction angles for each beam mode, as the propagating distance increases. If the propagation is sufficiently far away, the interference no longer occurs because the beam modes become completely separated from each other.

The experimental result was verified by performing Fourier optics simulation of the propagation of light using VirtualLab Fusion software. In this simulation, the experimental configuration and optical samples with a mm-scale periodic structure were considered. In the propagation of light in free space, the complex field amplitude at the output plane (*E _{out}*) is expressed as

*i*is the imaginary unit,

*ν*

_{x}= 1/

*λ*

_{x}and

*ν*

_{y}= 1/

*λ*

_{y}are spatial frequencies,

*E*is the complex field amplitude at the input plane, Δz is the propagation distance through an infinite homogeneous medium with a given refractive index

_{in}*n*(here,

*n*= 1 for free space),

*λ*is light wavelength,

*F*is Fourier transform, and

*F*

^{−1}is inverse Fourier transform. The output plane is placed along the axis (z) of beam propagation such as in the experiment, and the center intensities extracted at each output plane during the beam propagation are denoted in Fig. 4. Figure 4(a) denotes the distribution of the center intensity calculated by variation of the propagating distance; comparison between the experimental and simulation results is shown in Fig. 4(b). The simulation results almost match the experimental results shown in Fig. 3. In particular, it can be seen that the modulation period of the center intensity and the overall intensity distribution during the beam propagation are exactly reproduced for each sample.

The constructive and destructive interferences at the beam center are clearly observed by observing the 2D beam images, as shown in Fig. 5. Figure 5(a) shows the 2D beam images observed at the propagation distances of 35 cm and 50 cm after the reduced collimation for the sample with a waviness of 2 mm and a modulation depth of ± 15 nm. The distances of 35 cm and 50 cm respectively denote the positions at which constructive and destructive interferences of the center intensity are most evident (see blue curve in Fig. 3). Thus, the two interference patterns are exactly reversed with respect to each other along the radial direction of the beam, as shown in Fig. 5(b). The simulation results representing the 2D image and its line profile in Fig. 5(c) and (d) are in agreement with the experimental results shown in Figs. 5 (a) and (b).

## 4. Discussion

The aspect ratio (~10^{−5}) and the groove density (0.5 lines/mm) of waviness of the sample which was used in the experiment were extremely small compared to the usual periodic conditions used at which Bragg diffraction occurs. However, the diffraction angle of the 1st and −1st order beam modes with respect to 0th order can be quantitatively determined from the equation of the diffraction grating because the sample consists of 50 circular waviness patterns with a period of 2 mm. From the equation *d*(sin*θ*_{i} ± sin*θ*_{m}) = mλ (where *d* = period of waviness (2 mm), *θ*_{i} = angle of incidence (3.15°), *θ*_{m} = angle of the m^{th} mode, λ = wavelength (808 nm)), the angles of *θ*_{1} and *θ*_{-1} with respect to *θ*_{0} both become equal to Δ*θ* = 0.02° (0.4 mrad).

The experimental results can be described using the configuration of Fig. 6. The red lines of Fig. 6(a) present the wavefronts for the three beam modes, such as the 0th, 1st, and −1st order beam modes formed after the reflection on the surface of the sample. Even though the angles of the 1st and −1st orders with respect to 0th order are very small (Δ*θ* = 0.02°), as noted above, we set the angles to large values to describe our experimental results. The wavefronts of 1st and −1st orders are symmetrically placed in the opposite directions with respect to the wavefront of the 0th order, as shown in this figure. As the three wavefronts propagate, three types of propagation lines are formed which exhibit periodic constructive interference patterns. The formation of the first line is attributed to the 1st and the −1st modes, the second line is attributed to the 0th and the 1st modes, and the third line is attributed to the 0th and the −1st modes, which are denoted by α(1, −1), β(0, 1), and γ(0, −1), respectively, as shown in Fig. 6(a). It is important to note that the angles between α(1, −1) and β(0, 1) (or that between α(1, −1) and γ(0, −1)) are equal to half of the values of the angles between the 0th and the 1st propagation lines (or the 0th and the −1st propagation lines). In addition, the interval of the interference fringe between the 0th and the 1st orders (or the 0th and the −1st orders) in the radial direction are denoted by λ/(sinΔ*θ*), whereas that between the 1st and the −1st orders is denoted by λ/(sin2Δ*θ*), i.e., there is a difference in the angle arguments by a factor of 2 × (see the intervals of α(1, −1), β(0, 1), and γ(0, −1) in Figs. 6(a) and 6(b)) [35].

One of the many α(1, −1) lines follows exactly the propagation line of the 0th order (same as the propagation direction of the center intensity in the 2D image). With respect to the α(1, −1) lines, the β(0, 1) and γ(0, −1) lines cross symmetrically and form rhombic shapes, such as those shown in Fig. 6(b). As a result, the points at which all of the three propagation lines α(1, −1), β(0, 1), and γ(0, −1) overlap, forms the rhombus structure. The spacing between the top and bottom vertices of the rhombus corresponds to the period of constructive interference observed in Fig. 3. Conversely, the left and right vertices of the rhombus appear when the center intensity of the beam is the weakest, i.e., when destructive interference becomes most pronounced. Thus, the configuration of Fig. 6(c) corresponds to the experimental result shown in Fig. 5(b).

The periodic structure with a periodicity of ~30 cm observed based on the propagation of the beam was also associated with the rate of beam magnification based on the reduced collimation. In this experiment, we have reduced the beam size from 100 mm to 17 mm to directly measure the 2D beam image using a beam profiler. In this process, the scheme of the reduced collimation was applied using concave and convex mirrors. In this case, the diffraction angle Δ*θ* = 0.02° between the 1st (−1st) and 0th orders was determined from the reflection on the surface of the sample and can be changed. The configuration of the reduced collimation is denoted by the ray transfer matrix. Equation (2) expresses the multiplication of the three matrices that respectively denote the reflection from a concave mirror, the propagation in free space, and the reflection from a convex mirror,

*f*is the focal length of the concave mirror,

_{1}*d*is the distance between the concave and the convex mirror, and

*f*is the focal length of the convex mirror. Given that

_{2}*d*=

*f*-

_{1}*f*for the collimation condition, Δ

_{2}*θ*′ finally becomes equal to

*f*Δ

_{1}f_{2}*θ*. Based on this, the angle difference (Δ

*θ*′) of the 1st (−1st) order with respect to the 0th order after the reduced collimation becomes equal to 0.14°, which is seven times higher compared to the angle obtained before the reduced collimation.

Furthermore, we can compare the transfer matrix to the configuration of the geometric rhombus, as shown in Fig. 7. The geometric configurations before and after the reduced collimation are respectively shown in Figs. 7(a) and 7(b). As shown in Fig. 7(a), the single period of constructive interference associated with the center intensity becomes equal to 11.5 m when the angle (0.01°) between α(1, −1) and β(0, 1) and the interval (1 mm) between the interference fringes of the 1st and the −1st orders are considered. The angle (0.01°) is effectively equal to half of the angle between the 0th and the 1st propagation lines and the interval (1mm) in the radial direction is equal to λ/(sin2Δ*θ*). By contrast, in Fig. 7(b) it is shown that the angle between α(1, −1) and β(0, 1) becomes equal to 0.07° by considering the values of 0.15 m and 175 μm. The value of 0.15 m is the length which is equal to one half of the period (which is equal to 0.3 m in Fig. 3), and 175 μm is the interval between the interference fringes between the 1st and the −1st modes in the radial direction (175 μm is equal to half of the interval between the two peaks near the center of the line profile highlighted in red color, as shown in Fig. 5(b)). Given that the angle between the 0th and the 1st propagation lines (or 0th and −1st propagation lines) is equal to two times the angle (0.07°) between α(1, −1) and β(0, 1) (or α(1, −1) and γ(0, 1)), the angle differences (Δ*θ*′) of the 1st (−1st) order with respect to the 0th order after the reduced collimation becomes equal to 0.14°, and coincides with the result obtained based on the ray transfer matrix. Thus, the long period associated with constructive interference and the consequent significant intensity increase caused by Bragg scattering from an optical surface with periodicity at the mm-scale is one of the critical points in the large configurations, such as high-power lasers, space/astronomical optics, and accelerators. Accordingly, we ought to be careful when we determine the positions of the optical components located after the optical element, which includes mm-scale periodic structure. Bragg scattering is expected to provide an acceptable range of the optical element with periodic structures in the entire optical system by quantifying the effect of the periodic structure present on the optical surface.

## 5. Conclusions

In conclusion, we have observed the period of constructive interference formed from the beam propagation of the 0th, 1st, and −1st beam modes caused by Bragg scattering from the special optical elements in conditions which differed considerably from usual diffraction conditions. The period of the constructive interference observed in the beam propagation was associated with the rate of beam magnification given by the optical configuration. It is also interesting to note that the intensity increment based on the constructive interference formed by the waviness with modulation depths of ± 50 nm was 10 times higher than that of the flat mirror at the propagation distance at which constructive interference was most pronounced. Accordingly, it is necessary to investigate in subsequent research studies the effects of the higher-order beam mode above the 2nd order and the spatial separation between the beam modes.

A possible scenario of future use of the findings of our study may be summarized as follows. The period of the constructive interference observed by the beam propagation can be applied to estimate the curvatures of unknown optical surfaces based on the variations of the periods of constructive interferences. Additionally, because the extent of the increase of the intensity based on the interference between the beam modes can be related to the surface damage of optical elements, this intensity increase can be applied as a guideline to quantify specific characteristics, such as the period and modulation depth of the periodic structure to control the effect of interference between the beam modes caused by Bragg scattering during beam propagation.

## Funding

Creative Convergence Research Project in the National Research Council of Science and Technology of Korea (CAP–15–01–KBSI); Korea Basic Science Institute (D39615).

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