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Abstract
Optical coatings with extremely low scattering losses are critical in high-precision optical systems. In this study, the abnormal scattering of nodular defects in high-reflection multilayer coatings was investigated experimentally and theoretically. The measurements and finite-difference time-domain simulations showed that the total scattering does not vary monotonically with increasing nodular structure size, but rather oscillates. Field distribution analysis revealed that the anomalous scattering originates from the coupling of the incident light with the surface wave at the top of the defects. These findings contribute to the field of low-scattering-loss multilayer coatings and high-precision optical systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In optical systems, the performance, sensitivity, and signal-to-noise ratio heavily depend on the optical components, such as optical coatings. In some high-precision optical systems, including gravitational wave detection systems, as LIGO and laser gyroscopes, extremely low scattering loss is required for optical coatings [1–6]. In addition, scattering hinders the performance of optical systems containing laser optical resonant cavities [7].
Bulk inhomogeneity, roughness, and defects are the main sources of scattering in multilayer coatings. Scattering caused by bulk inhomogeneity originates from the inhomogeneity of the refractive index within the coatings. With the development of high-energy deposition technology, dense, amorphous, high-quality multilayer coatings can be achieved; thus, the scattering caused by bulk inhomogeneity in high-quality optical coatings is quite low and can be neglected. For example, Ta2O5/SiO2 multilayer coatings based on ion beam sputtering (IBS) are ideal for achieving quite low film loss [8,9]. Although the coating process has been improved, it is difficult to avoid scattering caused by inherent roughness and defects in coatings.
Surface roughness-induced scattering has been studied widely, including various calculation methods and multiple descriptions of physical effects [10–15]. It has been established that even with only a few defects in the thin film, the measured scattering intensity is significantly higher than that caused by roughness [12,16]. Local defect-induced scattering is considered to be one of the causes of discrepancies in the measurement results when international round-robin experiments exposed serious uncertainty problems in scattering studies [17]. Zhang et al. quantitatively proved that a single nodule per square millimeter grown from SiO2 microspheres with a diameter of 1.0 µan induce a scattering loss of 5 ppm, which is about two or three times the scattering loss caused by roughness (multilayer coating with 0.216 nm roughness-induced scattering value ∼2 ppm) [18]. It is evident that the presence of a few nodular defects can cause fatal scattering loss in high-precision laser measurement systems, because the presence of defects can increase the scattering by tens or hundreds of parts per million (ppm).
Nodules usually develop from contamination during substrate preparation or coating processes and are considered to be among the defects in dielectric high reflection (HR) multilayer coatings. The nodular structure manifests as an ellipsoid with a micrometer-sized dome protruding above the coating surface [19,20]. These structural properties can be understood by the mathematical expression of the nodule diameter D which can be expressed as a function of the seed diameter d as $D\textrm{ = }\sqrt {Cdt} $, where C and t are the geometric constant and seed depth [21,22]. The geometric constant C depends on the deposition technology and conditions, such as the geometry of the coating machine and the mobility of the deposited atoms [23–25]. It can be observed that the high uncertainty of the contamination source causes nodular defects with different structures. The nodular structure will change when seed diameter and position affect parameters d and t or the multilayer coating design and deposition affect parameters t and C.
Previous studies have shown the strong scattering intensity of the nodule, but they have all been conducted with a fixed nodule structure size, which is unrealistic, and have an important in limitation that only nodule scattering with a single seed diameter setting was considered [18,26]. Research has shown that the growth of nodular structures can be suppressed by the planarization method, reducing the scattering value simultaneously [18]. Therefore, it can be considered that the scattering induced by a nodule has a monotonic relationship with the size of the structure. In particular, the scattering value decreases with decreasing structure size and increases with the growth of the structure; however, it still lacks relevant arguments. Thus far, there have been no scattering analysis reports on the structural variability of nodules, making it difficult to determine the full scattering characteristics of nodular defects in reality.
Therefore, exploring the scattering characteristics of nodular defects of different sizes is necessary for comprehensive analysis of the actual multilayer coating scattering and optical system performance during operation. Furthermore, we needed to clarify the interaction between the coatings with the nodule structure and incident light during the scattering process. Understanding nodular defect-induced scattering can facilitate the formulation of faster and more effective methods to suppress scattering loss that meet the requirements of practical applications.
In this study, we investigated the scattering properties of nodular structures in HR coatings. We found that the total scattering cross-section does not vary monotonically with the size of the nodular structure, but rather has an oscillatory character. Abnormal light scattering was confirmed by measurements of the artificial nodules as well as finite-difference time-domain (FDTD) simulations. In addition, the field distributions obtained from these simulations indicated that the oscillation properties originate from the light interaction of the surface mode on the top of the nodules and the incoming plane wave. Further, we confirmed the origin of anomalous scattering by studying the scattering properties of a homogeneous nodular structure. Our findings can contribute to the development of laser films with low scattering losses and high-precision optical systems.
2. Results and analysis
We investigated the scattering properties of multilayer coatings with nodules by introducing an artificial seed structure on a dielectric substrate. A schematic diagram of the nodule structure is shown in Fig. 1(a), where we considered the nodules originating from seeds located on the substrate, which we covered with Ta2O5/SiO2 quarter-wavelength HR multilayer coatings. The multilayer coating system working at 1064 nm was designed as (Sub|(HL)13L|Air), where H and L represent the high- and low-refractive-index materials, respectively. The scattering schematic diagram of multilayer coatings with defects is shown in Fig. 1(b), where θi and θs are the incident and scattering polar angles, respectively; ΔSΩ represents the detector aperture used for measurement. In this study, we conducted analysis under normal incidence; thus, the incident angle θi was zero.
Fig. 1. (a) Nodular structure with seed grown from substrate. (b) Scattering caused by multilayer coatings with nodule defect.
The FDTD method is an important approach for the numerical calculation of electromagnetic fields. This technique is commonly used to simulate the electric field distributions and scattering of nodular structures in multilayer coatings, and accurate calculation results can be obtained with good correspondence to actual experiments [18,26,27]. The homemade FDTD software named Gallop was used to calculate the far-field scattering and electric field distributions of the high-reflection multilayer coatings with nodular defects. The size of the simulation region was set as 50 µm × 50 µm × 12.5 µm. The plane wave is incident on the multilayer coatings with nodules along the negative direction of x-axis, and the uniaxial perfectly matched layers (UPMLs) are adopted along x, y and z directions to absorb scattering waves at boundaries. In addition, the total-field/scattered-field technique was used for studying the scattering behavior of nodules. In the simulation, the diameter d of the seed was changed from 0.1 to 3.0 µm, and the geometric constant C of the nodule was 2.5 to simulate nodular structural growth in multilayer coatings by the IBS process. Because the far-field scattering intensity of a nodule was obtained, it could then be used to calculate the angle-resolved scattering (ARS), which reflects the scattering intensity properties at different scattering angles. The ARS is defined as
The total scattering (TS) is the integral of the ARS over the entire hemispherical space and is used to describe scattering loss. In this study, we treated the scattering distribution as axially symmetric and used the simplified formula shown in Eq. (2):
Figure 2 presents the relationship between the calculated TS and seed size; the results show that the scattering loss value caused by nodules in multilayer coatings does not increase monotonically with seed size as we intuitively anticipated, but rather is accompanied by oscillations with maxima and minima. More calculations show that the scattering oscillation properties of the nodular structure remain when the last layer of the multilayer structure is deposited using different materials, and the material effect of the ends layer of multilayer structure on light scattering can be neglected. This property is interesting and has never been reported or discussed in other literature.
Seed size is one of the main factors affecting the scattering of nodules. The relationship between scattering values and seed size contains two components, a linear component that increases with seed size and an oscillatory component that varies with seed size. The variation of TS with seed diameter is summarized in Eq. (3), describing the scattering characteristics of nodules.
where k1 and k2 represents the linear and oscillation parameters. The parameter k1 indicates the increase rate of TS with increasing seed diameter and parameter k2 indicates the characteristic of TS oscillating with the variation of the seed diameter. These two parameters are determined by the nodular structure, the coating material and some other factors.Electric field distributions were calculated for further investigation, which clearly reflected the interaction between the electromagnetic field and nodular structure. Figure 3 shows the Re(z) component of the electric field distribution during scattering, illustrating the amplitude and phase distribution in the nodular structure. The presence of spherical defect seeds deforms the regular stacking of high- and low-refractive-index materials, making the reflection mechanism more complicated and influenced by a variety of parameters. As the dome of the nodules protrudes from the coating surface, a coupling effect is produced during the interaction between the structure and light. A portion of the incident energy is captured and forms a guided wave near the top surface because the energy in the HR multilayer coating is concentrated in the top layers. Therefore, in addition to the microlens-like nodular focusing effect resulting in an enhanced electric field intensity at the center of the nodule, there is another electric field enhancement in the top region of the structure.
Figure 3 depicts the different amplitude and phase distributions at the dome containing the guided waves in nodular structures, which are indicated by the color values. The greater the seed diameter, the greater the curvature of the arc at the top of the nodule, the stronger the focusing effect, and the greater the electric field strength at the center of the structure. In addition, the guided wave near the top surface becomes complex with increasing nodular structure; the electric field intensity increases and exhibits a different phase distribution. The nodules with seed diameters of 0.4, 1.0, 1.6, and 2.1 µm show similar phase distributions in Figs. 3(a)–(d), where scattered waves are superposed coherently in the far field, corresponding to the maximum values of TS. The nodules with seed diameters of 0.7, 1.2, 1.8, and 2.4 µm show similar phase distributions as in Figs. 3(e)–(h), where scattered waves are suppressed in the far field, resulting in the minimum values of TS.
The ARS is an important parameter for describing scattering. It reflects the distribution of scattering at different angles, and the calculated integral value directly corresponds to the TS. Combined with the electric field properties mentioned above, the guided waves near the top surface become more complex, and the electric field strength increases with increasing structure size, which can be directly reflected in the ARS curve. Figure 4 shows the ARS curves of the nodule structure, inducing extreme TS values. Overall comparison and analysis of the ARS curves shows that the number of peaks increases and the central peak of the ARS becomes more concentrated with increasing power as the nodule structure increases, presenting a general trend of increasing TS values with increasing seed diameter.
Fig. 4. Comparison of ARS curves corresponding to different TS extremes, where the red and blue lines indicate the ARS curves for nodule scattering at the TS maxima and minima, respectively.
In each figure, the red and blue curves indicate ARS belonging to the TS maximum and minimum, respectively. It can be observed that the maximum value of TS has a stronger ARS central peak than the minimum value and that the overall ARS curve profile is higher, that is, the red curve is above the blue curve. This phenomenon corresponds to the consistent phase behavior of the different structures shown to produce the same extreme TS values in Fig. 3.
The scattering process can be concluded as follows. Because of the structural characteristics of nodule defects, incident light at the top surface is coupled to form a guide wave. Different seed diameters cause various nodular structures with dissimilar phase distributions, as shown in Fig. 3. When the guide wave of the nodular structure causes the superposition of scattered wave in the far-field, the ARS and TS increase, and when the structure causes the cancellation of scattered waves, the ARS in the far field becomes low, reducing the TS. Consequently, the TS of the increasing nodular structure has an oscillation characteristic different from the intuitive analysis that the scattering value monotonically increases with the size of the nodular structure.
For verification, we designed a special homogeneous nodular structure that maintains the dome with the same dimensions as the original nodule, but all the coating were changed to the same material. Three materials commonly used in coatings were selected to coat the simple structure, and the refractive index relationship between the three materials is tantalum pentoxide (Ta2O5) > hafnium dioxide (HfO2) > silicon dioxide (SiO2). Different homogeneous nodular structures are distinguishable by the size of “invisible seeds,” as the seed has the same material as the film. Figure 5(a) shows the TS of homogeneous structures with respect to the diameter of the “invisible seed”. The scattering results in Fig. 5(a) illustrate that the TS is quite low because of weak reflection from the homogeneous nodular structures compared to normal nodule defects in multilayer coatings. The higher the refractive index of coating, the stronger the reflection, inducing larger scattering values and stronger oscillation properties. The same conclusion can be obtained by comparing the scattering characteristic of single layer structures with different coating materials in Fig. 5(a). Figures 5(b)–(e) shows the electric field distribution in the structures with Ta2O5 material. In such a simplified structure, the internal focusing characteristics are enhanced, and the guided waves caused by the coupling effect of the surface mode on the top of nodule with the incident light still exist, as shown in Figs. 5(b)–(e). The structure that induces the maximum value of TS has the same phase distribution, as shown in Figs. 5(b) and 5(c). The structure that induces the minimum value of TS has the same phase distribution, as shown in Figs. 5(d) and 5(e). By maintaining the guide wave influence through the dome top coupling with incident light, homogeneous nodular structures maintain the far-field scattering oscillation properties, similar to nodular scattering.
Fig. 5. (a) Relationship between TS of a single homogeneous nodular structure and the diameter of the invisible seed of the structure. The red, green, and yellow lines represent the single layer structures coated with Ta2O5 material, SiO2 material, and HfO2 material, respectively. (b)–(e) Electric field distributions in different monolayer structures coated with Ta2O5 material, where the solid lines indicate the outline of the homogeneous nodular structure and the dashed lines indicate the shape and size of the “invisible seed”.
3. Experiments
Artificial nodules were used to measure and characterize the scattering induced by defects experimentally. Because the artificial nodule structure can be well controlled, the measurement results were compared with the simulation results. For accuracy and efficiency, SiO2 microspheres with diameters around the nodule TS extreme values in the simulation (0.5, 0.7, 1.0, 1.2, 1.5, 1.8, 2.0 µm) were chosen as the seeds of the artificial nodule in experiment. A certain density of approximately 100/mm2 of monodisperse SiO2 microspheres were dispersed on clean fused silica substrates with a roughness of approximately 0.2 nm randomly and uniformly. The Ta2O5/SiO2 coating results were consistent with those of the simulation deposited on the substrates using the IBS technique. For the generality of the results, three or more samples were prepared for each seed size under the same experimental conditions, and all samples participated in the same batch of coating process. Figure 6 shows an SEM image of the experimental samples of the artificial nodule.
The ARS of the samples was measured using the light scattering measurement system Albatross-TT as described by von Finck et al. [28]. The measured ARS was compared with the simulation results, as shown in Figs. 7(a)–(d). It can be observed that the ARS curve profiles and the peak positions of the experimental measurements agree well with the simulation results. Figure 7(e) shows the TS values derived from measurements of ARS with different artificial nodules, which are the average values of the samples with the same SiO2 seed diameter. The measured TS results of the artificial nodule clearly reflect the oscillation characteristics of scattering; that is, the TS value increases non-monotonically with the seed size variation owing to the complicated scattering process, which is consistent with the predictions.
Fig. 7. (a)–(d) ARS comparisons between the scattering measurement and simulation of different nodules. (e) Experimental TS results for a single nodule with different seed diameters.
It should be noted that an additional multilayer coating without artificial nodules was prepared as a comparison sample in the same production process. The measured TS of the multilayer coating without artificial nodules is about 10 ppm, indicating that the scattering loss of layer uniformity and coating roughness is very small and negligible compared to that of artificial nodules.
During the experiment, the growth of the artificial nodules could not be controlled as precisely as in the simulations. The inhomogeneity of the layer thickness and the deformation of the layer structure in the artificial nodular structures lead to differences in the scattering values between experiments and simulations. Nevertheless, the experimental results show the oscillation properties that scattering value of nodules does not increase linearly with increasing seed diameter, confirming the anomalous scattering properties of nodules.
4. Conclusion
Compared with the scattering caused by the roughness of the film interface, those of multilayer coatings containing nodular defects were proven to experience higher scattering loss, which significantly limits the performance and development of high-precision optical systems with low-loss requirements. In this study, we examined the scattering characteristics of nodular structures with different seed sizes and showed that the far-field scattering changes non-monotonically with increasing seed size. Further investigation of the electric field distribution in the structure revealed that the coupling effect of the surface mode on the top of the nodule of the incident light produce a complex guided wave, resulting in anomalous scattering. A simplified monolayer structure was proposed to confirm the influence of the waveguide. Furthermore, we established a series of artificial nodules to verify the scattering characteristics experimentally.
This research provides a systematic understanding of nodule scattering in dielectric HR multilayer coatings. In the future, we will continue to focus on the scattering characteristics of nodules and strive to gain a deeper understanding. In addition, we will study the scattering characteristics of nodules in different applications, such as HR multilayers coatings working at oblique incidence or other functional multilayer coatings, and different defect types like pits. On the basis of the systematic understanding of defect scattering, a solid foundation for more targeted guidance on the suppression of film defect scattering loss will be provided in future work.
Funding
National Natural Science Foundation of China (11874285, 61621001, 61925504, 6201101335, 62020106009, 62061136008, 62111530053); Science and Technology Commission of Shanghai Municipality (20JC1414604, 21JC1406100); Shanghai Municipal Education Commission, “Shu Guang” project (17SG22); Innovation Program of Shanghai Municipal Education Commission (2017-01-07-00-07-E00063); Fundamental Research Funds for the Central Universities.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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