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Abstract
Starting from the Rayleigh-Rice perturbation theory, this paper derives the empirical Harvey scatter model and ABg scatter model applied extensively in optical analysis software packages and verifies the shift-invariant behavior of the scattered radiance in direction cosine space. Using data obtained from multi-wavelength laser scatterometer on carbon nanotube black coating and pineblack coating, we establish the polynomial model based on the sine of the scattering angle plus the sine of the specular reflection angle, i.e., sin θs+sin θ0 and the dual-Harvey model based on sin θs-sin θ0 , respectively. The models are in good accordance with the experimental data and further extend the valid range of empirical models.
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1. Introduction
The modeling of scattering phenomena plays a critical role in stray light rejection systems used to view a relatively dim target in a much brighter background [1–4]. Stray light is defined as any light reaching the optical system focal plane that follows an unintended path. It is mainly caused by scattering from roughness, contamination or cosmetic defects in the optical surfaces, or unwanted reflections (ghosts) or diffraction in optomechanical elements [5–8]. Stray light can significantly affect the performance of optical systems, reduce the contrast of the image plane, increase the noise of the system, lead to the resolution reduction, and even cause system failure [9,10]. Therefore, optical scattering measurement and modeling are indispensable for accurate stray light assessment and minimization [11–13].
Approaches for describing the light scattering can be divided into theoretical models and empirical models. Theoretical models make use of certain approximation to Maxwell’s equations to solve the parameters, such as Rayleigh-Rice vector model, Beckmann-Kirchhoff model and Harvey-Shack model [14,15]. When the material surface meets their ranges of validity, theoretical models generally make good predictions for the surface scatter. However, these models have some disadvantages [15–17]: It is difficult to obtain the surface scattering parameters, such as the dielectric constant, power spectral density function to provide material input for the models and it is also not easy to calculate and represent the multiple scattering or bulk scattering from coated surfaces, new material or surface manufacturing process.
When there is insufficient information to develop or apply an accurate theoretical model for surfaces or materials, we can consider trying some technical approaches to establish useful empirical models to meet application requirements. For example, when measuring the scatter from polished and aluminized fused quartz sample, Harvey found that if the scattered radiance was plotted versus sin $\theta _s$-sin $\theta _0$, i.e., sine of the scattering angle minus the sine of the specular reflection angle, the curves for different incident angles coincided almost perfectly. Then he concluded that scattered radiance from well-behaved optical surfaces is shift-invariant in direction cosine space [18]. Based on Harvey’s experimental observation, it is efficient to incorporate large amounts of bidirectional reflectance distribution function data or models into ASAP, TracePro or FRED commercial optical software for stray light analysis [18–20]. In addition to the empirical Harvey model, a three-parameter mathematical model called ABg model for rotationally-symmetric BRDF is also widely used in these software packages. It is worth noting that the empirical Harvey model and ABg model are only applicable to smooth, clean, and front surfaces [21], and may be invalid for rough coated surfaces. To some extent, the lack of theoretical support limits the expansion and application of empirical models.
In this paper, we will present the theoretical derivation of empirical Harvey model and ABg model and compare the scattering measurements with empirical models predictions. Our goal is not only to demonstrate our empirical models have good consistency with the measurement results but also to provide a useful technical approach for scattering analysis when we lack information about the material scattering mechanism to meet the strong need for actual application. Therefore, in section two, we will review the scattering concept and derive two empirical models. In section three, we will introduce the measurement principle of multi-wavelength laser scatterometer and present the instrument signatures at different wavelengths and incident angles. In section four, we will compare and discuss scattering measurement and modeling results for carbon nanotube black coating and pineblack coating which are used for suppressing stray light in optomechanical systems. This paper is closed with the conclusion and the discussion.
2. Theory and definition
Since Nicodemus first introduced bidirectional reflectance distribution function (BRDF) in 1970, this function has been used to describe the hemispherical scatter of material. It is defined as the reflected (or scattered) radiance divided by the incident irradiance [2,21,22]
As one of the most widely used theories for clean, smooth surface, Rayleigh-Rice surface scatter theory is given as [21]
where $Q$ is the reflectivity polarization factor, and $PSD(f_x,f_y)$ is the 2-D power spectral density function. The $PSD(f_x,f_y)$ of real optical surface satisfies the inverse power law and can be described by the modified Lorentzian function, i.e., ABC function [24–26] where $A$ is the magnitude of the $PSD$ at low frequencies, $B$ is related to the roll-off frequency, and $C$ controls the slope of the PSD [23]. Based upon hemispherical grating equations, the spatial frequencies in the $x$ and $y$ directions are [21]And $f$ is expressed as
For isotropic surface, it is usually sufficient to measure the scattering within the incident plane $(\phi _i=\phi _s=0^{\circ })$, then Eq. (5) can be simplified as
Using Eqs. (2), (3), (5) and (6), BRDF can be written as
Equation (7) can be converted to empirical Harvey model or 3-parameter Harvey model [13,27]
In this paper, the incident angle $\theta _i$ and specular reflection angle $\theta _0$ are the same, satisfying $\theta _i=\theta _0$. Similarly, Freniere [28,29] proposed the ABg empirical model, which is expressed as follows:
We use $\mathcal {A}$ and $\mathcal {B}$ in ABg model to distinguish from the parameters of the ABC function. Similarly, $\mathcal {A}$ is the magnitude of the $BRDF$ at low $\left |\vec {\beta }-\vec {\beta _0}\right |$, $\mathcal {B}$ is related to the roll-off $\left |\vec {\beta }-\vec {\beta _0}\right |$, and $g$ controls the slope of BRDF. These three unknown parameters can be determined by the measured data. As shown in Fig. 2, $\vec {\beta }$ and $\vec {\beta _0}$ represent the projected vectors of scattered and specular reflection light onto the sample surface, respectively. They are given by
where $\vec {r}_i$, $\vec {r}_0$, and $\vec {r}_s$ are the unit vectors of incident light, specular reflection light, and scattered light, respectively. When the three vectors are simultaneously in the incident plane, then $\left |\vec {\beta }-\vec {\beta _0}\right |$ reduces to $\left |\sin \theta _s-\sin \theta _0 \right |$, and Eq. (10) can be written asToday, almost all the commercial optical software packages, such as ASAP, TracePro or FRED include these two scatter models to do the ray tracing for stray light analysis. There is a relationship between these two models as follows:
3. Experiment
Figure 3 is the schematic diagram of the multi-wavelength laser scatterometer used in this paper. The beam generated by one of the three lasers (1) operating at 520 nm, 640 nm, and 1064 nm passes through the optical modulator (2), the shutter (3) and the dichroic beam splitter (4) in turn, then it reaches the beam preparation system (5) with a specific frequency. The beam preparation system (5) includes the variable attenuator (6), the aperture (7), the pinhole (8) and the lens group, mainly for beam expansion, shaping, filtering, and coupling. Subsequently, the beam hits the sample (9) at incident angle $\theta _i$ and the scattered light is collected by the detector (10) mounted on the 3-D motion mechanism.
Based upon the reference signal from optical modulator (2), the lock-in amplifier (11) separates the detector signal from the background noise. Finally, BRDF can be obtained after the data processing system. The combination of motion mechanisms achieves the scattering measurement in the entire hemisphere around the sample.
Before the formal measurement, it is necessary to calibrate the power of laser beam by measuring the standard sample (Spectralon, Labsphere, USA). To further verify the instrument performance, the instrument signature is also required by measuring without sample, as shown in Fig. 4, to ensure the scattering caused by the instrument itself.
Figure 5 shows the instrument signatures at 520 nm, 640 nm and 1064 nm when the incident angle $\theta _i$ is zero. Due to the absence of sample on the fixture, the incident beam passes directly through the sample fixture to the surface of the beam trap, where it is reflected and received by the detection system. This part of the beam corresponds to the dashed line in Fig. 5, which is the specular reflection beam or backscattering from the beam trap, but does not belong to background noise.
When the sample is placed for formal measurement, the beam trap will not be directly illuminated, or will be illuminated after multiple reflections, and the corresponding energy has already decayed to the point where it can be ignored. Drop in the peak around the scattering angle of 0$^{\circ }$ is caused by the obscuration of the detector [30]. Rayleigh scattering from the laboratory atmosphere, residual scattering light from the instrument housing and the electronic noise from the system are the possible reasons for the generation of residual signal [31]. Among these three wavelengths, the shortest wavelength 520 nm has the highest background scattering. Typical instrument signatures measured at different incident angles with the wavelength 640 nm are shown in Fig. 6. The background noise at different incident angles exhibits similar "specular peak" and "peak drop" phenomena. Based upon the analysis above, the scatterometer has a maximum background noise of 10$^{-6}$ sr$^{-1}$ at the three different wavelengths when the incident angle ranged from 0$^{\circ }$ to 75$^{\circ }$.
4. Results and discussion
Blackening the surfaces of the optical systems is one of the common methods to control stray light. Carbon nanotube and pineblack as two effective coatings are frequently used to reduce scatter and suppress stray light by absorbing the unwanted light. We choose two kinds of samples for the scattering measurement, one is the aluminum substrate coated with carbon nanotube and the other is K9 glass substrate coated with pineblack. The measurement is conducted within the incident plane. The incident angle is from 0$^{\circ }$ to 75$^{\circ }$, and the scattering light is received by the moving detector ranged from −75$^{\circ }$ to 75$^{\circ }$, with an interval of 1$^{\circ }$. The incident wavelengths are 640 nm and 1064 nm, respectively.
4.1 Carbon nanotube black coating
Figure 7(a) shows the raw BRDF measurement data of the aluminum substrate coated with carbon nanotube at the wavelength of $\lambda$ = 640 nm for incident angles of 0$^{\circ }$, 15$^{\circ }$, 30$^{\circ }$, 45$^{\circ }$, 60$^{\circ }$, 70$^{\circ }$ and 75$^{\circ }$. The BRDF value of this sample is much greater than 10$^{-6}$ sr$^{-1}$, so the background noise can be ignored. The smoothness of the curves is mainly related to the manufacturing process of the substrate. Compared to the positive scatter angles, all the curves of negative scatter angles are relatively flat. There is no specular peak for this sample and all the BRDF curves gradually increase as the scattering angle increases. Notice that for the normal incidence, this material is closer to Lambertian scatter which represents the perfect diffuse reflectance. The BRDF function satisfies [32],
where $\rho _{d}$ is the diffuse albedo. The BRDF of Lambertian model is a constant and it does not vary as a function of incident or scatter angle.
Fig. 7. Measured data plotted versus (a) scattering angle $\theta _s$ and (b) |sin $\theta _s$-sin $\theta _{0}$|.
Therefore, we calculate the average BRDF at this incident angle as the approximate Lambertian model for comparison with other scattering model,
According to Harvey’s experimental observation, the BRDF data can be plotted as the function of |sin $\theta _s$-sin $\theta _0$|, i.e., the sine of the scattering angle minus the sine of the specular reflection angle, as shown in Fig. 7(b). The BRDF curve at the same incident angle splits in two, one is the forward scatter, and the other is the backward scatter. The forward and backward scatter approximately coincide at the incident angle of 0$^{\circ }$. As the incident angle increases, the two curves become more distinct.
Due to the good coincidence of the curves at negative scatter angles, there will be the tendency to overlap if they move to the right. So we re-plot the data as the function of |sin $\theta _s$+sin $\theta _0$|, i.e., the sine of the scattering angle plus the sine of the specular reflection angle, as shown in Fig. 8(a). It can be seen that the scattering curves at different incident angles can basically overlap, and the ratio of the two farthest points on the y-axis is less than 2.5. This result is also consistent with the shift-invariant behavior of the scattered radiance or BRDF in direction cosine space. We can get the mean curve by performing 1-D interpolation on discrete BRDF data, as the black dotted line shown in Fig. 8(a). Then the mean curve is fitted using the polynomial method, as shown in Fig. 8(b). The polynomial function expression is as follows
Fig. 8. (a) Measured BRDF data plotted versus |sin $\theta _s$+sin $\theta _{0}$| and (b) the polynomial fit to the mean curve.
The measured data of this material includes 2 orders of magnitude, so the relative root mean square error (RRMSE) is defined to evaluate the deviation of the model from the measured data:
4.2 Pineblack coating
Figure 10(a) shows the raw BRDF measurement data of the K9 glass substrate coated with pineblack at the wavelength of $\lambda$ = 1064 nm for incident angles of 0$^{\circ }$, 15$^{\circ }$, 30$^{\circ }$, 45$^{\circ }$, 60$^{\circ }$ and 75$^{\circ }$. The BRDF value of this sample is much greater than 10$^{-6}$ sr$^{-1}$, so the background noise can also be ignored. The smoothness of the scattering curves of this sample is better than the previous one. The reason is that the K9 glass substrate is smoother than the aluminum substrate. It even exhibits the usual specular peak of smooth surfaces near the direction of specular reflection, but the peak is not sharp enough. Strictly speaking, there is fairly narrow diffuse scatter around the specular direction, but no obvious specular beam. Furthermore, the BRDF near the specular direction gradually increases as the incident angle increases, because there is more absorption at small incident angles. Note that the BRDFs tend to increase at large scattering angles rather than the usual down-turn behavior. BRDFs tend to reduce according to the generalized Harvey-Shack theory and the Rayleigh-Rice theory [2]. At present, the majority [2,23,33] believes that it is caused by surface particulates or subsurface (non-topographic) scatter.
Fig. 10. Measured data plotted versus (a) scattering angle $\theta _s$ and (b) sin $\theta _s$-sin $\theta _{0}$.
When the BRDF data is plotted versus sin $\theta _s$-sin $\theta _0$, all the curves of smaller incident angles are approximately enveloped within the scattering curve of 75$^{\circ }$ incident angle, as illustrated in Fig. 10(b). We believe that there is probably a certain degree of diffuse reflection on the surface, but it is still dominated by the specular reflection. Therefore, the scattering curve at each incident angle is fitted by the dual-Harvey model, that is, the sum of two empirical Harvey model. The expressions are as follows:
Table 1. The fitting parameters for different incident angles.
The measured data of this material includes 4 orders of magnitude, so we still use the RRMSE to evaluate the deviation of the model from the measured data. Figure 11 gives the RRMSE as a function of incident angles and the single Harvey model is also added for comparison. It can be seen that RRMSE of the dual-Harvey model is less than 0.35 and the fitting result is better than that of single Harvey model.
Figure 12 shows the individual fittings for the measured data at different incident angles. The predicted results of the dual-Harvey model are more consistent with the measured data, especially for forward scattering. Thus, the dual-Harvey model does a good job of fitting the measured BRDF data.
Fig. 12. Individual fittings for the measured data at different incident angles (a) $\theta _i=0^{\circ }$, (b) $\theta _i=15^{\circ }$, (c) $\theta _i=30^{\circ }$, (d) $\theta _i=45^{\circ }$, (e) $\theta _i=60^{\circ }$, and (f) $\theta _i=75^{\circ }$.
5. Conclusion
In order to meet the demands of surface scattering modeling for stray light analysis, this paper establishes two empirical models by some useful technical approaches, making up for the shortcomings of the theoretical models with too many parameters, complex computational processes and expressions. Based on the Rayleigh-Rice vector perturbation theory, this paper firstly theoretically verifies the shift-invariant behavior of the scattered radiance in direction cosine space and proves "sin $\theta _s$-sin $\theta _0$" is proportional to the spatial frequency by deriving the empirical Harvey model and the ABg model. The equipment used in this experiment is the multi-wavelength laser scatterometer, and the performance of the scatterometer at different wavelengths and incident angles is analyzed by measuring the instrument signatures, i.e., without sample, before making a formal measurement. Within our measurement requirements, the background noise is lower than 10$^{-6}$ sr$^{-1}$, which has no effect on the measurement result of the samples. And finally, the polynomial model based on the "sin $\theta _s$+sin $\theta _0$" is used to fit the aluminum substrate coated with carbon nanotube, and the RRMSE is less than 0.15 which is better than that of the purely Lambertian model. The dual-Harvey model based on "sin $\theta _s$-sin $\theta _0$" is used to fit the K9 glass substrate coated with pineblack and the RRMSE is less than 0.35 which is better than that of the single Harvey model. Both of the models are in good agreement with the measured data.
Through the analysis of this paper, the empirical Harvey model is extended from the smooth surface to the rough coating surface and it provides the useful technical methods for the scattering modeling of complex or unknown materials. At the same time, the empirical models can be further applied to the optical analysis software packages, which has great significance for improving the accuracy of stray light analysis.
Funding
National Key Research and Development Program of China (2021YFC2202100, 2021YFC2202104, 2021YFC2203501).
Acknowledgments
Author Zhanpeng Ma thanks Dr. Tuochi Jiang for useful discussions.
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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