Harvey–Shack theory for a converging–diverging Gaussian beam

Zhanpeng Ma; Zhanpeng Ma; Zhanpeng Ma; Zhanpeng Ma; Poul-Erik Hansen; Hu Wang; Hu Wang; Mirza Karamehmedović; Qinfang Chen

doi:10.1364/JOSAB.478801

1. INTRODUCTION

Surface scattering phenomena is an important topic in diverse areas of science and engineering, especially in optical applications [1–4]. Researchers have investigated the prediction of the scattered light distribution from random rough surfaces and proposed various theories and methods [5–13]. The Phong and Ward models [5], for example, are phenomenological models with a small number of parameters that can be manipulated interactively to achieve the appearance of an object. In [6], the Torrance–Sparrow model was analyzed on the basis of geometrical optics, in which the roughened surfaces consist of specular v-grooves. The model has been extended to the Cook–Torrance model [7] to distinguish between metals and dielectrics. These models, as well as the Monte Carlo method and the adding–doubling method, have been widely used in computer graphics [8].

Another important category is physics-based modeling of the light–matter interaction using rigorous numerical methods based on Maxwell’s equations and approximate analytical models. Rigorous numerical methods include rigorous coupled-wave analysis, the finite-difference time-domain, and the finite-element method [3,8]. Approximate analytical models consist of the Rayleigh–Rice (RR) vector perturbation model, the Beckmann–Kirchhoff (BK) model, and the generalized Harvey–Shack (GHS) model. The application of the RR theory to random rough surfaces was studied in [9]; it was found that it is valid for ideal, clean, smooth surfaces. However, not all applications of interest satisfy the smooth-surface criterion. The authors in [10] applied the BK scattering theory to random rough surfaces with low and medium roughness and found that it agrees well with experimental measurements for a wide range of scattering and incident angles, but it does not agree with experiments for large incident and scattering angles. GHS scatter theory complements the applicability of each of the above theoretical treatments in which the scattering behavior is characterized by the surface transfer function [14,15]. Based upon Harvey’s experimental observation that scattered radiance is shift-invariant in direction cosine space, Freniere [16,17] presented the ABg empirical scattering model, which was applied to TracePro, FRED, and other stray light analysis software. However, the GHS theory has limitations. First, in all of these analyses, the incident beam is assumed to be a plane wave. To the authors’ knowledge, no paper mentions the converging–diverging Gaussian beam when using the GHS theory. Second, the GHS theory is based on scalar theory and cannot explain the polarization effect of scattered light. Finally, when the incident angle and scattering angle are large, the analysis results are not accurate [18].

Therefore, in this paper, the Gaussian beam Harvey–Shack (GBHS) scatter theory is derived by applying the conventional GHS theory in conjunction with 2D plane-wave decomposition (PWD) of the 3D converging–diverging Gaussian beam. Based on the specular reflectance measured at normal incidence, we calculate the polarization factor $Q$ for the $s$-polarized source and $s$-sensitive receiver and also use this factor in the GBHS theory to obtain more accurate and reasonable results.

2. CONVERGING–DIVERGING GAUSSIAN BEAM

As shown in Fig. 1, the converging–diverging spherical Gaussian beam with linear polarization angle ${\Psi _E}$ and wave vector ${\boldsymbol k}$ propagates along the $z^\prime $ direction of the beam coordinate system $({x^\prime},{y^\prime},{z^\prime})$ and passes through the sample surface in the coordinate system $(x,y,z)$ at any incident angle ${\theta _i}$ and any azimuth angle ${\phi _i}$, from the incident region with the refractive index ${n_I}$ to the transmission region with the refractive index ${n_s}$. The scattering zenith and azimuth angles are ${\theta _s}$ and ${\phi _s}$, respectively.

Fig. 1. Geometry of the sample surface illuminated by the converging–diverging Gaussian beam.

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The incident converging–diverging Gaussian beam in the beam-coordinate system $(x^\prime ,y^\prime ,z^\prime)$ can be represented as [19]

(1)$$\begin{split} {{{\textbf E}_{{\rm inc}}}}& ={{{\left[{\frac{{{w_{0{x^\prime}}}}}{{{w_{{x^\prime}}}({{z^\prime}} )}}} \right]}^{1/2}}{{\left[{\frac{{{w_{0{y^\prime}}}}}{{{w_{{y^\prime}}}({{z^\prime}} )}}} \right]}^{1/2}}\exp\! \left\{{- \left[{{{\left({\frac{{{x^\prime}}}{{{w_{{x^\prime}}}({{z^\prime}} )}}} \right)}^2}} \right.} \right.}\\&\quad+{\left. {\left. { {{\left({\frac{{{y^\prime}}}{{{w_{{y^\prime}}}({{z^\prime}} )}}} \right)}^2}} \right]} \right\}\exp\! \left\{{- j\frac{1}{2}k\left[{\frac{{{x^{^\prime 2}}}}{{{R_{{x^\prime}}}({{z^\prime}} )}} + \frac{{{y^{^\prime 2}}}}{{{R_{{y^\prime}}}({{z^\prime}} )}}} \right]} \right\}}\\&\quad\times {\exp\! \left\{{j\frac{1}{2}\left[{\mathop {\tan}\nolimits^{- 1} \!\left({\frac{{{z^\prime}}}{{{z_{0{x^\prime}}}}}} \right) + \mathop {\tan}\nolimits^{- 1} \!\left({\frac{{{z^\prime}}}{{{z_{0{y^\prime}}}}}} \right)} \right]} \right\}}\\&\quad\times {\exp\! \left({- jk{z^\prime}} \right)\hat e = {E^{{\rm inc}}}\hat e,}\end{split}$$

where $\hat e$ is the polarization unit vector of the central beam, and ${w_{0{u^\prime}}}$ is the beam waist radius in the ${u^\prime}({u^\prime} = {x^\prime}$) or (${y^\prime})$ direction. The beam radius ${w_{{u^\prime}}}({z^\prime})$ is

(2)$${w_{{u^\prime}}}({{z^\prime}} ) = {w_{0{u^\prime}}}{\left[{1 + {{\left({\frac{{{z^\prime}}}{{{z_{0{u^\prime}}}}}} \right)}^2}} \right]^{1/2}},$$

where ${R_{{u^\prime}}}$ is the beam radius of curvature of the phase front, and ${z_{0{u^\prime}}}$ is the Rayleigh range in the ${u^\prime}$ direction. They are

(3)$${R_{{u^\prime}}}({{z^\prime}} ) = {z^\prime}\left[{1 + {{\left({\frac{{{z_{0{u^\prime}}}}}{{{z^\prime}}}} \right)}^2}} \right],$$

(4)$${z_{0{u^\prime}}} = \frac{{\pi {n_I}}}{{{\lambda _0}}}w_{0{u^\prime}}^2.$$

The length of the incident wave vector is

(5)$$k = \frac{{2\pi {n_I}}}{\lambda},$$

where $\lambda$ is the incident wavelength. For the central beam specified by the polarization angle ${\Psi _E}$, the polarization unit vector $\hat e$ is expressed as

(6)$$\begin{split}{\hat e}& = {{e_x}\hat x + {e_y}\hat y + {e_z}\hat z = ({\cos {\Psi _E}\cos \phi {_i}\cos {\theta _i} - \sin {\Psi _E}\sin {\phi _i}} )\hat x}\\&\quad+ ({\cos {\Psi _E}\sin {\phi _i}\cos {\theta _i} + \sin {\Psi _E}\cos {\phi _i}} )\hat y - ({\cos {\Psi _E}\sin {\theta _i}} )\hat z.\end{split}$$

Figure 2(a) is the corresponding configuration of a converging–diverging Gaussian beam at an arbitrary incidence angle ${\theta _i}$ and at azimuthal angle ${\phi _i}={ 0^ \circ}$ focused on the input surface. Figure 2(b) is the 3D converging–diverging Gaussian beam.

Fig. 2. Geometry of the converging–diverging Gaussian beam. (a) 2D. (b) 3D.

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When the beam is at normal incidence, the beam coordinate system $({x^\prime},{y^\prime},{z^\prime})$ coincides with the sample surface coordinate system $(x,y,z)$. Combining Eqs. (1)–(4), the incident amplitude at $z = 0$ is

(7)$${E_{\rm{inc}}}(x,y,z = 0) = \exp\! \left\{{- \left[{{{\left({\frac{x}{{{w_{0x}}}}} \right)}^2} + {{\left({\frac{y}{{{w_{0y}}}}} \right)}^2}} \right]} \right\},$$

where ${w_{0x}}$ and ${w_{0y}}$ are the incident beam waist radii along the $x$ and $y$ directions, respectively.

3. GAUSSIAN BEAM HARVEY–SHACK SCATTER THEORY

Based on the 2D plane-wave decomposition (PWD) of the 3D converging–diverging Gaussian beam, we derived the Gaussian beam Harvey–Shack (GBHS) scatter model, which can be used for the general case of an obliquely incident Gaussian beam at an arbitrary incidence angle ${\theta _i}$ and at an arbitrary azimuthal angle ${\phi _i}$. In addition, the introduction of the reflectivity polarization factor $Q$ can effectively improve the accuracy of the GHS model; further, in the smooth surface approximation, we can obtain the power spectral density (PSD) function directly from the bidirectional reflectance distribution function (BRDF).

A. Two-Dimensional Plane-Wave Decomposition

The angular spectrum $F({k_x},{k_y})$ is obtained as the Fourier transform of the complex amplitude distribution ${E_{\rm{inc}}}$ [19,20]. The Fourier transformation may replace the angular spectrum theory in the vicinity of the Gaussian beam waist radius since the $z$-component of the wave vector is almost equal to the wavenumber ($k$). Consequently, a Gaussian beam can be expressed as a superposition of many plane waves by the inverse Fourier transformation, which is called the “angular spectrum of plane waves” [21]. The process can be expressed as

(8)$$\begin{split}{E_{\rm{inc}}}(x,y,z) &= \int_{- \infty}^\infty \int_{- \infty}^\infty F({k_x},{k_y})\\&\quad\times\exp\! \left[{j({k_x}x + {k_y}y + {k_z}z)} \right]{\rm d}{k_x}{\rm d}{k_y},\end{split}$$

where ${k_u}$ is the wave-vector component along the $u$ ($u = x$, $y$ or $z$) direction.

The 2D PWD [19] is to determine the plane-wave spectrum of the incident Gaussian beam by applying a 2D discrete Fourier transform. Since the incident electric field needs to be represented as a discrete set of plane waves in our implementation, we rewrite Eq. (8) as a discrete transformation. If the number of sampling points, along the direction $u$ ($u = x$ or $y$), over the interval ${-}{u_{{\max}}} \le u \le {u_{{\max}}}$ is ${M_u}$, then the incident amplitude at $z = 0$ can be expressed in terms of its plane-wave spectrum as

(9)$$\begin{split}{{E_{\rm{inc}}}(x,y,z = 0)}&= {\sum\limits_{{m_x} = - {M_x}/2}^{{M_x}/2 - 1} \sum\limits_{{m_y} = - {M_y}/2}^{{M_y}/2 - 1} F\big({{k_{x,{m_x}}},{k_{y,{m_y}}}} \big)}\\&\quad\times{\exp \big[{j\big({{k_{x,{m_x}}}x + {k_{y,{m_y}}}y} \big)} \big],}\end{split}$$

where ${k_{u,{m_u}}} = {m_u}{\Delta _k}$ is the wave-vector component along the $u$ ($u = x$ or $y$) direction, and ${\Delta _k}$ is the sampling spacing. For the $({m_x},{m_y})$ subbeam, the $z$ wave-vector component can be expressed as

(10)$${k_{z,{m_x},{m_y}}} = {\left({{k^2} - k_{x,{m_x}}^2 - k_{y,{m_y}}^2} \right)^{1/2}}.$$

Therefore, as shown in Fig. 2(b), each propagating component of the $({m_x},{m_y})$ subbeam can be specified by an incident angle, azimuthal angle, and plane-wave angular spectrum with the application of 2D PWD. Analytical integration of the normally incident Gaussian beam mode ${{\rm TEM}_{00}}$ in Eq. (7) is given by Eq. (11) and in Supplement 1 for higher-order modes:

(11)$$\begin{split} & F\!\left({{k_{x,{m_x}}},{k_{y,{m_y}}}} \right)\\& = \frac{{{w_{0x}}{w_{0y}}}}{{4\pi}}\exp\! \left\{{- \left[{{{\left({\frac{{{w_{0x}}}}{2}{k_{x,{m_x}}}} \right)}^2} + {{\left({\frac{{{w_{0y}}}}{2}{k_{y,{m_y}}}} \right)}^2}} \right]} \right\}.\end{split}$$

Parseval’s theorem tells us that the integral over real space is equal to the integral over Fourier space:

(12)$$\begin{split}&{\int_{- \infty}^\infty \int_{- \infty}^\infty {{\left| {{E_{\rm{inc}}}(x,y,z = 0)} \right|}^2}{\rm d}x{\rm d}y}\\&={ (2\pi {)^2}\int_{- \infty}^\infty \int_{- \infty}^\infty {{\left| {F\!\left({{k_{x,{m_x}}},{k_{y,{m_y}}}} \right)} \right|}^2}{\rm d}{k_x}{\rm d}{k_y}.}\end{split}$$

Clearly, Eqs. (7) and (11) are bounded such that they are zero outside the intervals $[- {x_{{\max}}},{x_{{\max}}}] \times [- {y_{{\max}}},{y_{{\max}}}]$ and $[- {k_{x,\max}},{k_{x,\max}}] \times [- {k_{y,\max}},{k_{y,\max}}]$. The positions ${u_{{\max}}}$ and ${k_{u,\max}}$ ($u = x$ or $y$) are found from numerical integration. We will be using the sampling spacing $\Delta$ and ${\Delta _k}$ along the two directions in real space and $k$-space. The Nyquist–Shannon sampling theorem, for one direction, requires that the following relation between the sampling spacing $\Delta$, ${\Delta _k}$ and ${u_{{\max}}}$, ${k_{u,\max}}$ are satisfied:

(13)$${\frac{\pi}{\Delta} \le {k_{u,\max}}}\quad{\frac{\pi}{{{\Delta _k}}} \le {u_{{\max}}}.}$$

In this work, we choose ${\Delta _k} = \pi /{u_{{\max}}}$. Figure 3 shows the normalized weight $({F({{k_{x,{m_x}}},{k_{y,{m_y}}}})})/{\rm max}({F({{k_{x,{m_x}}},{k_{y,{m_y}}}})})$ for the angular spectrum in Eq. (11).

Fig. 3. Normalized weight of plane waves in the angular spectrum for ${M_x} = {M_y} = 40$.

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The bidirectional reflectance distribution function (BRDF) defined by Nicodemus [22,23] as the differential surface radiance ($dL$) divided by the incident differential surface irradiance ($dE$),

(14)$${\rm BRDF} \equiv \frac{{dL}}{{dE}} \cong \frac{{d{P_s}/d{\Omega _s}}}{{{P_i}\cos {\theta _s}}} \cong \frac{{{P_s}/{\Omega _s}}}{{{P_i}\cos {\theta _s}}},$$

where $d{P_s}$ is the differential scattered power scattered through the differential projected solid angle $d{\Omega _s}$, ${P_i}$ is the incident power and ${\theta _s}$ is the scattering angle. Based on the definition of BRDF, the GBHS scatter model can be derived by combining the 2D PWD results for all subbeams, so the BRDF for the Gaussian beam in Eq. (7) can be derived as [19,24,25]

(15)$$\begin{split} {{{\rm BRDF}}_{{\rm Gau}}} & = \sum\limits_{{m_x}} \sum\limits_{{m_y}} \frac{{d{P_{s,{m_x},{m_y}}}/d{\Omega _{s,{m_x},{m_y}}}}}{{{P_i}\cos ({\theta _s})}}\\&={ \frac{{\sum\nolimits_{{m_x}} {\sum\nolimits_{{m_y}} {{{\left| {F({{k_{x,{m_x}}},{k_{y,{m_y}}}} )} \right|}^2}{\rm Re}({k_{z,{m_x},{m_y}}^*} ) \cdot {{{\rm BRDF}}_{{\rm GHS}}}}}}}{{\sum\nolimits_{{m_x}} {\sum\nolimits_{{m_y}} {{{\left| {F({{k_{x,{m_x}}},{k_{y,{m_y}}}} )} \right|}^2}{\rm Re}({k_{z,{m_x},{m_y}}^*} )}}}},}\end{split}$$

where the ${{\rm BRDF}_{{\rm GHS}}}$ represents the GHS model, and $d{P_{s,{m_x},{m_y}}}$ is the differential scattered power scattered by the subbeam $({m_x},{m_y})$ through the differential projected solid angle $d{\Omega _{s,{m_x},{m_y}}}$. For the Gaussian beam obliquely incident at the arbitrary angle of incidence ${\theta _i}$ and azimuth ${\phi _i}$, the corresponding wave vector can be given by

(16)$${{\textbf k}^\prime} = {{\textbf R}_{{\theta _i}}}{{\textbf R}_{{\phi _i}}}{\textbf k},$$

where

(17)$${{\textbf R}_{{\theta _i}}} = \left[{\begin{array}{*{20}{c}}{\cos {\theta _i}}&\;\;0&\;\;{\sin {\theta _i}}\\0&\;\;1&\;\;0\\{- \sin {\theta _i}}&\;\;0&\;\;{\cos {\theta _i}}\end{array}} \right],$$

(18)$${{\textbf R}_{{\phi _i}}} = \left[{\begin{array}{*{20}{c}}{\cos {\phi _i}}&\;\;{\sin {\phi _i}}&\;\;0\\{- \sin {\phi _i}}&\;\;{\cos {\phi _i}}&\;\;0\\0&\;\;0&\;\;1\end{array}} \right].$$

${\textbf k}$ represents the wave vector of the normally incident beam. Using Eqs. (10)–(18), we can obtain the general case of the GBHS scatter model.

B. Generalized Harvey–Shack Scatter Theory

After modification of the random phase variation in the original theory [11,15], the GHS scatter theory can make the scattering prediction for large incident and scattering angles. The surface transfer function is [15,26]

(19)$${H_s}\big({\hat x,\hat y;{\gamma _i},{\gamma _s}} \big) = \exp\big\{{- {{\big[{2\pi {{\hat \sigma}_s}({{\gamma _i} + {\gamma _s}} )} \big]}^2}\big[{1 - {{\hat C}_s}(\hat x,\hat y)/\hat \sigma _s^2} \big]} \big\},$$

where

(20)$${\gamma _i} = \cos {\theta _i},\quad{\gamma _s} = \cos {\theta _s}.$$

${\theta _i}$ and ${\theta _s}$ are the incident and scattering angles, respectively. All variables $x$, $y$, rms roughness ${\sigma _s}$, and autocovariance function ${C_s}(x,y)$ are scaled by the wavelength. Equation (19) can be written in the following form:

(21)$${H_s}\!\left({\hat x,\hat y;{\gamma _i},{\gamma _s}} \right) = A({{\gamma _i},{\gamma _s}}) + B({{\gamma _i},{\gamma _s}})G\!\left({\hat x,\hat y;{\gamma _i},{\gamma _s}} \right),$$

where

(22)$$A({{\gamma _i},{\gamma _s}}) = \exp\! \left\{{- {{\left[{2\pi ({{\gamma _i} + {\gamma _s}} ){{\hat \sigma}_s}} \right]}^2}} \right\},$$

(23)$$B({{\gamma _i},{\gamma _s}}) = 1 - \exp\! \left\{{- {{\left[{2\pi ({{\gamma _i} + {\gamma _s}} ){{\hat \sigma}_s}} \right]}^2}} \right\},$$

(24)$$G\big({\hat x,\hat y;{\gamma _i},{\gamma _s}} \big) = \frac{{\exp\big\{{{{\big[{2\pi ({{\gamma _i} + {\gamma _s}} )} \big]}^2}\hat C(\hat x,\hat y)} \big\} - 1}}{{\exp \big\{{{{\big[{2\pi ({{\gamma _i} + {\gamma _s}} ){{\hat \sigma}_s}} \big]}^2}} \big\} - 1}}.$$

If we take the Fourier transform of Eq. (21), we can obtain the angle spread function (ASF) as the sum of a specular response, $A$, and a scattering function $S$:

(25)$${\rm ASF}\!\left({{\alpha _s},{\beta _s}} \right) = {\left. {\left[{A\delta \!\left({\alpha - {\alpha _o},\beta} \right) + S\!\left({\alpha - {\alpha _o},\beta} \right)} \right]} \right|_{\alpha = {\alpha _s},\beta = {\beta _s}}}.$$

The scattering function is given by

(26)$$S\!\left({\alpha - {\alpha _o},\beta} \right) = B{\cal F}\left\{{G\!\left({\hat x,\hat y;{\gamma _i},{\gamma _s}} \right)\exp\! \left({- i2\pi {\alpha _o}\hat x} \right)} \right\},$$

where

(27)$${\alpha _0} = \sin {\theta _o},{\theta _o} = - {\theta _i},$$

(28)$$\alpha = \sin \theta \cos \phi ,$$

(29)$$\beta = \sin \theta \sin \phi .$$

${\cal F}$ represents the Fourier transform. BRDF is the product of ASF and scalar reflectance $R$:

(30)$${\rm BRDF}\!\left({{\theta _i},{\phi _i},{\theta _s},{\phi _s}} \right) = R \cdot {\rm ASF}\!\left({{\alpha _s},{\beta _s};{\gamma _i},{\gamma _s}} \right).$$

The rms roughness limit for the validity of the smooth surface approximation has been investigated in [27,28]. The authors studied the difference between the total integrated scatter expression and its first-order Taylor expansion from which the usual criteria, $g = (4\pi \sigma \cos {\theta _i}/\lambda) \ll 1$, originated and reported the wavelength-dependent criteria of the smooth surface approximation to $g \lt 0.365$ at 250 nm and $g \lt 0.107$ at 850 nm. After the approximation to Eqs. (22)–(24), the BRDF of GHS model for the smooth surface is expressed as [26,29]

(31)$${\rm BRDF} = R\frac{{4{\pi ^2}}}{{{\lambda ^4}}}{\left({\cos {\theta _i} + \cos {\theta _s}} \right)^2}{\rm PSD}{\left({{f_x},{f_y}} \right)_{2 - D}},$$

where $R$ is the scalar reflectance, and ${\rm PSD}{({{f_x},{f_y}})_{2 - D}}$ is the 2D power spectral density function. However, the reflectance $R$ in Eq. (31) is valid only when the scatter is in-plane and surface reflectance or transmittance is not a strong function of the angle of incidence or polarization [30]. After replacing the $R$ in Eq. (31) with the polarization factor $Q$, a more accurate expression can be obtained,

(32)$${{\rm BRDF}_{{\rm GHS}}} = Q\frac{{4{\pi ^2}}}{{{\lambda ^4}}}{\left({\cos {\theta _i} + \cos {\theta _s}} \right)^2}{\rm PSD}{({{f_x},{f_y}} )_{2 - D}},$$

where $Q$ represents the reflected Fresnel reflectivity of the incident light by the sample. It is dimensionless and is also a function of the sample dielectric constant $\varepsilon$ plus the angles of incidence and scatter; it also takes on different forms, depending on incident and scattered polarization states [22,26,30]. For an $s$-polarized source and an $s$-sensitive receiver, the individual expression for the $Q$ is [22]

(33)$$Q = {\left| {\frac{{(\varepsilon - 1)\cos {\phi _s}}}{{\left({\cos {\theta _i} + \sqrt {\varepsilon - \mathop {\sin}\nolimits^2 {\theta _i}}} \right)\left({\cos {\theta _s} + \sqrt {\varepsilon - \mathop {\sin}\nolimits^2 {\theta _s}}} \right)}}} \right|^2},$$

where $\theta$ and $\phi$ are the elevation and azimuth angles, subscripts $i$ and $s$ are used to denote incident and scattered rays, respectively. The relative dielectric constant $\varepsilon$ is the square of the refractive index $n$. Based on the Fresnel equations, the relationship between the specular reflectance and the refractive index at normal incidence is as follows:

(34)$$R = {\left({\frac{{n - 1}}{{n + 1}}} \right)^2}.$$

The smooth surface approximation of the GBHS formula is obtained by inserting Eq. (32) into Eq. (15) and by taking the incident angle as the subbeam angle ${\theta _{{m_x},{m_y}}}$ given by

(35)$${\theta _{{m_x},{m_y}}} = {\rm a \cos}\left({\frac{{{k_{z,{m_x},{m_y}}}}}{k}} \right).$$

C. Inverse Scattering for Smooth Surfaces

In our context, inverse scattering refers to estimating the surface topography from the BRDF data [31]. According to Eq. (32), we can also obtain the PSD from the measured BRDF data:

(36)$${\rm PSD}{\left({{f_x},{f_y}} \right)_{2 - D}} = \frac{{{\lambda ^4}}}{{4{\pi ^2}}}\frac{{{\rm BRDF}}}{{{{({\cos {\theta _i} + \cos {\theta _s}} )}^2}Q}}.$$

The modified Lorentzian function or K-correlation function has proven to be the best-fit function for the 2D PSD of the smooth surface [32,33]:

(37)$${\rm PSD}{(f)_{2 - D}} = K\frac{{AB}}{{{{\left[{1 + {{(Bf)}^2}} \right]}^{(C + 1)/2}}}},$$

where

(38)$$K = \frac{1}{{2\sqrt \pi}}\frac{{\Gamma [(C + 1)/2]}}{{\Gamma (C/2)}},$$

and A, B, and C are the fitting parameters. Also

(39)$$f = \sqrt {f_x^2 + f_y^2} ,$$

where

(40)$${f_x} = \frac{{\sin {\theta _s}\cos {\phi _s} - \sin {\theta _i}}}{\lambda},{f_y} = \frac{{\sin {\theta _s}\sin {\phi _s}}}{\lambda}.$$

4. COMPARISON OF SCATTER MEASUREMENT AND MODELING PREDICTION

To verify the accuracy of the GBHS model, we measured the BRDF of a set of mirrors, calculated the PSD from the BRDF, and compared the ability of the GHS and GBHS scattering models to reproduce the BRDF data.

A. PSD Calculation and Fitting

The black mirror with rms roughness 0.78 nm is the test sample, and a 0.64 µm $s$-polarized laser is used for the BRDF measurement. The reflected light from the sample surface can be received by the moving detector when the reflection angle ranged from $ - 85^\circ$ to 85°, with an interval of 1° [29]. After measuring the standard sample (Spectralon, Labsphere, USA), the accuracy of the equipment is better than 2%. The measurement is mainly affected by the detector, laser, mechanical scanning structure, and air scattering. The overall standard uncertainty of the system is 5.83%. The stated uncertainty does not include the contributions from aperture misalignment and solid angle error. Figure 4(a) shows the scatterometer system we used; Fig. 4(b) is the measurement result at different angles of incidence (AOI): 5°, 15°, 30°, 45°, 60°, and 75°.

Fig. 4. (a) Scatterometer system. (b) Measurement results with different AOIs.

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This sample satisfies the smooth surface criterion, so the smooth approximation GHS model in Eq. (32) is applicable to this sample. When the incident wavelength $\lambda$ is 0.64 µm, the specular reflectance $R$ measured at normal incidence is 5.815%. Thus, the reflectivity polarization factor $Q$ in the plane of incidence [$\sin (\phi) = 0$] can be determined by using Eqs. (33) and (34). Based on the inverse scattering in Eq. (36) and the measured BRDF data in Fig. 4(b), we calculated the PSDs as shown in Fig. 5. The symbols ($+$) and (−) represent forward and backward scatter, respectively. The PSD for 5° is close to the true material PSD, since it just has a small frequency shift from the incident light. We use the sum of two K-correlation functions to fit the true material PSD:

(41)$${{\rm PSD}_{{\rm total}}} = {{\rm PSD}_1} + {{\rm PSD}_2}.$$

Both ${{\rm PSD}_1}$ and ${{\rm PSD}_2}$ satisfy the form in Eq. (37). See [29] for more information on BRDF measurement of this material and the detailed calculation of PSD fitting.

Fig. 5. Illustration of PSD curves predicted by the inverse scattering. ${f_1}$ represents the frequency band limit due to the beam wait radius. Spatial frequencies greater than ${f_2} = 1/\lambda$ are nonpropagating for normal incident light.

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B. Comparison of BRDF Prediction

We can use the fitted PSD in Eq. (41) as input to the GHS model with scalar reflectance $R$ in Eq. (31) and with polarization factor $Q$ in Eq. (32) and GBHS model in Eq. (15) with Eq. (32) to predict the BRDFs in the incident plane, as shown in Fig. 6. We also plot the experimental data for comparison.

Fig. 6. BRDF predictions of GHS and GBHS models at incident angles of (a) 5°, (b) 15°, (c) 30°, and (d) 45°.

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Among the four different incident angles given in Fig. 6, all the beam waist radii ${w_0}$ are 5 µm. The fitting accuracies of the GHS and GBHS models using polarization factor $Q$ are nearly identical, and both are much better than the GHS model using scalar reflectance $R$. Figure 7 and 8 show the fitting effect of each model when the incident angles are 60° and 75°, respectively. Subfigures (a) and (b) of each figure correspond to two beam waist radii 5 and 15 µm. When the beam waist radius is 5 µm, the GBHS model is in better accordance with the data than the GHS model, and the predictions for BRDF are similar when the beam waist radius is 15 µm. Likewise, using the reflectivity $R$ gives the worst result.

Fig. 7. BRDF predictions of GHS and GBHS models at incident angle of 60° with different beam waist radii ${w_0}$. (a) 5 µm. (b) 15 µm.

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Fig. 8. BRDF predictions of GHS and GBHS models at incident angle of 75° with different beam waist radii ${w_0}$. (a) 5 µm. (b) 15 µm.

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We also observe that the agreement between experimental data and all the Harvey–Shack models get worse with increasing angle of incidence. This disagreement may be caused by sample quality, unaccounted contributions in the uncertainty budget and the ability of the Harvey–Shack models to represent the surface. Measurements performed at higher angles of incidence have a larger light–matter interaction area on the sample than measurements performed at lower angles of incidence. A perfect isotropic surface is completely characterized by one roughness and one correlation length. However, samples are seldom perfect; it is also well-known that small areas of a real sample are more similar to a perfect isotropic sample than larger areas. The insert in Fig. 7(a) shows a closeup around the specular peak together with the ${\pm}$3 standard uncertainty ($\sigma$) lines. We observe that only the specular GBHS value are within the ${\pm}3\sigma$ limits. Figure 7(b) shows that the GBHS model approaches the GHS model with increasing beam waist radius.

In order to study the influence of the beam waist radius, $w$, on the BRDF fitting, we use the chi-square ${\chi ^2}(w)$ estimation to evaluate the consistency of each model with the measured data:

(42)$${\chi ^2}(w) = \frac{1}{M}\frac{1}{N}\sum\limits_{{\theta _i}} \sum\limits_{{\theta _s}} {\left({{{{\rm BRDF}}_{{\rm model}}}(w) - {{{\rm BRDF}}_{{\rm data}}}} \right)^2},$$

where the incident angles ${\theta _i}$ are 5°, 15°, 30°, 45°, 60°, and 75°, and the scatter angle is from −85° to 85°, with an interval of 1°, and the beam waist radius $w$ is from 5 to 50 µm with an interval of 1 µm. $M$ and $N$ represent the number of incident and scatter angles, respectively. Figure 9 gives the chi-square ${\chi ^2}(w)$ values of the different models as a function of beam waist radius. We observe a minimum at 6 µm for the GBHS theory and no dependency on beam waist radius for the GHS (or plane-wave) method, as expected. We furthermore see that the chi-square value of the GBHS model gradually approaches the chi-square value of the GHS model with increasing beam waist radius, as it should.

Fig. 9. Illustration of chi-square ${\chi ^2}(w)$ estimation of different models with different beam waist radii.

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To sum, for all the considered incident angles, the fitting result of GBHS with polarization factor $Q$ is the best, followed by the GHS with polarization factor $Q$, and the GHS with fixed reflectivity $R$ is the worst.

5. CONCLUSION

Based on the conventional generalized Harvey–Shack theory and 2D plane-wave decomposition of the 3D converging–diverging Gaussian beam, we propose the GBHS scatter model with the reflectivity polarization factor $Q$ and obtain more accurate and reasonable results on sample satisfying the smooth surface criteria. The agreement between the GBHS model and data get worse with increasing angle of incidence. We argue that, to a large extent, this difference can be attributed to unaccounted contributions in the uncertainty budget and the fact that we do not have a perfect isotropic sample. The results also show that the GBHS becomes identical to the conventional GHS theory for large beam waist radii but is significantly different for smaller beam waist radii. The derivation of the Gaussian beam BRDF is analogous to the diffraction efficiency of gratings, which also justifies the view that the scattering characteristics of random rough surfaces are well-modeled by diffraction from gratings.

We believe that our physical-based GBHS scatter theory would be valuable in the fields of reflectometry, scatterometry, and ellipsometry. In ellipsometry and reflectometry, the state of the art is the effective medium theory, which is a mix of dielectric functions and which could be replaced by our GBHS theory. In scatterometry, our GBHS scatter theory may be used to incorporate roughness into the inverse modeling using a multistage scattering model, as shown in [13,28]. Finally, we also believe that the presented GBHS scatter theory will give more accurate results for rough surfaces than the GHS. This will be investigated in future work.

Funding

European Metrology Programme for Innovation and Research (20IND07, BxDiff, JRP 18SIB03, TracOptic); Danish Agency for Science and Higher Education; China Scholarship Council (CSC) (202104910354); National Key Research and Development Program of China (2021YFC2202100, 2021YFC2202104); Villum Fonden (25893).

Acknowledgment

Author Z. Ma thanks Dr. T. Jiang for useful discussions. P-E. Hansen was also supported by the Danish Agency for Institutions and Education. Z. Ma was supported by China Scholarship Council and National Key Research and Development Program of China. M. Karamehmedović was supported by The Villum Foundation.

Disclosures

The authors declare no conflicts of interest.

Data availability

The data obtained and used in this contribution can be provided by the corresponding author upon request. The data are available at [34].

Supplemental document

See Supplement 1 for supporting content.

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Harvey–Shack theory for a converging–diverging Gaussian beam

Abstract

1. INTRODUCTION

2. CONVERGING–DIVERGING GAUSSIAN BEAM

3. GAUSSIAN BEAM HARVEY–SHACK SCATTER THEORY

A. Two-Dimensional Plane-Wave Decomposition

B. Generalized Harvey–Shack Scatter Theory

C. Inverse Scattering for Smooth Surfaces

4. COMPARISON OF SCATTER MEASUREMENT AND MODELING PREDICTION

A. PSD Calculation and Fitting

B. Comparison of BRDF Prediction

5. CONCLUSION

Funding

Acknowledgment

Disclosures

Data availability

Supplemental document

REFERENCES

Supplementary Material (1)

Data availability

Cited By

Figures (9)

Equations (42)

Journal of the Optical Society of America B