Application of first-order nonparaxial scalar theory to determine surface scattering intensity of multilayer optical coatings

Kepeng Zhang; Wei Huang; Bin Zhang; Xiaoxi Tian; Yinhua Zhang; ChunLin Guan

doi:10.1364/OE.26.034592

1. Introduction

Light scattering in optical thin-films remains a crucial issue and is still limiting the development of advanced optical applications [1–3]. The scattering usually originates from the structural irregularities at the interfaces and in the bulk of the materials [4]. However, in most cases, the roughness-induced light scattering is predominant for scattering levels of stacks. To understand the origins and to further control the magnitude of off-specular optical losses in thin film technologies [5,6], a lot of theoretical tools in vector and scalar have been developed.

Based on the first-order vector perturbation approach, several vector theories have been presented by Bousquet et al. [7], Elson et al. [8,9], Amra et al. [10], etc. These theories have been widely used to predict the angular behavior of surface scattering in optical multilayers. It is worth noting that all these theories lead to identical results [11], even though the derivations are quite different. These theories exhibit some common advantages [12]: firstly, they are general without a prior restrictive hypothesis for the interfacial correlation properties. Secondly, they include the polarization properties of the incident and scattered light. On the basis of these theories, a lot of works have been carried out by Amra et al. for the sake of solving the inverse scattering problem [13–15], that is, retrieving interface irregularity information from light scatter data. Recently, there are numerous literatures on dealing with ellipsometry measurement [16], localized defects [17], multiscale roughness [18], multiple scattering [19], and light scattering reduction [20], which further promotes the understanding of light scattering in thin film systems.

However, there is by far no obvious progress of scalar methods. In general, scalar theories are derived from the classical Beckmann-Kirchhoff (BK) theory [21]. Inevitably, their paraxial assumption limits their abilities to handle wide-angle scattering and large angles of incidence accurately. For example, the most prominent scalar theory, presented by Eastman [22,23] and extended by Carniglia [24], only permits the calculation of the total amount of scattered light. In recent years, an empirically modified BK theory [25] that is nonparaxial has been developed by Krywonos et al. based on the typical generalized Harvey-Shack (GHS) theory [26], and its prediction is in good agreement with experimental wide-angle scatter measurement for surfaces with moderate roughness at arbitrary incident. This provides the motivation for developing a nonparaxial scalar scattering (NSS) theory to predict the angular surface scattering from optical multilayers, as reported in this paper. It is worth noting that the above studies are performed all for quite low-loss high-quality coatings. However, not all applications of interest satisfy the smooth-interface limitation especially at short wavelengths. To our knowledge, both existing vector and scalar theories adequately describe the behavior of scattering in multilayer coatings with moderately rough interfaces, which is also the interest of this paper.

This paper is organized as follows. In Sec. 2, the detail derivation of the NSS theory that is nonparaxial and able to handle wide-angle scattering at arbitrary incident angles for an optical multilayer with rough interfaces is given. The single surface and smooth approximations for the NSS theory are presented in Sec. 3, as well as the predictions for a narrow-band filter, an antireflective coating, and a 24-layer mirror. Modeling results for a single-layer coating and the 24-layer mirror at different roughness levels are presented and discussed in Sec. 5, followed by a conclusion in Sec. 6.

2. First-order NSS theory for surface scattering from typical p-layer design

2.1 Amplitudes of fields scattered by moderately rough interface within multilayer

A typical multilayer stack is shown in Fig. 1. There are s interfaces in this stack, forming the boundaries of p layers, where s = p + 1. Assume that all media i are homogeneous in their bulk, each bulk with a complex refractive index n_i = v_i – iχ_i and a thickness e_i. The surrounding medium such as air has an index of n₀ and the index of the substrate is n_s. The i_th interface separates media i and i + 1. Expect for interface i, all the interfaces are considered to be perfectly smooth. The i_th interface is rough and presents a random height irregularity h_i(x, y) that represents the displacement of the real interface from its mean position.

Fig. 1 Schematic illustration of the irregularity at interface i in a multilayer. The z axis is normal to all interfaces. The top-surface, interface 0, is assumed to be in air, and the substrate is semi-infinite.

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In our model, several assumptions for the interfacial irregularity are given as follows. Firstly, the rms value of h_i is not larger than the wavelength λ of the incident light. Secondly, the slope of h_i is much less than one. Thirdly, the presence of h_i does not affect the indices and thicknesses of media i and j where j = i + 1. Besides, scattering induced by local defects (dust, pits, scratches, etc.) as well as multiple scattering are not taken into consideration, whereas multiple reflections in multilayered media are taken into account.

Clearly, there would be scattered light produced in whole space when a monochromatic plane wave strikes the multilayer. To illustrate the definition of quantities for this phenomenon, a diagram is drawn in Fig. 2. Our objective in this section is to derive the formulas of a scalar method that are used for calculating the amplitudes of the fields, denoted by E_i⁽^d^{) ±} , which are scattered in the entrance and exit media by the microstructural irregularity at the i_th interface of the multilayer.

Fig. 2 Directions of waves of incidence, reflection and transmission scattering. θ₀ is the polar incident angle. θ_s and ϕ_s characterize a particular scattering direction. θ_s is from the sample normal and ϕ_s is the azimuthal angle.

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2.1.1 Theoretical method

To derive the expressions for the amplitudes of the scattered fields E_i⁽^d^{) ±} , we hereby follow a similar method that was used for Raman scattering by multilayers [27]. The foundation of the derivation is the initially scattered fields (Fig. 3) generated at interface i.

Fig. 3 Directions of the initially scattered fields at interface i. θ_i and θ_si are the propagation angles (given by Snell’s law) of the main fields and the scattered fields in medium i, respectively.

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As shown in Fig. 3, there are four initially scattered fields E_ik⁽^s⁾ (where k = 1,2,3,4 are the indices of the fields) at interface i that are induced by two main incident fields E_i₁⁽⁰⁾ and E_i₄⁽⁰⁾ on the interface. Based on the modified BK theory, the amplitudes of the initially scattered fields are calculated in Sec. 2.1.2, which is the key to the NSS theory. Regarding the main field, its calculation is carried out in Sec. 2.1.3. It is worth noting that when the rms value of h_i are far less than λ, the main field is determined as the stationary field inside a multilayer that is free of irregularities. However, for large roughness values, the adjustment of the main field for the scattering loss is extremely essential.

Four initially scattered fields are firstly added at the two sides of the i_th interface:

{\begin{cases} E_{i}^{(s) -} = E_{i 1}^{(s)} {E_{i 1}^{(0)}} + E_{i 2}^{(s)} {E_{i 4}^{(0)}} \\ E_{i}^{(s) +} = E_{i 3}^{(s)} {E_{i 1}^{(0)}} + E_{i 4}^{(s)} {E_{i 4}^{(0)}} \end{cases},

in which the bracketed main field is responsible for the excitation of the corresponding scattered field. In this paper, the superscript (s) refers to the scattered field at interface i, the superscript (0) always relates to the main field, the superscript (d) denotes the scattered fields in either of the two outer media that surround the multilayer, and the superscripts + and – correspond to the fields subtended by the exit and entrance media, respectively.

Next, the fields E_i⁽^s^{) ±} leave the i_th interface, continue to propagate through the rest of the film, and finally flow into the two outer semi-infinite media. In this process, the scattered fields obey the usual rules of field propagation in a multilayer. Fortunately, the rules have been formulized by Eastman in terms of a standard matrix solution [24]. Besides, multiple reflections, introduced by multilayered media, are also considered that cause portions of each field to pass out of the multilayer to the entrance and exit media (as shown in Fig. 4), i.e.,

{\begin{cases} E_{i}^{(d) -} = P_{i 1} {E_{i}^{(s) +}} + P_{i 2} {E_{i}^{(s) -}} \\ E_{i}^{(d) +} = P_{i 3} {E_{i}^{(s) +}} + P_{i 4} {E_{i}^{(s) -}} \end{cases},

where P_ik is the initially scattered field propagation function and calculated in Sec. 2.1.4. In this relation, P_i₁{E_i⁽^s⁾⁺} and P_i₂{E_i⁽^s⁾⁻} represent the portions of the fields E_i⁽^s⁾⁺and E_i⁽^s⁾⁻ that are reflected into the entrance medium; P_i₃{E_i⁽^s⁾⁺} and P_i₄{E_i⁽^s⁾⁻} denote the portions of the fields E_i⁽^s⁾⁺and E_i⁽^s⁾⁻ that are transmitted into the exit medium.

Fig. 4 Illustration for notation of the initially scattered field propagation in a multilayer.

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Note that the functions P_ik do not depend on the interfacial irregularities but on the ideal multilayer characteristics (thickness and index of each layer), as well as on the observation conditions (wavelength, incidence and scattering angles). After E_ik⁽^s⁾, E_ik⁽⁰⁾, and P_ik are obtained, the expressions for the amplitudes of the scattered fields E_ik⁽^d⁾ are easily given in Sec. 2.1.5.

In what follows we sum up every step of the calculation.

2.1.2 Amplitudes of initially scattered fields E_ik^(s)

Recalling the scattering coefficient, it is defined as the ratio of the scattered filed to the field reflected in the specular direction by a perfectly smooth surface [21]. Utilizing this definition, the amplitudes of the initially scattered fields E_ik⁽^s⁾ at interface i can be expressed as:

E_{i k}^{(s)} (θ_{s}, ϕ_{s}; θ_{0}) = a_{i k} (θ_{0}) ρ_{i k} (f_{x}, f_{y}),

where ρ_ik is the scattering coefficient associated with irregularity h_i, and based on the modified BK theory can be given by:

ρ_{i k} (f_{x}, f_{y}) = \frac{1}{A_{s}} F {\exp [- i \frac{2 π}{λ} η_{i k} h_{i} (x, y)]},

where A_s is the illumination area, and the bold F represents the Fourier transform. The spatial frequencies f_x and f_y are determined in terms of θ_s, ϕ_s, and θ₀:

\begin{matrix} f_{x} = \frac{\sin θ_{s} \cos θ_{s} - \sin θ_{0}}{λ}, & f_{y} = \frac{\sin θ_{s} \sin ϕ_{s}}{λ}, \end{matrix}

It should be pointed out that Eq. (5) is for reflection scattering, while sinθ₀ of Eq. (5) must be replaced by n₀sinθ₀/n_s for transmission scattering. Notice that here we are more concerned with the modification conditions in the modified BK theory rather than the improved closed-form solutions that only exist for surfaces with Gaussian autocovariance (ACV) functions. For this reason, the NSS theory does not depend on the kind of function (Gaussian, exponential, etc.) that describes the correlation function at interfaces.

In addition, a_ik in Eq. (3) refers to the reflected or transmitted field in the specular direction, and η_ikh_i in Eq. (4) labels the optical path difference between the incident and scattering rays. The respective terms a_ik and η_ik are written as:

\begin{matrix} {\begin{cases} a_{i 1} = r_{i 1}^{(0)} E_{i 1}^{(0)} \\ a_{i 2} = t_{i 2}^{(0)} E_{i 4}^{(0)} \\ a_{i 3} = t_{i 1}^{(0)} E_{i 1}^{(0)} \\ a_{i 4} = r_{i 2}^{(0)} E_{i 4}^{(0)} \end{cases}, & {\begin{cases} η_{i 1} = N_{i}^{(0)} + N_{i} \\ η_{i 2} = - (N_{j}^{(0)} - N_{i}) \\ η_{i 3} = N_{i}^{(0)} - N_{j} \\ η_{i 4} = - (N_{j}^{(0)} + N_{j}) \end{cases}, \end{matrix}

where r_i₁⁽⁰⁾ and r_i₂⁽⁰⁾ (or t_i₁⁽⁰⁾ and t_i₂⁽⁰⁾) represent the Fresnel reflectance (or transmittance) of the i_th interface for the fields next to the i_th interface in media i and j, respectively, and N_i⁽⁰⁾ and N_i rely on the refractive index of medium i. All these quantities can be calculated by Eqs. (7)-(9), i.e.,

\begin{matrix} r_{i 1}^{(0)} = \frac{N_{i}^{(0)} - N_{j}^{(0)}}{N_{j}^{(0)} + N_{i}^{(0)}}, & r_{i 2}^{(0)} = \frac{N_{j}^{(0)} - N_{i}^{(0)}}{N_{j}^{(0)} + N_{i}^{(0)}}, \end{matrix}

\begin{matrix} t_{i 1}^{(0)} = \frac{2 N_{i}^{(0)}}{N_{j}^{(0)} + N_{i}^{(0)}}, & t_{i 2}^{(0)} = \frac{2 N_{j}^{(0)}}{N_{j}^{(0)} + N_{i}^{(0)}}, \end{matrix}

\begin{matrix} N_{i}^{(0)} = {(n_{i}^{2} - n_{0}^{2} \sin^{2} θ_{0})}^{1 / 2}, & N_{i} = {(n_{i}^{2} - n_{0}^{2} \sin^{2} θ_{s})}^{1 / 2} \end{matrix} .

2.1.3 Amplitudes of the main fields E_ik⁽⁰⁾

To make it possible for our model to satisfy the law of energy conservation, it is quite necessary to adjust the main field amplitude for light scattering loss. Based on the well-established method of matrix calculation (see Eqs. (39)-(45) of [24]), the amplitudes of the main fields E_ik⁽⁰⁾ can be calculated by recurrence from the values E_s⁽⁰⁾, i.e.,

[\begin{matrix} E_{i 1}^{(0)} \\ E_{i 2}^{(0)} \end{matrix}] = I_{i}^{(0)} [\begin{matrix} E_{i 3}^{(0)} \\ E_{i 4}^{(0)} \end{matrix}] = I_{i}^{(0)} \prod_{m = i + 1}^{p} T_{m}^{(0)} I_{m}^{(0)} [\begin{matrix} E_{s}^{(0)} \\ 0 \end{matrix}],

where E₀₁⁽⁰⁾ is known, whereas E_s⁽⁰⁾ that is the field transmitted into the substrate is unknown. T_m⁽⁰⁾ satisfies:

T_{m}^{(0)} = [\begin{matrix} \exp (i φ_{m}^{(0)}) & 0 \\ 0 & \exp (- i φ_{m}^{(0)}) \end{matrix}],

and represents the propagation of the fields across the m_th layer, in which φ_m⁽⁰⁾ = 2πe_mN_m⁽⁰⁾/λ is the phase thickness of layer m, and

I_{m}^{(0)} = \frac{1}{d_{m 3} t_{m 1}^{(0)}} [\begin{matrix} 1 & d_{m 4} r_{m 1}^{(0)} \\ d_{m 1} r_{m 1}^{(0)} & d_{m 3} d_{m 2} t_{m 2}^{(0)} t_{m 1}^{(0)} + d_{m 4} d_{m 1} r_{m 1}^{(0)} r_{m 1}^{(0)} \end{matrix}]

denotes the propagation of the fields across interface m from media m to m + 1, in which based on the GHS theory [26], the terms d_mk that cause the reduction of the main field can be expressed as:

d_{m 1} = \sqrt{\exp {- {[\frac{4 π}{λ} N_{m}^{(0)} σ_{m, r e l}]}^{2}}},

d_{m 2} = d_{m 3} = \sqrt{\exp {- {[\frac{2 π}{λ} (N_{m}^{(0)} - N_{m + 1}^{(0)}) σ_{m, r e l}]}^{2}}},

d_{m 4} = \sqrt{\exp {- {[\frac{4 π}{λ} N_{m + 1}^{(0)} σ_{m, r e l}]}^{2}}} .

In Eqs. (13-15), the terms σ_m_,_rel are the bandwidth limited rms roughness of interface m, usually calculated by integrating the power specular density (PSD) of the interface [26].

σ_{m, r e l} = {[\int_{- 1 / λ}^{1 / λ} \int_{f_{y} = - \sqrt{1 / λ^{2} - f_{x}^{2}}}^{\sqrt{1 / λ^{2} - f_{x}^{2}}} P S D_{m} (f_{x}, f_{y}) d f_{x} d f_{y}]}^{1 / 2} .

Note that the functions PSD and ACV are Fourier transforms with each other.

2.1.4 Field propagation functions P_ik

By utilizing Eq. (10), the propagation function P_ik for the scattered field E_i⁽^s⁾ ^± can be given by:

P_{i k} (E) = b_{i k} E,

where

\begin{matrix} b_{i 1} = {\begin{matrix} 1 & i = 0 \\ \frac{1}{A_{i} W_{i}} & i \neq 0 \end{matrix}, & b_{i 2} = {\begin{matrix} \frac{D_{j}}{A_{j} C_{j} W_{j}} \exp (i φ_{j}) & i \neq p \\ 0 & i = p \end{matrix}, \end{matrix}

\begin{matrix} b_{i 4} = {\begin{matrix} \frac{1}{C_{j} W_{j}} & i \neq p \\ 1 & i = p \end{matrix}, & b_{i 3} = {\begin{matrix} 0 & i = 0 \\ \frac{B_{i}}{A_{i} C_{i} W_{i}} \exp (- i φ_{i}) & i \neq 0 \end{matrix} . \end{matrix}

Additionally, the terms W_i, A_i, B_i, and C_i satisfies:

W_{i} = 1 - \frac{B_{i} D_{i}}{A_{i} C_{i}} \exp (i 2 φ_{i}),

[\begin{matrix} A_{i} \\ B_{i} \end{matrix}] = \prod_{m = 1}^{i} \frac{1}{1 - r_{i - m}} [\begin{matrix} \exp (i φ_{i - m + 1}) & 0 \\ 0 & \exp (- i φ_{i - m + 1}) \end{matrix}] [\begin{matrix} 1 & - r_{i - m} \\ - r_{i - m} & 1 \end{matrix}],

[\begin{matrix} C_{i} \\ D_{i} \end{matrix}] = \prod_{m = i}^{p} \frac{1}{1 + r_{m}} [\begin{matrix} \exp (i φ_{m}) & 0 \\ 0 & \exp (- i φ_{m}) \end{matrix}] [\begin{matrix} 1 & r_{m} \\ r_{m} & 1 \end{matrix}],

where φ_m is the phase thickness of layer m, r_m is the amplitude reflectance, and their calculations can use the formulas for φ_m⁽⁰⁾ and r_i₁⁽⁰⁾ in which θ₀ is replaced by θ_s.

2.1.5 Amplitudes of scattered fields E_ik^(d)

Equations (1)-(2) and the knowledge of E_ik⁽^s⁾, E_ik⁽⁰⁾, and P_ik lead to the following relationship for the amplitudes of the scattered fields E_i⁽^d^{) ±} , i.e.,

E_{i}^{(d) \pm} = \sum_{k = 1}^{4} c_{i k}^{\pm} ρ_{i k},

where c_ik ^± is the optical factor of interface i, expressed by:

\begin{array}{l} \begin{matrix} c_{i 1}^{-} = a_{i 1} b_{i 1}, & c_{i 2}^{-} = a_{i 2} b_{i 1}, & c_{i 3}^{-} = a_{i 3} b_{i 2}, & c_{i 4}^{-} = a_{i 4} b_{i 2}, \end{matrix} \\ \begin{matrix} c_{i 1}^{+} = a_{i 1} b_{i 3}, & c_{i 2}^{+} = a_{i 2} b_{i 3}, & c_{i 3}^{+} = a_{i 3} b_{i 4}, & c_{i 4}^{+} = a_{i 4} b_{i 4} . \end{matrix} \end{array}

By solving the above-mentioned problems, we can calculate the E_i⁽^d^{) ±} field amplitudes and give a simple analytic expression of Eq. (23). Notice that Eq. (3) should be valid for moderately rough interfaces because it is directly gleaned from the modified BK theory, and the adjustment for the main field amplitude has been made in Eq. (12). Thus Eq. (23) should be suitable for a multilayer where only the i_th interface is moderately rough.

2.2 Expressions for angle-resolved scattering (ARS)

In the previous section we dealt with the amplitudes of the fields that are scattered by interface i in each of the two media that surround the multilayer. We now concentrate on the case of all interface i consisting of the roughness h_i(x, y). Based on the assumption that multiple scattering is neglected, the total scattered fields are simply the sum of the fields scattered by all interfaces in the p-layer stack, which are given by:

E^{(d) \pm} = \sum_{i = 0}^{p} E_{i}^{(d) \pm} = \sum_{i = 0}^{p} \sum_{k = 1}^{4} c_{i k}^{\pm} ρ_{i k},

where c_ik ^± is the factor associated with the initially scattered fields E_ik⁽^s⁾.

The energy scattered in a solid angle element around the direction (θ_s, ϕ_s) is proportional to:

{| E^{(d) \pm} |}^{2} = {| \sum_{i = 0}^{p} \sum_{k = 1}^{4} c_{i k}^{\pm} ρ_{i k} |}^{2} .

Since the expression for ρ_ik includes the random variable h_i, Eq. (26) cannot be directly used. Using the autocorrelation theorem of Fourier transform theory, Eq. (26) is transformed into:

\begin{matrix} {| E^{(d) \pm} |}^{2} = {| F {{\tilde{E}}^{(d) \pm}} |}^{2} = F {{\tilde{E}}^{(d) \pm} \otimes {\tilde{E}}^{(d) \pm}} \\ = F {\sum_{i = 0}^{p} \sum_{j = 0}^{p} \sum_{k = 1}^{4} \sum_{l = 1}^{4} {\tilde{E}}_{i k}^{(d) \pm} \otimes {\tilde{E}}_{j l}^{(d) \pm}}, \end{matrix}

where the circled cross denotes the autocovariance relationship, and _{${\tilde{E}}^{(d) \pm}$} is the inverse Fourier transform, labeled by F⁻¹, of E⁽^d^{) ±} . The use of Eq. (4) leads to:

{\tilde{E}}_{i k}^{(d) \pm} = c_{i k}^{\pm} F^{- 1} {ρ_{i k}} = \frac{1}{A_{s}} c_{i k}^{\pm} \exp [- i \frac{2 π}{λ} η_{i k} h_{i} (x, y)],

which is associated with E_ik⁽^s⁾.

Compared with Eq. (26), Eq. (27) directly describes the correlation properties of the various initially scattered fields. In order to remove random variables h_i, we take the expected value for Eq. (27) and obtain:

ε {{| E^{(d) \pm} |}^{2}} = F {\sum_{i = 0}^{p} \sum_{j = 0}^{p} \sum_{k = 1}^{4} \sum_{l = 1}^{4} ε {{\tilde{E}}_{i k}^{(d) \pm} \otimes {\tilde{E}}_{j l}^{(d) \pm}}},

where the symbol ε{} means the ensemble average over the entire area of illumination. Using Eq. (28), the expectation term in Eq. (29) can be written as:

ε {{\tilde{E}}_{i k}^{(d) \pm} \otimes {\tilde{E}}_{j l}^{(d) \pm}} = \frac{1}{A_{s}} c_{i k}^{\pm} c_{j l}^{\pm *} ε {\exp {- i \frac{2 π}{λ} [η_{i k} h_{i}^{'} + η_{j l} h_{j}^{' ​'}]}},

where h_i′ = h_i(x′, y′), h_j″ = h_j(x′ − x, y′ − y), and the asterisk is for the complex conjugate.

Further, we denote by H_ik_;_jl the expectation term in the right side of Eq. (30). Supposed that both h_i′ and h_j″ are normal random variables, H_ik_;_jl can be derived by:

\begin{matrix} H_{i k; j l} = ε {\exp [- i \frac{2 π}{λ} (η_{i k} {h^{'}}_{i} + η_{j l} {h^{″}}_{j})]} \\ = \exp {- i \frac{2 π}{λ} [η_{i k} ε {{h^{'}}_{i}} + η_{j l} ε {{h^{″}}_{j}}]} \exp [- \frac{2 π^{2}}{λ^{2}} (η_{i k}^{2} σ_{i, r e l}^{2} - 2 η_{i k} η_{j l}^{*} A C V_{i j} + η_{j l}^{2} σ_{j, r e l}^{2})], \end{matrix}

where ACV_ij is the covariance function of h_i′ and h_j″, that is, the autocovariance function (i = j) or the cross-covariance function (i ≠ j). Notice that Eq. (31) is in the same form as the transfer function in the GHS theory (see Eqs. (32)-(35) in [26]). Similar to the transfer function, Eq. (31) can be written in the from:

H_{i k; j l} (x, y; θ_{0}, θ_{s}) = A_{i k; j l} (θ_{0}, θ_{s}) + B_{i k; j l} G_{i k; j l} (x, y; θ_{0}, θ_{s}),

where

A_{i k; j l} = \exp [- 2 π^{2} (η_{i k}^{2} σ_{i, r e l}^{2} + η_{j l}^{2} σ_{j, r e l}^{2}) / λ^{2}],

B_{i k; j l} = 1 - \exp [- 2 π^{2} (η_{i k}^{2} σ_{i, r e l}^{2} + η_{j l}^{2} σ_{j, r e l}^{2}) / λ^{2}],

are the fraction of the total reflected radiant power contained in the specular and the scattered components, respectively, and

G_{i k; j l} = \frac{\exp {\frac{2 π^{2}}{λ^{2}} \frac{η_{i k}^{2} σ_{i, r e l}^{2} + η_{j l}^{2} σ_{j, r e l}^{2}}{η_{i k}^{2} σ_{i, t o t}^{2} + η_{j l}^{2} σ_{j, t o t}^{2}} η_{i k} η_{j l}^{*} A C V_{i j}} - 1}{\exp {2 π^{2} (η_{i k}^{2} σ_{i, r e l}^{2} + η_{j l}^{2} σ_{j, r e l}^{2}) / λ^{2}} - 1},

where σ_i_,_tot labels the intrinsic rms roughness of interface i and corresponds to the entire spatial frequency range, which differs from the band-limited roughness σ_i_,_rel.

When only consider the diffuse component of scattering, the A_ik_;_jl term in Eq. (32) is omitted. Using Eqs. (30)-(35), Eq. (29) can be rewritten as:

ε {{| E^{(d) \pm} |}^{2}} = \frac{1}{A_{s}} {\sum_{i = 0}^{p} \sum_{j = 0}^{p} \sum_{k = 1}^{4} \sum_{l = 1}^{4} c_{i k}^{\pm} c_{j l}^{\pm *} F {B_{i k; j l} G_{i k; j l}}} .

According to the modified BK theory, the square amplitude of the total scattered field equals to the scattered radiance defined as radiant power per unit solid angle per unit projected source area [25]. A quantity describing the angular scattered intensity is the angle-resolved scattering (ARS) defined as the power ΔP_s scattered into a small solid angle ΔΩ_s that is normalized to the solid angle and the incident power P_i [28]. By multiplying the radiance by cosθ_s and integrating over the illumination area A_s, the expressions for the ARS in reflection and transmission for the multilayer can be given by:

A R S^{\pm} (θ_{s}, ϕ_{s}; θ_{i}) = \cos θ_{s} {\sum_{i = 0}^{p} \sum_{j = 0}^{p} \sum_{k = 1}^{4} \sum_{l = 1}^{4} K_{i k} c_{i k}^{\pm} K_{j l}^{*} c_{j l}^{\pm}^{*} F {B_{i k; j l} G_{i k; j l}}},

where according to the GHS theory a renormalization constant K_ik is added to ensure the energy conservation, i.e.,

K_{i k} (θ_{i}) = \sqrt{B_{i k; i k} (θ_{i}) {(\int_{f_{x} = - 1 / λ}^{1 / λ} \int_{f_{y} = - \sqrt{1 - f_{x}^{2}}}^{\sqrt{1 - f_{x}^{2}}} F {B_{i k; i k} G_{i k; i k}} d f_{x} d f_{y})}^{- 1}},

and it only differs from one if evanescent waves are produced.

The Bidirectional scattering distribution function (BSDF) is related to the ARS by BSDF = ARS/cosθ_s. From Eq. (37), we can see that the roughnesses h_i of the different interfaces enter into the ARS expressions, partly through their autocorrelation and cross-correlation functions and partly through their total and band-limited rms roughness value. The c_ik ^± terms only rely on σ_i_,_rel and the ACV_ij terms only exist in G_ik;jl. In addition, Eq. (37) is more general compared to those of the typical scalar theories because it is not necessary to make a prior hypothesis on the ACV_ij terms. However, in order to deal with the calculation of Eq. (37), it is necessary to know ACV_ij in all i and j. The interface roughness of designs would be assumed to be fully correlated in the following sections for simplicity. This assumption means that all interfaces have the same shape as the substrate profile, yielding ACV_ij = ACV_ii = ACV for all i and j.

By comparing Eq. (37) with Eqs. (22)-(23) of [7] (for the popular Bousquet’s vector scattering (BVS) theory that are only applied to optical multilayers with smooth interfaces), one can notice that the scalar ARS distinctly differs from the vector ARS, owing to the fact that the scalar ARS, i.e., Eq. (37) is derived based on four initially scattered fields E_ik⁽^s⁾ (k = 1, 2, 3, 4) at each of the interfaces within a multilayer. For comparison, the vector ARS is derived based on two initially scattered fields on the two sides of each interface. It is also noticed that in the case of the fully correlated interface roughness, the scalar ARS does not meet the linear relationship with the interface PSD, which differs from the vector ARS. Besides, the polarization properties of the incident and scattering light are not included in the scalar ARS.

3. Single-surface and smooth-interface approximations to the first-order NSS theory

In previous works, researchers have derived analytic expressions for the ARS of a moderately rough surface (see Eqs. (32)-(36) of [25]) and that of an optical coating (see Eqs. (22)-(23) of [7]). However, their results are confined with the assumption that the number of interfaces is one or that the interfaces are smooth enough. It was not existent for such a theory that can be used to predict scattering behavior for multilayered moderately rough interfaces. Due to the fact that multiple scattering is very remarkable in such a multilayer, it is almost impossible to sufficiently confirm the first-order NSS theory by experimental measurement. In this section, Eq. (37) in the single-surface and smooth-surface approximations was discussed and analyzed.

3.1 Case of single surfaces

In this subsection, we are concerned with an uncoated random rough surface. The roughness is described by function h(x, y), whose rms value can be the same order of the incident wavelength λ. The media of indices n₀ and n_s are separated by this surface. The angular scattered intensity is still calculated by Eq. (37). For the case of s = 1, Eq. (37) in refection is approximated as:

\begin{matrix} A R S^{-} (θ_{s}, ϕ_{s}) = \cos θ_{s} {| r_{01}^{(0)} E_{01}^{(0)} |}^{2} K_{01} K_{01}^{*} F {B_{01; 01} G_{01; 01}} \\ = \cos θ_{s} R P_{i} K \cdot F {B_{01; 01} G_{01; 01}}, \end{matrix}

where R = |r₀₁⁽⁰⁾|² is the radiant reflectance, P_i = |E₀₁⁽⁰⁾|² is the power of incidence, and K = K₀₁K₀₁*.

Correspondingly, B_01;01 and G_01;01 are still calculated by Eqs. (34) and (35), i.e.,

B_{01; 01} = 1 - \exp [- \frac{4 π^{2}}{λ^{2}} n_{0}^{2} {(\cos θ_{i} + \cos θ_{s})}^{2} σ_{0, r e l}^{2}],

G_{01; 01} = \frac{\exp {\frac{4 π^{2}}{λ^{2}} n_{0}^{2} {(\cos θ_{i} + \cos θ_{s})}^{2} \frac{σ_{0, r e l}^{2}}{σ_{0, t o t}^{2}} A C V_{00}} - 1}{\exp {π^{2} n_{0}^{2} {(\cos θ_{i} + \cos θ_{s})}^{2} σ_{0, r e l}^{2} / λ^{2}} - 1} .

It is noticed that Eqs. (39)-(41) are exactly the same as Eqs. (32)-(36) of [25] given by the GHS theory. Consequently, the NSS theory can be applied for uncoated surfaces even if the roughness is moderate. Meanwhile, it is also validated that the initially scattered field amplitude is given by Eq. (3) for moderately rough interfaces.

3.2 Case of smooth-surfaces

By assuming the interfacial roughness to be small compared to the incident wavelength (i.e., h_i << λ for all i), the scattering function Eq. (37) can be approximated as:

A R S^{\pm} (θ_{s}, ϕ_{s}) = \frac{4 π^{2} \cos θ_{s}}{λ^{2}} {\sum_{i = 0}^{p} \sum_{j = 0}^{p} C_{i}^{\pm} C_{j}^{\pm}^{*} P S D_{i j} (f_{x}, f_{y})},

where C_i ^± is the factor that does not depend on the interface roughness and is given by:

C_{i}^{\pm} = \sum_{k = 1}^{4} K_{i k} η_{i k} c_{i k}^{\pm},

and PSD_ij is the Fourier transform of ACV_ij. Due to low scattering levels in this approximation, the damping factors d_ik in Eqs. (13)-(15) are omitted by setting d_ik = 1 at all i and k. In addition, σ_i_,_rel is set to be equal to σ_i_,_tot in Eq. (42) for simplicity.

By comparing Eq. (42) with Eqs. (22)-(23) of [7], it is found that the approximated scalar ARSs are in the same form as the vector ARSs given in the BVS theory. Especially in the case of the fully correlated interface roughness (PSD_ij = PSD_ii = PSD), Eq. (42) is simplified to:

A R S^{\pm} \propto {| \sum_{i = 0}^{p} C_{i}^{\pm} |}^{2} P S D (f_{x}, f_{y}),

which indicates the linear relationship between the ARS and the interface PSD, which is present in the vector ARSs as well. Based on Eq. (42) and Eqs. (22)-(23) of [7], Figs. 5(a)-5(c) are plotted for three representative design types (where the interfaces are isotropic and smooth, of which the rms roughnesses are in the range of validity of the BVS theory) to illustrate how close the predictions of the NSS and BVS theories are to each other. For the prediction of the BVS theory, both the incident and scattered light fields are assumed to be s polarized and remain in the plane of incidence.

Fig. 5 Angular surface scattering calculated for (a) a narrow-band filter, (b) an anti-reflective coating, and (c) a 24-layer mirror. The substrate and the incident medium have indices n_s = 1.52 and n₀ = 1.0. The design wavelength λ₀ is equal to the illumination wavelength: λ₀ = λ = 633 nm. The incident angles θ₀ are in Fig. 5(a), 0°; Fig. 5(b), 0°; Fig. 5(c), 30°. The design angles are equal to the incident angles: i₀ = θ₀. The angular range from −90° to 90°corresponds to reflection scattering and the region of 90° < θ_s < 180° and −180° < θ_s < −90°correspond to transmission scattering. Note that all interfaces within a design have the same roughness.

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Figure 5(a) is for a narrow-band filter SH(LH)²L⁶(HL)²H (cited from [13]); Fig. 5(b) is for an antireflective coating LH²SH²L; Fig. 5(c) is for a dielectric reflector S(HL)¹¹HL² (cited from [28]), where S represents the substrate, H and L are for high- and low-index quarterwave optical thickness layers, respectively. The parameters used for the calculation of Fig. 5(a) are as follows: n_H = 2.3 and n_L = 1.3. The interfaces are described by the sum of a Gaussian and an exponential [14], i.e.,

A C V (x, y) = σ_{s 1}^{2} \exp [- {(r / l_{c 1})}^{2}] + σ_{s 2}^{2} \exp (- | r / l_{c 2} |),

where r² = x² + y². The rms values of roughness are set to be σ_s₁ = 0.5 nm and σ_s₂ = 1 nm, and the correlation lengths l_c₁ = 0.2 μm, and l_c₂ = 2 μm. The basic parameters involved in Fig. 5(b) are identical to those of Fig. 5(a), except that n_H = 1.90 and n_L = 1.38. In Fig. 5(c), the parameters n_H and n_L are set to be n_H = 2.3 and n_L = 1.45. The autocovariance function of interface is [28]:

A C V (x, y) = σ_{s 1}^{2} \exp [- {(r / l_{c 1})}^{2}] + σ_{s 2}^{2} \exp [- {(r / l_{c 2})}^{2}],

where σ_s₁ = 0.65 nm, σ_s₂ = 0.25 nm, l_c₁ = 0.1 μm, and l_c₂ = 2 μm.

It can be seen from Fig. 5(a) that the NSS prediction agrees well with the prediction of the BVS, although there is a slight disagreement at near specular directions. The BVS curve here is consistent with the theoretical curve in Fig. 17 of [13]. The excellent agreement between the NSS and BVS curves is also found in Fig. 5(b) especially at the specular direction. In Fig. 5(c), the NSS method still predicts the curve that is almost exactly identical to the BVS curve. The difference can be seen in the regions of 60°~150° and −60°~−150° (that is, the angular range away from the specular direction), but their weight is negligible for the total scattering. Besides, it is noticed that the predicted curves in Fig. 5(c) agree well with the theoretical curve in Fig. 11 of [28]. However, compared to the measured curve in Fig. 11 of [28], the NSS curve of Fig. 5(c) is decreased at the near specular direction. The decreased scattering may be the result of neglecting scattering of surface defects or dust particles in the NSS model. In the following, we further give Fig. 6 for the reflector as the example to seek how the validity of the first-order NSS theory varies with (a) the incident angle θ₀ and (b) the correlation length l_c₂.

Fig. 6 Drawings for the ARS of the 24-layer reflector predicted by the NSS and BVS theories.

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It is clear from Fig. 6 that the NSS prediction agree well with the prediction of the BVS in all cases. However, there are slight differences mainly in the angular range that is away from the specular direction. These differences are acceptable owing to their negligible proportion for the amount of scattered light. And the differences are increased with either θ₀ or l_c₂ for different reasons. The reason why the differences increase with the increasing of θ₀ is that large values of θ₀ cause the lateral displacement of various reflected waves inside the mirror, which makes the interfacial correlation properties changed. Small values of l_c₂ result in the violation of the small roughness slope assumption, which makes the differences increased.

To summarize, the expression of the NSS theory in the case of smooth-interfaces, i.e., Eq. (42), is not only in the same form as those given by the BVS theory (that is, Eqs. (22)-(23) of [7]), but also their predictions for different stacks are in good agreement especially at small incident angles and large correlation lengths. This indicates the validity of the NSS theory for optical coatings with smooth interfaces. It is worth pointing out that the BVS theory is also first-order limited just like the NSS theory. The reason why the first-order theories are valid here is that high-quality coatings have quite low scattering levels.

4. Interface roughness dependence of the validity of the first-order NSS theory

In this section, we provided examples to show the dependence of the validity of the NSS theory on the interface roughness. To this effort, the results of ARS calculations using the NSS, BVS, and GHS theories in the following discussion were given for optical coatings at four different roughness levels. It is worth pointing out that the NSS theory is only used to calculate the scattering from bare substrate or top surface of a stack. The purpose is to show the regularity and magnitude of scattering from moderately rough surfaces as a reference to judge whether the prediction of the NSS theory is reasonable in the case of moderate roughness. Two typical designs were analyzed: a single low index layer and a 24-layer mirror.

4.1 Single low index layer coating

Consider a single-layer coating deposited on an opaque substrate. In this case, the impact of multiple scattering is relatively low because only two rough interfaces are involved. Besides, the parameters used for simulations are as follows: n₀ = 1.0, n₁ = 1.3, and n_s = 1.52. e₁ = λ/4 at i₀ = 0° and λ₀ = λ = 633 nm. For simplifying the calculation, we suppose that the interfaces are described by a one-dimension Gaussian autocovariance function:

A C V (x) = σ_{t o t}^{2} \exp [- {(x / l_{c})}^{2}],

where l_c/λ = 1.

The results of ARS calculations using the NSS theory for the single layer coating at different roughness levels (σ_tot/λ = 0.01, 0.05, 0.1, and 0.5) are given in Fig. 7. It is very intuitive that the intensity of scattering at the near specular direction increases firstly with the increasing of the roughness and then decreases slightly with the further increasing of the roughness, while the large angle scattering keeps increased with the increasing roughness. This redistribution of power is in conformity with the law of energy conservation.

Fig. 7 Angular surface scattering in reflection for a single layer coating at normal illumination.

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Moreover, the curves of ARS calculated by the NSS, BVS, and GHS theories for the single layer coating are plotted in Fig. 8.

Fig. 8 Results of (a) ARS and (b) normalized ARS calculations for the single layer coating with different interfacial structures represented as different roughness values.

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In Fig. 8(a1), for the smoothest interface case under investigations (σ_tot/λ = 0.01), i.e. the condition that the interfaces are smooth enough is satisfied, the prediction of the NSS agrees well with the BVS prediction. In theories of single surface scattering, the criterion to evaluate whether or not an optical surface is smooth is [29]: σ_totcosθ_i/λ≤ 0.02. We suppose that this criterion is valid likewise for interfaces inside a multilayer. The GHS theory predicts the higher ARS curve than the NSS and BVS theories, which implies that it is possible to reduce scattering in multilayers by taking a proper design. The reason why the scattering is reduced may be that the optical factors, c_ik^± , at different interfaces are in opposite phase, which leads to a destructive interference effect [20].

Figure 8(a2) for σ_tot/λ = 0.05 shows that the BVS predicts slightly stronger scattering than the NSS over all scattering angles. This is reasonable because the smooth interface lamination is slightly violated. Except this, the prediction in Fig. 8(a2) is quite similar to that in Fig. 8(a1) in terms of the curve trend. With further increasing of the roughness in Fig. 8(a3) for σ_tot/λ = 0.1 and Fig. 8(a4) for σ_tot/λ = 0.5, the differences between the BVS and NSS increase quickly. The reason is that the ARS predicted by the BVS is proportional to the PSD of interface and thereby to the square roughness while the predictions of the NSS at low scattering angles are decreased especially at large values of roughness. As a result, the BVS model predicts the highest but relatively reasonable curve in Fig. 8(a3), while it is notably failed for the BVS prediction in Fig. 8(a4) because of its TS value greater than one.

Although the smooth interface approximation is violated seriously in Fig. 8(a4), the reason why we consider that the NSS theory show reasonable accuracy is based on the fact that the NSS results should be close to those of the GHS. The reflection scattering of a multilayer is the superposition of the two components: the first consisting of reflection scattering of the top-surface and the second consisting of portion of transmission scattering of the top-surface that is then reflected back to the incident medium. Due to low reflectance in the single low-index layer coating, the scattering level is predominated by the reflection scattering at the top-surface, while the contribution of the other interfaces is relatively low. Moreover, it is noticed that the results in Figs. 8(a1)-8(a3) are consistent with the conclusion of [30] that a scattering-reduction effect can be obtained by coating a rough surface with an antireflection layer. However, the case of quite rough surfaces was not discussed in [30], and according to the results in Fig. 8(a4), we think that this method of reducing scattering is not suitable for very rough surfaces.

To seek how the ARS varies with the roughness, the results of normalized ARS are shown in Fig. 8(b). In Fig. 8(b1), similar to Fig. 8(b3), the width of the calculated curve increases with the increasing roughness. However, in Fig. 8(b2), the BVS result does not rely on the roughness. This means that the variation of roughness only causes the change in the magnitude of the ARS predicted by the BVS without affecting its profile. Besides, Fig. 8(b4) for the case of θ₀ = 45° and l_c = 2λ shows that different roughness levels lead to slightly shift in the location of ARS curve peaks corresponding to the specular direction.

4.2 24-layer dielectric mirror

To reveal the interface roughness dependency of the validity of the first-order NSS theory, it is necessary to discuss the influence of the roughness on the scattering properties of a system with several layers. The results of ARS and normalized-ARS for the 24-layer mirror are shown in Figs. 8(a) and 8(b), respectively. Note that a 1-D Gaussian ACV function was used to describe the interfaces within the mirror for simplicity purpose.

Figure 9(a1) shows that the NSS and BVS results agree well with each other, as expected. Since the validity of the BVS theory is largely restricted to smooth interface criteria, the differences between the BVS and NSS predictions are dramatically increased with the increasing roughness until extremely remarkable, as shown in Figs. 9(a2)-9(a4). Regarding the GHS, its prediction in Fig. 9(a) is much smaller than the predictions of the NSS and GHS, which differs from that in Fig. 8(a). This can be explained by the fact that the reflectance of the mirror is rather higher than that of the single layer coating, leading to scattering from interface i ≠ 0 that contributes mainly to scattering level instead of scattering from interface i = 0. Hence, we conclude that the NSS theory predicts ARS curves in reasonable values in Fig. 9. Also, it can be summarized some features for scattering of a mirror from Fig. 9(b). In Fig. 9(b2), similar to Fig. 8(b2), the predicted curve is not related to the roughness. In Fig. 9(b1), the angular width of the predicted curve decreases with the increasing of the roughness, which is opposite to that in Fig. 9(b3). In general, rougher surfaces contain larger slopes of the surfaces leading to wider angle ranges of scattering, which is for the case of single surface scattering. In Figs. 8(b1), for a surface having a single low index layer, the presence of this effect may be due to the predomination of the top-surface scattering for the reflected scattering. However, in Fig. 9(b1), for a surface with a high-reflection film, the opposite effect may be caused by the fact that the scattered field of which the propagation is along the specular direction is easier to be reflected back into the incident medium. Besides, in the case of the oblique incidence shown in Fig. 9(b4), there are still shift in the location of the normalized ARS curve peak.

Fig. 9 Results of (a) ARS and (b) normalized ARS calculations for the 24-layer mirror.

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5. Conclusion

In conclusion, a novel first-order nonparaxial scalar scattering theory for multilayer coatings in which the interfaces are assumed to be rough was proposed. In this theory, neglecting multiple scattering and based on the modified BK model, analytic expressions for the ARS of a typical p-layer design were derived. The theory, as a result of scalar treatment, first gives access to the angular scattering for multilayer coatings and does not contain a prior restrictive hypothesis for the correlation degree of the various interfaces. Moreover, we show that the formulas in the single surface approximations are exactly identical to those given by the GHS theory and they in the smooth-surface approximation are in the same form as those given in the BVS theory. Numerical graphs for three representative design types are plotted to confirm the validity of the NSS theory for multilayer coatings assuming slightly rough interfaces. Results indicate that the NSS theory is valid especially for small incident angles and large correlation lengths. Also, the simulations for two multilayer design types at four different roughness levels reveal that the first-order NSS theory enable to predict the magnitude or regularity of scattering in multilayer coatings with moderately rough interfaces, which demonstrates its unique advantage compared with typical vector and scalar theories. Besides, the prediction of the NSS theory fits measured data roughly for a 24-layer mirror with smooth interfaces. However, the NSS theory is first-order limited since multiple scattering is neglected, which makes it quite difficult to verify the usability of the NSS theory in the case of moderate roughness by experiment, which is the aim of our future work.

Funding

Sichuan Science and Technology Program (Grant No. 2018JY0550) and Anhui Province Key Laboratory of Non-Destructive Evaluation, Hefei ZC Optoelectronic Technologies Ltd (Grant No. CGHBMWSJC06).

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Application of first-order nonparaxial scalar theory to determine surface scattering intensity of multilayer optical coatings

Abstract

1. Introduction