Interference suppression of light backscattering through oblique deposition of high-reflectivity multilayers: a theoretical analysis

Jinlong Zhang; Shenghuan Fang; Igor V Kozhevnikov; Xinbin Cheng; Zhanshan Wang

doi:10.1364/OE.404097

1. Introduction

Radiation scattering due to surface and interface roughness is a crucial issue in advanced optical systems. In recent decades, considerable advances have been made to improve polishing and deposition technologies that have helped to minimize scattering by mirrors. State-of-the-art mirror fabrication techniques are discussed briefly in [1] with respect to mirrors operating at wavelengths spanning the visible to X-ray regions.

However, even the impressive optical performance of atomically smooth mirrors may not satisfy the needs of modern optical devices. Perhaps the best-known example in visible optics is the problem of backscattering toward the incident beam in laser gyroscopes. Backscattering may result in a strong coupling of contra-propagating gyro modes, and thus lead to potential errors in the rotation rate measurements and the total disappearance of the gyroscopic effect [2–4]. Ideally, the backscattering intensity should tend toward zero.

As current deposition technologies cannot provide a low enough backscattering intensity, it is necessary to analyze alternative approaches to facilitate scattering suppression. Justifications stating that the roughness of interfaces in single films and multilayer mirrors is replicated over a wide interval of spatial frequencies are manifold [5–9]. Therefore, a new possibility for scattering reduction arises: designing multilayer structures that enable the destructive interference of coherent waves scattered from correlated interfaces [5,10,11]. The idea was firstly suggested by C. Amra et al. in [10].

However, in a previous study [1], we demonstrated a fundamental problem preventing backscattering suppression in the direction toward the incident beam when interfacial roughness is totally conformal (identical); namely, that the only possible way to decrease the backscattering intensity is to decrease the mirror reflectivity. This is because the conditions conducive to backscattering suppression toward the incident beam are identical to those for the destructive interference of the waves specularly reflected by different interfaces.

Nevertheless, a modified approach exists that allows this problem to be solved; namely, the use of oblique deposition to skew roughness replication within the multilayer. In this case, the conditions for the destructive interference of the scattered waves are changed relative to the case of conformal roughness, producing an asymmetric scattering pattern even when the impinging beam is at normal incidence. This effect has been firstly proposed for the scattering suppression by C. Amra in [7], and has been observed in the visible [7] and extreme ultraviolet [12,13] regions.

The interference suppression of light backscattering toward the incident beam was analyzed theoretically and verified experimentally in our recent work [1], for which we studied a set of SiO₂-on-Ta₂O₅ bi-layers deposited onto glass substrates at different deposition angles. The bi-layers were designed to enable the observation of the total backscattering suppression of radiation (633 nm wavelength, 45° incident angle) at the scattering angle θ_s = −35°. This value of θ_s was chosen because we were unable to measure scattering in the direction towards the incident beam as it was blocked by the detector. Our experiments demonstrated that the scattering by the bi-layer fabricated at a deposition angle of 7.7° was suppressed by a factor of approximately 30 compared with the scattering by the sample deposited at normal incidence. The accuracy of the scattering suppression measurements was limited by the background noise associated with the measurements.

The paper [1] attracted academic interest as the demonstration of the interference suppression of the backscattering through oblique deposition of a bi-layer, the simplest case of a layered structure. The goal of the current study is to perform a theoretical analysis of the total scattering suppression toward the incident beam for the case of high-reflectivity multilayer mirrors, which will illustrate whether the effect can be applied practically in laser gyroscopes. Several multilayer mirror designs were analyzed to identify the design that offers the maximum practical benefit from a technological perspective.

2. General formulae of light diffraction from a multilayer with skewed roughness replication

The measured angle-resolved scattering (ARS) is defined as the ratio of the radiation power dQ_s, scattered into a small solid angle dΩ, to the incident power Q. The ARS is expressed via the amplitude of the scattered wave A as

(1)$$\textrm{ARS}({\theta _s},{\theta _i}) = \frac{1}{{{Q_i}}}\frac{{d{Q_s}}}{{d\Omega }} = \frac{{\left\langle {|A({\theta_s},{\theta_i}){|^2}} \right\rangle }}{{S \cdot \cos {\theta _i}}},$$

where the scattered field in a vacuum is written as ${E_s}(r \to \infty ) = A \cdot \exp (ikr)/r$, k = 2π/λ is the wave number in a free space, S is the illuminated area on the sample surface, triangular brackets indicate ensemble averaging, and the incidence and scattering angles ${\theta _i} \in [0,\pi /2]$ and ${\theta _s} \in [0,\pi /2]$ are measured from the Z-axis in anticlockwise and clockwise directions, respectively, in order that they are positive for the scattering geometry G1, shown in Fig. 1, and negative for geometry G2. Our approach considers light scattering in the incidence plane only, and thus will not take into account the azimuth scattering angle.

Fig. 1. Schematic of the multilayer mirror fabricated with oblique deposition. The multilayer consists of the main multilayer structure (ML) containing N bi-layers as well as additional bi-layer (design A) or tri-layer (designs B–D). Designs B–D differ in the tri-layer composition. Scattering geometries of contra-propagating waves (G1 and G2) are also shown. Materials with high, low, and medium optical density are denoted as H, L, and M, respectively, with the dielectric constant ε_L < ε_M < ε_H. The even interfaces that contribute significantly to the scattering are indicated in red.

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The goal of this analysis is to determine the conditions for the total scattering suppression toward the incident beam, i.e., to find multilayer mirror parameters providing $\textrm{ARS}({\theta _s} ={-} {\theta _i}) = 0$ at the fixed angle θ_i = 45°. Instead, we will consider the similar equation for the scattering amplitude $A({\theta _s} ={-} {\theta _i}) = 0$, because the direct analysis of the ARS, a quadratic function of A, is essentially more complex and is typically based on extensive computer simulations. As we demonstrate below, analyzing the scattering amplitude enables the condition for the backscattering suppression in an explicit analytic form to be obtained, resulting in a clearer understanding of the physics of the scattering suppression phenomenon.

Our analysis focuses specifically on s-polarized light (i.e., electric field vector is perpendicular to the incident plane), which is the dominant polarization state in the case of laser gyros, from a multilayer mirror deposited onto a rough substrate at oblique incidence. Several multilayer mirror designs are illustrated in Fig. 1. The main (lower) part of the multilayer mirror, corresponding to the period d = z_j₊₂ – z_j (j = 0,…, 2N-2), consists of N bi-layers, and is deposited at the deposition angle α_ML, while the uppermost bi- or tri-layer is deposited at either the deposition angle α_BL or α_TL. The incident flux of particles is assumed to be parallel to the ZY plane. To simplify the theoretical analysis, the plausible dependence of the growth angle on the deposited material is neglected. The growth angle is measured from the Z-axis in the clockwise direction, in order that β_ML > 0 and β_BL < 0 in Fig. 1. The boundaries between the layers are numbered in ascending order from the top of the sample to the bottom, and the interface z = z₀ = 0 corresponds to the top of the main multilayer in all designs studied.

The jth interface relief is described by the stochastic function $z = {z_j} + {\zeta _j}(\vec{\rho }),$ with the averaged value $\left\langle z \right\rangle = {z_j},$ where $\vec{\rho } = (x,y)$ is the 2D vector in the XY plane. For simplicity, we neglected the surface relaxation during the deposition process, as well as the development of intrinsic film roughness uncorrelated with the substrate relief. Then, a feature on an underlying surface is simply shifted along the Y-axis by $\Delta y = {d_j}\tan {\beta _j}$ during the deposition of jth layer with thickness d_j, as is shown in Fig. 1. Therefore, the reliefs of two neighboring interfaces are interrelated as follows:

(2)$${\zeta _{j - 1}}(\vec{\rho }) = {\zeta _j}(\vec{\rho } - {\vec{n}_y}{d_j}\tan {\beta _j}),\quad {\beta _j} = \left\{ {\begin{array}{c} {{\beta_{ML}}\textrm{, if }j = 1,\; \ldots 2N}\\ {{\beta_{BL}}\textrm{, if }j = 0, - 1} \end{array}} \right.,\quad {d_j} = {z_j} - {z_{j - 1}},$$

where ${\vec{n}_y}$ is the unit vector along the Y-axis, and the function ${\zeta _{2N}}(\vec{\rho }) \equiv {\zeta _{sub}}(\vec{\rho })$ describes the substrate surface. Subsequently, the amplitudes of the Fourier harmonics of the jth interface relief $\zeta _j^F(\vec{\nu })$ are written in the following form:

(3)$$\begin{aligned} &\zeta _{2j}^F(\vec{\nu }) = \;{e ^{i{\eta _{ML}} \quad (N - j)}}\zeta _{sub}^F(\vec{\nu }) = {e ^{ - i\eta j}}\zeta _{ML}^F(\vec{\nu }),\quad j = 0,\ldots ,N,\\ &\zeta _{2j - 1}^F(\vec{\nu }) = {e ^{i\eta _2^{ML}}}\zeta _{2j}^F(\vec{\nu })\;,\quad j = 1,\ldots ,N,\\ &\zeta _{ - 2}^F(\vec{\nu }) = {e ^{i\eta _{ - 2}^{BL}}}\zeta _{ - 1}^F(\vec{\nu }) = {e ^{i{\eta _{ML}}}}\zeta _{ML}^F(\vec{\nu }),\\ &{\eta _{ML}} = \eta _1^{ML} + \eta _2^{ML} ={-} 2\pi {\nu _y}d\tan {\beta _{ML}}\,,\quad \eta _j^{ML} ={-} 2\pi {\nu _y}{d_j}\tan {\beta _{ML}},\\ &{\eta _{BL}} = \eta _1^{BL} + \eta _2^{BL} ={-} 2\pi {\nu _y}d\tan {\beta _{BL}},\quad \eta _j^{BL} ={-} 2\pi {\nu _y}{d_j}\tan {\beta _{BL}}, \end{aligned}$$

where $\vec{\nu } = ({\nu _x},{\nu _y})$ is the 2D spatial frequency, while $\zeta _{sub}^F(\vec{\nu })$ and $\zeta _{ML}^F(\vec{\nu }) \equiv \zeta _0^F(\vec{\nu })$ are the Fourier harmonic amplitudes of the substrate surface and the top surface of the main multilayer deposited at the angle α_ML, respectively. In addition, we introduced the parameters η_ML and η_BL to shorten the formulae. These parameters characterize the shift of the Fourier-harmonic phase after the deposition of the single multilayer period or the uppermost bi-layer.

Assuming abrupt interfaces between layers and applying first order perturbation theory to the roughness height, the total scattering amplitude A can be written as a sum of the partial amplitudes corresponding to each interface [1] and, then, subdivided as a sum of the scattering amplitudes corresponding to the lower multilayer and the uppermost bi-layer:

(4)$$A({\theta _s},{\theta _i}) = {A_{ML}}({\theta _s},{\theta _i}) + {A_{BL}}({\theta _s},{\theta _i}),\quad \sin {\theta _s} = \sin {\theta _i} + \lambda {\nu _x},$$

(5)$${A_{ML}}({\theta _s},{\theta _i}) = \frac{{{k^2}}}{{4\pi }}\sum\limits_{j = 0}^{2N} {({\varepsilon _j} - {\varepsilon _{j + 1}}){E_j}({\theta _i}){E_j}({\theta _s})} \;\zeta _j^F(\vec{\nu }),$$

(6)$${A_{\,BL}}({\theta _s},{\theta _i}) = \frac{{{k^2}}}{{4\pi }}\sum\limits_{j ={-} 2}^{ - 1} {({\varepsilon _j} - {\varepsilon _{j + 1}}){E_j}({\theta _i}){E_j}({\theta _s})\,\zeta _j^F(\vec{\nu })} ,$$

where the scattering and incident angles are interrelated via the diffraction grating equation, ${E_j}(\theta ) \equiv E({z_j},\theta )$ is the field amplitude at the jth interface, and the function E(z,θ) is the solution of the wave equation describing the wave propagation inside a perfectly smooth multilayer, which accounts for the multitude of specular reflections of the waves scattered by the interfaces, while neglecting re-scattering effects.

The same treatment is applied to the scattering amplitude from the tri-layer. Then, the ARS [Eq. (1)], which can only be measured experimentally, is written in the following form:

(7)$$\begin{aligned} &\textrm{ARS}({\theta _s},{\theta _i}) = \frac{{{k^4}}}{{{{(4\pi )}^2}\cos {\theta _i}}}\sum\limits_{j,l ={-} 2}^{2N} {{F_j}F_l^\ast PS{D_{jl}}(\vec{\nu })} \;,\quad {F_j} = ({\varepsilon _j} - {\varepsilon _{j + 1}}){E_j}({\theta _i}){E_j}({\theta _s}),\\ &\qquad \qquad\textrm{ PS}{\textrm{D}_{jl}}(\vec{\nu }) = \textrm{PS}{\textrm{D}_{sub}}(\vec{\nu }) \cdot \exp \left( {i\sum\limits_{n = j}^{2N - 1} {{\eta_n}} - i\sum\limits_{m = l}^{2N - 1} {{\eta_m}} } \right), \end{aligned}$$

which considers Eq. (4) and the relation between the power spectral density (PSD) and the Fourier harmonics of rough interfaces, i.e., $\left\langle {\zeta_j^F(\vec{\nu }){{({\zeta_l^F(\vec{\nu }^{\prime})} )}^ \ast }} \right\rangle = \textrm{PS}{\textrm{D}_{jl}}(\vec{\nu })\;\delta ({\vec{\nu } - \vec{\nu }^{\prime}} )$, with the asterisk indicating the complex conjugate. PSD_sub denotes the PSD-function of the substrate roughness.

Next, we calculated the scattering amplitude [see Eq. (5)] corresponding to the main multilayer structure in an explicit form. To achieve this, we analyzed the case of scattering suppression toward the incident beam, i.e. setting θ_i = -θ_s = 45°, and the selection of the multilayer structure consisting of quarter-wavelength (QW) layers, i.e. the thickness of jth layer is set to ${d_j} = \lambda /4/{({\varepsilon _j} - {\sin ^2}{\theta _i})^{1/2}}$. Several multilayer mirror designs were considered, with the dielectric constant of high (H) and low (L) optical density materials denoted as ε_H and ε_L, respectively. The uppermost layer of the main multilayer consists of L-material in all designs studied.

A typical feature of high-reflectivity QW multilayers is that the field intensity is maximal at H-on-L interfaces (i.e. even numbered interfaces in all designs shown in Fig. 1) and it is extremely low at L-on-H interfaces. Therefore, the contribution of odd interfaces to the scattering amplitude can be neglected [see Eq. (6)]. Using the characteristic matrix that describes a single period in a multilayer comprising QW layers (for example, see Eq. (86) in monograph [14]) we obtain the following relation between the electric field amplitudes at two interfaces of the bi-layer:

(8)$$\frac{{{E_{j - 2}}({\theta _i})}}{{{E_j}({\theta _i})}} ={-} \frac{{{\kappa _j}}}{{{\kappa _{j - 1}}}}\;,\quad {\kappa _j} = \frac{{2\pi }}{\lambda }\sqrt {{\varepsilon _j} - {{\sin }^2}{\theta _i}}$$

and thus, we can express the field amplitude at all even interfaces of the main multilayer in terms of the amplitude on the top interface z = z₀:

(9)$${\left( {\frac{{{E_{2j}}({\theta_i})}}{{{E_0}(\theta {}_i)}}} \right)^2} = {\left( {\frac{{{\kappa_L}}}{{{\kappa_H}}}} \right)^{2j}} = {\left( {\frac{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}}} \right)^j},\quad j = 0, \ldots ,N,\quad {\kappa _{H,L}} = \frac{{2\pi }}{\lambda }\sqrt {{\varepsilon _{H,L}} - {{\sin }^2}{\theta _i}} .$$

Considering Eq. (4) for the Fourier harmonics at the even interfaces, we calculated the sum in Eq. (5) in an explicit form:

(10)$$\begin{aligned} &\qquad \qquad\textrm{ }{A_{ML}}({\theta _s} ={-} {\theta _i} = {45^ \circ }) \approx \frac{{{k^2}}}{{4\pi }}({\varepsilon _H} - {\varepsilon _L})\;\zeta _{ML}^F(\vec{\nu })\sum\limits_{j = 0}^N {E_{2j}^2({\theta _i}){\textrm{e}^{ - i\eta j}}} \\ &= \frac{{{k^2}}}{{4\pi }}({\varepsilon _H} - {\varepsilon _L})E_0^2({\theta _i})\;\zeta _{ML}^F(\vec{\nu }){\sum\limits_{j = 0}^N {\left( {\frac{{\kappa_L^2}}{{\kappa_H^2}}{\textrm{e}^{ - i\eta }}} \right)} ^j} \approx \frac{{{k^2}}}{{4\pi }} \cdot \frac{{({\varepsilon _H} - {\varepsilon _L})E_0^2({\theta _i})\;\zeta _{ML}^F(\vec{\nu })}}{{1 - {\textrm{e}^{ - i\eta }}\kappa _L^2/\kappa _H^2}}, \end{aligned}$$

where, for simplicity, we continued the geometrical progression (sum on j) up to infinity, which is reasonable, because the field intensity is extremely low at the interfaces located deep within the multilayer structure and, therefore, these interfaces contribute negligibly to the total scattering.

The next section considers specific features of the multilayer mirror designs illustrated in Fig. 1.

3. Mirror design with a topmost bi-layer

The scattering amplitude from the bi-layer located at the top of the main multilayer (design A in Fig. 1) can be expressed, similarly, via the field at the top of the main multilayer and the Fourier harmonic of its relief:

(11)$${A_{\,BL}}({\theta _s} ={-} {\theta _i} = {45^ \circ }) \approx \frac{{{k^2}}}{{4\pi }}(1 - {\varepsilon _L})E_0^2({\theta _i}){\left( {\frac{{{\kappa_H}}}{{{\kappa_L}}}} \right)^2}{\textrm{e}^{i{\eta _{BL}}}}\zeta _{ML}^F(\vec{\nu }).$$

Then, the condition for the scattering suppression can be written as:

(12)$$\begin{aligned} A({\theta _i} = &-{\theta _s} = {45^ \circ }) = \frac{{{k^2}}}{{4\pi }}({1 - {\varepsilon_L}} )\cdot E_0^2({\theta _i}) \cdot \frac{{\kappa _H^2}}{{\kappa _L^2}} \cdot \frac{{{e ^{2i{\eta _{BL}}}}}}{{1 - {e ^{ - i{\eta _{ML}}}}\kappa _L^2/\kappa _H^2}} \cdot \zeta _{ML}^F(\vec{\nu })\\ &\times \underbrace{{\left[ {1 - \frac{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}}\left( {{\textrm{e}^{ - i{\eta_{ML}}}} + \frac{{{\varepsilon_H} - {\varepsilon_L}}}{{{\varepsilon_L} - 1}}{\textrm{e}^{ - i{\eta_{BL}}}}} \right)} \right]}}_{F} = 0. \end{aligned}$$

The condition expressed in Eq. (12) is fulfilled if the expression F (enclosed in square brackets) is equal to zero. If we denote η_ML = x and η_BL = y, this expression is written in the form of Eq. (33) (see Appendix). Next, the solutions of Eq. (12) are written in the following form [see Eqs. (35) and (38)]:

(13)$${\eta _{ML}} = \arccos \left\{ {\frac{1}{2}\frac{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}}\left[ {1 - {{\left( {\frac{{{\varepsilon_H} - {\varepsilon_L}}}{{{\varepsilon_L} - 1}}} \right)}^2}} \right] + \frac{1}{2}\frac{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}} \right\},$$

(14)$${\eta _{BL}} ={-} \arcsin \left( {\frac{{{\varepsilon_L} - 1}}{{{\varepsilon_H} - {\varepsilon_L}}}\sin {\eta_{ML}}} \right),$$

and the growth angles β_ML and β_BL are found via Eq. (15):

(15)$$\tan {\beta _{ML}} = \frac{{\lambda {\eta _{ML}}}}{{4\pi ({d_H} + {d_L})\sin {\theta _i}}}\,,\quad \tan {\beta _{BL}} = \frac{{\lambda {\eta _{BL}}}}{{4\pi ({d_H} + {d_L})\sin {\theta _i}}},$$

where d_H and d_L are the thicknesses of the H- and L-layers, respectively. Only solutions with positive η_ML and β_ML are considered. According to the analysis performed in the Appendix, a second possible solution of Eq. (12) is obtained if we replace η_ML and η_BL with -η_ML and -η_BL. Note that the solutions given by Eqs. (13)–(15) provide the minimal absolute value of the growth angles (see Appendix).

Hereafter, SiO₂ and BK7 glass are assigned as the low optical density (L) and substrate materials, respectively. Choosing the solutions of Eqs. (13)–(15) with a positive β_ML, we calculated the dependence of the growth angles on the refraction index n_H of the H-material. The result is presented in Fig. 2. While the solutions of Eqs. (13) and (14) have physical meaning [Eq. (37)] in the limited interval of the ${n_H} = \sqrt {{\varepsilon _H}}$ values only, this interval is sufficiently wide in the case considered, extending up to ${n_H} \approx 3.18$. The refractive indices of HfO₂ and Ta₂O₅ are shown in Fig. 2 to illustrate the feasible values of growth angles for two materials that are widely used in visible optics. The refractive indices at λ = 633 nm of a selection of materials commonly utilized in multilayer mirror fabrication and their corresponding geometrical layer thicknesses in a QW multilayer structure are listed in Table 1. The optimal growth angles that result in the total suppression of backscattering toward the incident beam are presented in Table 2 for several multilayer mirrors that were considered in our analysis. The number of bi-layers used in the main multilayer, indicated in the raw “Composition”, was chosen to produce an approximately equivalent reflectivity for all the mirror designs, in order that $1 - R = (2.5 \div 3.2) \cdot {10^{ - 6}}$.

Fig. 2. Optimal growth angles for design A in Fig. 1 as a function of the refractive index of the H-material, with SiO₂ used as the L-material. The solution corresponding to a positive β_ML is shown.

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Table 1. Geometrical layer thickness (in nm) and refractive index (at λ = 633 nm) of several materials used in high-reflection multilayer mirror fabrication

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Table 2. Material composition and growth angles (in degrees) of multilayer mirrors with additional bi- or tri-layers on the top designed to suppress scattering toward the incident beam.

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The calculated ARSs for SiO₂/Ta₂O₅ and SiO₂/HfO₂ multilayer mirrors (A1 and A2 in Table 2) are shown in Fig. 3. The ARS was calculated using the computer program described in [15]. We applied the following model of the two-dimensional PSD function of an isotropic substrate when calculating the ARS with Eq. (7):

(16)$$\textrm{PS}{\textrm{D}_{sub}}\textrm{(}\nu \textrm{)} = \frac{{{\sigma ^2}{\xi ^2}h}}{{\pi {{({1 + {\xi^2}{\nu^2}} )}^{1 + h}}}},$$

where the root-mean-squared (rms) roughness σ = 3 nm, the correlation length ξ = 100 µm, and the Hörst parameter h = 0.5. Note that the conditions of the interference suppression of the backscattering are independent of the type and parameters of the PSD-function.

Fig. 3. ARSs in the incident plane for SiO₂/Ta₂O₅ (A1 in Table 1) and SiO₂/HfO₂ (A2) mirrors calculated for the G1 (curve 1) and G2 (curve 2) scattering geometries. For comparison, the ARS for the similar mirrors, but fabricated using deposition at normal incidence is also shown (curve 3).

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The ARSs from the SiO₂/Ta₂O₅ and SiO₂/HfO₂ mirrors (A1 and A2 in Table 2, respectively) are shown in Fig. 3. Curve 1 demonstrates the total suppression of the backscattering at θ_s = -θ_i = −45° by the mirrors fabricated with oblique deposition. For comparison, the curve 3 shows the ARS from the similar multilayer fabricated with normal incidence deposition.

Evidently, interference suppression is a useful property for applications in laser gyroscopes, if only it is observed at once for beams incident to mirror from both the left and the right side (geometries G1 and G2 in Fig. 1). The radiation diffraction mechanism in geometry G2 can be equated to that in geometry G1 by changing the sign of both growth angles, i.e. β_ML and β_BL. Equations (13) and (14) are invariant with respect to the change of η_ML and η_BL to -η_ML and -η_BL. Hence, if the pair of angles β_ML and β_BL are a solution of Eq. (12), the pair -β_ML and -β_BL having opposite sign is another solution of this equation, and thus the A1 and A2 mirror designs provide the total backscattering suppression for both the G1 and G2 diffraction geometries. Indeed, curve 2 in Fig. 3 shows the ARS from these mirrors in the G2 geometry, demonstrating zero backscattering intensity for both contra-propagating waves, although we did not consider the G2 geometry when designing mirrors A1 and A2.

This result is explained by the fact that all parameters in the expression F [enclosed by the square brackets in Eq. (12)] are real numbers. Therefore, the transition from the G1 geometry to the G2 geometry (i.e. change of η_ML and η_BL sign) occurs via the complex conjugation of F. Evidently, if F = 0, then F * = 0 as well. In turn, the real values of the parameters in expression F are the direct consequence of the fact that the ratio of the electric field at two subsequent even or odd interfaces is a real number [Eq. (8)].

Next, the ARS curves corresponding to the G1 and G2 geometries are both totally symmetric for the SiO₂/Ta₂O₅ mirror (A1), while this is not the case for SiO₂/HfO₂ mirror (A2), for which the difference in the shape of the curves is observed clearly for scattering angles |θ_s| < 20°. This behavior can be explained using Fig. 4. As the dielectric constant of HfO₂ is less than that of Ta₂O₅, the angular interval in which the reflectivity resonance occurs is narrower for the mirror containing HfO₂. Indeed, Fig. 4(a) shows that the resonant reflection region of the SiO₂/Ta₂O₅ mirror extends over the entirety of the angular interval spanning from 0 to 90°, and therefore, according to the general theory of linear differential equations with periodic coefficients [16], the imaginary part of the ratio of the field amplitudes at two even interfaces is negligibly small for any incident angle [Fig. 4(b)]. This means that the transition from the G1 geometry to the G2 geometry, i.e. the change of the sign of the growth angles β_ML and β_BL, does not change the absolute value of the scattering amplitude, and thus the ARS remains unchanged.

Fig. 4. Reflectivity (a) and the imaginary part of the ratio of the field amplitudes (b) at two subsequent even interfaces for the SiO₂/Ta₂O₅ (A1 in Table 1) and SiO₂/HfO₂ (A2) mirrors as a function of the angle of incidence.

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However, this is not the case for the SiO₂/HfO₂ mirror. In this case, the incident angles for which resonance occurs are limited by the condition |θ_i| > 20° and the reflectivity drops sharply at small incident angles [Fig. 4(a)]. The ratio of the field amplitudes ${E_{2j}}({\theta _s})/{E_0}({\theta _s})$ becomes a complex function here [Fig. 4(b)], to ensure that the change of the sign of the growth angles β_ML and β_BL is not equivalent to the complex conjugation of the expression for F, which would result in asymmetry of the ARS curves for the G1 and G2 scattering geometries for |θ_i| < 20°.

Finally, we discuss the geometrical interpretation of the scattering suppression toward the incident beam. In the first-order perturbation theory approximation, the total scattering amplitude is merely the sum of the partial scattering amplitudes $\sum {{A_j}}$ from the individual interfaces [see Eqs. (4)–(6)], where

(17)$${A_j}\sim ({\varepsilon _j} - {\varepsilon _{j + 1}})E_j^2({\theta _i})\exp \left( {i\sum\limits_{n = j}^{2N - 1} {{\eta_n}} } \right).$$

In essence, the suppression of backscattering means that the sum of these partial scattering amplitudes is equal to zero at θ_s = -θ_i because of destructive interference of the scattered waves.

Following the approach adopted in our previous work [1], we consider the scattering amplitudes as vectors on a complex A-plane. Then, the zero sum signifies that these vectors form a closed polygon. The vector length |A_j| decreases quickly with the increasing depth of the multilayer structure, so that substantial contributions to the total scattering amplitude are limited to the several uppermost even interfaces only. The moduli of A_j for the SiO₂/Ta₂O₅ and SiO₂/HfO₂ mirrors (A1 and A2 in Table 2) are shown by the red and blue data, respectively, in Fig. 5(a). The partial scattering amplitudes at a complex plane are shown in Fig. 5(b) for the SiO₂/HfO₂ mirror. The vector length coincides with that in Fig. 5(a). The point O in Fig. 5(b) indicates the scattering amplitude from the substrate. Each subsequent vector A_j is added to the end of the vector preceding it and is turned by the angle η_j, which is counted from the real axis in an anticlockwise direction. The point B corresponds to the total scattering amplitude from the main (lower) multilayer, while the vector BO represents the scattering amplitude contribution from the upper bi-layer. Subsequently, the vector BO is turned by the angle π + η_BL (where η_BL < 0), because the difference ${\varepsilon _j} - {\varepsilon _{j + 1}}$ in Eq. (17) is positive at all interfaces except the uppermost one, where the difference $1 - {\varepsilon _L} < 0$ results in an additional turning angle π. This process explains the small absolute value of the angle η_BL = −29°, and thus the feasible value of the growth angle β_BL = −8.24°.

Fig. 5. (a) Absolute value of the partial scattering amplitudes from each interface of the SiO₂/Ta₂O₅ (curve 1) and SiO₂/HfO₂ (curve 2) mirrors (A1 and A2 in Table 2), and (b) geometrical interpretation of the backscattering suppression phenomenon based on a representation of partial scattering amplitudes as vectors on a complex A-plane (for the case of the SiO₂/HfO₂ mirror).

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Consequently, we designed SiO₂/Ta₂O₅ and SiO₂/HfO₂ mirrors with a topmost SiO₂ layer (samples A1 and A2 in Table 1), which provided total backscattering suppression toward the incident beam for both contra-propagating waves. The mirrors were characterized by practicable values of the growth angles β_ML and β_BL.

However, these designs are not without shortcomings. First, the electric field intensity at the interfaces of the QW multilayer with a topmost L-layer is several times higher than that in a multilayer with a topmost H-layer. This circumstance results in an increase in the radiation absorption, i.e. a decrease in the mirror’s reflectivity, and the enhancement of the scattering by intrinsic film roughness uncorrelated with the substrate roughness. The latter may be a crucial factor resulting in the decay or even the total disappearance of backscattering suppression due to interference [1]. Next, following surface cleaning, surfaces exposed to air become contaminated within several hours. The contamination consists both of dust particles and the natural adhesion layer (∼ 1 nm thickness and ∼ 1 g/cm³ density), which comprises mainly hydrocarbons, other organics, and water molecules glued to the surface [17]. As the maximal field intensity occurs at the uppermost mirror interface, both additional radiation absorption and parasitic scattering intensity achieve maximal values for mirrors whose topmost layer comprises an L-material. Therefore, we adapted the mirror designs by incorporating a topmost layer comprising an H-material, thus creating a tri-layer at the top of the multilayer structure.

4. Mirror design with a topmost tri-layer

First, we consider design B in Fig. 1, where the tri-layer consists of the same materials as the main multilayer. Then, the partial scattering amplitude from the tri-layer has the following form:

(18)$${A_{\,TL}}({\theta _s} ={-} {\theta _i} = {45^ \circ }) \approx \frac{{{k^2}}}{{4\pi }}({\varepsilon _H} - {\varepsilon _L})E_0^2({\theta _i}){\left( {\frac{{{\kappa_H}}}{{{\kappa_L}}}} \right)^2}{\textrm{e}^{i{\eta _{TL}}}}\zeta _{ML}^F(\vec{\nu }),$$

and the condition for the scattering suppression toward the incident beam can be written in the form of Eq. (33):

(19)$$F = 1 - \frac{{{\varepsilon _L} - {{\sin }^2}{\theta _i}}}{{{\varepsilon _H} - {{\sin }^2}{\theta _i}}}({{\textrm{e}^{ - i{\eta_{ML}}}} - {\textrm{e}^{ - i{\eta_{TL}}}}} )= 0.$$

The solutions of Eq. (19), with η_ML > 0, are found via Eqs. (35) and (39):

(20)$${\eta _{ML}} = \arccos \left( {\frac{1}{2} \cdot \frac{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}} \right),$$

(21)$${\eta _{TL}} = \pi - {\eta _{ML}},$$

and, if the necessary condition (37) is fulfilled:

(22)$${\varepsilon _H} \le 2{\varepsilon _L} - {\sin ^2}{\theta _i}.$$

The condition expressed in Eq. (22) means that the argument of the arccosine in Eq. (20) does not exceed unity. The growth angles are found via Eq. (15), where η_BL is replaced by η_TL. The dependence of the optimal growth angles β_ML and β_TL on the refractive index of the H-material is shown in Fig. 6 (curves 1 and 2), where the L-material was chosen as SiO₂. Equation (22) demonstrates that the refractive index of the H-material should not exceed n_H = 1.966; therefore, HfO₂ can be used as the H-material in design B. The parameters of the HfO₂/SiO₂ mirror (design B1) are presented in Table 2.

Fig. 6. Optimal growth angles for design B in Fig. 1 as a function of the refractive index of the H-material, where the L-material is SiO₂. Design B2 differs to design B1 in that it incorporates a threefold increase in the thickness of the SiO₂ layer and the underlying H-layer in the tri-layer. The solution providing a positive β_ML is chosen.

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Figure 7 demonstrates the ARS for the QW HfO₂/SiO₂ multilayer mirror deposited at normal incidence (curve 1), while curve 2 shows the ARS for the same mirror design (B1 in Table 2) fabricated with oblique deposition. As in section 3, both growth angle pairs (β_ML, β_TL) and (-β_ML, -β_TL) are solutions of Eq. (19), i.e. the designed mirror suppresses the backscattering toward the incident beam for both contra-propagating waves completely. However, the deposition angle of the tri-layer β_TL ≈ 44° is too large for practical multilayer structure fabrication because of the columnar growth of layers and the quick development of roughness [18].

Fig. 7. ARS in the incident plane for the HfO₂/SiO₂ mirrors (designs B1 and B2 in Table 1) in the G1 (curves 2 and 3) and G2 (curve 4) scattering geometries. For comparison, the ARS for the B1 design mirror fabricated with normal incidence deposition is also shown (curve 1).

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One solution to this problem is to increase the thickness of both the SiO₂ layer and the underlying H-layer in the tri-layer by a factor of 3 compared to the values presented in Table 1. For this case, the dependence of the optimal growth angle β_TL on the refractive index of the H-material is shown in Fig. 6 (curve 3). The HfO₂/SiO₂ mirror design incorporating the aforementioned thickness increases in the SiO₂ and H-material layers is labeled as B2 in Table 2. In response to these modifications, Eqs. (20) and (21) are unchanged, but the threefold increase in the layer thicknesses results in the same decrease in the value of $\tan {\beta _{TL}}$ (i.e. the second term of Eq. (15), where d_H,L is replaced by 3d_H,L). The resulting growth angle β_TL ≈ 17.9° looks more feasible [18]. The corresponding ARSs for both contra-propagating waves are shown in Fig. 7 (curves 3 and 4); note the asymmetry in the ARS curves for contra-propagating waves occurring for |θ_s| < 20°.

In contrast to HfO₂, Ta₂O₅ is not suitable for use as the H-material for design B. This fact is illustrated by curve 1 in Fig. 8(a), which shows the absolute value of the scattering amplitude |A_j| at the interfaces of the Ta₂O₅/SiO₂ multilayer structure, i.e. the length of the vector A_j on the complex A-plane. The sum of all the scattering amplitudes |A_j| from the main multilayer is approximately 0.8 in the units indicated in Fig. 8(a), which is exceeded by the scattering amplitude from the upper tri-layer (|A₁| ≈ 1.2). Evidently, such a set of vectors cannot form a closed polygon in the complex A-plane. In contrast, |A₁| ≈ 0.96 in the case of the HfO₂/SiO₂ mirror, while the sum of all the amplitudes from the main multilayer is slightly more this value, equaling approximately 1.04. Therefore, the backscattering can be suppressed using the HfO₂/SiO₂ mirror, as illustrated by Fig. 8(b). However, the vector BO must be turned through a large angle relative to the vector AB to form a closed polygon, thus resulting in a too large growth angle β_TL. Note that, in this case, the difference ${\varepsilon _{2j - 1}} - {\varepsilon _{2j}} = {\varepsilon _H} - {\varepsilon _L}$ matches the same sign in Eq. (17), and therefore the additional angle π does not arise as was observed in Fig. 5(b).

Fig. 8. (a) Absolute value of the partial scattering amplitudes from each interface of the Ta₂O₅/SiO₂ (curve 1) and HfO₂/SiO₂ (curve 2) mirrors, and (b) geometrical interpretation of the backscattering suppression phenomenon based on a representation of partial scattering amplitudes as vectors in a complex A-plane (for the case of HfO₂/SiO₂ mirror).

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Thus, we demonstrated that the total scattering suppression toward the incident beam can be realized using a high optical density material for the topmost layer. However, the refractive index of the H-material is limited by the condition expressed in Eq. (22), implying that scattering suppression cannot be achieved with a Ta₂O₅/SiO₂ mirror. Moreover, the necessary growth angle of the upper tri-layer β_TL proved to be too large for practical fabrication of the HfO₂/SiO₂ QW mirror. Therefore, the thickness of the SiO₂ layer and underlying HfO₂ layer in the tri-layer should be increased threefold. As a result, the total thickness of mirror B2 (L ∼ 4.7 µm) exceeds that of mirror A1 (L ∼ 3.2 µm) by a factor of 1.5. This factor may be crucial because of the development of intrinsic film roughness is uncorrelated with the substrate roughness during film growth. These shortcomings arose because the scattering amplitude from the upper tri-layer [Fig. 8(a), curve 1] was sufficiently large that the cumulative scattering amplitudes from all of the other interfaces could not compensate for it. Consequently, we refined the multilayer mirror designs further in order to decrease the scattering from the upper tri-layer.

Equation (18) shows that one possible way to decrease scattering from the tri-layer is to decrease the difference between the dielectric constants of the neighboring layers. Therefore, we considered two modified designs of the tri-layer, replacing (a) the uppermost H-layer with an M-layer with the mean value of the dielectric constant ε_L < ε_M < ε_H, and (b) the L-layer inside the tri-layer with an M-layer.

In the first case (design C in Fig. 1), the partial scattering amplitude from the tri-layer is written as:

(23)$${A_{\,TL}}({\theta _s} ={-} {\theta _i} = {45^ \circ }) \approx \frac{{{k^2}}}{{4\pi }}({\varepsilon _M} - {\varepsilon _L})E_0^2({\theta _i}){\left( {\frac{{{\kappa_H}}}{{{\kappa_L}}}} \right)^2}{\textrm{e}^{i{\eta _{TL}}}}\zeta _{ML}^F(\vec{\nu }),$$

with the condition for the scattering suppression expressed using the following form:

(24)$$F = 1 - \frac{{{\varepsilon _L} - {{\sin }^2}{\theta _i}}}{{{\varepsilon _H} - {{\sin }^2}{\theta _i}}}\left( {{\textrm{e}^{ - i{\eta_{ML}}}} - \frac{{{\varepsilon_H} - {\varepsilon_L}}}{{{\varepsilon_M} - {\varepsilon_L}}}{\textrm{e}^{ - i{\eta_{TL}}}}} \right) = 0.$$

In the second case (design D in Fig. 1), we have:

(25)$${A_{\,TL}}({\theta _s} ={-} {\theta _i} = {45^ \circ }) \approx \frac{{{k^2}}}{{4\pi }}({\varepsilon _H} - {\varepsilon _M})E_0^2({\theta _i}){\left( {\frac{{{\kappa_H}}}{{{\kappa_M}}}} \right)^2}{\textrm{e}^{i{\eta _{TL}}}}\zeta _{ML}^F(\vec{\nu }),$$

and

(26)$$F = 1 - \frac{{{\varepsilon _L} - {{\sin }^2}{\theta _i}}}{{{\varepsilon _H} - {{\sin }^2}{\theta _i}}}\left( {{\textrm{e}^{ - i{\eta_{ML}}}} - \frac{{{\varepsilon_H} - {\varepsilon_L}}}{{{\varepsilon_H} - {\varepsilon_M}}} \cdot \frac{{{\varepsilon_M} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}{\textrm{e}^{ - i{\eta_{TL}}}}} \right) = 0.$$

If we set ε_M = ε_H in Eq. (24) and ε_M = ε_L in Eq. (26), these equations are transformed to the form expressed in Eq. (19). The solutions of Eq. (24), with positive η_ML (design C), have the following form:

(27)$${\eta _{ML}} = \arccos \left[ {\frac{1}{2} \cdot \frac{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}} - \frac{1}{2} \cdot \frac{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}} \cdot \left( {{{\left( {\frac{{{\varepsilon_H} - {\varepsilon_L}}}{{{\varepsilon_M} - {\varepsilon_L}}}} \right)}^2} - 1} \right)} \right],$$

(28)$${\eta _{TL}} = \pi - \arcsin \left( {\frac{{{\varepsilon_M} - {\varepsilon_L}}}{{{\varepsilon_H} - {\varepsilon_L}}}\sin {\eta_{ML}}} \right).$$

As for the previous mirror designs, the growth angles β_ML and β_TL are found via Eq. (15). The solutions of Eq. (26) are written as:

(29)$${\eta _{ML}} = \arccos \left[ {\frac{1}{2} \cdot \frac{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}} - \frac{1}{2} \cdot \frac{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_H} - {{\sin }^2}{\theta_i}}} \cdot \left( {{{\left( {\frac{{{\varepsilon_H} - {\varepsilon_L}}}{{{\varepsilon_H} - {\varepsilon_M}}} \cdot \frac{{{\varepsilon_M} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}} \right)}^2} - 1} \right)} \right],$$

(30)$${\eta _{TL}} = \pi - \arcsin \left( {\frac{{{\varepsilon_H} - {\varepsilon_M}}}{{{\varepsilon_H} - {\varepsilon_L}}}\frac{{{\varepsilon_L} - {{\sin }^2}{\theta_i}}}{{{\varepsilon_M} - {{\sin }^2}{\theta_i}}}\sin {\eta_{ML}}} \right),$$

and the corresponding growth angles β_ML and β_TL are:

(31)$$\tan {\beta _{ML}} = \frac{{\lambda {\eta _{ML}}}}{{4\pi ({d_H} + {d_L})\sin {\theta _i}}}\,,\quad \tan {\beta _{TL}} = \frac{{\lambda {\eta _{TL}}}}{{4\pi ({d_H} + {d_M})\sin {\theta _i}}}.$$

The dependence of the growth angles on the refractive index of the M-material n_M is shown in Fig. 9 for designs C and D, which use SiO₂ and Ta₂O₅ as the L- and H-materials, respectively. As Fig. 9 shows, the range of possible values of n_M is rather narrow, with HfO₂ and Al₂O₃ appropriate M-material choices for designs C and D, respectively.

Fig. 9. Optimal growth angles for designs C (a) and D (b) as a function of the refractive index of the M-material, where SiO₂ and Ta₂O₅ are the L- and H-materials, respectively. Designs C2 and D2 differ to designs C1 and D1 in that they incorporate a threefold increase in the thickness of the layers in the upper tri-layer. The solutions providing a positive β_ML are chosen.

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The ARSs for these mirrors (C1 and D1 in Table 2) are shown in Fig. 10 (curves 1). To decrease the growth angles for mirrors C1 (β_TL ≈ 46°) and D1 (β_TL ≈ 48°), the thickness of the layers placed in the upper tri-layer should be increased threefold relative to the values listed in Table 1. The dependence of β_TL on n_M for the two designs is also shown in Fig. 9 (curves 3). The growth angle β_TL decreases to 19° for mirror C2 in Table 2 for which HfO₂ was used as the M-material, and to 20° for mirror D2 for which Al₂O₃ was used as the M-material. These values are much more practicable. Figure 10 shows the ARSs for mirrors C2 and D2 for both contra-propagating waves (geometries G1 and G2). The curves 4 show the corresponding ARSs for mirrors designed using the same materials, but deposited at normal incidence. Note that the total thickness of multilayers C2 and D2 is close to that of mirror A1.

Fig. 10. ARSs in the incident plane for mirrors C1 and C2 (a), and D1 and D2 (b) for the G1 (curves 1 and 2) and G2 (curves 3) scattering geometries. For comparison, the ARSs for the mirrors with equivalent material structures to C1 and D1, but fabricated using deposition at normal incidence, are also shown (curves 4).

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Thus, we demonstrated that mirrors C2 and D2 perform best, in terms of both fabrication and practical use, because they provide scattering suppression toward the incident beam for both contra-propagating waves and, importantly, are characterized by practicable values of both growth angles, negligible electric field at the top of the mirror, and a small total thickness of the multilayer structure.

5. Discussion and conclusions

In this paper, we analyzed the problem of backscattering suppression in the direction toward the incident beam upon the reflection of light from a rough multilayer mirror fabricated using oblique deposition, in order that the interface relief is replicated at a certain angle to the sample surface. The ability to suppress backscattering is of extreme importance for laser gyroscopes, because the backscattering may result in strong coupling of the contra-propagating gyro modes, which produces errors in rotation rate measurements and can even cause the total disappearance of the gyroscopic effect. Therefore, we considered the case of s-polarized radiation (electric field vector is perpendicular to the incidence plane) at a wavelength of λ = 633 nm and an incident angle of θ_i = 45° specifically, as these are typical parameters relating to laser gyroscope applications.

Mirror designs comprised two parts: the main (lower) multilayer consisting of N identical bi-layers growing at an angle β_ML to the mirror normal, with an additional bi- or tri-layer forming the topmost section of the mirror growing at another angle β_BL (or β_TL). The optical thickness of all layers was set to λ/4 to provide maximal reflectance from the fixed number of bi-layers. We analyzed four possible multilayer mirror designs that differed with respect to the upper part of the multilayer structure (Fig. 1). We showed that for all of these designs the appropriate choice of the growth angles β_ML and β_BL (β_TL) results in the disappearance of backscattering toward the incident beam because of the destructive interference of the waves scattered from the main multilayer and from the uppermost bi- or tri-layer. Moreover, we proved that the backscattering is suppressed simultaneously for both contra-propagating waves incident onto the gyroscope mirror from opposite sides.

The parameters of the various mirror designs are collated in Table 2. In our opinion, the mirrors C2 and D2 offer the best solution both for fabrication and practical use, because they are characterized by practicable values of both growth angles, a negligible electric field at the top of the mirror, and a small total thickness of the multilayer structure. However, for realizing backscattering suppression in the fabrication of laser gyroscope mirrors, there are two crucial problems that remain unsolved. First, only the deposition angle α is well-controlled during the experiment and, to date, the relation between the growth angle β and the deposition angle α is not evident. For example, considering the simplest linear model of film growth the deposition angle is equal to the growth angle, i.e. α = β [12]. However, in columnar film growth models [18] the so-called “tangent rule” is valid, i.e. tan(α) = 2tan(β).

In our experiments, upon observing the backscattering suppression due to the SiO₂/Ta₂O₅ bi-layer [1], we found that the suppression effect is well-pronounced if the deposition angle α is close to the optimal value of the growth angle β. In contrast, Trost et al. [13] concluded that the growth of Mo/Si multilayer mirrors at oblique deposition occurred according to the “tangent rule”. Therefore, it is necessary to perform specialized experiments using oblique deposition to establish the dependence β =β(α) for materials, which will be used for gyroscope mirrors fabrication. In addition, although we assumed in our analysis that the growth angle β is independent of the layer material if the deposition angle α is fixed, we can speculate that this assumption is generally incorrect. The planned experiments investigating the oblique deposition of materials will allow us to refine this statement as well. Note that the conditions expressed in Eqs. (13), (14), (20), (21), and (27)–(30) that were imposed on the η parameters, and guaranteed that the backscattering suppression was confined entirely toward the incident beam, remain valid for the arbitrary dependence β = β(α), which, moreover, may be different for the H- and L-materials. Then, instead of using Eq. (15) or Eq. (31) to identify the growth angles β, we should determine the deposition angle α, which is the same for the H- and L-materials, from the equation

(32)$${d_H}\tan {\beta _H}(\alpha )+ {d_L}\tan {\beta _L}(\alpha )= \frac{{\lambda \eta }}{{4\pi \sin {\theta _i}}},$$

where the parameter η characterizes the shift of an interface relief along the Y-axis after the deposition of a single bi-layer. It is determined by the above-mentioned equations, and the dependences on the deposition angle α of the growth angle of the H- and L-materials (β_H(α) and β_L(α)) will be found using electron microscopy based on preliminary experiments investigating the oblique deposition of materials.

Second, there is a further crucial factor influencing the scattering suppression, which is poorly controlled in experiments. Film growth is always associated with the development of short-scale intrinsic film roughness that is uncorrelated with the substrate roughness. The incoherent scattering from uncorrelated roughness cannot be reduced in consequence of the interference effect. As demonstrated in our previous study [1], uncorrelated roughness of only a fraction of a nanometer rms height may result in a sharp increase in the backscattering, and thus the subsequent disappearance of the interference suppression effect.

Finally, our analysis is based on the first-order perturbation theory, which was largely validated by most experiments. However, the authors of the recent papers [19,20] emphasized that there are some rare exceptions where the first-order theory fails, and this includes the case of the backscattering suppression effect. Indeed in this case the first-order scattering is suppressed, but the second-order scattering may be nonzero and thus constitutes the ultimate limit of the scattering suppression.

Therefore, the experimental determination of optimal deposition parameters that provide the minimal development of intrinsic film roughness will form an important part of the future study of oblique deposition of materials. We believe that an acceptable method for the quantitative characterization of the vertical correlation between the substrate and the film surface roughness should be based on the analysis of a set of scattering patterns (ARS) measured at different incident angles of a primary beam. Then, it is possible to determine the PSD-function of the film surface PSD_f and, most importantly, the cross-correlation complex PSD-function PSD_sf. An approach to studying substrate and film roughness conformity has been developed and applied successfully in the X-ray spectral region [8,9], which, as we do not foresee any fundamental limitations associated with applying this approach to visible optics, might serve as inspiration future investigations.

Appendix

Here we present a brief analysis of the real solutions (x,y) of the complex equation:

(33)$$1 - A{\textrm{e}^{ - ix}} - B{\textrm{e}^{ - iy}} = 0,$$

where A and B are nonzero real numbers. Equation (33) is equivalent to the following system of two real equations:

(34)$$\left\{ {\begin{array}{c} {1 - A\cos x = B\cos y}\\ { - A\sin x = B\sin y} \end{array}} \right..$$

Taking the square and summarizing these equations we find two sets of possible values for the unknown variable x:

(35)$${x_ \pm } ={\pm} \arccos \left( {\frac{{{A^2} - {B^2} + 1}}{{2A}}} \right) + 2\pi n,\quad n = 0, \pm 1, \ldots \quad .$$

Substituting Eq. (35) into the first expression of Eq. (34), we find two sets of possible values for the unknown variable y:

(36)$${y_ \pm } ={\pm} \arccos \left( {\frac{{{B^2} - {A^2} + 1}}{{2B}}} \right) + 2\pi m,\quad m = 0, \pm 1, \ldots \quad .$$

Then, considering the second expression of Eq. (34), we conclude that there are two pairs of solutions to Eq. (33), namely, $({x_ + },{y_ - })$ and $({x_ - },{y_ + })$, if sign(B) = sign(A), as well as $({x_ + },{y_ + })$ and $({x_ - },{y_ - })$, if sign(B) ≠ sign(A), i.e., if the pair (x, y) is the solution of Eq. (33), another pair (-x -y) is also the solution of the equation.

The solutions to (35) and (36) have a physical sense only if the argument modulus of the arccosines does not exceed unity, i.e., if

(37)$$|A - 1|\, < \,|B|\, < \,|A + 1|.$$

If the value of x > 0 is known (Eq. (35)), the unknown y can be written via the second of Eq. (34) in a more compact form:

(38)$$y ={-} \arcsin \left( {\frac{A}{B}\sin x} \right)\; + 2\pi m,\quad m = 0, \pm 1, \ldots ,\textrm{ if sign}(A) = \textrm{sign}(B),$$

(39)$$y = \pi + \arcsin \left( {\frac{A}{B}\sin x} \right)\; + 2\pi m,\quad m = 0, \pm 1, \ldots ,\textrm{ if sign}(A) \ne \textrm{sign}(B).$$

As the arccosine is changed within the (0, π) interval, the solutions to Eqs. (35) and (36) with n = m = 0 provide the smallest absolute values of x and y, i.e. the smallest growth angles, that are the most suitable for practical mirror fabrication. This is the only case analyzed in the main text.

As an illustration, Fig. 11 shows the map of several possible values of the growth angles β_ML and β_BL that provide backscattering suppression toward the incident beam. Calculations are performed for the SiO₂/ HfO₂ multilayer mirror (A2 in Table 2) assuming the topmost layer of the mirror is SiO₂. Solutions are found via Eqs. (13)–(15) expressed in the form of Eqs. (35) and (38)–(39). The values of (n, m) are indicated in the map. As Fig. 11 shows, the smallest growth angles are indeed achieved at n = m = 0. Increasing |n| or |m| value above 1 results in |β_ML| and |β_BL| approaching the impracticable value of 90°. Only half of the possible values of the growth angles are shown in the figure. The other half can be obtained by replacing β_ML and β_BL with -β_ML and -β_BL_.

Fig. 11. Map of possible values of the growth angles β_ML and β_BL that provide backscattering suppression toward the incident beam. Calculations are performed for the SiO₂/HfO₂ multilayer mirror (A2 in Table 2) assuming the topmost layer of the mirror is SiO₂, and for different values of (n, m), which change between −1 and +1, as indicated in the map. Only half of the possible values of the growth angles are shown. The other half can be obtained by replacing β_ML and β_BL with -β_ML and -β_BL.

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Funding

National Natural Science Foundation of China (61621001, 61675156, 61925504, 61975155); Shanghai Municipal Education Commission (2017-01-07-00-07-E00063); Science and Technology Commission of Shanghai Municipality (17JC1400800).

Acknowledgments

One of the authors (IVK) acknowledges the Russian Ministry of Science and Higher Education for the support of the work within the State assignment FSRC «Crystallography and Photonics» RAS.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Layer material	Ta₂O₅	HfO₂	Al₂O₃	SiO₂	BK7 glass
Thickness (nm)	76.98	88.12	102.36	121.96	-
Refractive index	2.1739	1.93	1.70	1.4777	1.5148

Design	Composition	β_TL	β_BL	β_ML
A1	SiO₂/Ta₂O₅ / [SiO₂/Ta₂O₅]₁₅	-	−8.24	19.81
A2	SiO₂/HfO₂ / [SiO₂/HfO₂]₂₂	-	−9.73	13.03
B1	HfO₂/SiO₂/HfO₂ / [SiO₂/HfO₂]₁₉	44.02	-	5.65
B2	HfO₂/SiO₂(3L)/HfO₂(3H) / [SiO₂/HfO₂]₁₉	17.85	-	5.65
C1	HfO₂/SiO₂/Ta₂O₅ / [SiO₂/ Ta₂O₅]₁₃	45.99	-	8.59
C2	HfO₂/SiO₂(3L)/Ta₂O₅(3H) / [SiO₂/ Ta₂O₅]₁₃	19.04	-	8.59
D1	Ta₂O₅/Al₂O₃/Ta₂O₅ / [SiO₂/ Ta₂O₅]₁₃	47.65	-	16.31
D2	Ta₂O₅/Al₂O₃(3L)/Ta₂O₅(3H) / [SiO₂/ Ta₂O₅]₁₃	20.08	-	16.31

Layer material	Ta₂O₅	HfO₂	Al₂O₃	SiO₂	BK7 glass
Thickness (nm)	76.98	88.12	102.36	121.96	-
Refractive index	2.1739	1.93	1.70	1.4777	1.5148

Design	Composition	β_TL	β_BL	β_ML
A1	SiO₂/Ta₂O₅ / [SiO₂/Ta₂O₅]₁₅	-	−8.24	19.81
A2	SiO₂/HfO₂ / [SiO₂/HfO₂]₂₂	-	−9.73	13.03
B1	HfO₂/SiO₂/HfO₂ / [SiO₂/HfO₂]₁₉	44.02	-	5.65
B2	HfO₂/SiO₂(3L)/HfO₂(3H) / [SiO₂/HfO₂]₁₉	17.85	-	5.65
C1	HfO₂/SiO₂/Ta₂O₅ / [SiO₂/ Ta₂O₅]₁₃	45.99	-	8.59
C2	HfO₂/SiO₂(3L)/Ta₂O₅(3H) / [SiO₂/ Ta₂O₅]₁₃	19.04	-	8.59
D1	Ta₂O₅/Al₂O₃/Ta₂O₅ / [SiO₂/ Ta₂O₅]₁₃	47.65	-	16.31
D2	Ta₂O₅/Al₂O₃(3L)/Ta₂O₅(3H) / [SiO₂/ Ta₂O₅]₁₃	20.08	-	16.31

Interference suppression of light backscattering through oblique deposition of high-reflectivity multilayers: a theoretical analysis

Abstract

1. Introduction

2. General formulae of light diffraction from a multilayer with skewed roughness replication

3. Mirror design with a topmost bi-layer

4. Mirror design with a topmost tri-layer

5. Discussion and conclusions

Appendix

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (11)

Tables (2)

Equations (39)

Optics Express