Interference suppression of light backscattering through oblique deposition of a layered reflecting coating: bi-layer on a substrate

Jinlong Zhang; Han Wu; Igor V. Kozhevnikov; Shuaikai Shi; Xinbin Cheng; Zhanshan Wang

doi:10.1364/OE.27.015262

1. Introduction

Radiation scattering by surface and interface roughness is one of the crucial issues in advanced optical systems operating in the infrared to X-ray region. As an illustration, we mention only two examples from quite different branches of optics. The first example is the mirror reflection of coherent and extremely high-power beams of X-ray free electron lasers (XFELs). The long-scale (low-frequency) roughness of the mirror surface creates undesirable speckles in the focal plane [1] resulting in the appearance of artifacts and can even make it impossible to correctly reconstruct the object structure from analysis of the diffraction pattern. The short-scale (high-frequency) roughness increases scattering into the mirror depth leading to a significant enhancement (by a factor of several times) of the absorbed power [2], and thus a proportional increase in the thermal deformation of the mirror, thereby spoiling the optical performance of the outgoing beam.

The second example is the high-quality interferometer used for gravitational wave detection (GWD) [3]. The long-scale roughness distorts the ideal eigenmode of the interferometer, while the small-scale roughness scatters light out of the interferometer, thus limiting the power buildup in the resonant cavity. Both factors result in imperfect interference, distortion in the cavity field, and thus a decrease in the sensitivity of gravitational wave detection.

Over the past few decades, considerable advances have been made to improve polishing and deposition technologies allowing to minimize scattering from mirrors. Today, the root-mean-squared (rms) roughness of substrates of different materials can be as small as 0.07–0.1 nm in the [0.5, 100 μm⁻¹] range of spatial frequencies [4]. The RMS roughness of the mirror installed in a European XFEL beamline and fabricated with elastic emission machining technology [5] is only 0.84 nm over the whole measurable range of spatial frequencies spanning eight orders of magnitude from 1.1·10⁻⁶ to 100 μm⁻¹, where the lowest frequency corresponds to the mirror length of 900 mm [2].

According to the results presented in [3], the peak-to-valley value (PVV) of the surface relief variations of one of the mirrors for the advanced GWD interferometer does not exceed 1.5 nm over the central mirror area with a diameter of about 140 mm. The author of [3] indicated that the PVV is one order of magnitude lower than that of the mirrors for the previous generation of GWD interferometers.

To date, essential progress has also been achieved in the deposition of multilayer coatings. As an example, we can mention the prototype of the 500-mm-long mirrors used for synchrotron beamlines. The silicon substrate was coated with an Ru/Si multilayer structure with a period of approximately 2.8 nm, and the total number of bi-layers was equal to 200 [6]. The rms roughness of the substrate before deposition was about 0.1 nm, as determined by atomic force microscopy (AFM). After deposition of the multilayer structure, the rms roughness was unchanged over the whole mirror surface area. This result confirmed the high quality of the multilayer deposition process and gives evidence of the conformal growth of interfacial roughness, at least in the measurable range of the spatial frequencies.

Nevertheless, the optical performance of even atomically smooth mirrors may not be high enough for the needs of modern optics. The best known example in visible optics is probably the problem of backscattering toward the incident beam in laser gyroscopes. Backscattering results in a strong coupling of counter-propagating gyro modes, and thus results in errors in the rotation rate measurements and even the total disappearance of the gyroscopic effect [7–9]. The sensitivity and accuracy of gyroscopes essentially depend on the backscattering intensity; thus, it should be as small as possible and tend towards zero in the ideal case.

If the current deposition technologies cannot provide a low enough backscattering intensity, it is necessary to analyze alternative approaches to scattering suppression. The fact is that the scattering intensity is determined by both statistical parameters of interfacial roughness and the stationary electric field distribution inside a multilayer structure. Therefore, by choosing the proper multilayer structure design leading to a shift of the field intensity maxima away from the rough interfaces, it is possible to reduce the total scattering, at least in the case of fully uncorrelated interfacial roughness [10].

At the same time, there are manifold justifications that the roughness of different interfaces is well-correlated. The comprehensive study of the roughness replication was performed in [11–13] as applied to single films and multilayer mirrors operating in the visible spectral range. The authors demonstrated that the substrate roughness was always replicated at each interface in the whole range of measurable spatial frequencies. Then a new possibility of scattering reduction arises; designing multilayer structures to provide destructive interference with coherent waves scattered from different correlated interfaces.

The possibility was firstly analyzed in [14], where the light scattering suppression in the specular direction as well as the scattering reducing in the whole back hemisphere was demonstrated theoretically and experimentally through a single quarter-wavelength or half-wavelength layer deposition on a substrate. While the paper was devoted to single layers, it initiated extensive studies of destructive interference in scattering from multilayer structures.

Similar idea was also proposed in [15] as applied to the scattering suppression in the specular direction in X-ray domain, when X-ray beam falls onto a substrate at extremely small grazing angle within the total external reflection region.

More recently, the idea suggested in [14] was applied to rough multilayer structures. Adding half-wavelength layer with properly chosen dielectric constant on a top of a multilayer structure was demonstrated in [11] to reduce scattering as the scattering amplitudes from the interfaces of the added layer were in phase opposition with those of the initial coating.

Generalization of this approach was realized in [16], where the destructive interference of the scattered waves was provided by adding Fabry–Perot cavity structures on top of the multilayer with conformal interfacial roughness. A 30% reduction in the total scattering from a high-reflectivity Ta₂O₅/SiO₂ multilayer mirror was experimentally observed at normal incidence with 808-nm wavelength radiation.

The final goal of our long-term work is the development of highly reflective multilayer mirrors with suppressed backscattering in towards the incident beam rather than decreasing the total scattering. The problem is of extreme importance for laser gyroscope physics, while any attempts to solve it are unknown to us. As it is demonstrated in the next section, in the case of fully correlated (conformal) interfacial roughness, the only possible way to decrease the backscattering intensity toward the incident beam is to reduce the mirror reflectivity, which is unacceptable for laser gyros.

Therefore, we will analyze another approach to solve this problem, namely, the use of oblique deposition of materials resulting in the skewed replication of roughness within the multilayer [13,17–20]. In this case, the conditions of the destructive interference of the scattered waves are changed compared to the case of conformal roughness, so that the scattering pattern is asymmetric even at the normal incidence of the outgoing beam. The effect was firstly observed in the visible range [13] and, more recently, in the extreme ultraviolet region [20]. The effect of the skewed roughness was successfully used in [21] to decrease the total scattering of extreme ultraviolet radiation (13.5 nm wavelength, 10° incidence angle) from a Mo/Si multilayer mirror. The authors demonstrated a 70% reduction in the total scattering, and based on their data presented in [21] we can conclude that they also observed backscattering suppression towards the incident beam by a factor of about 5-7.

The present paper is only the first step in our study, and we limit our investigations to the analysis of scattering suppression from a single bi-layer with skewed roughness. The bi-layer can be considered as the simplest example of a layered structure demonstrating specific features of scattering due to oblique deposition. In Section 3, we demonstrate theoretically the possibility of entirely suppressing (down to zero) the backscattering in the direction towards the incident beam, if the development of uncorrelated roughness during multilayer structure growth is neglected. The necessary conditions for the complete suppression of backscattering towards the incident beam are also analyzed. The results of our first experimental observations of backscattering suppression are presented in Section 4, where, in addition, we discuss a number of factors that can adversely affect the scattering suppression. Finally, the main results obtained are summarized in Section 5.

2. Backscattering from conformal roughness

In this section, we demonstrate the fundamental problem preventing backscattering suppression in the direction toward the incident beam if the interfacial roughnesses are conformal (identical), and they are replicated perpendicular to the substrate surface during the deposition of a multilayer structure. We will show that in this case, the only possible way to decrease the backscattering intensity is to decrease the mirror reflectivity.

We will analyze the mostly general case of a random layered media with conformal irregularities. Suppose that $ε_{0} (z)$ and $ε (\vec{r})$ is the dielectric permittivity of an ideal homogeneous layered medium and the same medium with random irregularities, respectively. Z-axis is directed into the medium depth, so that $ε (\vec{r}) \to 1$ , if $z \to - \infty$ , and $ε (\vec{r}) \to ε_{s u b}$ , if $z \to + \infty$ . An s-polarized plane wave (the electric field vector is perpendicular to the incidence plane) is assumed to fall onto a surface of inhomogeneous medium. Below we will analyze the case of s-to-s scattering only, what is of the prime interest for laser gyroscope physics and corresponds to the condition of our experiment described below. Notice that s-to-p scattering intensity is typically very small compared with the s-to-s one.

The 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_i; this is expressed via the amplitude of the scattered wave A as

A R S (θ_{s}, φ_{s}, θ_{i}) = \frac{1}{Q_{i}} \frac{d Q_{s}}{d Ω} = \frac{〈 | A (θ_{s}, φ_{s}, θ_{i}) |^{2} 〉}{S \cdot \cos θ_{i}} .

where the scattered field in vacuum far from the sample is expressed as

E_{s c a t} (r \to \infty) = A \cdot \exp (i k r) / r

(k = 2π/λ is the wave number), S is the illuminated area on the sample surface, triangular brackets mean ensemble averaging, the incidence and scattering angles

θ_{i} \in [0, π / 2]

and

θ_{s} \in [0, π / 2]

are measured from the Z-axis, and

φ_{s} \in [0, 2 π]

is the azimuth scattering angle measured from the incident plane XZ.

The scattering amplitude in the approximation of one-fold scattering (the first order of the Distorted Wave Born Approximation) has the following form (see, e.g., [22])

\begin{array}{l} A (θ_{s}, φ_{s}, θ_{i}) = \frac{k^{2}}{4 π} \int e^{i ({\vec{q}}_{0} - \vec{q}) \vec{ρ}} Δ ε (\vec{r}) E_{0} (z, θ_{i}) E_{0} (z, θ_{s}) d^{3} \vec{r}, Δ ε (\vec{r}) = ε (\vec{r}) - ε_{0} (z), \\ {\vec{q}}_{0} = k (\sin θ_{i}, 0), \vec{q} = k (\sin θ_{s} \cos φ_{s}, \sin θ_{s} \sin φ_{s}), \vec{ρ} = (x, y), k = 2 π / λ . \end{array}

Here, $Δ ε (\vec{r})$ is a perturbation of the dielectric permittivity, ${\vec{q}}_{0}$ and $\vec{q}$ are the projections of the incident and diffracted wave vectors in the XY-plane, λ is the wavelength in vacuum, and $E_{0} (z, θ)$ is the amplitude of an unperturbed wave obeying the one-dimensional wave equation and conventional boundary conditions

\begin{array}{l} {E^{″}}_{0} (z, θ) + k^{2} (ε_{0} (z) - \sin^{2} θ) E_{0} (z, θ) = 0, \\ E_{0} (z \to - \infty, θ) = e^{i k z \cos θ} + r (θ) \cdot e^{- i k z \cos θ}, E_{0} (z \to + \infty, θ) = t (θ) \cdot e^{i k z \sqrt{ε_{s u b} - \sin^{2} θ}} . \end{array}

where r and t are the reflected and transmitted amplitudes, respectively.

We consider an inhomogeneous random medium with the specific dielectric permittivity $ε (\vec{r}) = ε_{0} (z - ζ (\vec{ρ}))$ , where the random function $ζ (\vec{ρ})$ describing the irregularities of the medium is independent of the z-coordinate. These irregularities are called conformal, because a surface of constant value of ε is determined by the same equation $z - ζ (\vec{ρ}) = c o n s t$ over the whole medium depth. Assuming the scattering from the medium irregularities to be low and expanding the perturbation of the dielectric permittivity as $Δ ε (\vec{r}) = ε (\vec{r}) - ε_{0} (z - ζ (\vec{ρ})) \approx - ζ (\vec{ρ}) {ε^{'}}_{0} (z)$ , we obtain from Eqs. (1)-(2) that in frame of the first order perturbation theory, the ARS distribution is determined by the following equation

\begin{array}{l} A R S (θ_{s}, φ_{s}, θ_{i}) = \frac{k^{4}}{{(4 π)}^{2} \cos θ_{0}} {| \int E_{0} (z, θ_{i}) E_{0} (z, θ_{s}) {ε^{'}}_{0} (z) d z |}^{2} PSD (\vec{ν}), \\ PSD (\vec{ν}) = \int 〈 ζ (\vec{ρ}) ζ (0) 〉 \exp (2 i π \vec{ρ} \vec{ν}) d^{2} \vec{ρ}, 2 π \vec{ν} = \vec{q} - {\vec{q}}_{0} . \end{array}

where PSD is the power spectral density of the medium irregularities.

Equation (4) is also valid for multilayer structures with step-like variation of the dielectric constant at interfaces. The appearance of delta functions in the expression for ${ε^{'}}_{0} (z)$ results in no problems, as the field functions $E_{0} (z)$ are continuous.

Next, we demonstrate that the integral contained in Eq. (4) can be calculated exactly for arbitrary $ε_{0} (z)$ in the case of backward and forward scattering, when $θ_{s} = θ_{i}$ , while φ_s = π or 0, and $E_{0} (z, θ)$ is independent of φ_s. Integrating by parts, we obtain

S \equiv \int_{- \infty}^{+ \infty} E_{0}^{2} {ε^{'}}_{0} d z = \lim_{L_{1, 2} \to \infty} {{(E_{0}^{2} ε_{0})}_{z = - L_{1}}^{z = L_{2}} - \int_{- L_{1}}^{L_{2}} ε_{0} \frac{d E_{0}^{2}}{d z} d z} .

Multiplying Eq. (3) by $d E_{0} / d z$ , we find that

ε_{0} \frac{d E_{0}^{2}}{d z} = \sin^{2} θ_{i} \frac{d E_{0}^{2}}{d z} - \frac{1}{k^{2}} \frac{d}{d z} {(\frac{d E_{0}}{d z})}^{2} .

Then we can integrate Eq. (5) for arbitrary ε₀ and obtain

\begin{array}{l} S = \lim_{L_{1, 2} \to \infty} {(ε_{0} - \sin^{2} θ_{i}) E_{0}^{2} + \frac{1}{k^{2}} {(\frac{d E_{0}}{d z})}^{2}}_{z = - L_{1}}^{z = L_{2}}, \\ = \frac{1}{k^{2}} \lim_{L_{1, 2} \to \infty} {- E_{0} \frac{d^{2} E_{0}}{d z^{2}} + {(\frac{d E_{0}}{d z})}^{2}}_{z = - L_{1}}^{z = L_{2}} . \end{array}

Substituting an asymptotic representation from Eq. (3) for the field function, we find that $S = - 4 \cos^{2} θ_{i} \cdot r (θ_{i})$ and the final expression for the backscattering intensity (θ_s = θ_i, φ_s = π) is written as

A R S (θ_{i}, π, θ_{i}) = \frac{k^{4}}{π^{2}} \cos^{3} θ_{i} \cdot | r (θ_{i}) |^{2} \cdot PSD (2 k \sin θ_{i}) .

An expression for the forward scattering intensity (θ_s = θ_i, φ_s = 0) is formally the same, if the argument of the PSD-function is set to zero, although we do not discuss the physical sense of the PSD-function at a zero spatial frequency here.

Equation (8) demonstrates that, in the case of conformal roughness, the backscattering intensity could be only reduced through a decrease in the reflectivity. Therefore, it is impossible to design a highly reflective multilayer mirror (periodic or aperiodic, having abrupt or smooth interfaces, consisting of two or more materials) providing even a slight decrease in the backscattering intensity, if the interfacial roughnesses are conformal.

Notice that our consideration is not in a contradiction with the results obtained in [14,15], where the total scattering suppression in the specular direction was predicted. The fact is that Eq. (8) has been deduced assuming the both interfaces roughness to be fully conformal (identical) in contrast to [14,15], where a deposited film was supposed to smooth substrate roughness, while the roughness of both interfaces was still correlated. The roughness smoothening was necessary to guarantee an equality of the amplitudes of waves scattered from different interfaces.

An example is presented in Fig. 1, where the calculated ARS in the incident plane is shown for two high-reflectivity Ta₂O₅/SiO₂ multilayer mirrors operating at an incident angle of θ_i = 45° and a wavelength of 633 nm characterized by the same model PSD function. The first mirror (HR45) is a standard quarter-wavelength multilayer mirror, while the second one (LSHR45) is an aperiodic mirror designed to decrease the scattering intensity caused by the conformal roughness. The approach used for designing this mirror is described in [16]. While the scattering was significantly reduced for the LSHR45 mirror, the backward and forward scattering intensities (at $| θ_{s} | = θ_{i} = 45^{\circ}$ ) are the same for both mirrors according to Eq. (8), because their reflectivity is almost unity.

Fig. 1 The calculated ARS (at λ = 633 nm) in the incident plane (φ_s = 0) from two high-reflectivity Ta₂O₅/SiO₂ multilayer mirrors: HR45 and LSHR45 designed to suppress scattering into the back hemisphere. Here and below the ARS was calculated with the computer program described in [23], Elson et al.

Abstract

1. Introduction

2. Backscattering from conformal roughness

3. Backscattering from the bi-layer fabricated with oblique deposition

4. Experiment

5. Conclusion

Funding

References

Cited By

Figures (15)

Equations (19)

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