General optimization of tapered anti-reflective coatings

Brandon L. Good; Shaun Simmons; Mark Mirotznik

doi:10.1364/OE.24.016618

1. Introduction

Removing unwanted reflections from contrasting interfaces is a design challenge experienced by both the radiofrequency [1,2] and optical [3–6] communities. Various design methods have evolved that reduce these reflections by applying antireflective (AR) surface coatings. In general, these coatings are designed using one of two methods [7]. One approach is to design a stack of thin dielectric layers. As a result of constructive and destructive interference reflected energy can be reduced at particular wavelengths and at specific incident angles. This is the traditional thin film or dielectric transformer approach and includes a number of common profiles including; quarter-wave transformers [8], binomial transformers [8] and Chebyshev transformers [9,10]. A second approach is the use of a continuously graded or tapered refractive index. Unlike thin film methods the index of refraction is smoothly varied from the incident to exit region such that minimal reflection occurs during the transition. Since this approach is based on the theory of small reflection and not on resonant interference it is effective over a much broader band of wavelengths and incident angles. Some of these common profiles include the exponential taper [8], triangular taper [8], and Klopfenstein taper [11].

Even though the continuous profiles can expand the anti-reflective wavelengths and incident angles, fabricating a continuously graded index of refraction, particularly at optical wavelengths, can be challenging. Consequently, most investigators create “effective” index of refraction gradients by fabricating subwavelength surface textures. A number of investigators have studied various graded index profiles resulting in a library of canonical shapes that exhibit attractive AR properties [2,4–7,12]. These “effective” index of refraction gradients through canonical shapes make continuous tapers more attractive to practical fabrication methods.

Recently, taper formulations not based on the theory of small reflections are being explored for more targeted applications [13–15]. The goal of these methods is to produce optimal AR properties over a selected band of wavelengths and incident angles. This approach can be used to improve AR performance or reduce the overall coating thickness. For example, Zhang et. al. [14] presented a general analytical approach for designing tapered index profiles by solving a non-linear ordinary differential Eq.. This approach was found to be equivalent to the problem of side-lobe suppression in signal processing and antenna arrays. Zhao et. al. [15] presented an iterative optimization approach based on genetic programming (GP). Here the index of refraction profile was assumed to be polynomial in form (e.g. linear, cubic or quantic) with a number of undetermined coefficients. The GP algorithm was used to find optimal coefficients that minimize a fitness function over a desirable wavelength band and angles of incidence. More recently, Kim and Park’s [16] formulated an analytical approach for designing tapered index profiles that result in perfect antireflective properties over any wavelength band. In their approach, described in more detail in the next section, analytical expressions were derived in terms of a wave amplitude function that guarantees perfect antireflective properties. Although the dispersive materials that result from this approach are in general not realizable it was nevertheless a significant achievement in the field of AR coating design, and is useful for finding approximate, yet realizable, solutions over a finite band of wavelengths.

In this paper we present an iterative optimization algorithm that leverages the theoretical contributions of Kim and Park’s [16]. Specifically, we will show that by optimizing the shape wave amplitude function it is possible to efficiently arrive at index tapers that minimize reflection over any desirable wavelength band, range of incident angles or polarization state. We validate this approach both numerically and with experimental results within the microwave frequency band. This article is organized as follows. Section 2 describes the Kim and Park perfect AR grading of a lossless, dielectric medium, then expands the method for oblique incidence. Section 3 describes the general optimization routine to modify Kim and Park antireflection wave amplitude while maintaining perfect antireflection at a selected frequency, incidence angle, and linear polarization combination. Section 4 presents two numerical design examples: the first is a low pass filter as a function of wavelength and the second optimizes broadband oblique angle performance. Section 5 describes how to implement the tapers into a canonical shaper by using effective medium theory. Section 6 shows experimental validate of a subwavelength taper using the general optimization in Sections 2 and 3 and of the effective medium theory subwavelength taper approach of Section 5. A brief conclusion will summarize the work and discuss potential applications.

2. Analytic approach for perfect antireflective graded index

2.1 Normal incidence

The optimization approach described in this article leverages an analytic formulation for the design of perfect antireflection dielectric gradings reported by Kim and Park [16]. For completeness, we briefly describe the formulation in [16] before presenting our new methodology. In [16] a perfect broadband AR solution was derived by first expressing the electric and magnetic fields, Eq. (1), within a graded surface layer as

\tilde{E} (x) = {\hat{a}}_{y} P (x) e^{j Q (x)}, \tilde{H} (x) = \hat{a} {}_{z}Y (x) P (x) e^{j Q (x)}

where P(x) and Q(x) denote the spatially varying wave amplitude and phase respectively and Y(x) represents the wave admittance. The incident field is assumed to be a normally incident plane wave traveling in the + x direction. The graded surface layer is assumed to be lossless and non-magnetic. From these two assumptions, the dispersive index of refraction distribution n(x,ω) can be determined as a function of the wave amplitude function, P(x), as

n (x, ω) = \sqrt{\frac{4 μ_{o} S^{2}}{P {(x)}^{4}} - \frac{1}{ω^{2} μ_{o} P (x)} \frac{\partial^{2} P (x)}{\partial x^{2}}}

where S denotes a factor proportional to the electric field intensity and ω denotes angular frequency. It was shown in [16] that the only requirement on the wave amplitude function to achieve perfect antireflection is that P(x) and its first derivative, must satisfy the specific boundary conditions given in Eq. (3).

P (0) = \frac{2 μ_{o} c S}{n_{i n c}}, P (d) = \frac{2 μ_{o} c S}{n_{e x i t}}, \frac{\partial P}{\partial x} (0) = \frac{\partial P}{\partial x} (d) = 0.

There are an infinite number of wave amplitude functions that satisfy these boundary conditions for a lossless and non-magnetic medium. One such function, presented in [16], is given as

P (x) = \sqrt{2 μ_{o} c S} [(\frac{1}{\sqrt{n_{i n c}}} - \frac{1}{\sqrt{n_{e x i t}}}) (2 \frac{x^{3}}{d^{3}} - 3 \frac{x^{2}}{d^{2}}) + \frac{1}{\sqrt{n_{i n c}}}]

\frac{\partial P (x)}{\partial x} = \sqrt{2 μ_{o} c S} (\frac{1}{\sqrt{n_{i n c}}} - \frac{1}{\sqrt{n_{e x i t}}}) (6 \frac{x^{2}}{d^{3}} - 6 \frac{x}{d^{2}})

Substituting Eqs. (4) and (5) in Eq. (2), results in an inhomogeneous, dispersive and lossless permittivity distribution that achieves perfect antireflective properties at normal incidence.

2.2 Extension to oblique incidence

To extend the analysis above to the case of oblique incidence is a relatively straightforward exercise when employing transverse index notation. Specifically, the index refraction for the incident and exit regions, n_inc and n_exit, in Eqs. (3) through (5) are replaced by their transverse index expressions, n^T_inc and n^T_exit given by

\begin{array}{l} n_{i n c}^{T} = {\begin{matrix} n_{i n c} \cos (θ_{i n c}) T E \\ n_{i n c} / \cos (θ_{i n c}) T M \end{matrix} \\ n_{e x i t}^{T} = {\begin{matrix} n_{e x i t} \cos (θ_{e x i t}) T E \\ n_{e x i t} / \cos (θ_{e x i t}) T M \end{matrix} \end{array}

In Eq. (6) θ_exit is found via Snell’s law as

\cos (θ_{e x i t}) = \sqrt{1 - {(\frac{n_{i n c} \sin (θ_{i n c})}{n_{e x i t}})}^{2}}

After substitution of Eqs. (6) and (7) into Eqs. (2) through (5) we arrive at the graded transverse index distribution in terms of the transverse wave amplitude function P^T,

n^{T} (x) = \sqrt{\frac{4 μ_{o} S^{2}}{P^{T} {(x)}^{4}} - \frac{1}{ω^{2} μ_{o} P^{T} (x)} \frac{\partial^{2} P^{T} (x)}{\partial x^{2}}}

The graded index profile is found by solving for n(x) in Eq. (9).

n^{T} (x) = {\begin{matrix} n (x) \cos (θ (x)) T E \\ n (x) / \cos (θ (x)) T M \end{matrix}

where cos(θ(x)) is given by

\cos (θ (x)) = \sqrt{1 - {(\frac{n_{i n c}}{n (x)} \sin θ_{i n c})}^{2}}

Lastly, to ensure that the proper phase is assigned to the fields as they propagate along the curved optical path, shown in Fig. 1, the graded index profile needs to be warped according to the spatial coordinate transformation, given by

x^{'} = {\begin{matrix} x T E \\ \int_{ξ = 0}^{x} \frac{d ξ}{\cos {(θ (ξ))}^{2}} T M \end{matrix}

The final graded index profile is found by substitution of Eq. (11) into the n(x) solution found using Eqs. (9) and (10).

Fig. 1 Perfect antireflection using transverse wave amplitude function.

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3. Finding an optimal wave amplitude function for the case of non-dispersive medium

Any continuous wave function, P(x), that satisfies the boundary conditions given in Eq. (3) will result in a dispersive permittivity distribution that possesses perfect broadband antireflection properties. However, as a practical matter, realizing the resulting dispersive material is not possible due to violating Kramers-Kronig relations [17]. Consequently, it is advantageous to explore a design space that assumes only non-dispersive media. Here we present an optimization algorithm, illustrated in Fig. 2, for targeting an AR reflection response at specific frequency bands and incidence angles using non-dispersive media.

Fig. 2 Iterative optimization algorithm.

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The algorithm begins by parametrizing the wave amplitude function P(x), or more specifically its derivative, dP(x)/dx, such that the boundary conditions given in Eq. (3) are always satisfied. To this end, we chose the following strategy. We begin by expanding the derivative of a new wave amplitude function, $\tilde{P} (x)$ , into a product of an initial function, P_o(x), and M exponentially modulated polynomials. The addition of the exponential term was found useful when trying to ensure monotonically varying tapers. This is written mathematically as

\frac{d \tilde{P} (x)}{d x} = \frac{d P_{o} (x)}{d x} \cdot \prod_{m}^{M - 1} (A_{m} {(\frac{2}{d})}^{m} {(x - \frac{d}{2})}^{m} + 1) e^{A_{M} x}

where A_m denote a set of undetermined coefficients, d is the thickness of the grading layer and P_o(x) represents the initial wave amplitude function, such as those given in Eqs. (4) and (5). The new amplitude wave function, P^new(x) is determined by integrating Eq. (12) as given in Eq. (13).

P^{n e w} (x) = P_{o} (0) + \frac{P_{o} (d) - P_{o} (0)}{\tilde{P} (d) - \tilde{P} (0)} \int_{0}^{x} \frac{d \tilde{P} (x)}{d x} d x

In Eq. (13) normalization factors have been added to ensure that boundary conditions are satisfied. A new index of refraction profile is then calculated at a specific frequency, ω_o, using either Eq. (14) for the case of normal incidence or Eqs. (6) through (12) for the case of oblique incidence.

n^{n e w} (x, ω_{o}) = \sqrt{\frac{4 μ_{o} S^{2}}{P^{n e w} {(x)}^{4}} - \frac{1}{ω_{o}^{2} μ_{o} P^{n e w} (x)} \frac{\partial^{2} P^{n e w} (x)}{\partial x^{2}}}

Lastly, Eqs. (12) through (14) are embedded within an iterative optimization algorithm to determine the optimal P(x) function given a specific cost function and constraints. It is important to highlight that by employing the design approach described above within an iterative algorithm we are ensuring that for any combination of coefficients, A_m, (i.e. for each iteration) a tapered index profile results that has perfect antireflective properties at the selected frequency ω_o and incident angle θ_inc. The optimization algorithm is then used to optimize bandwidth or range of incidence angles around these set points. We found that this approach results in both better AR performance and faster convergence over more traditional methods. The objective function we chose to minimize was simply the maximum reflectance, R, at a discrete number of frequencies within the band of interest, f_i, and a discrete number of angles of incidence, θ_j, as given by Eq. (15).

F = \min (\max [\frac{1}{2 M N} \sum_{j = 1}^{M} \sum_{i = 1}^{N} [| R_{T E} (f_{i}, θ_{j}) | + | R_{T M} (f_{i}, θ_{j}) |]])

A number of iterative optimization algorithms could be employed including traditional derivative-based algorithms, genetic algorithms or direct pattern search algorithms. We choose to implement a particle swarm optimization algorithm [18].

As last comment on the finding the optimal wave amplitude function, it should be noted that various flags or Boolean outcomes can and sometimes must be implement for real world application. For example a flag could be set when the distribution is not monotonic, or a flag could be set for materials restrictions such as n>1. When the flag is triggered, the objection function can be set to make the wave amplitude function unattractive to the optimizer. In section 6, a monotonic restriction and a material restriction will be implemented to be able to fabricate the grading.

4. Design examples

To illustrate the effectiveness of this method two numerical design examples are presented. The first is a low pass wavelength filter. The second has a design objective with a range of transmissive wavelength at a range of transverse electrically polarized incident wave. These two examples are a subset of the potential design examples using this general optimization of tapered anti-reflective coatings. Parameters for these examples were chosen without any particular application and are for academic purposes.

Example #1: low pass filter as a function of wavelength

The first example of a low pass filter as a function of wavelength was chosen to show the broadband capability of the optimization method. Visible wavelengths were chosen to be set as a pass band. At a wavelength of 900 nm and thickness of one third of the perfect anti-reflective wavelength, t = 300 nm, the taper will be optimized to match an index of one to an index of four (n_inc = 1.0 to n_exit = 4.0). For example #1, the incidence angle was zero, θ_j = 0, removing the need to account for both polarizations. The discrete number of wavelengths (frequencies, f_i) within the band of interest were equally spaced every 30 nm from 300 nm to 900 nm. The number of undetermined coefficients was set to eight (M = 8) for this solution. The number of particles was set to 200 and the particle swarm was iterated 100 times. The result in Fig. 3 show a monochromatic taper and a low pass filter with transmission greater than 99% over the pass band.

Fig. 3 Example 1 low pass filter as a function of wavelength.(top) The index of refraction distributions for example one of the optimized taper and the corresponding Klopfenstein taper. (bottom) The reflection response of the optimized taper and corresponding Klopfenstein taper.

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Additionally analysis was done on this example due to the known Klopfenstein taper [11]. Since the Klopfenstein taper is the optimal low filter as a function of wavelength [7], the general optimization of tapered anti-reflective coatings was compared. A similar taper profile and performance should be expected and, in fact, is shown in Figs. 3(a) and 3(b). The optimized solutions inability to replicate the Klopfenstein is an error in the polynomial expansion and particle swarm optimization. If the peaks and troughs of the Klopfenstein were known, the optimizer would arrive much closer to the Klopfenstein solution. This was tested, but not shown here for conciseness. The primary purpose of this example is to present the overall effectiveness of the optimizer even when solving for a large range and number of frequencies.

Example #2

The second example of a broadband oblique AR grading was chosen to illustrate the angular potential of the optimization approach. The example setup is a grading to match from n_inc = 1.0 to n_exit = 10.0. The design parameter were a bandwidth 8.0 µm to 14.0 µm and an angular range of 35 to 55 degrees for transverse electromagnetic incident wave and a thickness targeting 3.0 µm. The perfect anti-reflection solution was located at center of the wavelength, angle range (11.0 µm, 45 degrees). The number of undetermined coefficients was set to four (M = 4) for this solution. The number of particles was set to 200 and the particle swarm was iterated 100 times. The optimized taper is shown in Fig. 4(a) and the transmission power is shown in Fig. 4(b). The taper of 3.0 µm transmits over 96.6% or more of the power within 8.0 µm to 14.0 µm and 35 to 55 degrees for transverse electromagnetic. The mean value over the solution space is 99.0% transmitted power. For the case of TM polarization, not shown in the Fig. 4, the character of the transmittance is similar to the case of TE polarization. Specifically, low reflectance is achieved over a similar wavelength band and range of incident angles.

Fig. 4 Example 2 Oblique Incidence (top) shows the index of refraction profile of the optimized design. (bottom) Transmitted power for all TE incident waves from 5 to 20 µm. Color scale indicates optical power.

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It should be noted that for this example, the index of refraction profile does not vary monotonically. Previously this indicated that the taper was not able to be fabricated. With new additive manufacturing methods, that can resolve millimeter and micrometer features, these profiles are manufacturable for applications in the microwave or millimeter wave spectrum. In the next section, a subtractive fabrication method will be described that directly relates the optimization to an inverse moth-eye surface.

5. Realization of graded index profiles using subwavelength texturing

In this section, we present a practical approach for realizing the graded index profiles, designed in the previous section, by exploiting the effective properties of subwavelength periodic structures. Subwavelength texturing has been utilized for many years to create antireflective surfaces, commonly referred to as moth-eye surfaces [19, 20]. As illustrated in Fig. 5, the method approximates a continuous grading of n(x) by fabricating a periodic array of subwavelength tapered holes in a bulk dielectric substrate. Assuming that the distance between the holes, Λ, is much smaller than the material wavelength, λ, then effective electromagnetic properties, n_eff(x), can be realized that closely match the original grading over a reasonably wide band of frequencies. It should be noted, however, that for any textured surface there is a maximum frequency at which the grating remains zeroth order. For frequencies above this value higher order diffraction effects will occur and the AR solution will no longer be valid.

Fig. 5 Illustration of using a periodic array of subwavelength tapered holes to realize a continuous magneto-dielectric grading.

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To model the effective properties as a function of the material properties and geometry of the tapered holes we utilized the Maxwell-Garnett mixture formulas similar to [20]. Assuming circularly cylindrical tapered holes the effective properties are approximated by

n_{e f f} (x) = n_{b} \sqrt{(1 + \frac{2 v_{f} (x) α}{1 - v_{f} (x) α})}

where α is given by

α = \frac{n_{h}^{2} - n_{b}^{2}}{n_{h}^{2} + n_{b}^{2}}

In Eqs. (16) through (18), n_b and n_h denote the bulk index of refraction of the base material and hole material respectively (illustrated in Fig. 5). The local volume fraction of the hole material to base material is denoted by v_f(x) and varies in the direction of the taper (i.e. x axis). For tapered holes packed in a square lattice, as shown in Fig. 3, the volume fraction is specifically given by

v_{f} (x) = \frac{π a {(x)}^{2}}{Λ^{2}}

where Λ is the lattice period and a(x) describes the radius of the tapered holes as a function of hole depth. It should be noted that Eqs. (16) and (18) assume that the incident field is polarized in the plane of the square lattice. Given a desired effective index profile and a bulk material to fabricate the tapered hole array the geometry of the tapered hole pattern is easily found by solving for a(x) is Eqs. (16) through (18). This is given specifically as

a (x) = \frac{Λ}{\sqrt{π α}} \sqrt{\frac{{(\frac{n_{e f f} (x)}{n_{b}})}^{2} - 1}{{(\frac{n_{e f f} (x)}{n_{b}})}^{2} + 1}}

6. Experimental validation of general optimization and subwavelength texturing

To validate the performance of our algorithm experimentally we fabricated a taper designed using the general optimization routine in Sections 2 and 3 and Eq. (19) in Section 5. Due to easier fabrication, the taper was designed in radiofrequency spectrum and measured within the 4-18 GHz frequency band at normal incidence. The optimization targeted a two interface design; meaning the taper was applied to both sides of the slab. The perfect anti-reflection grading should remain perfectly AR at normal incidence at the desired frequency for a symmetric two interface design. With a perfect AR solution at one frequency, the entire two interface taper can be optimized with regard to the cost function in Eq. (15). Due to the simplicity of the design, a rigorous description of the optimization and index profile are not addressed. The primary objective was to minimize the reflection from 12 to 16 GHz with a perfect AR frequency at 12 GHz.

The experimental sample, shown in Fig. 6(a), was fabricated from a 240 mm x 240 mm x 12.5 mm plate of the thermoplastic Rexolite. The dielectric permittivity of the material was measured to be ε_r = 2.56 with a negligible loss tangent. The tapered hole-arrays, shown in Fig. 6(a), were machined into the plate using a 3-axis computerized numerical control (CNC) router. The router machined the tapered hole-array on both the top and bottom surfaces of the plate. An image of the fabricated surface and a micro-CT scan of a single tapered hole are shown in Fig. 6(c). The micro-CT image provided a quantitative measurement of the fabricated taper geometry verses our desired profile. A comparison of the measured verses desired profiles are shown in Fig. 6(b). As shown in the Fig. 6 the fabricated part closely matched our designed profile.

Fig. 6 Illustration of using a periodic array of subwavelength tapered holes to realize a continuous grading.

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To measure the properties of the samples in both reflection and transmission we used the experimental setup shown in Fig. 7(a). At the heart of this setup is an Agilent PNA Vector Network analyzer with a frequency range from 20 MHz to 40 GHz. On the transmit side a focused Gaussian beam [21] was generated and illuminated the sample at normal incidence. On the receive side an identical focused beam antenna system was used to detect the transmitted fields. Frequency averaging and time-domain gating were used to remove extraneous reflections. Experimental results for the sample shown in Fig. 7(a) are presented in Fig. 7(b) for normal incidence illumination. Here the measured reflectance is plotted as a function of frequency for the AR machined sample. Also presented in the Fig. 6 are the simulated results, using the rigorous coupled wave method. The reflectance of the sample machined with the AR surface was less than −33dB over the designed frequency band (i.e. 12 GHz – 16 GHz). The predicted resonance near 16.5 GHz was not measured due to the tightness of the beam waist with respect to the period size at higher frequencies. There was generally good agreement between the measured and simulated results.

Fig. 7 (a) Illustration experimental radiofrequency focus beam system. (b) Plot of expected reflection power of the optimized taper and the subwavelength textured moth-eye taper with the measured results using the focused beam system of the fabricated part.

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7. Conclusion

In this work, we outlined a new method to optimize antireflective tapers. The method leverages the wave amplitude distribution from Kim and Park to minimize the potential index of refraction gradings. This approach increases the efficiency of the optimization and guarantees perfect antireflection at one frequency, incidence angle, and linear polarization. Additionally, any frequency, incident angle, and linear polarization combination can target a particular reflection power. Three examples, broadband, oblique incidence, and two boundary optimization, were described to show the capabilities of this method. The last example was fabricated using an inverse moth-eye technique and validated the optimization method. Future work will focus on magneto-dielectric and non-linear polarization tapers.

Funding

This work was funded by Office of Naval Research.

Acknowledgments

We would like to thank University of Delaware Center Composite Materials for the use of the CT scan.

References and links

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General optimization of tapered anti-reflective coatings

Abstract

1. Introduction

2. Analytic approach for perfect antireflective graded index

2.1 Normal incidence

2.2 Extension to oblique incidence

3. Finding an optimal wave amplitude function for the case of non-dispersive medium

4. Design examples

Example #1: low pass filter as a function of wavelength

Example #2

5. Realization of graded index profiles using subwavelength texturing

6. Experimental validation of general optimization and subwavelength texturing

7. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (7)

Equations (19)

Optics Express