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Abstract
This paper describes a systematic design study of periodic gold-nanostrip arrays placed on a thin film aimed at enhancing the electric field inside the film when irradiated by light. Based on the study, a “selection rule” is proposed, which provides optimization-based design methods with an a priori choice between field-enhancement dominated by coupling to guided modes, by plasmonic near-field enhancement or by a mix hereof. An appropriate choice of wavelength and grating period is shown to selectively suppress or include waveguiding effects for the optimized designs. The validity of the selection rule is demonstrated through a numerical topology optimization study in which gold nanostrips are optimized for electric-field enhancement in an erbium-doped TiO2 thin film, targeting increased spectral upconversion in the erbium ions. The obtained designs exhibit waveguide excitation within the predicted intervals and, for light polarized perpendicularly to the strips, plasmonic response outside.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Localized enhancement of the electric field under external irradiation is desirable for a range of applications. Examples include, near-field optical microscopy [1], optically assisted data storage [2], and enhanced ionic luminescence [3, 4]. Obtaining high enhancement factors is possible through careful optimization of the system under design [5–7]. The application, considered in the following is enhanced spectral upconversion of sunlight, which is a promising way of improving the limited absorption in photovoltaic devices [8]. Spectral upconversion relies on an optically active medium (e.g. erbium ions) to convert low-energy photons below the band gap of the absorber into photons of higher energy within the absorption band, thereby contributing to the current production. For efficient upconversion it is, however, necessary to concentrate the electric field in local areas of the active medium, e.g. using photonic devices [3,4,6]. Photonic devices can in many cases be viewed as the introduction of surface modulations to an existing system, purposely designed to exploit field-enhancing resonant phenomena such as guided-mode resonances and/or plasmonic effects. Grating structures, designed to phase match incident light with modes supported by an underlying core material, enable wave propagation confined to the regions defined by the core boundaries [9,10]. Similarly, tuning the plasmonic response of metal nanostructures by tailoring their design, allows for extreme local field enhancement [11].
A complete upconverter-assembly comprising, among others, erbium ions and optimized photonic devices embedded into a suitable host, could be placed on the rear side of a solar cell. The rear-placement ensures that the photonic devices only interact with the infrared light transmitted through the cell, thus not interfering with the intrinsic photon-absorption.
Such an embedded setup is one potential end goal for designing field-enhancing devices, however, considering instead a setup with photonic device placed on top of thin film is also of great interest, e.g. for photonic crystal biosensors [12] and polarization-tuned color filters [13]. For such a configuration, one question is, how to efficiently couple incident light into the film and how to systematically target different field-enhancing mechanisms?
This work addresses these questions by utilizing an optimization-based design method and formulating a selection rule for selecting the dominant field-enhancing mechanism, of a periodic array of identical metallic nanostrips place on top of a erbium-doped thin film. The nanostrip design is optimized for electric-field enhancements in the film, targeting enhanced spectral upconversion, when excited by normally incident infrared light. The underlying physical mechanisms responsible for the observed field-enhancements are investigated by optimizing for varying periodicity and interpreting the results using waveguide analysis. The topology optimization method is used as a numerical synthesis tool to design the 2D nanostrip cross section (assuming the strips are extruded infinitely in the out-of-plane direction).
2. The model problem
The 2D model problem considered in this work is shown in Fig. 1. It is comprised of a top domain of air (ΩTop) and an erbium-doped TiO2 (TiO2:Er) thin film (ΩFilm) deposited onto a SiO2 substrate (ΩSub). The system is excited by a time-harmonic plane wave at normal incidence, polarized in either the x- or z-direction (in-plane and out-of-the-plane of Fig. 1, respectively). The electric-field distribution, E, in the model is obtained by solving the electric wave equation, for each polarization using the finite element method, assuming linear, isotropic, and non-magnetic materials without any external charges or currents. To enhance the field in ΩFilm, in turn increasing the upconversion luminescence, incident light is focused into the film by an optimized distribution of air and Au (i.e., the nanostrips) in the design domain, (Ωd ⊆ ΩTop), placed on top of the film.
Fig. 1 Model problem. Domain truncation at the top and bottom using perfectly matched layers (PML) [14,15], periodicity at ΓP imposed using Floquet-Bloch conditions [14,15].
3. The selection rule
Photonic devices, exploiting waveguide excitation, offer field-enhancements spanning large areas but they are generally limited by a narrow spectral band of operation. Plasmonic structures, on the contrary, are typically associated with a much broader operation band and strong local enhancements. Control of these enhancement mechanisms enable devices designed for specific use cases.
The proposed selection rule is based on a map showing the possibility for coupling to guided modes as a function of incident wavelength, λ, and grating period, Λ, such as the one in Fig. 2. This map enables the user, prior to performing the optimization, to determine the field-enhancing mechanism of the optimized design by carefully selecting values of (λ, Λ). The map is calculated from waveguide and phase-matching analysis based on the materials chosen for ΩTop, ΩFilm, and ΩSub as well as the designable material used in Ωd. The selection rule is therefore not limited to the materials or specific dimensions stated in section 2. The conditions for waveguide excitation is based on using the two extreme material distributions obtainable in Ωd, which in turn is shown to provide valuable insight about the field-enhancing mechanisms for all investigated material distributions intermediate to the two extrema. In this work the extrema are: (I) Ωd full of air and (II) Ωd full Au. It is noted that the analysis depends on the incident angle, ϕ, of the exciting wave (here only normal incidence, ϕ = 0°, is considered).
Fig. 2 Black and blue lines show incident wavelength, λ, versus phase-matching period, Λ, for extrema (I) and (II). Lines: black for x-polarization, blue for z-polarization, dashed for extremum (I) and solid for extremum (II). Highlighted regions to the left represent diffractive coupling to the m = 1 mode, at n = 1 and regions to the right at n = 2. The red horizontal line mark the wavelength used in the optimizations, λ = 1520 nm.
The following steps outline the procedure for obtaining the map, for the device shown in Fig. 1, as well as how to select (λ, Λ).
- Choose the materials for the substrate, film, top domain and the designable material. A high refractive index, η, of the film relative to the top domain and substrate is a prerequisite for mode confinement in the film, i.e. max {ηTop, ηSub} < ηFilm [16].
- Assuming a 2D infinitely wide slab waveguide (independent of Λ), calculate the supported guided modes for a range of λ, by solving the polarization-dependent mode equations for the propagation-constant βm(λ) at the mth mode [16]. The mode number, m, describes the mode shape/electric-field distribution in the film, see e.g. [16]. The calculated βm(λ) is the x-component of the propagating (guided) waves k-vector. This step has to be performed for both material extrema (I) and (II) and for all considered polarizations, e.g. x- and/or z-polarization.
- Supported modes are excited efficiently by an incoming plane wave if phase matching is fulfilled, i.e. if the mismatch between incident the wave vector k and βm is balanced by a grating structure. Thus, for each βm(λ) obtained in step (b), solve the phase-matching condition: ‖k‖ sin(ϕ) = βm(λ) + 2πnΛ−1 for Λ, considering an interval of n ∈ {−N, . . ., −2, −1, 0, 1, 2, . . ., N}, with N being an integer. The order of diffraction, n, describes, the direction/order of the diffracted wave with respect to the grating normal [9]. For ϕ = 0°, the matching condition reduce to: βm(λ) = −2πnΛ−1. This step leads to a value of Λ for each considered βm(λ) and n.
- Create the lines in the (λ, Λ)-map by plotting the solutions found from step (c), see the black and blue lines in Fig. 2. For each polarization, every m and, n : highlight the regions spanned by the lines for extrema (I) and (II). In Fig. 2, these regions are shown in light- and dark-gray between pairs of solid dashed lines (black and blue).
- Use the map to select (λ, Λ)-values to target the desired field-enhancement mechanism: for designs exploiting waveguide excitation select (λ, Λ) within a highlighted region and for pure plasmonic designs select (λ, Λ) outside these regions. As shown in section 5, it is, for the presented case, possible to select (λ, Λ) obtaining a mix of plasmonic and waveguide enhancement.
- For the specified (λ, Λ)-values, design the nanostrips using an optimization method of choice.
Figure 2 shows a map of the phase-matching condition for the fist two diffracted orders n ∈ {1, 2} for Λ ∈ {500 nm, 2000 nm} and λ ∈ {1400 nm, 1600 nm}. In the considered interval of λ, the 320 nm thick film supports only the first waveguide mode, m = 1 (even mode), see e.g. [16]. Black and blue represent x- and z-polarization, respectively. The dashed lines (excluding the red line) indicate phase-matching for extremum (I) and solid lines for extremum (II). These lines, thus, represent the upper and lower bounds for phase-matching when Ωd consists only of air or Au. The highlighted regions, between pairs of solid and dashed lines, indicates the (λ, Λ)-span where phase-matching is possible for intermediate air and Au distributions in Ωd. The map is created following steps (a)–(d) and calculated using numerical methods similar to those reported in [17].
4. The optimization based design problem
The investigated design problem concerns the maximization of upconversion luminescence under infrared irradiation at λ = 1520 nm, which has been found to be proportional to ‖E‖3 [3,4,18]. The purpose of step (f) in the selection rule thus becomes to identify a nanostrip design/material distribution of air and Au which maximizes ‖E‖3 in the film, thereby increasing the spectral upconversion in the erbium. The design of the nanostrips is performed using the topology optimization method.
The goal of topology optimization in its simplest form is to identify a spatial distribution of materials A (in this work air) and B (in this work Au), which maximizes the performance of the system in question expressed through a scalar objective function. The design is represented in discrete raster format via identical unit cells, pixels (2D) or voxels (3D), each of which is assigned a design variable. The distribution of material is expressed through interpolation in the relevant material properties of A and B using the design variables. In this work, material properties are described by the complex relative permittivity entering the wave equation, using the interpolation scheme reported in [6,7]. For a detailed description of the topology optimization framework the reader is referred to [19,20] and the references therein.
Using an objective function, Φ, similar to those used in [3,4,6,21] the optimization problem is formulated as,
Where Φ is dimensionless and evaluates the enhancement of ‖E‖3 in the film, relative to the background field, Eb, i.e. without any Au in Ωd. Thus, Φ is a measure of enhanced upconversion luminescence. Equation (1) is solved for either x- or z-polarized light as indicated by the index i ∈ {x, z}. The optimization is performed for a range of Λ across the different regions of the map in Fig. 2, to investigate if the optimization conforms to the predicted upper and lower bounds on Λ found in the waveguide and phase-matching analysis, i.e. steps (b)–(d). To avoid mapping poor local maxima in the investigation, a multi-start approach is used where for each Λ, the optimization is started from multiple randomized initial material distributions. All designs are forced mirror symmetric along a vertical line though the center of Ωd. Material data for Au and SiO2 are taken from [22] and [23], respectively, while data for TiO2:Er is obtained experimentally via spectral ellipsometry measurements.The model and optimization problems are implemented in, and solved using COMSOL Multiphyics [15]. The optimization is performed using the built-in implementation of the Globally Convergent Method of Moving Asymptotes (GCMMA) [24]. A custom MATLAB code controls and automates the entire process through the LiveLink API. The model is discretized by linear non-uniform triangular elements with a maximum size of λ/ηj/40 for the domains j ∈ {ΩTop, ΩFilm, ΩSub} and 2nm uniform linear quadrilateral elements in Ωd. The PML regions are each 2 μm thick and discretized using a maximum element size of 200 nm.
5. Results and discussion
The objective function value for the best designs at each period, obtained by the multi-start procedure, are shown in Fig. 3. The vertical lines represent the upper and lower bounds on Λ for phase-matching at λ = 1520 nm (see Fig. 2). Designs with the highest performance are obtained at periods matching those for guided-mode excitation. This trend is in agreement with the map presented in Fig. 2, and indicates that at these specific periods the optimization exploits field enhancements from excitation of guided modes. Plasmonic resonance is normally associated with highly localized field enhancements, hence when evaluated over large areas as in Eq. (1), the contribution to Φi is small relative to the enhancement from a guided mode spanning the entire ΩFilm. This explains the drop in Φi at intermediate periods where, according to the map in Fig. 2, waveguide coupling is suppressed. A likely explanation as to why the overall picture is less clear for Φx, is the possibility for both plasmonic and waveguide field enhancements, which creates additional local maxima for field enhancement compared to Φz. (To remove the possibility of a plasmonic response a study optimizing Φx using a dielectric material (SiO2) instead of Au was performed. This study resulted in a curve similar to that of Φz in Fig. 3 (SiO2-curve not shown) supporting the conclusions). Lastly, it is noted that for both polarizations, Φi increases drastically near the dashed-vertical lines representing Ωd filled with air.
Fig. 3 Objective function value for the best performing designs optimized at λ = 1520 nm. Top: Φx. Bottom: Φz. The vertical lines indicate the theoretical bounds on Λ for phase-matching to the m = 1 mode at λ = 1520 nm (see Fig. 2). Dots indicate the designs optimized at periods Λ ∈ [780 nm, 900 nm, 1220 nm, 1850 nm] (top) and Λ ∈ [820 nm, 880 mm, 1220 nm, 1640 nm] (bottom). Designs at periods indicated by red dots are shown in Fig. 4.
Fig. 4 Field enhancement from designs optimized at periods indicated by the red dots in Fig. 3. Left: optimized for Φx showing . Right: optimized for Φz showing . Green lines shows the design contour and magenta lines indicate the interface interface between Ωd and ΩFilm. ΩTop is partly shown and ΩSub is excluded.
Figure 4 shows the field enhancement for selected designs optimized for Φx (left) and Φz (right) at periods indicated by the red dots in Fig. 3. Designs in the first row are optimized at the intermediate period Λ = 1220 nm where no waveguide-coupling should be possible. The detailed features of the Φx-optimized design create distinct localized enhancements typically attributed to plasmonic resonances. The lack of enhancement from the Φz-optimized design, is believed to stem from the assumption of infinite extrusion in the z-direction which makes a plasmonic response impossible when the field excitation is in the z-direction. The second row shows the field enhancement for designs at Λ = 900 nm (Φx) and Λ = 820 nm (Φz) both of which are periods exciting the waveguide mode at n = 1 near the air-line (dashed vertical line in Fig. 3). The field characteristics from these designs resemble those normally accredited to mode confinement. Similar behavior is observed for all designs within the bounds for phase-matching (field distributions not show). Finally, designs in the third row are also optimized for periods in the n = 1 waveguide region, but offset further from the air-line. The field-enhancement from the Φx-design at Λ = 780 nm carries characteristics of both mode confinement and plasmonic enhancement. An explanation for this mixed enhancement-profile is possibly a weaker waveguide coupling, leaving the plasmonic enhancement more pronounced. The Φz-design at Λ = 880 nm shows similar enhancement as the design at Λ = 820 nm, but with a much reduced Φz-value, which, again, is assumed due to a less effective waveguide coupling.
To further investigate the contributing field-enhancing mechanisms, the spectral performance, Φi(λ), of the six designs is evaluated across a 200 nm wavelength span, the results of which are shown in Fig. 5. The wide peak of the Φx-design optimized at Λ = 1220 nm adds to the notion of plasmon-dominated enhancement outside the waveguide-regions [25]. The narrow and high-valued peaks from the three designs with the periods Λ ∈ {900 nm, 820 nm, 880 nm} (Φx, Φz, Φz) indicates diffractive coupling with the designs acting as transmission gratings exciting the guided mode in the film. The mixed enhancement characteristic claimed for the Λ = 780 nm-design (Φx) is supported as a wide peak similar to the profile of the Λ = 1220 nm-design (Φx) but with 3 × the amplitude. Hence a mixture of the waveguide and plasmonic spectral profiles is seen. Similar mixed performance has been observed for other designs optimized for Φx, at periods in the waveguide-regions but off-set from the air-line (plots not shown), indicating a subset of periods in the waveguide-regions where one can obtain designs exploiting both plasmonic and wave-guiding effects.
Fig. 5 Performance of the optimized designs shown in Fig. 4, evaluated for wavelengths λ ∈ [1400 nm, 1600 nm]. Left: Designs optimized for Φx. Right: Designs optimized for Φz. The green vertical line indicates the optimization wavelength λ = 1520 nm. Peak values are given in Fig. 3.
It is noted that, the monochromatic optimization at normal incidence does not ensure an enhanced field across a broad wavelength and angular spectrum. The influence of optimizing Au nanostrips for field-enhancement at multiple wavelengths and/or incident angles were investigated in a recent study by the authors [6]. Here, it was found that simultaneously optimizing for multiple wavelength increases the robustness towards changes in incident angle.
The designs in Fig. 4 contain non-realizable features (e.g. floating Au), an aspect which is not addressed in this work as the focus is on evaluating the conditions for waveguide coupling. Several other works have, however, demonstrated methods for including production constraints directly into the topology optimization framework [6,26,27].
6. Conclusion
Conditions for phase-matched waveguide-exaction were utilized to formulate a selection rule, which enables the designer to a priori select a desired field-enhancing mechanism obtained when designing metallic nanostrip-arrays on thin-films utilizing optimization-based design tools. The validity of the rule was investigated using topology optimization to design the 2D nanostrip cross section at varying periodicity for enhancing 1520 nm infrared light in an TiO2:Er thin film, targeting increased spectral upconversion. Optimizing at periods predicted to couple to the guided mode resulted in designs with enhancement profiles attributable to guided modes, i.e. high enhancement factors due to strong confinement of the electric fields but also with a wavelength-sensitive performance. Designs optimized at non-coupling periods resulted in a plasmon-dominated performance with lower overall enhancement but a spectral-enhancement profile less sensitive to changes in excitation wavelength. Mixed waveguide and plasmonic performance were obtained for designs optimized for x-polarized light, at phase-matched periods off-set from the theoretical air line.
The proposed selection rule is generally applicable to design problems where the goal is to obtain field enhancements in a thin film, using an optimization based design tool to distribute material on top of the film.
Funding
Innovation Fund Denmark under the SunTune project (4106-00002B) and by Villum Fonden through the NATEC (NAnophotonics for Terabit Communications) project.
Acknowledgment
The authors thank Ph.D. Fellows; Søren H. Møller for valuable inputs and Harish Lakhotiya for providing the TiO2:Er ellipsometry measurements.
References and links
1. H. G. Frey, F. Keilmann, A. Kriele, and R. Guckenberger, “Enhancing the resolution of scanning near-field optical microscopy by a metal tip grown on an aperture probe,” Appl. Phys. Lett. 81, 5030 (2002). [CrossRef]
2. M. Mansuripur, A. R. Zakharian, A. Lesuffleur, S.-H. Oh, R. J. Jones, N. C. Lindquist, H. Im, A. Kobyakov, and J. V. Moloney, “Plasmonic nano-structures for optical data storage,” Opt. Express 17, 14001–14014 (2009). [CrossRef] [PubMed]
3. S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. N. Larsen, “Upconversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106, 053101 (2015). [CrossRef]
4. H. Lakhotiya, A. Nazir, S. P. Madsen, J. Christiansen, E. Eriksen, J. Vester-Petersen, S. R. Johannsen, B. R. Jeppesen, P. Balling, A. N. Larsen, and B. Julsgaard, “Plasmonically enhanced upconversion of 1500 nm light via trivalent Er in a TiO2 matrix,” Appl. Phys. Lett. 109, 263102 (2016). [CrossRef]
5. E. Wadbro and C. Engström, “Topology and shape optimization of plasmonic nano-antennas,” Comput. Methods Appl. Mech. Eng. 293, 155–169 (2015). [CrossRef]
6. J. Vester-Petersen, R. E. Christiansen, B. Julsgaard, P. Balling, O. Sigmund, and S. P. Madsen, “Topology optimized gold nanostrips for enhanced near-infrared photon upconversion,” Appl. Phys. Lett. 111, 133102 (2017). [CrossRef]
7. R. E. Christiansen, J. Vester-Petersen, O. Sigmund, and S. P. Madsen, “A non-linear material interpolation for design of metallic nano-particles using topology optimization,” Submitted (2018).
8. J. C. Goldschmidt and S. Fischer, “Upconversion for photovoltaics - a review of materials, devices and concepts for performance enhancement,” Adv. Opt. Mater. 3, 510–535 (2015). [CrossRef]
9. K. Ogawa, W. Chang, B. Sopori, and F. Rosenbaum, “A theoretical analysis of etched grating couplers for integrated optics,” IEEE J. Quantum Electron. 9, 29–42 (1973). [CrossRef]
10. T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977). [CrossRef]
11. S. Nie, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science 275, 1102–1106 (1997). [CrossRef] [PubMed]
12. D. Threm, Y. Nazirizadeh, and M. Gerken, “Photonic crystal biosensors towards on-chip integration,” J. Biophotonics 5, 601–616 (2012). [CrossRef] [PubMed]
13. I. Koirala, V. R. Shrestha, C.-S. Park, S.-S. Lee, and D.-Y. Choi, “Polarization-Controlled Broad Color Palette Based on an Ultrathin One-Dimensional Resonant Grating Structure,” Sic. Rep. 7, 40073 (2017). [CrossRef]
14. J.-M. Jin, The Finite Element Method in Electromagnetics (Wiley-IEEE, 2014), 3rd ed.
15. COMSOL AB, “COMSOL Multiphysics v. 5.3”,.
16. A. Yariv and P. Yeh, Photonics (Oxford University, 2006).
17. R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Express 17, 24112–24129 (2009). [CrossRef]
18. M. Pollnau, D. R. Gamelin, S. R. Lüthi, H. U. Güdel, and M. P. Hehlen, “Power dependence of upconversion luminescence in lanthanide and transition-metal-ion systems,” Physical Review B 61, 3337–3346 (2000). [CrossRef]
19. M. P. Bendsøe and O. Sigmund, Topology Optimization (Springer, 2004). [CrossRef]
20. J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev. 5, 308–321 (2011). [CrossRef]
21. S. P. Madsen, S. R. Johannsen, B. R. Jeppesen, J. V. Nygaard, P. B. Jensen, J. Chevallier, B. Julsgaard, P. Balling, and A. N. Larsen, “Optimizing Plasmonically Enhanced Upconversion,” Energy Procedia 77, 478–486 (2015). [CrossRef]
22. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices”, Appl. Opt. 37, 5271–5283 (1998). [CrossRef]
23. I. H. Malitson, “Interspecimen Comparison of the Refractive Index of Fused Silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965). [CrossRef]
24. K. Svanberg, “MMA and GCMMA - two methods for nonlinear optimization,” 1–15 (2007).
25. J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002). [CrossRef]
26. O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sinica 25, 227–239 (2009). [CrossRef]
27. R. E. Christiansen, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Creating geometrically robust designs for highly sensitive problems using topology optimization: Acoustic cavity design,” Struct. Multidiscip. Optim. 52, 737–754 (2015). [CrossRef]
