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macrostructure for 3D functional gradient systems based on additive manufacturing
Optics Communications 414 (2018) 195–201
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Design and fabrication of integrated micro/macrostructure for 3D functional gradient systems based on additive manufacturing Ming Yin *, Luofeng Xie, Weifeng Jiang, Guofu Yin School of Manufacturing Science and Engineering, Sichuan University, Chengdu, 610065, China
a r t i c l e
i n f o
Keywords: 3D functional gradient system Metamaterials Additive manufacturing Photonic crystals
a b s t r a c t Functional gradient systems have important applications in many areas. Although a 2D dielectric structure that serves as the gradient index medium for controlling electromagnetic waves is well established, it may not be suitable for application in 3D case. In this paper, we present a method to realize functional gradient systems with 3D integrated micro/macrostructure. The homogenization of the structure is studied in detail by conducting band diagram analysis. The analysis shows that the effective medium approximation is valid even when periodicity is comparable to wavelength. The condition to ensure the polarization-invariant, isotropic, and frequency-independent property is investigated. The scheme for the design and fabrication of 3D systems requiring spatial material property distribution is presented. By using the vat photopolymerization process, a large overall size of macrostructure at the system level and precise fine features of microstructure at the unit cell level are realized, thus demonstrating considerable scalability of the system for wave manipulation. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Systems with designed and engineered distribution of gradient material property offer unprecedented opportunities to achieve required functions, which have important implications for a multitude of disciplines in today’s scientific and engineering world. In many functional devices, the integration of dissimilar materials could result in an incompatible interface. Engineered material property transitions across the interface can reduce the mismatches in electromagnetic, mechanical, or thermal properties. Graded transitions can also enhance the mechanical performance of the interface to prevent premature failure by reducing stress concentrations, redistributing thermal stresses, or improving interfacial bonding [1,2]. Controlled material property gradients over volume has also been used in structural engineering applications, for example, to produce artificial body for soft robot or biomaterial for medical applications [2–4]. Moreover, devices with specific gradient distribution of material properties enable the control of electromagnetic, thermodynamic, or mechanical wave propagation in desired ways [5–9]. Such functional gradient systems would usually require complex material parameters and precise control of property distribution. The distribution of material properties can be engineered by continuously varying the composition or microstructure of the constituent materials
over volume. Such functionally graded materials can be realized by the composite material synthesis approach, which normally involves complex manufacturing techniques. The development of new fabrication techniques such as the multimaterial additive manufacturing process offers the ability to fabricate functionally graded parts with controllable shape and material performance [3,4]. However, the state of the art is still limited for the real application aspect. More available materials and comprehensive processing capabilities would be required to develop functional parts with precise control of complex structure and gradient property from laboratory samples with relatively simple morphology and discrete material property distribution. Metamaterial provides a promising approach to construct functional gradient systems. By using the principle of metamaterial, the required gradient property distribution can be realized by spatially changing the subwavelength structure rather than the composition or microstructure of constituent materials, which would then ease the challenge for fabrication. The principle of metamaterial can be applied to different areas [7–9], but in the present study, we mainly focus on the 3D electromagnetic gradient systems. In electromagnetism, the planar gradient metamaterials with 2D or 2.5D geometry are well established [10–14]. However, when they are applied to complex 3D systems, the out-ofplane propagation cannot be ignored, and the incident angle or polarization of wave may be limited. Moreover, metamaterials constructed
* Corresponding author.
E-mail address: [email protected] (M. Yin). https://doi.org/10.1016/j.optcom.2017.12.088 Received 12 October 2017; Received in revised form 28 December 2017; Accepted 30 December 2017 0030-4018/© 2018 Elsevier B.V. All rights reserved.
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Optics Communications 414 (2018) 195–201
Fig. 2. Band diagrams for the lowest first and second bands. Upper left inset: symmetry k-points in the Brillouin zone of the FCC lattice. Lower right inset: illustration of structural parameters for the woodpile structure.
techniques such as the direct laser writing or the vat photopolymerization process. In the structure design, the available fabrication capabilities should be considered. Dielectric structures with self-supporting topologies, without the need of a substrate, would ease the fabrication and be suitable as a 3D gradient medium. Fig. 1(a) shows the structure of dielectric spheres arranged in the lattice. The radius of each sphere embedded in air should be large enough to cause overlap and create connected networks. A structure that allows a larger range of dielectric volume fraction is preferred to achieve more flexible control of effective material property. However, in this case, low dielectric volume fraction cannot be realized. Fig. 1(b) shows the structure of air spheres arranged in the lattice. For the photopolymerization-based processes, the unexposed material that exists in the enclosed structure cannot be removed during or after fabrication. Therefore, in this case, the radius of each air sphere should be large enough to avoid undesirable enclosed cavities in the integrated structure, and high dielectric volume fraction cannot be realized. Fig. 1(c) and (d) present the so-called rod-connected and woodpile structures, respectively. For these two cases, the dielectric volume fraction can be theoretically tuned from 0 to 1 because the dielectric rods would always remain connected and make the structure self-supporting. Compared with the rod-connected structure, the woodpile structure is simpler in geometry and easier to fabricate. It is studied as a proof-of√ principle. When the rod thickness is 𝑎∕4 and the rod spacing 𝑏 is 𝑎∕ 2, the lattice is strictly FCC with the lattice constant of 𝑎. If the rod width is set to be 𝑤, then the dielectric volume fraction can be controlled by varying the 𝑤∕𝑏 ratio. Lossless and non-dispersive material with permittivity of 3 (a typical value for photopolymer [22]) is applied as the constituent material. The band diagram-based calculations are conducted using BandSOLVE. The calculated band diagrams for the lowest bands are shown in Fig. 2. In general, when the normalized frequency is relatively low, the lowest first and second bands are very closely spaced and compromise a degenerate pair, indicating that the unit cell with high symmetry is polarization invariant at long wavelength. Moreover, the dispersion relationship 𝜔(𝐤) is linear, indicating that the structure can be homogenized as a non-dispersive effective medium in a broadband as expected. The dispersion curves for the wave vector k moving from the origin 𝛤 to the special symmetry points at the irreducible Brillouin zone (𝐾, 𝑊 , 𝑋, 𝑈 , and 𝐿, respectively) are plotted in Fig. 3. As shown in Fig. 3(a) and (b), for 𝐤𝛤 −𝑋 and 𝐤𝛤 −𝐿 , where k moves from 𝛤 towards the face center of the Brillouin zone, the first and second bands nearly remain degenerate for the whole Brillouin zone. As shown in Fig. 3(c)–(e), for 𝐤𝛤 −𝑈 , 𝐤𝛤 −𝐾 , and 𝐤𝛤 −𝑊 (where 𝑈 , 𝐾, and 𝑊 are the symmetry points
Fig. 1. Typical 3D dielectric structures. (a) Dielectric spheres arranged in the lattice. (b) Air spheres arranged in the lattice. (c) Rod-connected structure. (d) Woodpile structure.
using dielectrics without metallic inclusions can enable to realize broadband and low-loss property at optical frequencies. Therefore, dielectric structures with 3D topology are expected to be good candidates for 3D systems. In this paper, we present a method to realize 3D functional gradient system with 3D integrated micro- and macrostructure. The homogenization of the structure is studied using band diagram calculations. The condition to validate the effective medium approximation and to ensure the polarization-invariant, isotropic, and frequency-independent property is investigated and discussed in detail. The required spatial material property distribution is realized with integrated micro/macrostructure. The scheme for the design and fabrication of such systems is presented. The vat photopolymerization-based additive manufacturing process is used in the implementation. Large overall size at the system level and fine features at the unit cell level are realized, thus demonstrating considerable scalability of the system for wave control. 2. Homogenization of the 3D periodic dielectric structure Artificial dielectric structure can be homogenized and can serve as a gradient index medium in the long-wavelength regime. A 2D case in this regime has been studied in detail [14–18], and a 3D case has also been previously realized [19]. For a 3D periodic dielectric structure to serve as an effective medium, properties such as polarization independence, isotropy, and non-dispersion are expected in unit cells with highsymmetry geometry. To satisfy the effective medium approximation, it is usually required that the periodicity should be one order of magnitude smaller than the free-space wavelength. Moreover, when the ratio of periodicity to wavelength is not negligible, it will reach the photonic crystal regime, where the Bragg diffraction mechanism becomes significant. However, the effective medium limit for the 3D dielectric structure is not that strict [19]. We conduct band diagrambased calculations to study this point in detail. Typical 3D dielectric structures with high-symmetry face-centered cubic (FCC) lattice [20,21] are presented in Fig. 1. As discussed in Section 3, such periodic structures with gradient topology can be realized by photopolymerization-based 3D fabrication 196
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Fig. 3. Dispersion curves of the first and second bands with different 𝑤∕𝑏 ratios for (a) 𝐤𝛤 −𝐾 , (b) 𝐤𝛤 −𝑊 , (c) 𝐤𝛤 −𝑋 , (d) 𝐤𝛤 −𝑈 and (e) 𝐤𝛤 −𝐿 .
From the band diagrams, the effective refractive index 𝑛𝑒𝑓 𝑓 for a given wave vector k is calculated by
at the edge or corner of the Brillouin zone), as the frequency becomes higher, the degenerate bands start to split slightly, especially in the case of midrange volume fraction. Moreover, the dispersion relationship is no longer linear near the Brillouin zone edge, indicating that it reaches the Bragg diffraction limit. However, in general, we find that when the normalized frequency (𝜔𝑎∕2𝜋𝑐) is below 0.49, the first and second bands remain degenerate for the woodpile structure with volume fraction ranging from 0 to 1, and the dispersion relationship 𝜔(𝐤) is also linear. Therefore, the group velocity remains constant in this regime. It should be noted that for a constituent material with higher refractive index, the Bragg diffraction tends to appear at lower normalized frequency because the wave scattering becomes more significant. The degeneracy lifting of the first two bands and group-velocity dispersion also start with a smaller magnitude of k. We focus on the analysis and discussion of normalized frequency below 0.49 where the first two bands remain degenerate. In this regime, as plotted in Fig. 4(a) in detail, the dispersion curves of the first band for 𝐤𝛤 −𝐾 , 𝐤𝛤 −𝑊 , 𝐤𝛤 −𝑋 , 𝐤𝛤 −𝑈 , and 𝐤𝛤 −𝐿 are in good agreement with each other. The group velocity is given by 𝐯 (𝐤) = ∇𝐤 𝜔 (𝐤) ,
𝑛𝑒𝑓 𝑓 = 𝑐 |𝐤| ∕𝜔 (𝐤) ,
(2)
where 𝑐 is the velocity of light in free space. As shown in Fig. 5(a), 𝑛𝑒𝑓 𝑓 is frequency independent when 𝜔𝑎∕2𝜋𝑐 < 0.49, which further validates the broadband property of the effective medium. The calculated 𝑛𝑒𝑓 𝑓 for different volume fractions 𝑓 (𝑓 = 𝑤∕𝑏 for the woodpile structure) is shown in Fig. 5(b). The result shows almost a linear relationship between the effective refractive index 𝑛𝑒𝑓 𝑓 and the volume fraction 𝑓 , which can be approximated by √ √ 𝑛𝑒𝑓 𝑓 = (1 − 𝑓 ) 𝜀𝑏 + 𝑓 𝜀𝑖 ,
(3) 𝑛2 )
where 𝜀𝑏 and 𝜀𝑖 represent the dielectric constant (𝜀 = of the background and constituent material, respectively. The result is also compared to that obtained from the common effective medium theory by using analytical equations [23]. The averaging approximation result is calculated by 𝑛𝑒𝑓 𝑓 =
(1)
√ (1 − 𝑓 )𝜀𝑏 + 𝑓 (𝜀𝑖 − 𝜀𝑏 ).
(4)
Moreover, the Maxwell–Garnett approximation result is calculated by √ ( ) 2𝜀𝑏 + 𝜀𝑖 + 2𝑓 (𝜀𝑖 − 𝜀𝑏 ) 𝑛𝑒𝑓 𝑓 = 𝜀𝑏 . (5) 2𝜀𝑏 + 𝜀𝑖 − 𝑓 (𝜀𝑖 − 𝜀𝑏 )
where ∇𝐤 is the gradient with respect to k. For wave vector k with the same magnitude, the magnitude of the group velocity 𝐯 (𝐤) is a constant value, which is direction invariant. Therefore, for normalized frequency below 0.49, the equi-frequency surface would be spherical in the Brillouin zone. As a proof-of-principle, the equi-frequency surface at the normalized frequency of 0.49 (𝑤∕𝑏 = 0.4𝑎) is shown in Fig. 4(b). The value of the effective refractive index is then demonstrated to be independent of direction, indicating that the homogenized effective medium is isotropic in this regime. For higher frequencies, isotropic property is not strictly ensured, even if the equi-frequency surface is almost spherical.
Comparison with the result calculated from the dispersion curves reveals that the averaging approximation result is generally larger and the Maxwell–Garnett result is generally smaller. Therefore, the accurate value of the effective refractive index should be calculated from the dispersion curves. However, in applications where the precise local control of refractive index is not strictly required, the effective index can be calculated using the above analytical equations to simplify the design. The calculated effective index of the non-resonant structure ranges from 1.0 (air) to 1.73 (bulk material). Practical applications 197
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example, the rod radius), while keeping the lattice constant unchanged, the global control of the effective material property is realized. The transition of the microstructure is at the boundary of adjacent unit cells, and the self-supporting topology is preserved to provide the required mechanical strength. Thus, the microstructure at the unit cell level is integrated, and the gradient macrostructure is constructed at the system level. For material property distribution profile that is more regular, the discretization can be simplified and realized with layer systems. As a conceptual example, Fig. 6(b) shows a typical spherical system with radially varying material property realized with the woodpile structure. For layer systems, the material property distribution can be controlled by changing the structural parameters (in this example, the rod width) within each layer. The transition of the microstructure is at the boundary of adjacent layers. The samples presented in Fig. 6 were fabricated using the stereolithography-based vat photopolymerization process. At the system level, the macrostructure captures the desired global effect of material property gradation in the relevant region. The length scale of the macrostructure should be designed large enough to allow significant spatial variations in the microstructure for required function. At the unit cell level, the microstructure determines the local effective material property and the operating frequencies of the system. The length scale of the unit cell should be small enough compared with the free-space wavelength to satisfy the effective medium approximation, while ensuring a sufficient resolution of the discrete material property distribution profile. Moreover, the features of the sub-wavelength microstructure should be fine and accurate to realize the precise target value of effective material property and smooth control of the gradation. For real applications, sufficiently large overall size of the macrostructure at the system level and precise fine features of the microstructure at the unit cell level are expected. Scalability of such integrated 3D gradient micro/macrostructure can extend the available working frequency range and enhance the applicability. Such 3D gradient structure in the metamaterial regime has previously been experimentally realized by using the direct laser writing technique. This structure was first applied in the implementation of 3D carpet cloak at optical wavelength [19]. A similar approach was also used recently to realize the 3D Luneburg lens [24]. The twophoton polymerization process enables the production of microscale and nanoscale 3D subwavelength microstructures; yet, the achievable length scale of the macrostructure and the materials available are relatively limited. The development of additive manufacturing techniques offers the opportunity to extend the scalability of such sophisticated structures. We realize the proposed structure based on the vat photopolymerization process with a stereolithography (SL) system (SPS600B, Shaanxi Hengtong). The fabricated sample of a typical spherical gradient system is presented in Fig. 7 as an example. The macrostructure consists of more than 50,000 FCC unit cells with a lattice constant of 3 mm. The diameter of the system is as large as 30 cm, and the smallest rod width of the woodpile structure in the sample is 100 μm, thus demonstrating a rather large ratio of overall size to feature size. As shown in Fig. 7(a), an accurate external shape of the system is preserved in the macrostructure. The achievable length scale of the macrostructure depends on the maximum build area of the fabrication process; therefore, the overall size can be further extended. For SL systems with multiple laser sources, the build area can be quite large (e.g., 1500 × 750 × 550 mm for ProX 950, 3D Systems). The surface features of the macrostructure are shown in Fig. 7(b). In general, for typical additive manufacturing techniques, a support structure is needed to avoid distortion and movement of the overhanging features. However, adding support can dramatically complicate the presented 3D gradient structure and increase difficulty in design and fabrication. Even if the support can be removed by post-processing, small features of the microstructure could be damaged. However, as demonstrated, for the vat photopolymerization process, the presence of the cross-stacking features at the unit cell level makes the integrated structure as self-supported to some extent; therefore, it can be
Fig. 4. (a) Dispersion curves of the first band for 𝐤𝛤 −𝐾 , 𝐤𝛤 −𝑊 , 𝐤𝛤 −𝑋 , 𝐤𝛤 −𝑈 . and 𝐤𝛤 −𝐿 with different 𝑤∕𝑏 ratios for comparison. (b) Spherical equi-frequency surface in the Brillouin zone at the normalized frequency of 0.49 (𝑤∕𝑏 = 0.4𝑎).
requiring higher indices can be realized using a photo-curable material with higher dielectric constant, as discussed in Section 3. It should be noted that in the current calculation, the lattice constant of the cubic cell is adopted as the period 𝑎. For the FCC lattice, the smallest repeating unit is a rhombohedron of volume 𝑎3 ∕4. If the primitive cell √ is considered as the unit cell, the period size can be regarded as 𝑎/ 2. Then, the ratio of the period to the wavelength, which satisfies the effective medium approximation, will be 0.35 instead. As demonstrated, for such 3D periodic dielectric structure with high symmetry, the effective medium limit is rather lenient. The effective medium approximation is validated even when the periodicity is comparable to wavelength, and the polarization-invariant, isotropic and frequencyindependent property is ensured in a broad bandwidth. 3. Design and fabrication of the integrated 3D gradient micro/macrostructure The scheme to realize the required spatial distribution of material property is shown in Fig. 6. The original material distribution profile is discretized into unit cells of certain size, for example, the cubic lattice unit cells as shown in Fig. 6(a). It should be noted that the resolution of the discrete model should be fine enough to ensure that it approximates the original ideal design sufficiently well. The required local material property is achieved by the microstructure at the unit cell level (the rod-connected structure as a conceptual example here). By spatially varying the unit cell microstructure parameters (in this 198
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Fig. 5. Calculated effective refractive index. (a) Relationship between 𝑛𝑒𝑓 𝑓 and the normalized frequency for different volume fractions. (b) 𝑛𝑒𝑓 𝑓 vs volume fraction for different calculation methods.
directly fabricated without additional support structure. Furthermore, by arranging the fabrication orientation to ensure that each layer is solidified successively along the height direction of the rods, the undesired staircase effect in features commonly seen for the layer-bylayer process is also avoided. Fig. 7(c) shows the internal gradient structure of the system. The presented cross-section is perpendicular to the height direction of the rods. The accuracy of the microstructure at the unit cell level is reflected by the quality of the features. The inset in Fig. 7(c) presents the smallest features of the sample (i.e., rods with the smallest width). Smooth spatial transition of the microstructure is realized, which would reduce the scattering in the internal system. By adjusting the fabrication parameters (e.g., power, spot radius and scanning speed of the laser), the feature size can be tuned and further reduced. However, acceptable mechanical strength needs to be considered to avoid the defect of microstructure under the large weight of the system. The wave manipulating ability of such gradient structure has been demonstrated by both simulation and experimental results in our previous work [25,26]; this proves that the designed permittivity profile can be achieved by the presented method. Moreover, as previously validated by us in the microwave regime, the performance of such prototype is robust, even if the homogenization
was approximately implemented according to the effective medium theory rather than by the dispersion analysis [25,26]. The operating frequencies for the system are determined by the unit cell size. Arranging dielectrics in a lattice with higher symmetry (e.g., FCC rather than simple cubic lattice) would satisfy the effective medium approximation at higher frequency. For current setting (𝑎 = 3 mm), the theoretical upper limit of the operating frequency for the woodpile structure is 49 GHz, as demonstrated in Section 2. By scaling down to a smaller lattice, the upper limit may be theoretically extended to terahertz regime within current fabrication capability. However, it should be noted that when the smallest feature size is set, arranging dielectrics in a smaller lattice would make smaller volume fraction unattainable. A variety of materials with a wide range of properties are available for the vat photopolymerization process. Moreover, the intrinsic material properties can be even further tuned by the mixed-material approach (e.g., the PolyJet Digital Materials, Stratasys). For the constituent material of the 3D electromagnetic gradient systems discussed in the paper, the nondispersive and lossless property is desired in most cases. Such property is available for common vat photopolymerization materials [22]. The material we used has a dielectric constant of 3.0 when cured (measured with a well-established waveguide method [27]), 199
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Fig. 6. Scheme for the design and fabrication of the integrated 3D gradient micro/macrostructure. (a) Discrete approximation of spatial material property distribution with cubic lattice unit cells. (b) Layer approximation for a more regular system.
which is almost frequency independent and has low loss within a broad bandwidth in the microwave regime. For applications that require higher dielectric constant, photocurable materials incorporated with ceramic can be used (e.g., ceramic SL). In addition to the photo-curable materials, other types of constituent materials can be indirectly realized by post-processing based on the polymer structure template [28]. The resolution of the vat photopolymerization process can be further increased by using the mask-image-projection process (e.g., Projection Microstereolithography), thus allowing for the creation of microscale 3D
dielectric structure that can operate at higher frequencies [29]. Current state-of-the-art vat photopolymerization system, which integrates the SL and digital light processing photopolymerization process by using both the laser scan mechanism and the mask-image-projection mechanism [30], enables the production of microscale features with large scale overall size. In general, more materials can be expected to be developed for the vat photopolymerization-based process. It also offers the ability to produce microscale features while preserving scalability to larger macro size. 200
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Fig. 7. A fabricated sample of the proposed system. (a) Overall view of the macrostructure showing the accurate external shape of the system. (b) Features at the surface of the highlighted part in the system. (c) Cross-section perpendicular to the height direction of rods, which shows the internal gradient microstructure with smooth spatial transition. The inset presents the features of the rods.
4. Conclusion In conclusion, the design and fabrication method of the integrated micro/macrostructure for 3D functional gradient systems is introduced. Band diagram-based calculation demonstrates that the homogenized 3D periodic dielectric structure is suitable as an effective medium for 3D gradient systems. For the woodpile structure, the effective medium approximation is validated when the ratio of the lattice constant to the wavelength is smaller than 0.49. Moreover, the polarizationinvariant, isotropic, and frequency-independent property is ensured. A typical sample fabricated by SL-based vat photopolymerization process demonstrates the large overall size of macrostructure and fine features of microstructure. As discussed, the scalability of the structure can be extended within current fabrication capability, which can expand the operating frequency range and widen the applications for wave manipulation. The scheme of using 3D gradient micro/macrostructure to realize spatial material property distribution can also be extended to mechanics [31,32] and can be further applied in systems for other functions [2,3]. Acknowledgments This work is supported by the National Natural Science Foundation of China (No. 51705347) and the Department of Science and Technology of Sichuan Province (Nos. 2016GZ0161 and 2016GZ0195). References [1] S. Suresh, Graded materials for resistance to contact deformation and damage, Science 292 (2001) 2447–2451. [2] R. Libanori, R.M. Erb, A. Reiser, H. Le Ferrand, M.J. Suess, R. Spolenak, A.R. Studart, Stretchable heterogeneous composites with extreme mechanical gradients, Nat. Commun. 3 (2012) 1265. [3] N.W. Bartlett, M.T. Tolley, J.T.B. Overvelde, J.C. Weaver, B. Mosadegh, K. Bertoldi, G.M. Whitesides, R.J. Wood, A 3D-printed, functionally graded soft robot powered by combustion, Science 349 (2015) 161–165.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design and fabrication of integrated micro/macrostructure for 3D functional gradient systems based on additive manufacturing Ming Yin *, Luofeng Xie, Weifeng Jiang, Guofu Yin School of Manufacturing Science and Engineering, Sichuan University, Chengdu, 610065, China
a r t i c l e
i n f o
Keywords: 3D functional gradient system Metamaterials Additive manufacturing Photonic crystals
a b s t r a c t Functional gradient systems have important applications in many areas. Although a 2D dielectric structure that serves as the gradient index medium for controlling electromagnetic waves is well established, it may not be suitable for application in 3D case. In this paper, we present a method to realize functional gradient systems with 3D integrated micro/macrostructure. The homogenization of the structure is studied in detail by conducting band diagram analysis. The analysis shows that the effective medium approximation is valid even when periodicity is comparable to wavelength. The condition to ensure the polarization-invariant, isotropic, and frequency-independent property is investigated. The scheme for the design and fabrication of 3D systems requiring spatial material property distribution is presented. By using the vat photopolymerization process, a large overall size of macrostructure at the system level and precise fine features of microstructure at the unit cell level are realized, thus demonstrating considerable scalability of the system for wave manipulation. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Systems with designed and engineered distribution of gradient material property offer unprecedented opportunities to achieve required functions, which have important implications for a multitude of disciplines in today’s scientific and engineering world. In many functional devices, the integration of dissimilar materials could result in an incompatible interface. Engineered material property transitions across the interface can reduce the mismatches in electromagnetic, mechanical, or thermal properties. Graded transitions can also enhance the mechanical performance of the interface to prevent premature failure by reducing stress concentrations, redistributing thermal stresses, or improving interfacial bonding [1,2]. Controlled material property gradients over volume has also been used in structural engineering applications, for example, to produce artificial body for soft robot or biomaterial for medical applications [2–4]. Moreover, devices with specific gradient distribution of material properties enable the control of electromagnetic, thermodynamic, or mechanical wave propagation in desired ways [5–9]. Such functional gradient systems would usually require complex material parameters and precise control of property distribution. The distribution of material properties can be engineered by continuously varying the composition or microstructure of the constituent materials
over volume. Such functionally graded materials can be realized by the composite material synthesis approach, which normally involves complex manufacturing techniques. The development of new fabrication techniques such as the multimaterial additive manufacturing process offers the ability to fabricate functionally graded parts with controllable shape and material performance [3,4]. However, the state of the art is still limited for the real application aspect. More available materials and comprehensive processing capabilities would be required to develop functional parts with precise control of complex structure and gradient property from laboratory samples with relatively simple morphology and discrete material property distribution. Metamaterial provides a promising approach to construct functional gradient systems. By using the principle of metamaterial, the required gradient property distribution can be realized by spatially changing the subwavelength structure rather than the composition or microstructure of constituent materials, which would then ease the challenge for fabrication. The principle of metamaterial can be applied to different areas [7–9], but in the present study, we mainly focus on the 3D electromagnetic gradient systems. In electromagnetism, the planar gradient metamaterials with 2D or 2.5D geometry are well established [10–14]. However, when they are applied to complex 3D systems, the out-ofplane propagation cannot be ignored, and the incident angle or polarization of wave may be limited. Moreover, metamaterials constructed
* Corresponding author.
E-mail address: [email protected] (M. Yin). https://doi.org/10.1016/j.optcom.2017.12.088 Received 12 October 2017; Received in revised form 28 December 2017; Accepted 30 December 2017 0030-4018/© 2018 Elsevier B.V. All rights reserved.
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Fig. 2. Band diagrams for the lowest first and second bands. Upper left inset: symmetry k-points in the Brillouin zone of the FCC lattice. Lower right inset: illustration of structural parameters for the woodpile structure.
techniques such as the direct laser writing or the vat photopolymerization process. In the structure design, the available fabrication capabilities should be considered. Dielectric structures with self-supporting topologies, without the need of a substrate, would ease the fabrication and be suitable as a 3D gradient medium. Fig. 1(a) shows the structure of dielectric spheres arranged in the lattice. The radius of each sphere embedded in air should be large enough to cause overlap and create connected networks. A structure that allows a larger range of dielectric volume fraction is preferred to achieve more flexible control of effective material property. However, in this case, low dielectric volume fraction cannot be realized. Fig. 1(b) shows the structure of air spheres arranged in the lattice. For the photopolymerization-based processes, the unexposed material that exists in the enclosed structure cannot be removed during or after fabrication. Therefore, in this case, the radius of each air sphere should be large enough to avoid undesirable enclosed cavities in the integrated structure, and high dielectric volume fraction cannot be realized. Fig. 1(c) and (d) present the so-called rod-connected and woodpile structures, respectively. For these two cases, the dielectric volume fraction can be theoretically tuned from 0 to 1 because the dielectric rods would always remain connected and make the structure self-supporting. Compared with the rod-connected structure, the woodpile structure is simpler in geometry and easier to fabricate. It is studied as a proof-of√ principle. When the rod thickness is 𝑎∕4 and the rod spacing 𝑏 is 𝑎∕ 2, the lattice is strictly FCC with the lattice constant of 𝑎. If the rod width is set to be 𝑤, then the dielectric volume fraction can be controlled by varying the 𝑤∕𝑏 ratio. Lossless and non-dispersive material with permittivity of 3 (a typical value for photopolymer [22]) is applied as the constituent material. The band diagram-based calculations are conducted using BandSOLVE. The calculated band diagrams for the lowest bands are shown in Fig. 2. In general, when the normalized frequency is relatively low, the lowest first and second bands are very closely spaced and compromise a degenerate pair, indicating that the unit cell with high symmetry is polarization invariant at long wavelength. Moreover, the dispersion relationship 𝜔(𝐤) is linear, indicating that the structure can be homogenized as a non-dispersive effective medium in a broadband as expected. The dispersion curves for the wave vector k moving from the origin 𝛤 to the special symmetry points at the irreducible Brillouin zone (𝐾, 𝑊 , 𝑋, 𝑈 , and 𝐿, respectively) are plotted in Fig. 3. As shown in Fig. 3(a) and (b), for 𝐤𝛤 −𝑋 and 𝐤𝛤 −𝐿 , where k moves from 𝛤 towards the face center of the Brillouin zone, the first and second bands nearly remain degenerate for the whole Brillouin zone. As shown in Fig. 3(c)–(e), for 𝐤𝛤 −𝑈 , 𝐤𝛤 −𝐾 , and 𝐤𝛤 −𝑊 (where 𝑈 , 𝐾, and 𝑊 are the symmetry points
Fig. 1. Typical 3D dielectric structures. (a) Dielectric spheres arranged in the lattice. (b) Air spheres arranged in the lattice. (c) Rod-connected structure. (d) Woodpile structure.
using dielectrics without metallic inclusions can enable to realize broadband and low-loss property at optical frequencies. Therefore, dielectric structures with 3D topology are expected to be good candidates for 3D systems. In this paper, we present a method to realize 3D functional gradient system with 3D integrated micro- and macrostructure. The homogenization of the structure is studied using band diagram calculations. The condition to validate the effective medium approximation and to ensure the polarization-invariant, isotropic, and frequency-independent property is investigated and discussed in detail. The required spatial material property distribution is realized with integrated micro/macrostructure. The scheme for the design and fabrication of such systems is presented. The vat photopolymerization-based additive manufacturing process is used in the implementation. Large overall size at the system level and fine features at the unit cell level are realized, thus demonstrating considerable scalability of the system for wave control. 2. Homogenization of the 3D periodic dielectric structure Artificial dielectric structure can be homogenized and can serve as a gradient index medium in the long-wavelength regime. A 2D case in this regime has been studied in detail [14–18], and a 3D case has also been previously realized [19]. For a 3D periodic dielectric structure to serve as an effective medium, properties such as polarization independence, isotropy, and non-dispersion are expected in unit cells with highsymmetry geometry. To satisfy the effective medium approximation, it is usually required that the periodicity should be one order of magnitude smaller than the free-space wavelength. Moreover, when the ratio of periodicity to wavelength is not negligible, it will reach the photonic crystal regime, where the Bragg diffraction mechanism becomes significant. However, the effective medium limit for the 3D dielectric structure is not that strict [19]. We conduct band diagrambased calculations to study this point in detail. Typical 3D dielectric structures with high-symmetry face-centered cubic (FCC) lattice [20,21] are presented in Fig. 1. As discussed in Section 3, such periodic structures with gradient topology can be realized by photopolymerization-based 3D fabrication 196
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Fig. 3. Dispersion curves of the first and second bands with different 𝑤∕𝑏 ratios for (a) 𝐤𝛤 −𝐾 , (b) 𝐤𝛤 −𝑊 , (c) 𝐤𝛤 −𝑋 , (d) 𝐤𝛤 −𝑈 and (e) 𝐤𝛤 −𝐿 .
From the band diagrams, the effective refractive index 𝑛𝑒𝑓 𝑓 for a given wave vector k is calculated by
at the edge or corner of the Brillouin zone), as the frequency becomes higher, the degenerate bands start to split slightly, especially in the case of midrange volume fraction. Moreover, the dispersion relationship is no longer linear near the Brillouin zone edge, indicating that it reaches the Bragg diffraction limit. However, in general, we find that when the normalized frequency (𝜔𝑎∕2𝜋𝑐) is below 0.49, the first and second bands remain degenerate for the woodpile structure with volume fraction ranging from 0 to 1, and the dispersion relationship 𝜔(𝐤) is also linear. Therefore, the group velocity remains constant in this regime. It should be noted that for a constituent material with higher refractive index, the Bragg diffraction tends to appear at lower normalized frequency because the wave scattering becomes more significant. The degeneracy lifting of the first two bands and group-velocity dispersion also start with a smaller magnitude of k. We focus on the analysis and discussion of normalized frequency below 0.49 where the first two bands remain degenerate. In this regime, as plotted in Fig. 4(a) in detail, the dispersion curves of the first band for 𝐤𝛤 −𝐾 , 𝐤𝛤 −𝑊 , 𝐤𝛤 −𝑋 , 𝐤𝛤 −𝑈 , and 𝐤𝛤 −𝐿 are in good agreement with each other. The group velocity is given by 𝐯 (𝐤) = ∇𝐤 𝜔 (𝐤) ,
𝑛𝑒𝑓 𝑓 = 𝑐 |𝐤| ∕𝜔 (𝐤) ,
(2)
where 𝑐 is the velocity of light in free space. As shown in Fig. 5(a), 𝑛𝑒𝑓 𝑓 is frequency independent when 𝜔𝑎∕2𝜋𝑐 < 0.49, which further validates the broadband property of the effective medium. The calculated 𝑛𝑒𝑓 𝑓 for different volume fractions 𝑓 (𝑓 = 𝑤∕𝑏 for the woodpile structure) is shown in Fig. 5(b). The result shows almost a linear relationship between the effective refractive index 𝑛𝑒𝑓 𝑓 and the volume fraction 𝑓 , which can be approximated by √ √ 𝑛𝑒𝑓 𝑓 = (1 − 𝑓 ) 𝜀𝑏 + 𝑓 𝜀𝑖 ,
(3) 𝑛2 )
where 𝜀𝑏 and 𝜀𝑖 represent the dielectric constant (𝜀 = of the background and constituent material, respectively. The result is also compared to that obtained from the common effective medium theory by using analytical equations [23]. The averaging approximation result is calculated by 𝑛𝑒𝑓 𝑓 =
(1)
√ (1 − 𝑓 )𝜀𝑏 + 𝑓 (𝜀𝑖 − 𝜀𝑏 ).
(4)
Moreover, the Maxwell–Garnett approximation result is calculated by √ ( ) 2𝜀𝑏 + 𝜀𝑖 + 2𝑓 (𝜀𝑖 − 𝜀𝑏 ) 𝑛𝑒𝑓 𝑓 = 𝜀𝑏 . (5) 2𝜀𝑏 + 𝜀𝑖 − 𝑓 (𝜀𝑖 − 𝜀𝑏 )
where ∇𝐤 is the gradient with respect to k. For wave vector k with the same magnitude, the magnitude of the group velocity 𝐯 (𝐤) is a constant value, which is direction invariant. Therefore, for normalized frequency below 0.49, the equi-frequency surface would be spherical in the Brillouin zone. As a proof-of-principle, the equi-frequency surface at the normalized frequency of 0.49 (𝑤∕𝑏 = 0.4𝑎) is shown in Fig. 4(b). The value of the effective refractive index is then demonstrated to be independent of direction, indicating that the homogenized effective medium is isotropic in this regime. For higher frequencies, isotropic property is not strictly ensured, even if the equi-frequency surface is almost spherical.
Comparison with the result calculated from the dispersion curves reveals that the averaging approximation result is generally larger and the Maxwell–Garnett result is generally smaller. Therefore, the accurate value of the effective refractive index should be calculated from the dispersion curves. However, in applications where the precise local control of refractive index is not strictly required, the effective index can be calculated using the above analytical equations to simplify the design. The calculated effective index of the non-resonant structure ranges from 1.0 (air) to 1.73 (bulk material). Practical applications 197
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example, the rod radius), while keeping the lattice constant unchanged, the global control of the effective material property is realized. The transition of the microstructure is at the boundary of adjacent unit cells, and the self-supporting topology is preserved to provide the required mechanical strength. Thus, the microstructure at the unit cell level is integrated, and the gradient macrostructure is constructed at the system level. For material property distribution profile that is more regular, the discretization can be simplified and realized with layer systems. As a conceptual example, Fig. 6(b) shows a typical spherical system with radially varying material property realized with the woodpile structure. For layer systems, the material property distribution can be controlled by changing the structural parameters (in this example, the rod width) within each layer. The transition of the microstructure is at the boundary of adjacent layers. The samples presented in Fig. 6 were fabricated using the stereolithography-based vat photopolymerization process. At the system level, the macrostructure captures the desired global effect of material property gradation in the relevant region. The length scale of the macrostructure should be designed large enough to allow significant spatial variations in the microstructure for required function. At the unit cell level, the microstructure determines the local effective material property and the operating frequencies of the system. The length scale of the unit cell should be small enough compared with the free-space wavelength to satisfy the effective medium approximation, while ensuring a sufficient resolution of the discrete material property distribution profile. Moreover, the features of the sub-wavelength microstructure should be fine and accurate to realize the precise target value of effective material property and smooth control of the gradation. For real applications, sufficiently large overall size of the macrostructure at the system level and precise fine features of the microstructure at the unit cell level are expected. Scalability of such integrated 3D gradient micro/macrostructure can extend the available working frequency range and enhance the applicability. Such 3D gradient structure in the metamaterial regime has previously been experimentally realized by using the direct laser writing technique. This structure was first applied in the implementation of 3D carpet cloak at optical wavelength [19]. A similar approach was also used recently to realize the 3D Luneburg lens [24]. The twophoton polymerization process enables the production of microscale and nanoscale 3D subwavelength microstructures; yet, the achievable length scale of the macrostructure and the materials available are relatively limited. The development of additive manufacturing techniques offers the opportunity to extend the scalability of such sophisticated structures. We realize the proposed structure based on the vat photopolymerization process with a stereolithography (SL) system (SPS600B, Shaanxi Hengtong). The fabricated sample of a typical spherical gradient system is presented in Fig. 7 as an example. The macrostructure consists of more than 50,000 FCC unit cells with a lattice constant of 3 mm. The diameter of the system is as large as 30 cm, and the smallest rod width of the woodpile structure in the sample is 100 μm, thus demonstrating a rather large ratio of overall size to feature size. As shown in Fig. 7(a), an accurate external shape of the system is preserved in the macrostructure. The achievable length scale of the macrostructure depends on the maximum build area of the fabrication process; therefore, the overall size can be further extended. For SL systems with multiple laser sources, the build area can be quite large (e.g., 1500 × 750 × 550 mm for ProX 950, 3D Systems). The surface features of the macrostructure are shown in Fig. 7(b). In general, for typical additive manufacturing techniques, a support structure is needed to avoid distortion and movement of the overhanging features. However, adding support can dramatically complicate the presented 3D gradient structure and increase difficulty in design and fabrication. Even if the support can be removed by post-processing, small features of the microstructure could be damaged. However, as demonstrated, for the vat photopolymerization process, the presence of the cross-stacking features at the unit cell level makes the integrated structure as self-supported to some extent; therefore, it can be
Fig. 4. (a) Dispersion curves of the first band for 𝐤𝛤 −𝐾 , 𝐤𝛤 −𝑊 , 𝐤𝛤 −𝑋 , 𝐤𝛤 −𝑈 . and 𝐤𝛤 −𝐿 with different 𝑤∕𝑏 ratios for comparison. (b) Spherical equi-frequency surface in the Brillouin zone at the normalized frequency of 0.49 (𝑤∕𝑏 = 0.4𝑎).
requiring higher indices can be realized using a photo-curable material with higher dielectric constant, as discussed in Section 3. It should be noted that in the current calculation, the lattice constant of the cubic cell is adopted as the period 𝑎. For the FCC lattice, the smallest repeating unit is a rhombohedron of volume 𝑎3 ∕4. If the primitive cell √ is considered as the unit cell, the period size can be regarded as 𝑎/ 2. Then, the ratio of the period to the wavelength, which satisfies the effective medium approximation, will be 0.35 instead. As demonstrated, for such 3D periodic dielectric structure with high symmetry, the effective medium limit is rather lenient. The effective medium approximation is validated even when the periodicity is comparable to wavelength, and the polarization-invariant, isotropic and frequencyindependent property is ensured in a broad bandwidth. 3. Design and fabrication of the integrated 3D gradient micro/macrostructure The scheme to realize the required spatial distribution of material property is shown in Fig. 6. The original material distribution profile is discretized into unit cells of certain size, for example, the cubic lattice unit cells as shown in Fig. 6(a). It should be noted that the resolution of the discrete model should be fine enough to ensure that it approximates the original ideal design sufficiently well. The required local material property is achieved by the microstructure at the unit cell level (the rod-connected structure as a conceptual example here). By spatially varying the unit cell microstructure parameters (in this 198
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Fig. 5. Calculated effective refractive index. (a) Relationship between 𝑛𝑒𝑓 𝑓 and the normalized frequency for different volume fractions. (b) 𝑛𝑒𝑓 𝑓 vs volume fraction for different calculation methods.
directly fabricated without additional support structure. Furthermore, by arranging the fabrication orientation to ensure that each layer is solidified successively along the height direction of the rods, the undesired staircase effect in features commonly seen for the layer-bylayer process is also avoided. Fig. 7(c) shows the internal gradient structure of the system. The presented cross-section is perpendicular to the height direction of the rods. The accuracy of the microstructure at the unit cell level is reflected by the quality of the features. The inset in Fig. 7(c) presents the smallest features of the sample (i.e., rods with the smallest width). Smooth spatial transition of the microstructure is realized, which would reduce the scattering in the internal system. By adjusting the fabrication parameters (e.g., power, spot radius and scanning speed of the laser), the feature size can be tuned and further reduced. However, acceptable mechanical strength needs to be considered to avoid the defect of microstructure under the large weight of the system. The wave manipulating ability of such gradient structure has been demonstrated by both simulation and experimental results in our previous work [25,26]; this proves that the designed permittivity profile can be achieved by the presented method. Moreover, as previously validated by us in the microwave regime, the performance of such prototype is robust, even if the homogenization
was approximately implemented according to the effective medium theory rather than by the dispersion analysis [25,26]. The operating frequencies for the system are determined by the unit cell size. Arranging dielectrics in a lattice with higher symmetry (e.g., FCC rather than simple cubic lattice) would satisfy the effective medium approximation at higher frequency. For current setting (𝑎 = 3 mm), the theoretical upper limit of the operating frequency for the woodpile structure is 49 GHz, as demonstrated in Section 2. By scaling down to a smaller lattice, the upper limit may be theoretically extended to terahertz regime within current fabrication capability. However, it should be noted that when the smallest feature size is set, arranging dielectrics in a smaller lattice would make smaller volume fraction unattainable. A variety of materials with a wide range of properties are available for the vat photopolymerization process. Moreover, the intrinsic material properties can be even further tuned by the mixed-material approach (e.g., the PolyJet Digital Materials, Stratasys). For the constituent material of the 3D electromagnetic gradient systems discussed in the paper, the nondispersive and lossless property is desired in most cases. Such property is available for common vat photopolymerization materials [22]. The material we used has a dielectric constant of 3.0 when cured (measured with a well-established waveguide method [27]), 199
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Fig. 6. Scheme for the design and fabrication of the integrated 3D gradient micro/macrostructure. (a) Discrete approximation of spatial material property distribution with cubic lattice unit cells. (b) Layer approximation for a more regular system.
which is almost frequency independent and has low loss within a broad bandwidth in the microwave regime. For applications that require higher dielectric constant, photocurable materials incorporated with ceramic can be used (e.g., ceramic SL). In addition to the photo-curable materials, other types of constituent materials can be indirectly realized by post-processing based on the polymer structure template [28]. The resolution of the vat photopolymerization process can be further increased by using the mask-image-projection process (e.g., Projection Microstereolithography), thus allowing for the creation of microscale 3D
dielectric structure that can operate at higher frequencies [29]. Current state-of-the-art vat photopolymerization system, which integrates the SL and digital light processing photopolymerization process by using both the laser scan mechanism and the mask-image-projection mechanism [30], enables the production of microscale features with large scale overall size. In general, more materials can be expected to be developed for the vat photopolymerization-based process. It also offers the ability to produce microscale features while preserving scalability to larger macro size. 200
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Fig. 7. A fabricated sample of the proposed system. (a) Overall view of the macrostructure showing the accurate external shape of the system. (b) Features at the surface of the highlighted part in the system. (c) Cross-section perpendicular to the height direction of rods, which shows the internal gradient microstructure with smooth spatial transition. The inset presents the features of the rods.
4. Conclusion In conclusion, the design and fabrication method of the integrated micro/macrostructure for 3D functional gradient systems is introduced. Band diagram-based calculation demonstrates that the homogenized 3D periodic dielectric structure is suitable as an effective medium for 3D gradient systems. For the woodpile structure, the effective medium approximation is validated when the ratio of the lattice constant to the wavelength is smaller than 0.49. Moreover, the polarizationinvariant, isotropic, and frequency-independent property is ensured. A typical sample fabricated by SL-based vat photopolymerization process demonstrates the large overall size of macrostructure and fine features of microstructure. As discussed, the scalability of the structure can be extended within current fabrication capability, which can expand the operating frequency range and widen the applications for wave manipulation. The scheme of using 3D gradient micro/macrostructure to realize spatial material property distribution can also be extended to mechanics [31,32] and can be further applied in systems for other functions [2,3]. Acknowledgments This work is supported by the National Natural Science Foundation of China (No. 51705347) and the Department of Science and Technology of Sichuan Province (Nos. 2016GZ0161 and 2016GZ0195). References [1] S. Suresh, Graded materials for resistance to contact deformation and damage, Science 292 (2001) 2447–2451. [2] R. Libanori, R.M. Erb, A. Reiser, H. Le Ferrand, M.J. Suess, R. Spolenak, A.R. Studart, Stretchable heterogeneous composites with extreme mechanical gradients, Nat. Commun. 3 (2012) 1265. [3] N.W. Bartlett, M.T. Tolley, J.T.B. Overvelde, J.C. Weaver, B. Mosadegh, K. Bertoldi, G.M. Whitesides, R.J. Wood, A 3D-printed, functionally graded soft robot powered by combustion, Science 349 (2015) 161–165.
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