1. Introduction

The fabrication of three-dimensional (3D) periodic micro- and nano-structures is of interest to a variety of fields, including photonic crystals [1, 2], microfluidics [3], drug-delivery [4, 5], and tissue engineering [6]. The potential benefits obtained from creating tailored structures at these length-scales have fueled the rapid development of various fabrication techniques, such as self-assembly [7], layer-by-layer lithography [8], and holographic lithography [9, 10]. When selecting a fabrication technique, it is often necessary to assess the various strengths and weaknesses of competing approaches to determine the one best suited for a particular application. Often times a balance is struck between the following considerations: materials and processing conditions; cost and efficiency; defect control; potential for large area patterning; control over and versatility of resultant 3D structures.

Phas mask lithography (PML) has recently emerged as a promising technique for the fabrication of 3D periodic structures [3]. Generally speaking, PML employs the interference generated when coherent light passing through a grating diffracts and forms a periodic intensity distribution within a film of photoresist. In terms of the attributes mentioned above, PML is simple, inexpensive, and capable of rapidly producing a wide variety of defect-free structures over large areas. Furthermore, the structures produced in photoresist serve well as host templates for a broad array of materials that can be introduced using techniques like CVD [11–13] and electrodeposition [14].

While there is good agreement between experimentally demonstrated and theoretically predicted structures [3], the absence of a generally applicable design strategy has hindered widespread use of PML. This is especially true for applications where functionality depends strongly on a structure’s morphology. At issue is the relationship between a grating and the structure produced. While the forward relationship between a grating and the intensity distribution it produces is thoroughly understood from diffraction theory [15, 16], the corresponding inverse relationship is not. As a result, fabrication of structures using PML has been limited to simple designs based on intuition and fundamental diffraction optics [3, 17, 18]. In order to fabricate a variety of complex 3D structures, a general method for phase mask design is needed. Recently, Chan and coworkers [19] have taken steps in this direction with their proposal of a bi-level phase mask that generates a diamond-like structure. However, their approach relies on exhaustive search about an initial guess, making it inefficient and difficult to implement. Moreover, their method is specific to the diamond structure and does not generalize to other structures. Thus, the need for an effective design strategy persists.

In this paper, we describe a computational design strategy that will enable the fabrication of premeditated 3D structures using PML. Our approach makes use of a class of computational inverse solvers known as genetic algorithms (GAs). As their name suggests, GAs are based on the principles of evolution [20, 21]. They use an objective function to assign fitness values to a population of trial solutions. Those solutions having superior fitness are selected to crossover and mutate with greater frequency, enabling the discovery and exploitation of more productive search domains. Because they adeptly avoid convergence to local optima, GAs are ideal for complex problems requiring a global solution. In previous work, we successfully demonstrated how to implement GAs for use with holographic lithography [22], an interference-based technique closely related to PML. As such, we expect the application of GAs, or related approaches, to be an effective method for designing PML experiments.

 

Fig. 1. The illustration in part (a) depicts the helix structure used as the target model. For clarity, we plot two turns of the helix and outline in bold a single primitive cell with dimensions a×a√3/2×c. Here, c/a describes the helices’ relative elongation and has a value of 2.2. Parts (b) and (c) depict the interference based structure that is produced by the optimized grating shown in (d). Comparison between the structure in (b) and the target in (a) results in a fitness of 93%. By plotting several repeat units in (c) the full 3D hexagonal periodicity becomes apparent.

Download Full Size | PDF

To demonstrate this approach, we targeted a hexagonal array of helices (Fig. 1(a)). Such helical structures are useful for a variety of applications, including circular polarization filters [23], photonic band gap materials [24], and optical diodes [25]. Using GAs, we optimized a phase mask’s 2D relief profile and the incident polarization state to produce a helical structure that closely resembles the target cell outlined (Fig. 1(a)).

2. Theoretical framework for PML

Taking a Cartesian frame of reference described by the unit vectors x̂, ŷ, and ẑ, we represent the phase mask to be optimized with the function n(x, y), which describes the refractive index modulation in the region between een z=0 and z=h. The grating is a 2D binary step function that varies discontinuously between the values n ₀ and n ₁. It is sandwiched above (z>h) and below (z<0) by uniform dielectrics having a refractive index of n ₊ and n _- respectively. Its periodicity, which we take here to be hexagonal, is described by two primitive vectors a ₁=(a, 0, 0) and a ₂=(a/2, a√3/2, 0). A normally incident, monochromatic plane-wave having a wavelength λ₀ in free-space, and wave-vector k ₀=(0, 0, -2π/λ₀) exposes the grating and produces transmitted diffracted beams having wave-vectors k′ _pq=2π/a(q-p, -(p+q)/√3, γ_pq), where p and q are integers indexing the diffracted beams. From conservation of energy we know that |k′_pq|=2πn-/λ₀, thus for the diffracted wave-vector’s component in the z-direction,

For propagation into the far-field, γ_pq in Eq. (1) must be real, otherwise the mode indexed by (p, q) evanescently decays in the near-field. Therefore, propagating modes must satisfy the condition:

Along with k′ _pq, the complex field vector, Ẽ, for each diffracted beam must be calculated to determine the full intensity distribution, I(r). For this, the polarization state of the incident beam must be known. In general, elliptically polarized light is a function of two angles, ψ and χ, and can be written as the locally defined Jones vector, J=(cos ψ, e ^iχ sinψ) [26]. With well defined J and n(x, y), we can apply rigorous coupled wave theory (RCWT) [15, 16] to determine Ẽ. Knowing both Ẽ and k′ _pq, I(r) can now be calculated using Eq. (3),

where integers i and j index over the N diffracted beams having wave-vectors k ₁ through k_N. The structure produced in negative tone photoresist is determined by applying an intensity threshold, I _th, to I(r) and is given by the Heaviside step-function,

In Eq. (3) the set of wave-vector differences, expressed as g _ij=k _i-k _j, determine the translational symmetry possessed by I(r). While we know from symmetry that I(r) is 2D periodic with a ₁ and a ², full 3D periodicity is only obtained when each spatial frequency in the z-direction (belonging to the set g _ij·ẑ) is an integer multiple of the lowest frequency in z. In particular, if g _ij contains exactly one out-of-plane frequency, the distribution is necessarily 3D periodic. This is satisfied by a beam configuration possessing only two polar angles. Since there will always be a zeroth order transmission having k ₀₀=(0, 0, -2πn-/λ₀), the remaining diffracted wave-vectors (in the region z<0) must all possess the same z-component, given by k′ _pq·ẑ=(2π/a)×γ_pq. This is achieved by a seven beam configuration consisting of the (0, 0) mode flanked by six equally spaced first order modes, (1, 0), (0, 1), (1, 1), (-1, 0), (0, -1), and (-1, -1). Using Eq. (1), we confirm that ${\gamma }_{\mathrm{pq}}=-\sqrt{{\left(\frac{an\_}{{\lambda }_0}\right)}^2-\frac{4}{3}}$ for each first order diffracted mode, verifying that each propagates at the same polar angle. To ensure exactly seven transmitted beams, we confine the dimensionless ratio an-/λ₀ in Eq. (2) to the interval (2/√3, 2). Here, the lower limit ensures transmission of first order beams while the upper limit prevents transmission of second order beams. The resultant lattice, R, consists of linear combination of a ₁, a ₂, and a ₃, where a ₃=(0, 0, c) describes the modulation in z. Here, c is a function of a, n_, and λ₀, and is determined by comparing wave-vector differences in the z-direction (g _ij·ẑ) to the reciprocal lattice of R, resulting in

In view of this analysis, a, n_, and λ₀ are chosen so that I(r) coincides with the lattice of the desired target structure. This enables he evaluation of each phase mask design by considering only the structure produced within a single primitive cell of volume V=|(a ₁×a ₂)·a ₃|. Here, we choose an orthorhombic polyhedron with a length, width, and height of a, a√3/2, and c, respectively. This cell, outlined in Fig. 1(a), contains one full turn of a helix and is characterized by a path through space, r=(R cos (2πz/c), R sin (2πz/c), c), and a cross-sectional width, ρ. Here, R and ρ are set to 4a/15 and 2a/15 respectively. The target cell is then discretized into voxels representing either “polymer” (1) or “air” (0), taking care to sample the x, y, and z directions uniformly (approximately 30 voxels per length a). The fitness function, F, is then calculated as the fraction of matching voxels between the target and a trial structure. Additionally, we use a numerically constrained I _th to ensure a filling fraction equal to that of the target structure (9.3%).

3. Results and discussion

From a practical point of view, the optimized design should consist of readily available materials, possess attainable feature sizes, and produce a 3D structure that transfers into photoresist. To this end, we adopt a recently developed PML technique that calls for two-photon absorption within the photoresist [27]. Because larger gradients in the absorption profile are produced, more well-defined structures are obtained upon development. The photoresist is assumed to be commercially available SU-8 with a refractive index of n-=1.58 at 810 nm, roughly corresponding to its two-photon absorption peak. As a corollary, 3D periodicity is achieved for larger relief structures (a=592-1025 nm) alleviating some fabrication demands. We also assume that the grating relief profile is imprinted into the photoresist, as this has been shown to enhance the intensity distribution’s spatial gradient, resulting in more robust structures [18]. As such, we assume n ₀=1, n ₁=1.58, and n ₊=1 throughout the simulations.

Prior to optimization, the remaining experimental parameters ψ, χ, h, and n(x, y) are encoded into a binary format representing a trial design. Since a ₁ and a ₂ are given, it is only necessary to encode for the profile’s (n(x, y)) 2D primitive cell (Fig. 1(d)). Here, it is depicted as a 5×5 binary matrix, where each element represents either a raised or recessed region corresponding to refractive indices n ₁ or n ₀ respectively. Each remaining parameter, y, bound to the interval [α, β] is then encoded into a binary substring, y′, of length, l, using the transformation y′=(y-α)/(β-α)×2^l. Here, l is made sufficiently large (10–20 bits) to provide ample resolution of the parameter space. Once each parameter has been converted to binary, they are integrated into a single string called a “chromosome”.

A population of chromosomes is then randomly generated and optimized via GA. Here, we use a modified version of the algorithm described in [22], where a uniform crossover function is used instead of a single-point method. In this approach, each bit is given a 50% chance of exchanging during crossover, eliminating length-dependant representation effects that hinder performance [28]. Quantitatively, various run parameters are changed due to differences in the design task and the crossover function. In this work, a population of 100 chromosomes was optimized for 40 generations; the selection probability scaled linearly with fitness; and the two best (“elite”) chromosomes per generation were passed to the subsequent generation unaltered. All other aspects of the GA, qualitative and quantitative, remain unchanged from [22].

 

Fig. 2. Helices with various aspect ratios are obtained for phase masks optimized via GA. Parts (a)–(f) correspond c/a values of 1.2, 1.4, 1.6, 1.8, 2.0, and 2.2 respectively.

Download Full Size | PDF

Because a structure’s periodicity in the z-direction is affected by an_/λ₀, the helix’s normalized pitch, c/a, ranges between 0.87 and 2.72 (when a is 592 and 1025 nm respectively). With this in mind, we separately targeted multiple helices having different values for c/a. Consequently, a was adjusted to produce 9 evenly spaced normalized pitch values between 1.0 and 2.6. In all, 10 simulations were performed at each interval. For each simulation, the field expansion used to calculate the diffracted beams (via RCWT) retained only the first seven terms, i.e. those representing propagating modes. While finer details of the calculated structures are potentially obscured, the critical features are preserved and a significant decrease in run time is achieved. Approximately 90% of the simulations yielded helix-like structures while the remaining yielded non-helical structures. Figure 2 shows a sampling of designs resulting in helical structures throughout the range of c/a. We observed a large contingent of experimentally promising designs for c/a=2.2, prompting an additional batch of simulations at this aspect ratio. Figure 3 plots the fitness versus generation curve for the best all-around solution found at c/a=2.2. After 40 generations, convergence was achieved and a fitness of 94% obtained. Following optimization using GA, the parameters ψ, χ, and h were converted back into floating point numbers (while n(x, y) stayed binary). The solution in its new format underwent local optimization using a hybrid approach formulated to operate on both floating point and binary parameters. This technique alternately applied random search to optimize the relief profile, n(x, y), followed by simplex search [29] to optimize the floating point parameters, ψ, χ, and h. The number of terms retained in the field expansion was increased from 7 to 61 in order to more accurately calculate F. As a result, the starting fitness for local optimization decreased to 92%. Because this hybrid algorithm is still susceptible to localized convergence, a series of runs were performed using helices of different ρ (between a/10 and a/6). Such variation provided sufficient agitation to free the solution from local “traps” in the fitness landscape. The final design achieves a fitness of 93% and produces the structure illustrated in Fig. 3. The incident polarization state is given by ψ=3.71 and χ=5.86 radians and the optimized relief profile (Fig. 1(d) and Fig. 3) has a thickness h=257 nm.

 

Fig. 3. (a) Fitness of the best result from each generation plotted for a single GA run. (b) Grating relief profiles represented by a single unit cell of raised (light) and recessed (dark) elements with the polarization state depicted by the path a field vector traces at a point in space. (c) Corresponding interference structures produced during the generations are indicated by i–v in (a). Following GA optimization, the design produced at v was optimized further using a local search algorithm, producing the design and structure in vi. Because the diffraction is more accurately calculated during this step, the fitness at v was computed to be 92% (versus 94% as shown in (a)). After local optimization of the structure in v, a final fitness of 93% was achieved for the final design and structure in vi.

Download Full Size | PDF

From an experimental point of view, it is insufficient to evaluate a design on the basis of fitness alone; its experimental viability must also be considered. As such, we assess whether the smallest features of the surface relief are attainable using current fabrication techniques. Given the assumptions made thus far, the design requires a minimum feature size of 174×150×257 nm, which is attainable using electron beam lithography and PDMS (polydimethylsiloxane) micro-molding [23]. Another important aspect to consider is whether I(r) has sufficient contrast (V=I _max/I _min) to transfer into photoresist (SU-8). For the fully optimized design, a contrast of 13.5 is obtained. However, since we assume two-photon absorption during the exposure, an effective contrast (V_eff=V ²) of 182 is achieved, which we believe to be suitable for proper transfer.

4. Conclusions

In summary, we have developed a comprehensive design strategy for phase mask lithography. Using this approach, we have demonstrated a phase mask capable of producing an array of helices. In principle, this approach can be extended to other structures as well, making phase mask lithography a more versatile tool for 3D micro- and nano-fabrication.