1.

Background

Multi-beam interference (MBI), sometimes referred to in the literature as “holographic” or “interferometric” lithography, provides the ability to form a wide variety of submicron periodic optical-intensity distributions in one, two, and three dimensions. Accordingly, MBI has been used in a wide variety of application areas including nano-electronics, photonic crystals, metamaterials, subwavelength structures, optical trapping, and biomedical structures.^1–6 As a result of the broad application of MBI, research has demonstrated numerous periodic and quasiperiodic patterns with specific space-group symmetries by careful selection of individual beam amplitudes, polarizations, and wavevector configurations.^7–12 These same parameters are also used to optimize the contrast of the resulting interference lattice, providing lithographically useful patterning possibilities.^13–15

Numerous optical configurations and lithographic techniques have been developed to incorporate MBI, providing the potential for simple, rapid, wafer-scale, and low-cost fabrication.^5,16–21 These configurations may be broadly categorized as wavefront-dividing or amplitude-splitting methodologies.²² Perhaps the most common MBI configuration employs a Lloyd’s mirror in a wavefront-dividing scheme to reflect a portion of an expanded source beam to intersect with the transmitted portion forming an interference pattern of one-dimensional (1-D) fringes.²³ With multiple exposures, the Lloyd’s mirror configuration can produce two-dimensional (2-D) and three-dimensional (3-D) patterns.²⁴

To fabricate 2-D periodic patterns in a single exposure, multiple beams must be simultaneously generated. Two common three-beam wavevector configurations, as depicted in Fig. 1(a) and 1(c), are used to form square and hexagonal periodic lattices, respectively. For a lattice with square translation symmetry as depicted in Fig. 1(a) and 1(b), the three beams, represented by wavevectors ${\mathbf{k}}_1$, ${\mathbf{k}}_2$, and ${\mathbf{k}}_3$, intersect at the $x\text{-}y$ plane at a common incidence angle, $\theta$, with respect to the $z$ axis, where ${\mathbf{k}}_3$ is contained in the $y\text{-}z$ plane while ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ are contained in the $x\text{-}z$ plane. The periodicity or lattice constant, $a_{\mathrm{sq}}$, of the resulting interference pattern is given by

Fig. 1

Example of three-beam interference wavevector configurations. A square-lattice interference pattern with (a) p4m or (b) cmm plane-group symmetry is produced by three beams, where ${\mathbf{k}}_3$ is contained in the $y\text{-}z$ plane while ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ are contained in the $x\text{-}z$ plane, each at a common incidence angle, $\theta$. A hexagonal-lattice interference pattern with (c) p6m or (d) cmm plane-group symmetry is produced by three beams, where ${\mathbf{k}}_1$ is contained in the $y\text{-}z$ plan and ${\mathbf{k}}_2$ and ${\mathbf{k}}_3$ are arranged such that the projections of all three vectors are separated by exactly 120 deg in the $x\text{-}y$ plane.

Several optical configurations are available to create the wavevector schemes in Fig. 1. For example, to generate a lattice with hexagonal translational symmetry as depicted in Fig. 1(c) and 1(d), one wavefront-dividing method employs three gratings in a single photo-mask to diffract an incident expanded beam, such that the first order diffracted beams intersect and interfere at the sample plane as depicted in Fig. 2(a).^25–27 Similarly, a single compound diffractive optical element or phase mask may be placed in close proximity to the sample plane to produce a near-field self-interference pattern of the zero-, positive-, and negative-diffracted orders.^28,29 Another wavefront-dividing configuration incorporates a prism designed to divide and refract the single expanded beam into multiple beams as depicted in Fig. 2(b).^30,31

Fig. 2

Three-beam interference configurations. (a) A diffractive-grating mask diffracts portions of the incident-expanded beam such that the first-order diffracted beams intersect and interfere at the sample plane.²⁶ (b) A prism is used to divide and refract different portions of an incident collimated beam.²¹ (c) A three-beam pattern-integrated interference exposure system provides individual control of beam amplitude and polarization allowing for a single-step formation of a two-dimensional periodic lattice with nonperiodic functional elements integrated into the periodic pattern.²⁰

Amplitude-splitting configurations may also be used to generate the three interfering beams required for a hexagonal lattice, typically splitting a single common source beam into multiple beams through the use of beam splitters or diffractive elements.^16,19,20 The beams are then directed to intersect at the plane of interference through the use of mirrors, lens, and/or prisms, such as the pattern-integrated interference exposure system (PIIES) depicted in Fig. 2(c).^32,33 In this system, the three beams are generated and conditioned for specific amplitudes and linear polarizations via a combination of half-wave plates and beam-splitter cubes. In general, beam-splitting configurations typically offer increased opportunity to condition individual-beam parameters.

Each of the configurations used for MBI offer varying levels of interferometric stability, reconfigurability, and control over individual-beam amplitudes and polarizations. The important characteristics of the two main MBI configuration categories are listed in Table 1. As a result of the relatively short optical path lengths in wavefront-dividing configurations, these methods are essentially phase-locked, representing the most interferometrically stable option for MBIL. However, in most wavefront-dividing schemes, control over individual beam amplitudes and polarizations is typically limited and wavevector configurations are generally fixed.

Table 1

Comparison of MBIL configuration categories.

A multiple-beam-splitting configuration, such as the PIIES configuration depicted in Fig. 1(d), provides the ability to easily reconfigure the wavevector configuration for a wide range of lattice constants and translational symmetries, while allowing for individual control over beam parameters. However, a common drawback of beam-splitting-based configurations is the potential for interferometric instability. Any perturbations to the optical components, optical path lengths, or relative phases of the interfering beams may result in a translation of the pattern or change in the symmetry of the unit cell. Thus, pattern stability is generally low for amplitude-splitting schemes.

From Table 1, it is clear that the two MBI configuration categories have advantages and disadvantages that must be considered. Even though most MBI configurations are not easily reconfigured, they may all be designed to provide one of a wide range of periodicities and translational symmetries based on a fixed wavevector configuration. However, these same systems are characteristically limited in ability to condition the individual beams. In this case, it is not clear that full patterning capability remains as constraints are placed on beam amplitudes and polarizations. In some cases, the specific plane-group symmetries play a significant role in the performance characteristics of a device fabricated using MBI. For example, the plane-group symmetry and lattice point geometry have been shown to affect the photonic-bandgap characteristics in photonic crystals,^{7,8,10–12,34} selective plasmonic excitation in plasmonic crystals,³⁵ photonic crystal laser beam pattern³⁶ and polarization-mode control,³⁷ birefringence of photonic crystal fibers,³⁸ cell behavior in tissue engineering,³⁹ tuning of surface textures,⁴⁰ magnetization switching in periodic magnetic arrays,⁴¹ and negative refraction and superlensing in metamaterials.^42,43

Even if a particular plane-group symmetry is possible, it is not clear that sufficient contrast is attainable for optical lithography purposes. In the optimized design of the specific space-group symmetries and motif geometries, most research assumes individual control over beam amplitudes and polarizations. Some research suggests that sufficient contrast may be possible when beam parameters are perturbed.⁴⁴ In one study, the effects of a single linear polarization was analyzed for a system of cube beamsplitters and right-angle prisms used to generate three interfering beams from a common linearly polarized source.⁴⁵ However, this study was limited to a 2-D hexagonal lattice for a specific configuration. In another study, parametric constraints were considered for square-lattice optimized motif geometries.⁴⁶ The present work expands on these efforts, providing a general methodology and systematic study of the effects of amplitude and polarization constraints on the full range of MBI patterning possibilities in linearly polarized three-beam interference for both square- and hexagonal-lattice symmetries, under the conditions for primitive-lattice-vector-direction equal contrast.¹⁵

2.

Parametric Constraints

For three interfering beams, the general form of the total time-independent intensity distribution, $I_T(r)$, may be given as

Table 2

Constraints for optimal absolute contrast under the conditions for primitive-lattice-vector-direction equal contrasts.

Under the conditions for primitive-lattice-vector-direction equal contrasts, the individual linear polarization vectors, ${\widehat{\mathbf{e}}}_i$ are defined according to the polarization vector basis depicted in Fig. 3. Using this basis, each polarization vector is given by Ref. 47

Fig. 3

Orientation of basis vectors to define linear polarizations. ${\theta }_i$ and ${\phi }_i$ are the spherical coordinates of the wavevector ${\mathbf{k}}_i$. The polarization vector ${\widehat{\mathbf{e}}}_i$ is contained in the plane ${\mathbf{N}}_i$ orthogonal to the wavevector, and defined by the counterclockwise angular rotation, ${\psi }_i$ of the polarization vector (looking antiparallel to the wavevector).

To find the optimal set of polarization vectors for a given plane-group symmetry, objective functions were developed for each of the optimization constraints listed in Table 2 to solve for the conditional maximum. In this method, Lagrangian multipliers, $\mathrm{\Omega }$, are used to implement constraints to the objective function¹³ ($\lambda$ is normally used as the symbol for a Lagrangian multiplier; however, $\mathrm{\Omega }$ is used here to differentiate the multiplier from the freespace wavelength, $\lambda$). The resulting objective functions for maximum absolute contrast under the conditions for primitive-lattice-vector-direction equal contrasts are given in Table 3.

Table 3

Optimization functions for optimal absolute contrast and primitive-lattice-vector-direction equal contrasts assuming individual control over beam amplitudes and polarizations.

2.2.

Polarization Constraints

In addition to the ability to set the individual beam amplitudes, the configuration of Fig. 1(c) allows for control over individual-beam linear polarizations as required for maximum absolute contrast and primitive-lattice-vector-direction equal contrasts. This is accomplished with the final half-wave plate in the path of each beam. However, in most MBI configurations it is usually difficult to set the polarization for each beam individually. Typically, a single linearly polarized beam is divided into the multiple beams as depicted in Fig. 2(a) and 2(b), with each beam retaining the original linear polarization angle of the source beam ${\psi }_B$ The individual-beam polarization is then defined by Eq. (7) where ${\psi }_1={\psi }_2={\psi }_3={\psi }_B$ representing the first polarization constraint considered here for equal individual beam-set polarizations.

Alternatively, it may be possible to improve patterning possibilities or absolute contrast through the use of a single polarizer just prior to the sample plane to set a common polarization for each beam as depicted in Fig. 4(a). For this case, the individual beam-polarization vectors are determined by finding the vector ${\mathbf{e}}_i$ perpendicular to the wavevector ${\mathbf{k}}_i$ that is contained in the polarization plane $\mathbf{R}$ defined by the origin $(0,0,0)$, the vector defining the pass axis of the polarizer $\mathbf{P}$ and the wavevector ${\mathbf{k}}_i$, as depicted in Fig. 4(b). Here, the polarization pass axis vector is defined as

Eq. (8)

$$P=\sin (-{\alpha }_P)\widehat{x}+\cos (-{\alpha }_P)\widehat{y},$$

where $-{\alpha }_P$ is the angle of the pass axis of the polarizer with respect to the $y$ axis. The polarization vector ${\mathbf{e}}_i$ is then the intersection of the polarization plane ${\mathbf{R}}_i$ and the plane orthogonal to the wavevector ${\mathbf{N}}_i$. This intersection is found by taking the cross product of the normals ${\mathbf{n}}_R$ and ${\mathbf{n}}_N$ for each of the planes. With this polarization vector basis, the individual-beam polarization vectors for each wavevector are given by

Eq. (9)

$${\mathbf{e}}_i={\mathbf{k}}_i\times ([\sin (-{\alpha }_i)\widehat{x}+\cos (-{\alpha }_i)\widehat{y}]\times {\mathbf{k}}_i),$$

where ${\alpha }_1={\alpha }_2={\alpha }_3={\alpha }_P$ for the second polarization constraint considered in this study, equal sample-plane-set polarization. For the two polarization constraints, the new polarization vector definitions and required constraints are presented in Table 5.

Fig. 4

Orientation of basis vectors for sample-plane-set polarization. (a) A single linear polarizer, with a pass axis angle $-{\alpha }_P$, may be placed just prior to the sample plane to improve contrast. (b) The plane $\mathbf{R}$ contains the wavevector $\mathbf{k}$ and the polarization pass axis vector $\mathbf{P}$. The polarization vector $\widehat{\mathbf{e}}$ is then the intersection of the polarization plane $\mathbf{R}$ and the plane orthogonal to the wavevector $\mathbf{N}$.

Table 5

Polarization vector definitions and constraints for optimal absolute contrast and primitive-lattice-vector-direction equal contrasts for unconstrained, equal individual beam-set polarization, and equal sample-plane-set polarization.

Polarization case	Polarization vector definition	Constraints
Optimal individual beam-set polarization	${\widehat{\mathbf{e}}}_i={\mathbf{R}}_z(-{\phi }_i){\mathbf{R}}_y(-{\theta }_i){\mathbf{R}}_z(-{\psi }_i){\mathbf{R}}_y({\theta }_i){\mathbf{R}}_z({\phi }_i)(\widehat{z}\times {\mathbf{k}}_i)$	None
Equal individual beam-set polarizations		${\psi }_1={\psi }_2={\psi }_3={\psi }_B$
Equalsample-plane-setpolarizations	${\widehat{\mathbf{e}}}_i={\mathbf{k}}_i\times ([\sin (-{\alpha }_i)\widehat{x}+\cos (-{\alpha }_i)\widehat{y}]\times {\mathbf{k}}_i)$	${\alpha }_1={\alpha }_2={\alpha }_3={\alpha }_P$

3.

Constrained Optimization Results

The constrained optimization functions in Table 3 and 4 were solved using the polarization definitions and constraints given in Table 5 for both square- and hexagonal-lattice space-group symmetries across the full range of common incidence angles. The optimized constrained parameter values were then used to evaluate the resulting optical-intensity distribution and maximum absolute contrast given by Eqs. (3) and (6) as a function of the common sample-plane incidence angle. For the unconstrained case of optimal individual beam amplitudes and optimal individual beam-set polarization, unity absolute contrast $V_{\mathrm{abs}}=1$ is achieved across the full range of possible lattice constants for all plane-group symmetries as predicted by the conditions for primitive-lattice-vector-direction equal contrast.¹⁴ For the cmm(sq) and p6m symmetries, unity contrast is possible when the intensity profile is inverted, that is, when the points of intensity maxima become intensity minima. This result is obtained when the product of the three interference coefficients given by Eq. (4) are allowed to be negative in value. If the cmm(sq) and p6m symmetries are optimized for intensity maxima at the lattice points, the optimal absolute contrast varies from 0.6 to unity. Typically, optical lithography requires an absolute contrast 0.4 to 0.8 or higher to resolve the required features based on the photoresist properties, feature sizes, and coherence of the source.⁴⁸

3.2.

Polarization Constraints

A significant reduction in patterning possibilities and absolute contrasts occurs when polarization constraints are considered. These limitations are made worse when combined with the amplitude constraint. To demonstrate the effect of polarization constraints, Fig. 5(a) to 5(f) plot the maximum absolute contrast, $V_{\mathrm{abs}}$, for the unconstrained case against the various constrained case combinations for p4m, cmm(hex), p6m, and cmm(sq) plane-group symmetries, respectively. In each graph, the absolute contrast is plotted as a function of the common wavevector incidence angle at the sample plane. The unconstrained and the amplitude-constrained case (labeled “Equal Amplitudes” for equal individual beam amplitudes) are plotted as a baseline for comparison.

Fig. 5

Optimized absolute contrasts for square and hexagonal lattices. The unconstrained maximum absolute contrast is compared to the those for the amplitude and polarization constraints as a function of the common wavevector incidence angle at the sample plane for (a) p4m, (b) cmm(hex), (c) p6m, (d) p6m (negative pattern), (e) cmm(sq), and (f) cmm(sq) (negative pattern) plane-group symmetries.

Beginning with the p4m plane-group in Fig. 5(a), both polarization constraints (labeled “Equal Polarizations” for equal individual beam-set polarizations and “Sample-Plane-Set Polarizations” for the equal sample-plane-set polarizations) limit the range of incidence angles to 45 deg and higher in order to produce this symmetry. Furthermore, the absolute contrasts drop off sharply from a unity value at 45 deg. In the case of equal individual beam-set polarizations, the contrast drops to zero at a common incidence angle near $\theta \approx 66\mathrm{deg}$. If a conservative absolute contrast of $V_{\mathrm{abs}}=0.4$ is used as a minimum threshold for optical lithography purposes, the range of angles for p4m symmetry is reduced to $45\le \theta <57\mathrm{deg}$ or $76<\theta <90\mathrm{deg}$, and $45\le \theta <68\mathrm{deg}$ for the two polarization constraints, respectively. If a common ultraviolet (UV) i-line source is used at $\lambda =363.8\mathrm{nm}$, the resulting range of periodicities, using Eq. (1), is then $363.8\ge a_{\mathrm{sq}}>306.7\mathrm{nm}$ or $265.1>a_{\mathrm{sq}}>257.2\mathrm{nm}$, and $363.8\ge a_{\mathrm{sq}}>277.4\mathrm{nm}$. Finally, if the equal individual beam amplitudes are combined with either of the polarization constraints, the ability to achieve p4m symmetry is limited to a single incidence angle of $\theta =45\mathrm{deg}$ ($a_{\mathrm{sq}}=\lambda$). Similar results are obtained for the cmm(hex) symmetry as depicted in Fig. 5(b). However, when the equal individual beam amplitudes is combined with either of the polarization constraints, the cmm(hex) symmetry (with $V_{13}=V_{23}$ and $V_{12}=0$) is not possible.

For the p6m plot in Fig. 5(c) the maximum absolute contrast begins near unity at a 0 deg incidence angle for both the equal individual beam-set polarizations and equal sample-plane-set polarizations cases and gradually reduces to lower contrast values, falling below $V_{\mathrm{abs}}=0.4$ near $\theta \approx 48\mathrm{deg}$ and 54 deg for the two cases, respectively. When negative (or inverted) intensity distributions (labeled “negative pattern”) are considered, higher absolute contrasts may be achieved for the larger incidence angles as depicted in Fig. 5(d). When combined with equal individual beam amplitudes, the p6m symmetry is no longer possible (with $V_{13}=V_{23}=V_{12}$). Similar results are obtained for the cmm(sq) symmetry as depicted in Fig. 5(e) and 5(f). However, when the equal individual beam amplitudes are combined with equal individual beam-set polarizations, the cmm(sq) symmetry remains possible at a single incidence angle of 60 deg with unity absolute contrast.

4.

Summary and Discussion

A systematic and comprehensive analysis of the effects of constraints on individual-beam amplitudes and polarizations has been presented for the unique 2-D plane-group symmetries possible using three linearly polarized beams configured to produce an interference pattern with square- or hexagonal-lattice translational symmetry. These patterns were optimized for maximum uniform contrast while satisfying the conditions for primitive-lattice-vector-direction equal contrasts.

When only amplitude constraints are considered, the results demonstrate that all plane-group symmetries remain possible over the full range of lattice constants. While the absolute contrast is reduced to a minimum value of $V_{\mathrm{abs}}=0.88$, this value is still sufficient for optical lithography purposes.

When polarization constraints are introduced, significant reductions were noted for all plane-group symmetries. Based on the specific application requirements, it may be necessary to use an MBIL configuration, such as the multiple-beam-splitting configuration in Fig. 2(c), that allows individual conditioning or beam polarizations to ensure a robust patterning capability.

Although numerous additional parametric constraint combinations are possible and may merit consideration, the cases presented here provide useful insight into the general design requirements for an MBI system. However, the optimization methodology presented here may be modified as necessary to model and predict the patterning possibilities of other constrained configurations to ensure that a given MBI configuration meets the requirements for a specific patterning application.

As efforts to incorporate MBI into commercial fabrication processes continue, a complete understanding of the patterning possibilities is required in the systematic design of future MBI configurations. While the PIIES configuration in Fig. 2(c) allows for the widest range of high-contrast periodic pattern symmetries, future research will focus on increasing interferometric stability, to provide the potential for relatively simple, subwavelength, and cost-effective periodic patterning with integrated nonperiodic functional elements.