In this paper, we study tunable holographic lithography using an electrically addressable spatial light modulator as a programmable phase mask. We control the phases of interfering beams diffracted from the phase pattern displayed in the spatial light modulator. We present a calculation method for the assignment of phases in the laser beams and validate the phases of the interfering beams in phase-sensitive, dual-lattice, and two-dimensional patterns formed by a rotationally non-symmetrical configuration. A good agreement has been observed between fabricated holographic structures and simulated interference patterns. The presented method can potentially help design a gradient phase mask for the fabrication of graded photonic crystals or metamaterials.
© 2013 Optical Society of America
Periodic photonic structures such as photonic crystals (PhCs) provide new opportunities in device applications for the control and manipulation of photons [1,2]. One-dimensional PhCs such as fiber Bragg gratings have been fabricated in optical fiber, using phase mask based two-beam interference, for applications in fiber optical communications . Two-dimensional PhCs have been used for the integrated laser on chip  and all-optical circuit . Low threshold lasers in three-dimensional PhCs have been observed in PhC nanocavities . Various fabrication methods have been used for making PhCs, such as e-beam lithography , self-assembly of colloidal microspheres , two-photon lithography , and laser holographic lithography [10,11].
Multi-beam interferences can generate one, two or three dimensional patterns and have been used for fabricating a large number of photonic devices such as gratings and PhCs [3,10–15]. The optical setup for a precise interference pattern is usually complicated if bulky mirrors, polarizers, and beam splitters are used [10,11]. Using a static phase mask as a multi-beam generation tool can replace a large number of bulk optical elements for the same purpose. Static diffractive (phase mask) or deflective optical elements have successfully been used for the generation of one, two and three dimensional patterns [11–16]. A phase tuning of interfering beams is sometimes necessary in order to generate a desired interference pattern [14,15]. Very recently, spatial light modulators (SLMs) encoded with computer-generated-holograms or patterns have been used for the holographic fabrication of complex periodic or quasi-periodic photonic structures [17–19]. By engineering the phase pattern, the phase of individual interfering beams can be designed for helically stacked multi-layered microrotors realized in vortex-embedded three dimensional optical twister patterns  or for the formation of photonic chiral photonic lattices . Usually the first order was chosen by the Fourier filter for the reconstruction of the desired pattern. By embedding nondiffracting defect sites  or including higher Fourier orders in the phase pattern , simultaneous fabrication of functional defects in photonic lattices can be achieved  or multiperiodic staircase lattice can be formed . On the other hand, the recent appearance of transformation optics theory [24,25] can address the inverse problem of electromagnetism with respect to what nanostructures will perform a requested functionality. Theoretically, transformation optics can prescribe the path of light in nanostructures at will, by controlling the index of refraction as a function of position, potentially to any desired profile [24,25]. Using a SLM, it is possible to generate a gradient refractive index in holographic structures by controlling the local filling factor of the dielectric material where the filling factor is controlled by the phase of the interfering beams [26,27]. From a designer’s point of view, it is necessary to have a mathematical formula that can calculate the phase of interfering beams from the gray levels or gradient gray levels in the SLM for the design and fabrication of transformation optics devices [26,27].
In this paper, we worked out detailed calculation formulas showing how the gray scale value in the displayed pattern in SLM is converted to tune the phase of diffracted beams. The study was performed in an interference pattern that has a visually observable phase-dependent dual-lattice structure. We were able to digitally assign the phase for interfering beams when the laser beam was modulated by the programmed phase pattern displayed in the SLM. This phase assignment method was validated by comparing the fabricated holographic structures with theoretical simulations.
2. Experimental setup
The experimental setup for the interference lithography is shown in Fig. 1. A 532 nm laser beam (Cobolt Samba) was expanded, collimated, and reflected by an SLM (Holoeye Pluto phase-only SLM). A phase pattern as a dynamic phase mask was displayed on the SLM, and the wavefront of the reflected laser beam was modulated by the SLM. The interfering pattern was transferred and de-magnified through a 4f imaging system (f1 = 400 mm and f2 = 200 mm). Further demagnification was realized using a focusing objective lens. A thin film sample or CCD camera was mounted in a translational stage for the holographic fabrication or image recording, respectively.
3. Design of programmable phase mask and test of phase control capability
We used a hexagonal phase pattern as an example to understand how we can digitally control the relative phases in the interfering beams for the interference lithography. A designed phase pattern as a programmable phase mask is shown in Fig. 2(a). The phase pattern consisted of hexagons, and each hexagon was divided into six equal areas of equilateral triangles by drawing the lines between the center and the vertices of the hexagon. The gray levels of the six triangles can be set digitally. In Fig. 2(a), four out of six triangles in each hexagon were set to be black (gray level = 0). One of the remaining triangles was set to a gray level of 51(dark gray in the figure), and the other was set to a gray level of 153 (light gray in the figure). As shown in Fig. 2(a), the diffraction of a laser beam illuminated on the phase pattern on the SLM can be in vertical directions due to the periodic arrangement of “dark gray, black, dark gray, black” or “light gray, black, light gray, black” patterns, and it can also be in 60-degree-rotated directions relative to the vertical direction. The intersections of these diffraction direction lines indicated the locations of overall diffraction patterns. In the diffraction pattern drawn in Fig. 2(a), the red lines (for eye guidance) link the first order diffraction spots and form a hexagon with the 0th order in the center. The higher order diffraction patterns can be drawn and can form hexagonal structures (the hexagons were drawn for eye guidance).
Figure 2(b) shows the diffraction pattern after the laser beam was diffracted by the phase pattern in SLM and focused by the first lens as shown in Fig. 1. The diffraction pattern was recorded by a CCD camera (Edmund Optics). Surrounding the 0th order diffraction in the center, the six 1st order diffraction beams exhibited hexagonal symmetry. Outside the 1st order diffraction, 2nd and higher order diffractions were recorded. These diffraction spots were also at the vertices of hexagons. The recorded diffraction pattern looked very similar to the one predicted in Fig. 2(a). The designed side length of equilateral triangles was 0.706 mm. The displayed pattern in SLM was zoomed to 3.12% of the original. Thus the side length became 22.0 μm. From the diffraction pattern, we calculated the side length of the equilateral triangle in Fig. 2(a) to be 22.1 μm, which is very close to the designed value.
Based on the fact that the symmetry and location of the predicted diffraction pattern were the same as the experimentally observed one, we drew a hexagonal unit cell (hexagon in red lines) in the Fig. 2(c) for the phase pattern in Fig. 2(a) and assigned phases for each first-order diffracted beam using the pre-assigned gray levels. The hexagon was divided into six equal areas, and each area formed a kite-type four-sided polygon. Each kite had the same area as a light gray or dark grey triangle in Fig. 2(c). The phases of the diffracted beams at the head of each kite were determined by the overall gray level covered by the kite. The phases of beams 1, 3 and 5 in Fig. 2(c) should be the same because their phases were determined by the same gray level which is
The phases of beams 2, 4 and 6 were determined by the same gray level which is
We can test and understand the phase tunability of the laser beams by studying their interference pattern. Because we wanted to have visually observable phase-related structure changes, a four-beam interference pattern was generated by a rotationally non-symmetric configuration . This was accomplished by placing a Fourier filter in the Fourier plane of Fig. 1 to block other beams but let beams 1, 3, 4 and 5 in Fig. 2(b) pass through. After passing through the second lens and the de-magnifying objective lens, these four beams formed an28]:
Figure 3(a1) shows a portion of a programmed pattern where the gray level in the equilateral triangles was set to be 153, or 0 for black. In Figs. 3(b1) and 3(c1), the gray level in one set of equilateral triangles was adjusted to be 102 and 51, respectively. When these patterns were displayed on the SLM, the gray level was converted into a phase level based on the gamma curve used. The gamma curve can convert the gray level between 0 and 255 into a phase level between 0 and 2π, respectively. The phase for beams 1, 3 and 5 was the same in all Figs. 3(a1, b1, c1). The phase of beams 2, 4 and 6 in Figs. 3(b1) and 3(c) became smaller than the one in Fig. 3(a1). Based on the Eq. (3) and Eq. (4), the interference pattern involving beam 4 will have a pattern shift if beam 4 has a relative phase shift of 0.13 π and 0.27 π in Figs. 3(b) and 3(c), respectively:
Where m = 0.13 or 0.27. The phase change in beam 4 will shift the portion of the interference pattern in the (−1, 3) or (−1, −3) directions, as shown in Fig. 3. Figure 3(a2,b2,c2) shows the recorded CCD images of four-beam interference patterns formed by beams 1, 3, 4, and 5. The change of the interference pattern from Fig. 3(a2) to 3(b2) to 3(c2) was very clearly recorded. The interference pattern changed from fenced cylinder structures in Fig. 3(a2) to paired dual-lattice structures (dumbbell at each lattice), indicating the phase control capability in using the SLM as a phase mask.
Figures 3(a3,b3,c3,a4,b4,c4) are 2D and 3D views of simulated four-beam interference patterns assuming the relative phases of interfering beams are determined by Eqs. (1) and (2). These patterns clearly show the structural changes in the figures when the gray level in one set of triangles was changed from 153 to 102, and further to 51. The intensity strength was shifted to other regions in the (−1,3) or (−1,-3) directions as indicated in Fig. 3(b3).The simulated patterns in Fig. 3(c) are very similar to the CCD recorded patterns in Fig. 3(b), approving the phase determination by Eqs. (1) and (2).
4. Holographic fabrication
As a proof of concept, we fabricated the holographic structures formed by interfering four beams diffracted from the phase pattern displayed in Fig. 3(a1). The total laser (532 nm) power was 50 mW. The photoresist was photo-sensitized to 532 nm by preparing a mixture with the following components in the specified weight concentrations: dipentaerythritol penta/hexaacrylate (DPHPA) monomer (Aldrich, 90.36%), a photo initiator rose bengal (0.11%), co-initiator N-phenyl glycine (NPG, 0.65%), and chain extender N-vinyl pyrrolidinone (NVP, 8.88%). The mixture was spin-coated on a glass slide with a speed of 3000 rpm for 30 seconds. The film thickness was approximately 3.17 microns. The photoresist film was exposed to the interference pattern for a typical exposure time of 2 seconds. The exposed sample was developed in PGMEA for 3 minutes, rinsed by isopropanol for one minute and left to dry in air.
Figures 4(a1,a2,a3) show the CCD image of the interference patterns recorded at different locations. No changes are observed as the interference pattern is a 2D periodic structure. Figure 4(b) shows a scanning electron microscope (SEM) image of the holographically formed photonic structures in DPHPA. An enlarged view of the SEM is shown in Fig. 4(c). Because the photoresist, DPHPA, is in liquid form, aggregation during polymerization might cause the dome shape in the fabricated structure’s surface. Figure 4(d) shows atomic force microscope (AFM) image of the fabricated structures in DPHPA. The agreement among these SEM, AFM images, CCD recorded images, and simulated pattern in Fig. 3(a3) is very good. A surface profile along the line in the AFM is shown in Fig. 4(e), and the lattice period along the red line in Fig. 4(d) is measured to be 4.07 microns. By comparing the period of the CCD recorded patterns with and without the objective lens, a de-magnification of 4.1 was observed. The interfering angle after lens 2 was measured to 1.83 degrees. Based on the de-magnification factor, a theoretical calculation gave a lattice constant of 4.06 microns. The agreement between measured and theoretical values is very good.
We can test phase formulas of Eq. (1) and Eq. (2) in a more complicated phase mask where six triangles in the hexagon are assigned with different gray levels. As an example, Fig. 5(a) shows a portion of a programmed pattern where the gray level in the equilateral triangles (from a-f) were set to be 255, 202, 150, 99, 49, and 0, respectively. Figure 5(b) is a simulated interference pattern by diffracted beams (1, 3, 4, 5) using phases determined by a similar way as shown in Fig. 2. Figures 5(c) and 5(d) are the CCD image of the inference pattern and SEM of holographic fabricated structures in DPHPA. The agreement between the experimental pattern and the simulated one is very good. It further approves the method for the assignment of phases in the diffracted beams.
By comparison, computer generated holograms in SLM have shown powerful capability of generating an interference pattern from multi-beams up to 23 , and non-diffracting beams with same kz component [19–22]. However the presented method can be potentially used to set designed gray levels for certain triangles at specific locations for a spatially varying, pixelated phase mask. Such a phase mask can be potentially used for the fabrication of graded photonic crystals where the effective refractive index at a certain location is determined by the filling fraction in holographic structures [24–27,29]. Figures 5(e), 5(f),and 5( g) show simulated four-beam interference patterns formed by beams 1, 3, 4 and 5 with phases assignments from the gray levels shown in Figs. 3(a1, b1, c1), respectively. The patterns were obtained by setting the iso-intensity surface to be a half of maximum intensity. If a negative photoresist is exposed to these interference patterns, the high intensity region becomes polymerized. The filling fractions of dielectric materials versus the whole volume are 0.123, 0.143, and 0.176 in the patterns of Figs. 5(e),5(f),and 5(g), respectively. The effective refractive index is 1.09, 1.11, and 1.13, respectively, assuming a refractive index of 1.6 for DPHPA. By spatially varying the filling fraction and consequently the effective refractive index, graded index devices can be fabricated [24–27,29]. The accumulated knowledge in this paper will be used to investigate the potential design of a gradient phase mask.
In summary, we were able to understand and assign the phase for the diffracted beams from the phase pattern displayed in the spatial light modulator. The phases of the interfering beams can be tuned by the gray levels in the phase pattern and verified by the holographically fabricated structures. The presented method can be potentially used to set designed gray levels for certain triangles at specific locations for spatially varying phase mask.
This work is supported by research grants from the U.S. National Science Foundation under Grant Nos. DMR-0934157, CMMI-1109971, -1266251, and ECCS-1128099.
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