1. Introduction

Photonic crystals are attracting a great deal of attention due to the presence of a band gap in the electromagnetic wave spectrum. The occurrence of the band gap can be exploited in a number of ways such as to control the flow of light in devices, inhibit spontaneous emission, suppress vacuum fluctuations and light localizations [1,2]. Photonic band gap materials are often viewed as the optical analogue to electronics and the term photonic crystals has been coined to represent the artificial material displaying spatial variations in the dielectric. For electronic materials, research activities progressed from perfect periodicity, to studying the effects due to defects and surfaces, to the study of non-crystalline materials. The photonic crystal material research road map has followed that of electronics. The 2-D and 3-D photonic materials are well researched in theory and significant experimental results are available confirming expected optical performance [3,4]. The properties of defects and surface states have been well documented and exploited in many photonic band gap device designs. It had been suggested that quasi-crystalline materials could also possess photonic band gaps [5]. Quasi-crystals have a non-periodic lattice with long-range rotational order, thus they have only rotational symmetry, while a periodic photonic crystal have both rotational and long-range symmetries. Recently, several research groups have published the behaviour of light in the non-crystalline material and results indicate that full photonic band gaps are possible in planar 2-D structures [6,7]. An attractive feature of the quasi-crystal photonic band gap structures lies in the fact that the high degree of symmetry achievable in quasi-crystal patterns results in a relaxation of the high index contrast required to open up the band gap. As such quasi-crystal band gap materials can be fabricated from dielectrics whose indices more closely match those used for optical waveguides and optical fibers. This would reduce the coupling losses associated with the light transition across the waveguide-photonic crystal interface and enable a closer geometrical match between conventional waveguide dimensions and photonic crystal waveguide dimensions. In fact several holey fiber patterns can be viewed as 2-D quasi-crystal planar patterns extending the length of the fiber axis [8]. The study of defects and waveguides in quasi-crystal band gap materials may show interesting and unexpected properties due to the higher degree of symmetry present and number of different sites suitable for defect and waveguide implantation. It has been shown experimentally that the optical transmission properties of a waveguide introduced in a quasi-crystal structure are dependent on the location of the guide relative to the quasi-crystal pattern [9,10]. They also demonstrated that the light transmission properties are frequency dependant and as such the transmission can be higher for a waveguide with 90-degree bends as compared to the straight waveguide at certain frequencies.

Photonic crystal fabrication techniques lend themselves quite well to the production of 2-D and 3-D structures due to the periodicity that exists in the patterns. Defects and waveguides can be easily introduced since they represent a local alteration of the repetitious pattern. Quasi-crystal patterns on the other hand are more difficult to produce since they possess N-fold symmetry about a central feature. In this paper we present a dual beam multiple exposure technique that is suitable for the production of quasi-crystal patterns. The technique is similar to the multi-beam optical systems used to produce periodic photonic crystal patterns [11,12], and most recently quasi-crystal patterns¹³, but differs in the number of beams used (two) and number of exposures the photosensitive sample is subjected to. Reorientation of the photosensitive sample between each laser irradiation step enables the exposure of the quasi-crystal pattern. In Section 2, the experimental system used to produce quasi-crystal patterns is presented. In Section 3, the details of producing N-fold rotational symmetric quasi-crystal patterns are presented along with patterns produced and etched into silicon. In Section 4, we propose a technique for the production of 3-D quasi-crystal structures using the dual beam multiple exposure technique.

2. Optical system

The basic optical system required to produce photonic quasi-crystal patterns is shown in Fig. 1. The blue line (488 nm) of the Argon Ion laser is divided into two equal intensity beams (-1 and +1 order) and separated from the other laser lines using a diffraction grating. The individual beams are allowed to expand and are directed onto a three-axis rotation and translation platform where the photosensitive sample is mounted. In the sample the beams recombine producing the characteristic interference pattern of high and low intensity lines. The central undeviated beam is used in the initial alignment of the sample and laser beams. This beam is used to define the z-axis of the reference coordinate system. A shutter is placed close to the laser in order to control the on time of the exposing beams. At the sample the interference pattern is given by:

where E _j1,_j2, k _⃗j1,j2, and φ _oj1,oj2 are the plane waves amplitude, wave vector and phase factor for each of the interfering beams, 1 and 2, and j represents an index indicating the exposure step. The angle θ _j12 is the angle between the polarization directions of the interfering beams. The cosine of this angle is often related to the contrast of the interference fringes produced. In the dual beam technique this angle can be set to zero, beams polarized in the same direction, giving the largest fringe contrast for equal amplitude beams. The intensity in Eq. (1) attains its maximum value when the argument of the second cosine function is equal to an integer multiple of 2π. The ensemble of lines in 2-D and planes in 3-D can be expressed in simplified form as:

 

Fig. 1. Experimental dual beam multiple exposure optical system. The blue line of the Argon Ion laser is linearly polarized and divided into two equal intensity beams. These beams are recombined producing an interference pattern in a photosensitive material. The three rotation and translation stages allow the accurate orientation and positioning of the interference pattern relative to previous exposure steps.

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where the set (A_j,B_j,C_j) are normalized to unity and related to the differences in the x, y and z direction cosines, (ℓ _j1,2,m _j1,2,n _j1,2), of the interfering beam wave vectors. The parameter D_j is related to the plane family integer w_j and wavelength λ in the medium through:

Through manipulation of the denominator in (3) the minimum spacing between high intensity planes is $\frac{\lambda }{2}$ and corresponds to having the two beams counter propagating, relative beam propagation angle Δθ_j=θ ₁-θ ₂=180° as defined in Fig. 1. There is no upper limit on the plane spacing and in principle the desired plane spacing can be obtained by decreasing the relative beam propagation angle. Either rotating the individual beams or rotating the sample can achieve the orientation of the interference planes relative to the photosensitive material. Experimental ease dictates that the sample is rotated relative to fixed beams for each exposure step. In this way a crisscross set of high and low intensity regions selectively exposes the photosensitive material producing the desired cumulative dose. Development of the photosensitive material produces the desired quasi-crystal pattern.

3. Quasi-crystal patterns

In the dual beam multiple exposure technique, the photosensitive sample is illuminated for a fixed period of time by the interfering beams before being rotated for the next exposure. The production of a quasi-crystal pattern using this technique is best described through the production of a representative 2-D pattern containing 8-fold rotational symmetry. The photosensitive sample consists of a thin layer of photoresist (Shipley S1818, 3 µm at 2000 rpm, bake at 105°C for 1 minute, develop 1 min no post bake) spun onto a glass slide (we also produced patterns in silicon or silicon on insulator). The slide is mounted onto the stage configuration of Fig. 1 such that the light sensitive layer lies in the (x, y) plane. The two beam propagation directions are adjusted such that they are incident onto the photosensitive sample at an angle of θ ₁=4.2° and θ ₂=-4.2° respectively, k⃗_j1(0.07323,0,-0.99731) and k⃗ _j1=(-0.07323,0,-0.99731). This combination produces high intensity planes of 3.3 µm spacing confined to the (y, z) plane in the photosensitive material. These particular beam angles were selected such that large features are produced and easily observed in a microscope. Patterns have been produced using larger beam angles with features below the micron level and photographed in the SEM. In the production of the 2-D planar quasi-crystal patterns reported here the interference plane spacing is kept constant and the sample is rotated about the z-axis. Four equal time exposures are required in the production of an 8-fold symmetric quasi-crystal pattern. The sample’s z-axis rotation angles are (0°, 45°, 90°, 135°). At the sample location the individual beams are expanded to 4 mm in diameter and have a power of 1.88 mW each.

 

Fig. 2. Interface planes used in the production of an 8-fold quasi-crystal pattern. (A) Extend view of the intersecting planes indicating zones where the 4 planes overlap. (B) Enlarged view of the central region of (A) and (C) is a 3-D view of the interference planes extending through the thickness of the exposed material.

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Figure 2 shows the family of planes present in the photosensitive sample when viewed looking down the z-axis, (A) and (B), and in a 3-D view (C). Examination of this figure indicates that at the central position all 4 sets of interference planes intersect giving the highest exposure. At several other locations in the (x, y) view the 4 planes come into coincidence giving an exposure environment very similar to the central region of the pattern. The accumulated exposure dose in the photosensitive material is obtained by adding the exposure dose for each step and will result in certain region being highly exposed, while other regions receive a lower exposure and certain regions receiving no exposure. The maximum exposure dose can be obtained knowing the exposure time, individual laser beam power and beam dimension in the sample for each of the steps. It is convenient to normalizing the beam amplitudes to unity for each exposure in Eq. (1) and taking equal exposure times for each step indicates that the maximum exposure level of 16 is possible when 4 exposures are used, each exposure contributing 4 units to the total dose.

The properties of the photosensitive material would dictate which level, in the range between 0 and 16, corresponds to the threshold level for local exposure. If the threshold level were set to 8 out of 16 then half of the photosensitive material would be exposed while the other half would not. Using a positive photoresist, as the light sensitive material, would result in the exposed photoresist remaining where the exposure dose is above the threshold. For our experimental system an exposure time of 19.5 minutes is required to achieve the mid-threshold level, here 8 out of 16 for 4 exposures.

 

Fig. 3. Simulation of the exposed pattern in the photosensitive material when a threshold level of 8 out of a maximum of 16 selected. All exposure levels below 8 are in black and levels above 8 are in white. The 8-fold symmetry about the central point and other plane coincidence locations is clearly visible in the figure.

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Figure 3 is obtained when the threshold level of 8 out of a maximum of 16 is selected for the four exposures of Fig. 2. The 8-fold symmetry about the center is visible in the figure and highly resembles the 8-fold quasi-crystal pattern reported by Cheng et al., which they showed, contained a photonic band gap, and in which they fabricated waveguide structures and examined defect properties [10]. A photo of the exposed and developed photoresist for an 8-fold rotational symmetric pattern produced using the experimental system of Fig. 1 is shown in Fig. 4. The pattern obtained resembles that predicted by the modeling of the experimental system. One advantage of producing quasi-crystal patterns from equal interference plane spacing, as opposed to periodic photonic crystal patterns, using the dual beam multiple exposure technique lies in the fact that accurate positioning of the beams between exposure steps is not required. Only the rotational symmetry is important for a quasi-crystal and is governed by the accurate rotation of the sample between exposure steps. The quasi-crystal pattern shown demonstrates only long range periodicity and as such inaccurate positioning of the beams between steps results in an in plane (x, y) shift of the pattern produced and not a distortion or destruction of the pattern. Equivalently stated, the phase term (φ _oj1-φ _oj2) of Eq. (3) serves to introduce an (x, y) plane offset in the pattern. Positioning accuracy does become important in situations where the quasi-crystal pattern must be placed relative to other feature of the substrate. In this instance the quasi-crystal pattern generated using the dual beam multiple exposure technique could be used to produce a mask pattern then have the mask actively aligned in a second step.

 

Fig. 4. Image of the 8-fold symmetry pattern produced using 4 exposures of the dual beam multiple exposure experimental system. The intersected projection of the two arrows indicates the center of the pattern. About the center 8-fold rotational symmetry is observed. Dimension bar shown inclined to correspond to orientation of the quasi-crystal pattern.

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Figure 5 shows the 10-fold symmetric pattern obtained using 5 exposures at sample rotation increments of 36 degrees about the z-axis and Fig. 6 shows the 12-fold symmetric pattern obtained using 6 exposures at sample rotation increments of 30 degrees about the z-axis. The total exposure time for the production of the patterns in Figs. 4, 5 and 6 are the same. The exposure time for each individual step is reduced as the number of exposure steps is increased. A simple relationship exists between the rotational order, R, of the quasi-crystal pattern and the exposure parameters (number of exposures, N, rotation angle increment, θ, and exposure step time, t):

 

Fig. 5. Image of the 10-fold rotational symmetry pattern produced through 5 equal time exposures. (A) Extended view of the 10-fold pattern, experimental. (B) Orientation of the family of exposure planes taken about the center. (C) Computed exposure pattern about the center (exposure threshold of 10 out of a maximum of 20).

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The 10-fold pattern shown here closely resembles the 10-fold patterns reported by Kaliteevski et al. for which they computed a band gap [14]. A band gap has also been shown to be present in 12-fold quasi-crystal patterns [15,16] and we expect that a band gap will exist for our 12-fold pattern due to the high rotational symmetry present. We are in the process of developing Finite Element analysis (FEM) tools suitable for the computation of band structures for the quasi-crystals presented here. Simulation results are not available at the time of publication and when available will be reported in conjunction with experimentally determined quasi-crystal properties. We have also produced the quasi-crystal patterns on silicon and silicon on insulator wafers. Figure 7 shows a 8-fold quasi-crystal pattern, produced using the dual beam multiple exposure technique and subsequently etched into silicon. The image in (A) shows an etched wafer with 8-fold symmetry and (B) shows the etched 12-fold symmetry.

 

Fig. 6. 12-fold rotational symmetry pattern produced using 6 equal time exposures. (A) Extended view of the 12-fold pattern, experimental. (B) Orientation of the family of exposure planes taken about the center. (C) Computed exposure pattern about the center (exposure threshold of 12 out of a maximum of 24). (D) Expanded view of the central region of the experimentally obtained 12-fold pattern of (A).

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The fill factor of the quasi-crystal patterns produced can be controlled through the accumulated exposure dose deposited in the photosensitive sample. Figure 8 demonstrates the effect of changing the exposure dose threshold level to 6 (A) and 10 (B) out of a maximum of 16 for the 8-fold symmetry pattern. It is worthwhile to notice that the central patterns produced by exposure thresholds of 6 and 10 are not inverse patterns of each other. Inverse patterns can be obtained using photoresists of opposite polarity or by searching throughout the quasi-crystal pattern and finding inverse regions. The 6 of 16 pattern displays a highly connected pattern while that of 10 of 16 displays a highly disconnected pattern. This is an important aspect of the dual beam multiple exposure technique since it is well known that a TM gap primarily opens up in a disconnected structure (as in the threshold 6 pattern) while the TE gap primarily open up in an interconnected structure (as in the threshold 10 pattern) [17]. Experimental runs of the 8-fold symmetric quasi-crystal pattern under various fill factor values are also shown in Fig. 8.

 

Fig. 7. Eight-fold (left) and 12-fold (right) quasi-crystal patterns produced and etched into a silicon wafer. Etch is 1 µm deep.

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The diffraction patterns for 8-fold, 10-fold and 12-fold quasi-crystal patterns produced using the dual beam multiple exposure technique are shown in Fig. 9. The patterns demonstrate a series of high intensity points symmetrically distributed about the center characteristic of a crystalline material but containing the 8, 10, and 12 fold rotational symmetry. As shown by Kaliteevski et al. these points can be related to the reciprocal vectors of the structure and used to compute band gap properties of the quasi-crystal [14,15].

 

Fig. 8. Quasi-crystal patterns produced using a lower threshold of 6 out of 16 for the 8-fold symmetry (A) theory, (B) experimental. A highly interconnected pattern results. Quasi-crystal patterns produced using a lower threshold of 10 out of 16 for the 8-fold symmetry (C) theory, (D) experimental. A disconnected pattern results.

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4. 3-D quasi-crystals

Recently, interest has been expressed in whether the features observed for 2-D quasi-crystal can be extrapolated to 3-D quasi-crystals structures. A 3-D quasi-crystal would consist of an N-fold symmetric pattern simultaneously present about three or more non-coplanar axis. The resulting pattern would possess central sites of N-fold planar symmetry similar to those demonstrated in Figs. 4–8 and also extending into the third dimension. The simplest of the 3-D quasi-crystal patterns would be to extend the N-fold symmetry to three mutually perpendicular axes. The dual beam multiple exposure technique presented here may be used to produce 3-D quasi-crystal template patterns in a thick layer of photosensitive material. Developing and employing a backfill process would render the required stable dielectric profile. Depending on the particular exposure step and quasi-crystal pattern required the photosensitive material must be carefully orientated and positioned relative to the interfering beams. This positioning may be difficult to achieve when sub-micron linear and arc second rotational accuracy is required. However, positioning inaccuracy will only result in a shift in 3-D of the pattern and not in a complete destruction of the quasi-crystal pattern. There are two key system performance issues that need be addressed when considering the laser beam exposure of 3-D patterns in a thick photosensitive material weather for the production of periodic or aperiodic photonic crystals. The usual high absorption rate of light by the photoresist results in the beams losing considerable intensity as they penetrate a few microns into the photoresist. This results in the pattern fading into the unexposed background with exposure depth. This problem can be overcome, to some extent, by rendering the photoresist more transparent to the exposing beams [18] or by detuning the laser beam wavelength from the absorption band. The other issue to address results from the refraction of the laser light as it traverses the air photoresist boundary. Any modeling tool used to predict the exposure profile inside the photoresist would require that the incident beams be adjusted for the reflective loss and propagation direction change of the laser light. Techniques to overcome this problem have been proposed in systems employing multiple, 4 or more, beams by introducing an index match layer above the photosensitive material and sculpting the surfaces such that each beam penetrate the air index match material at normal incidence [19]. Using this technique beams can propagate in the photoresist at high angles with respect to the photoresist surface normal. The feasibility of such corrective techniques for the production of 3-D photonic quasi-crystals remains to be determined. As an example of the use of the dual beam technique we present the parameters required to produce a 8-fold 3-D quasi crystal.

 

Fig. 9. Experimentally obtained diffraction patterns for the (A) 8-fold, (B) 10-fold and (C) 12-fold quasi-crystal patterns experimentally produced and displayed in Fig. 4, 5, and 6 respectively. The diffraction patterns display the rotational symmetry associated with the corresponding quasi-crystal pattern.

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Fig. 10. View of the interference planes required producing the three axes 8-fold symmetric 3-D quasi-crystal pattern. There are 4 sets of intersection planes oriented at angles of 0, 45, 90 and 135 degrees about each of the three mutually perpendicular axes.

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In order to describe the interference beams and sample rotation required for each exposure step, the interference planes are initially aligned with the (y, z) plane and then rotated through the three Euler angles. This is achieved by having the two beams directed towards the origin of the reference coordinate system with wave vectors contained in the (x, z) plane, inclined at +45 and -45 degrees to the z-axis respectively. The wave vector direction cosines are ${\overrightarrow{k}}_{j1}=\left(\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}}\right)$ and ${\overrightarrow{k}}_{j2}=\left(-\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}}\right)$. The rotation of the planes is accomplished by rotating the sample using the three rotation stages indicated of Fig. 1. Nine equal time exposures are required for the production of the 8-fold three axis quasi-crystal pattern. The exposure step and sample orientation Euler angles (ϕ,θ,φ) are given below:

 

Fig. 11. 3-D view and (x, y) plane slices of the 8-fold 3 axis quasi-crystal pattern produced using the interference planes of Fig. 10. Slices are in 0.1-micron increments along the z-axis starting at A1 (z=0 microns) to C3 (z=1.0 microns). Increment sequence is A1, A2, A3, A4, B1, B2, B3, B4, C1, C2 and C3. The center of the pattern in A1 corresponds to the coordinate origin.

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In (5), exposure steps 1 to 4 produce the 8-fold symmetric pattern in the (x, y) plane, exposures 1, 5, 6 and 7 produce the 8-fold symmetric pattern in the (y, z) plane, and exposures 1, 6, 8 and 9 produce the 8-fold symmetric pattern in the (x, z) plane. Only nine exposures are required instead of 12 (3 axis of symmetry requiring 4 exposures each) since exposure planes produced by steps 1 and 6 are common to more than one symmetry axis. Figure 10 shows a plot of the exposure planes from (5) used in generating the 8-fold 3-D quasi-crystal pattern of Fig. 11. Figure 11 shows the interconnected exposed material for an exposure threshold of 18 out of a maximum of 36. The 8-fold symmetry of the pattern produced is clearly visible in the C1 (x, y) plane slice also shown in Fig. 11. The pattern C1 closely resembles the 8-fold planar patterns observed in Fig. 3 and 4. The underlying cubic nature of the highly exposed region of the 3-D structure results from the selection of three mutually perpendicular axes as the axis of 8-fold symmetry. It is expected that the volume diffraction pattern produced by this structure would display cubic as well as 8-fold rotational symmetry.

5. Conclusion

In this paper we presented a dual beam multiple exposure technique suitable for the production of N-fold symmetric quasi-crystal 2-D patterns. Due to the nature of the quasi-crystal pattern production technique, accurate positioning of the sample between exposure steps is not required making this system simple and practical. We have shown that a simple relationship exists between the rotational symmetry order and the exposure parameters (number of exposures, sample rotation angle and exposure time). We demonstrate the use of the pattern generating capabilities by producing 8, 10 and 12-fold symmetric planar quasi-crystal patterns. We also discuss a simple technique for varying the fill factor of the quasi-crystal patterns produced by adjusting the exposure time and accumulated exposure dose. We close the paper by describing how the dual beam multiple exposure technique may be used to produce 3-D quasi-crystal patterns and give details on the parameters required to produce and 8-fold, 3-axis, 3-D quasi-crystal template structure and the anticipated difficulties to be overcome.