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Fabrication of size scalable three-dimensional photonic structures via dual-beam multiple exposure and its robustness study Xuelian Zhu, Guanquan Liang, Yongan Xu, Shih-Chieh Cheng, and Shu Yang* Department of Materials Science and Engineering, University of Pennsylvania, 3231 Walnut Street, Philadelphia, Pennsylvania 19104, USA *Corresponding author:
[email protected] Received June 28, 2010; revised September 24, 2010; accepted September 29, 2010; posted September 29, 2010 (Doc. ID 130754); published November 5, 2010 We theoretically designed dual-beam triple exposure interference lithography to fabricate three-term diamondlike structures in SU-8 photoresist with scalable size and investigated the robustness of the optical setup against potential experimental errors. Minimal distortion could be achieved by careful selection of the angle between the bisector of the two beams and the normal of the sample surface to precompensate the anisotropic shrinkage. A small deviation of incident beam angles, however, would lead to a significant change in structural size when the angle between the two incident beams was small for a large sized structure, whereas the translational symmetry of the SU-8 structure remained reasonably close to face-centered cubic. We then experimentally demonstrate size scalable diamond-like photonic structures with the lattice symmetry and size close to the theoretical design. © 2010 Optical Society of America OCIS codes: 160.5298, 090.0090, 260.3160.
1. INTRODUCTION Interference lithography (IL) is an efficient and flexible technique to fabricate a wide range of defect-free threedimensional (3D) photonic crystal (PC) templates [1–5]. During exposure, the interference intensity profile is transferred to a photoresist film, and the shape of the resulting structure is determined by the isodose surface of the lithographic threshold value, which can be described by the corresponding level surface [6]. In the level-set approach, the surface of a porous structure is represented by a scalar-valued function F, which satisfies F共x , y , z兲 = t, where t is a constant. While many of 3D photonic structures with complete bandgaps have been proposed theoretically [2,7], in experiments the final structure may be distorted from the original design due to the refraction effect at the air/film interface and the shrinkage of the photoresist after crosslinking [3,4], thus lowering the structural symmetry and degrading the photonic bandgap (PBG) quality. To minimize the structural distortion, it is essential to design a robust optical setup to precompensate the refraction effect and film shrinkage. There are many possible optical configurations in IL to create 3D photonic structure, including four-beam single exposure [1,4,7–10], dual-beam multiple exposure [2,3,11–14], phase shifts and multiple exposure of three or four beams [6,15], and a hybrid five-beam configuration [16]. Among them, single exposure of four interference beams is most widely practiced, because it is straightforward and the relative phases between beams do not change the shape of the interference pattern [4,7]. However, not all structures can find a simple solution of an optical setup that can both eliminate unnecessary interfer0740-3224/10/122534-8/$15.00
ence and precompensate the structural distortion by this approach (see the detailed discussion in the Appendix on a single exposure four-beam umbrella setup to create three-term diamond-like SU-8 PCs). In addition, the size of the 3D structure is not changeable for a fixed exposure wavelength. In comparison, the multi-exposure of twobeam IL approach [2,3,12,13] is more versatile to create desired structure symmetries with tunable sizes. This is because any arbitrary structure can be expanded as a sum of Fourier terms, each of which could correspond to a grating arising from the interference of two beams [2]; the lattice size can be tuned by varying the beam angles. To tackle the shrinkage issue Meisel et al. utilized an elongated interference pattern during the dual-beam triple exposure to create a precompensated single cubic PC in SU-8 photoresist [3]. Nevertheless, little attention has been paid to quantitatively study the robustness of the optical setup by multiple exposures against the experimental deviations (e.g., beam misalignment and sample rotation) and unnecessary interference between beams as seen in the multi-beam single exposure setup. Using the three-term diamond-like SU-8 structure as a target, here we showed that dual-beam triple exposure would not only allow the flexibility to modulate the lattice size, but also eliminate the unwanted interference between the beams of different gratings while precompensating the distortion. First, we theoretically designed a triple exposure dualbeam IL to create a precompensated diamond-like structure and evaluated the influence of the deviation of beam angles on the lattice constant and symmetry. It was found that the lattice symmetry of the SU-8 structure was quite robust to the experimental errors at different film shrink© 2010 Optical Society of America
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Vol. 27, No. 12 / December 2010 / J. Opt. Soc. Am. B
age levels: even at shrinkage of ⬃41%, the face-centered cubic (fcc) symmetry was maintained. However, for creating a large size structure a small misalignment of the beam angles could lead to a significant variation of the lattice constant. We confirmed the theoretical study in experimentally fabricated SU-8 three-term diamond-like structures.
2. OPTICAL SETUP DESIGN It has been shown that in IL fabricated 3D structures, because the photoresist film is confined on a substrate, where the lateral dimension of the film 共l兲 is much larger than the film thickness 共t兲 (e.g., l / t ⬵ 5mm/ 8 m = 625 for our sample), the in-plane shrinkage is suppressed and negligible, leading to anisotropic shrinkage along the film thickness [3,4]. That is different from the case of twophoton lithography, where l is comparable to t (typically hundreds of microns). Therefore, the lateral shrinkage in the latter case should be considered [17,18], and a prefabricated rigid support around the film is often used to minimize the shrinkage and prevent pattern collapse [19]. Since the unidirectional shrinkage in the IL fabricated film could distort the structure and thus degrade the PBG properties [3], it is critical to design proper beam geometry such that the shrinkage of photoresist would be precompensated by a spatially elongated interference intensity profile. As a proof-of-concept, we studied the threeterm diamond-like structure, which belonged to space ¯ m (No. 166) [20]. Its level surface is group R3
冋
F共rជ 兲 = cos
2 d
冋
+ cos
册 冋 册
共− x + y + z兲 + cos
2 d
共x + y − z兲 ,
2 d
共x − y + z兲
册
共1兲
where d is the lattice constant of the fcc lattice, and x, y,
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z axes are along the [100], [010], and [001] directions, respectively, and the [111] direction is perpendicular to the substrate. Assuming that the film is uniformly crosslinked and the shrinkage across the film depth is consistent, which is reasonable for SU-8 films with film thicknesses less than 100 m, we reconstruct the level surface for the elongated interference pattern in a new reference ¯ ¯12兴, 关11 ¯ 0兴, and [111] frame with the ˜x, ˜y, ˜z axes along 关1 directions, respectively (Fig. 1):
再冋 再冋 再冋
⬘ Felongated 共rជ 兲 = cos
+ cos
+ cos
2 d
2 d
2 d
−
2冑2
˜x +
共1 − sr兲
˜z
册冎
冑3 冑3 冑2 共1 − sr兲 ˜+ ˜x − 冑2y ˜z 冑3 冑3 冑2 共1 − sr兲 ˜x + 冑2y ˜+ ˜z 冑3 冑3
册冎 册冎
, 共2兲
where sr is the anticipated photoresist shrinkage. The level surface described by Eq. (2) can be considered as a combination of three independent gratings. A dualbeam triple exposure setup was designed (see Fig. 1) to fabricate the diamond-like structure, because (1) the beams of different gratings would not interfere with one another, (2) the structure size could be tuned by modulating the incident beam angles, and (3) the relative phases of the laser beams could be simultaneously eliminated by a translation of the origin [4,7]. It is worth mentioning that the phase difference between the beams for each exposure has to be considered for four or more exposures [21]. The sample was placed on a rotational sample stage and was rotated around the ˜z axis by an angle of 120° between every two exposures. The highest contrast between the minimal and maximal intensities of interference pattern can be achieved by setting the two laser beams with the same polarized direction and intensity.
Fig. 1. (Color online) (a) Illustration of two-beam interference lithography with the sample fixed on a rotation stage. The black dashed line shows the normal of the photoresist film and the red dashed line is the bisector of the two laser beams. (b)–(f) Construction of three-term diamond-like lattice via dual-beam triple exposure and unidirectional photoresist shrinkage. (b)–(d) A set of one-dimensional lattice planes obtained from the corresponding three dual-beam exposures. (e) By combining the three exposures from (b) to (d), an elongated three-term diamond-like rhombohedral light intensity distribution with the (111) plane parallel to the substrate is generated in the photoresist. (f) After shrinkage in the [111] direction, the final structure has an fcc translational symmetry.
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A. Derivation of Incident Beam Angles The interference intensity profile of the dual-beam triple exposure (Fig. 1) can be constructed as 3
兺
I共rជ 兲 =
ជ i1 ជ i2 ជ i1 ជ i2 ⬘ 兩2 + 兩E ⬘ 兩2 + 2E ⬘ ·E ⬘ cos关共kជ i2 ⬘ − kជ i1 ⬘ 兲 · rជ 兴其, 兵兩E
i=1
共3兲 where i = 1 , 2 , 3 is the exposure sequence. For the ith exជ i1 posure, E ⬘ and Eជ i2 ⬘ are the electric fields of the beams, ជ and ki1 ⬘ and kជ i2 ⬘ are the wave vectors. A solution to realize the three-term diamond symmetry shown in Eqs. (1) and ជ 11 (2) is E ⬘ · Eជ 12 ⬘ = Eជ 21 ⬘ · Eជ 22 ⬘ = Eជ 31 ⬘ · Eជ 32 ⬘ . To maximize the interference contrast, the two beams were set at the same inជ i1 tensity and were s-polarized (i.e., E ⬘ and Eជ i2 ⬘ are perpendicular to the plane of incidence and parallel to each other). For any single exposure,
⬘ A兩 = 兩kជ i2 ⬘ 兩⌬kin
in A
⬘ − kជ i1
in A兩
=
冏 冋 冉 冊 2nr
sin ⌫⬘ +
⬘
− sin ⌫⬘ −
⬘ ˜ − kជ i1z ⬘ ˜兩 = 兩⌬k˜z⬘兩 = 兩kជ i2z =
4nr
2nr
sin ⌫⬘ sin
2
cos ⌫⬘ sin
⬘
cos ⌫⬘ +
⬘
=
2
4nr
⬘ 2
− cos ⌫⬘ −
2
⬘ 2
共5兲
,
where A denotes the ˜x˜y plane parallel to the photoresist film, and the ˜z axis is the direction perpendicular to the sample surface. The symbol of prime indicates the parameters in the photoresist. ⬘ is the angle of the two laser beams in the photoresist, and ⌫⬘ is the angle between the normal and the bisector. By comparing the intensity profile [Eq. (3)] with the desirable level surface [Eq. (2)], we obtain
⬘ A兩 = tan ⌫⬘ = 兩⌬k˜z⬘兩/兩⌬kin
冑
⬘ A兩 2 = 兩⌬k兩 = 兩⌬k˜z⬘兩2 + 兩⌬kin
4nr
sin
共1 − sr兲
⬘ 2
2冑2
共6兲
,
2 冑8 + 共1 − sr兲2
=
冑3
d
.
共7兲 Therefore
冋 册
⌫⬘ = arctan
冋
⬘ = 2 arcsin
共1 − sr兲 2冑2
2⬘ = ⌫⬘ +
⬘ 2
.
共10兲
j = arcsin共nr sin j⬘兲,
j = 1,2.
共11兲
If either 1⬘ or 2⬘ is higher than the critical angle of total internal reflection, 38.68° for SU-8 共nr = 1.6兲, a prism and/or index matching liquids will be needed to precompensate the refractive effect. B. Shrinkage Precompensation and Lattice Size Scalability To study the influence of beam angle deviation on lattice symmetry and constant, we rewrite Eqs. (8) and (9) as
d=
冑8 + 共1 − sr兲2 . 2nr 冑3 sin共⬘/2兲
共12兲
共13兲
Clearly, ⌫⬘ is only dependent on sr, and ⬘ determines the lattice constant d [Figs. 2(a) and 2(c)]. Here, we focus on 0 ⱕ sr ⬍ 1 although the above relationship is applicable to sr ⬍ 0, corresponding to the film expansion. The shrinkage sr is highly dependent on specific material systems, for example, ⬃41% for SU-8 [3,4], ⬃48% for epoxy-functionalized polyhedral oligomeric silsesquioxanes [22], and ⬃6% for poly(glycidyl methacrylate) [8] obtained from 3D cubic structures. We note that sr and ⌫⬘ almost follow a linear relationship within the region of 0 ⱕ sr ⬍ 1 (i.e., 0 ⬍ ⌫⬘ ⬍ 19.47°). The deviation of precompensated shrinkage can be calculated from Eq. (12) as ⌬sr ⌬⌫⬘
=−
2冑2 cos2 ⌫⬘
.
共14兲
As seen in Fig. 2(b), 兩⌬sr兩 would be ⬃0.05 if ⌫⬘ has a deviation of 1°. Here, the positive and negative signs of ⌬sr correspond to under- and over-compensation, respectively. Figure 2(c) demonstrates the relationship of the lattice constant d versus ⬘ and sr. In principle, d can vary from ⬃ / nr to infinity when ⬘ approaches zero, while the accessible structure size in experiment will be limited by the film thickness, the light absorption along the film depth, and the resolution and contrast of the photoresist. We note that when ⬘ → 0 a small deviation of the beam angles can lead to a significant variation of the lattice constant. The detailed discussion can be found later.
共8兲
,
冑8 + 共1 − sr兲2 2dnr 冑3
2
,
sr = 1 − 2冑2 tan ⌫⬘ ,
共4兲
,
⬘
When prisms and/or index matching liquids are not applied, to precompensate the refraction at the air/ photoresist interface, it is necessary to convert the parameters into the air using Snell’s law:
2
冉 冊册冏 冏 冋 冉 冊 冉 冊册冏 ⬘
1⬘ = ⌫⬘ −
册
3. STUDY OF ROBUSTNESS OF OPTICAL SETUP .
共9兲
The beam incidence angles in the photoresist, 1⬘ and 2⬘ , are
Although the anisotropic film shrinkage and refraction effect have been considered here, misalignment of two incident beams and/or accuracy of the rotation stage in multiple exposure of dual-beam IL could result in structural deviation from the theoretical design. The commercial ro-
Zhu et al.
Vol. 27, No. 12 / December 2010 / J. Opt. Soc. Am. B
(a)
0.8
sr 0.6 0.4 0.2
(∆ ∆ d / ∆ θ’)/ d ( 1/Degree)
0 0
5
10
15
Γ’ (Degree)
20
(b) -0.05
(c)
-0.052
-0.054
-0.056 0
5
10
15
20
Γ’ (Degree) 2
0
(d) az / az0
-0.2 -0.4 -0.6 -0.8
1.8
∆ Γ’=1°
-1°
1.6
0.5°
-0.5°
1.4
0.1°
-0.1°
(e)
1.2
sr
1
-1
-1.2 0
-0.048
d (λ λ /nr)
∆ sr / ∆ Γ’(1/Degree)
1
θ’ (Degree)
0.8 30
60
90
120
150
0
180
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
θ’(Degree)
sr
Fig. 2. (Color online) (a) Angle between the normal and the bisector, ⌫⬘, as a function of the photoresist shrinkage during fabrication, sr. (b) Error of the compensated shrinkage caused by the deviation of ⌫⬘ for materials at different ⌫⬘ values. (c) Lattice constant d versus two-beam angle ⬘ and shrinkage sr. (d) Error of lattice constant caused by the deviation of ⬘ at different ⬘ values. (e) az / az0 versus sr for various values of ⌬⌫⬘ within the region of sr 苸 关0 , 0.9兴.
tation stage (PRM1-Z7E, Thorlabs) used in our experiments has a high resolution of 1 arcsec 共=1 ° / 3600 ⬵ 0.000 278°兲 [23], which is negligible compared to the angle of rotation between two exposures, 120°. Errors from the deviation of the normal direction of the photoresist surface from the ideal ˜z axis due to wobbling of the rotation stage, so called wobble effect, could also be ignored since most commercial motorized rotation stages have a wobble angle⬍ 0.01° [21]. Therefore, here we focus on quantitative study of the errors caused by beam misalignment and decouple its effect on the lattice constant and symmetry of the final structure. First, we assume that the air/photoresist interface is perpendicular to the rotational axis (i.e., ˜z axis) [see Fig. 1(a)], the shrinkage is perfectly compensated, and the deviation of the angle between the two beams is ⌬⬘. The relative error of the lattice constant d can be derived from Eq. (13) as ⌬d/⌬⬘ d
=−
cot共⬘/2兲 2
.
共15兲
As seen in Fig. 2(d), 共⌬d / ⌬⬘兲 / d approaches zero for a large ⬘, but dramatically increases to infinity when ⬘ → 0, that is, a small deviation of ⬘ can result in a significant variation of the lattice constant at a small ⬘. Previously, we have shown that fcc lattice would change to rhombohedra by unidirectional shrinkage of the photoresist film [4]. To study the beam misalignment effect to the symmetry of the final structure, we calculate the change of the periodicity in the [111] direction (i.e., ˜z axis): az az0
=
=
1 − sr 1 − sr − ⌬sr
=
2冑2共1 − sr兲
2冑2共1 − sr兲 + 共s2r − 2sr + 9兲⌬⌫⬘
sin ⌫⬘ cos ⌫⬘ sin ⌫⬘ cos ⌫⬘ + ⌬⌫⬘
,
共16兲
where az0 and az represent the theoretical and experi-
mental values, respectively. For az / az0 = 1 (i.e., ⌬sr = 0 and ⌬⌫⬘ = 0) the structure is perfectly compensated, whereas az / az0 ⬍ 1 (i.e., ⌬sr ⬍ 0 and ⌬⌫⬘ ⬎ 0) corresponds to spatially shrinkage, and az / az0 ⬎ 1 (i.e., ⌬sr ⬎ 0 and ⌬⌫⬘ ⬍ 0) corresponds to the spatial elongation. Figure 2(e) shows the relationship of az / az0 versus sr for various values of ⌬⌫⬘ within the region of sr 苸 关0 , 0.9兴, which is reasonable for most of photoresist materials. For SU-8, to precompensate unidirectional film shrinkage 共sr = 0.41兲, ⌫⬘ is set to 11.78°; and az / az0 苸 共0.92, 1.09兲 for 兩⌬⌫⬘兩 ⬍ 1°, az / az0 苸 共0.96, 1.05兲 for 兩⌬⌫⬘兩 ⬍ 0.5°, and az / az0 苸 共0.99, 1.01兲 for 兩⌬⌫⬘兩 ⬍ 0.1° [Fig. 2(e)], suggesting that the final structure could be maintained reasonably close to the fcc lattice as long as 0 ⬍ 兩⌬⌫⬘兩 ⬍ 1°. The setup would become more robust for the photoresist system with lower shrinkage.
4. EXPERIMENTAL VALIDATION To validate the theoretical prediction, we chose the two pairs of incident beam angles, (1) 1⬘ = 0, 2⬘ = 23.6° (1 = 0, 2 = 39.8° in air) and (2) 1⬘ = 4.9°, 2⬘ = 18.7° (1 = 7.8°, 2 = 30.9° in air), to fabricate three-term diamond-like structures with the theoretical prediction of lattice constants as 1.36 and 2.30 m, respectively (Fig. 3). The detailed discussion of how to control the incident angles can be found in the Appendix. The photoresist solution 共⬃58 wt. %兲 from mixture of Epon SU-8 and 2.0 wt. % photoinitiator, Irgacure 261 (from Ciba Specialty Chemicals) in ␥-butyrolactone (GBL, Aldrich) was spin-coated on a clean glass substrate at 1500 rpm for 30 s. The photoresist film was pre-exposure baked at 65° C for 2 min and 95° C for 10 min. The film was exposed to the superimposed dual laser beams ( = 532 nm, Coherent Verdi V6, laser output of 1.5W) for ⬃2.4 s for each exposure. The laser beam size was expanded to ⬃6 mm in diameter to ensure a considerable size of intersection of all three exposures. After post-exposure bake at 65° C for 3 min and 95° C for 3 min, respectively, the film was developed in propylene glycol monomethyl ether acetate (Aldrich), fol-
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Fig. 3. (Color online) SEM images of three-term diamond-like SU-8 structures. (a),(c) top surface of the structure with lattice constants of 1.34 and 2.38 m, respectively. Insets: the simulation structures. (b),(d) the corresponding cross sections of (a) and (c) with the top surface tilted by 52°. The cross sections are FIB milled perpendicular to the (111) plane. The green dashed lines indicate the adjacent lattice planes in [111] direction. Taking into account the viewing angle (38°), the distances between the adjacent lattice planes in the [111] direction are (b) h111 = 0.73 m and (d) h111 = 1.36 m, and hence the periodicities in [111] direction 共az = 3h111兲 are (b) az = 2.19 m and (c) az = 4.08 m. Scale bars: 2 m.
lowed by drying in a supercritical CO2 dryer (SAMDRIPVT-3D, tousimis). The fabricated structures were characterized using the scanning electron microscope (SEM), and the cross section was ion milled perpendicular to the top surface [i.e., (111) plane] using the focus-ion-beam (FIB, FEI Strata DB235) (Fig. 3). The distance between the holes in the (111) plane 共axy兲 and the periodicity in the [111] direction 共az兲 were measured from the SEM images and compared with calculation. As seen from Table 1, the lattice parameters of fab-
ricated structures closely match the theoretical design with slight deviations of the lattice constant, 1.6% and 3.4% for d = 1.34 and 2.38 m, respectively. Importantly, the fcc symmetry was nearly maintained, az / az0 = 0.99 for d = 2.38 m and az / az0 = 0.92 for d = 1.34 m.
5. CONCLUSION We theoretically designed dual-beam multiple exposure IL to fabricate 3D photonic structures with scalable sizes
Table 1. Comparison of Theoretical and Experimental Parameters for Three-Term Diamond-Like SU-8 Structures under Different Incident Anglesa Incident Angles
Pitch in (111) plane, axy 共m兲 Periodicity in [111] direction az 共m兲 Lattice constant d 共m兲 Deviation of lattice symmetry 共az / az0兲
1 = 0 , 2 = 39.8° ⇒ 1⬘ = 0 , 2⬘ = 23.6°
1 = 7.8° , 2 = 30.9° ⇒ 1⬘ = 4.9° , 2⬘ = 18.7°
Theoretical Prediction
Experimental Value
Relative Deviation (%)
Theoretical Prediction
Experimental Value
Relative Deviation (%)
0.96 2.35 1.36 1
0.97 2.19 1.34 0.92
1.0 6.9 1.6
1.63 3.98 2.30 1
1.69 4.08 2.38 0.99
3.7 2.4 3.4
a For fcc lattice, the pitch in the 共111兲 plane axy = d / 冑2, and the periodicity in the 关111兴 direction az = 冑3d. For the rhombohedral lattice, the lattice constant d = 共4a2xy / 3 + az2 / 9兲1/2. The relative deviation is defined as 兩ctheo − cexp兩 / ctheo. To study the deviation of lattice symmetry from fcc, the experimental value of periodicity in the 关111兴 direction az is compared to the desirable one az0 = 冑6axy.
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Vol. 27, No. 12 / December 2010 / J. Opt. Soc. Am. B
and investigated the robustness of the optical setup against errors. Minimal distortion could be achieved by carefully selecting the angle between the bisector of the two beams and the normal of the sample surface to precompensate the anisotropic shrinkage. A small deviation of incident beam angles, however, might lead to a significant change in structural parameters when the angle between the two incident beams is small for a large lattice constant. The lattice symmetry of the SU-8 structure is relatively robust to the deviation of beam angles. We then fabricated the three-term diamond-like structures with different periodicities through modulation of the angle between the incident beams and demonstrated that the lattice size and symmetry were reasonably close to the theoretical design. We believe that the quantitative study presented here can be applied to designing other 3D photonic structures of interests while offering both flexibility in optics and robustness against experimental errors.
APPENDIX A: SUPPORTING INFORMATION 1. Difficulty of Minimizing Structure Distortion Using Single Exposure Four-Beam Umbrella Setup We attempted to minimize the structural distortion of three-term diamond-like structure using the single exposure four-beam umbrella setup. If we rotate Eq. (2) back to a conventional reference frame with the x, y, z axes along [100], [010], and [001] directions, the level surface for the elongated interference pattern could be reconstructed as
再 冋冉 冊 冉 冊 冉 冊 册冎 再 冋冉 冊 冉 冊 冉 冊 册冎 再 冋冉 冊 冉 冊 冉 冊 册冎
⬘ Felongated 共rជ 兲 = cos −
sr 3
2 d
z
+ 1−
−
sr 3
− 1+
3
z
y− 1+
3
1−
d
+ cos
sr 3
z
sr
x+ 1−
2
+ cos
sr
sr
2 d
,
3
sr 3
y+ 1
x− 1+
1−
sr 3
sr 3
y
x+ 1
共A1兲
where d is the lattice constant of the fcc lattice for the desirable three-term diamond-like structure, and sr is the anticipated unidirectional shrinkage of the photoresist film in the [111] direction, for example, sr ⬵ 41% in the 3D SU-8 film [3,4]. The level surface described by Eq. (A1) can be considered as a combination of three gratings, each of which is generated by the interference between the central beam and one of the side beams. It is critical that each set of beams should not interfere with each other during the exposure process [2], that is, the polarizations between any two side beams must be perpendicular to each other in a single exposure. Here, we start with the single exposure and four-beam umbrella configuration [4]. The interference intensity profile can be written as
3
I共rជ 兲 =
兺 i=0
3
Ei2 + 2
兺 l=1
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3
cl cos关共kជ l − kជ 0兲 · rជ 兴 + 2
兺
clm cos关共kជ l
l⬎m=1
− kជ m兲 · rជ + lm − l + m兴 = I0 + Ics + Iss ,
共A2兲
where cl = E0El兩ជ 0ⴱ · ជ l兩, clm = ElEm兩ជ lⴱ · ជ m兩, l = arg共ជ 0ⴱ · ជ l兲, 3 lm = arg共ជ lⴱ · ជ m兲, I0 = 兺i=0 Ei2 is the intensity background, 3 ជ ជ Ics = 2兺l=1cl cos关共kl − k0兲 · rជ 兴 is the interference between the 3 central beam and side beams, and Iss = 2兺l⬎m=1 clm cos关共kជ l ជ − km兲 · rជ + lm − l + 共m兲兴 is the interference of three side beams. From Eq. (A1), we calculate the reciprocal lattice vectors as bជ 1 = 共2 / d兲关−共1 + sr / 3兲 , 共1 − sr / 3兲 , 共1 − sr / 3兲兴, bជ 2 = 共2 / d兲关共1 − sr / 3兲 , −共1 + sr / 3兲 , 共1 − sr / 3兲兴, and bជ 3 = 共2 / d兲 ⫻关共1 − sr / 3兲 , 共1 − sr / 3兲 , −共1 + sr / 3兲兴. Without losing the generality, the reciprocal lattice vector bជ i can be defined as bជ i = kជ 0 − kជ i 共i = 1 , 2 , 3兲, and kជ 0 is assumed to be along the [111] direction. Given = 532 nm, nr = 1.60, and sr = 41% [1,2], we obtain the new wave vectors in SU-8 as showed in Table 2 while keeping the four-beam umbrella configuration. Here, our intent is not to use any prism nor any index matching liquid to precompensate the effects of refraction. Thus, it is necessary to convert the calculated wave vectors and polarization vectors from SU-8 into the air by applying Snell’s law and Fresnel’s equations [24]. All parameters are summarized in Table 2. Comparing Eq. (A2) with the level surface illustrated in Eq. (A1), the beams should be constructed to meet the constraints, cl = ccs ⫽ 0, clm = css = 0, and ជ i · kជ i = 0, where ccs and css are constants, and i = 0 , 1 , 2 , 3. In initial attempts, however, we found no solution that would perfectly match all constraints. Although we can increase the ratio of E0 / Ej 共j = 1 , 2 , 3兲 such that css Ⰶ ccs and Iss becomes negligible versus Ics as a compromise, it will also degrade the contrast between Ics and I0. As a result, the large background intensity will overwhelm the intensity variation ⌬I = Ics + Iss. We then engaged in a trial-and-error method to optimize the polarization directions and reduce the css / ccs ratio. At the first run, we kept the central beam as right-circularly polarized, and the surrounding beams elliptically polarized. Since the level surface described by Eq. (A1) is symmetric with respect to x, y, and z, for simplicity, the polarizations of the three side beams are assumed to maintain the same symmetry. The set of polarization vectors (in SU-8) that led to the lowest css / ccs value was selected as the optimum solution (see Table 2), where ccs = 0.656E0Es and css = 0.261Es2. The resulting css / ccs ratio decreases only by ⬃13% in comparison to that from the four-beam single exposure approach in Table 1 in [4]. Nevertheless, it is nontrivial to align the beams with elliptical polarizations. Therefore, it is usually difficult to eliminate unnecessary interference terms using a four-beam single exposure method. 2. Alignment of Incident Beam Angles The control of the incident beam angles in air, i 共i = 1 , 2兲, is illustrated in Fig. 4, which is a twodimensional (2D) projection of the deal-beam setup [Fig. 1(a)] on an optical table. Here, the pivot axis of the rotational stage and the two incident beams are parallel to the surface of the optical table. In practice, we first ap-
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Zhu et al.
Table 2. Wave Vectors and Polarization Vectors in Air and SU-8 Using Interference Lithography with Visible Light „ = 532 nm… Medium Refractive Index
Refraction Beams in SU-8 1.6
Wave vectors
kជ 0⬘ = kជ 1⬘ = kជ 2⬘ = kជ 3⬘ =
Polarization vectors for minimizing css / ccs in four-beam single exposure configuration
2 ⫻ 1.6
2 ⫻ 1.6 2 ⫻ 1.6 2 ⫻ 1.6
冋
Incident Beams in Air 1.0 1
1 ,
1 ,
冑3 冑3 冑3
kជ 0 =
关0.856,0.366,0.366兴
kជ 1 =
关0.366,0.856,0.366兴
kជ 2 =
关0.366,0.366,0.856兴
kជ 3 =
ជ 0⬘ = E0关circular polarization兴 E ជ 1⬘ = ES关0.073+ i0.512, −0.090− i0.569, −0.081 E − i0.628兴 ជ 2⬘ = Es关−0.081− i0.628, 0.073+ i0.512, −0.090 E − i0.569兴 ជ 3⬘ = Es关−0.090− i0.569, −0.081− i0.628, 0.073 E + i0.512兴
plied the fluorescing alignment disk (FAD, Thorlabs), which had a small hole at the center, to determine the position of the rotating center of the photoresist film (point O in Fig. 4). Next, we carefully marked down the projection of the sample surface, its rotating center, and the pivot axis of the rotational stage, which is perpendicular to the sample surface and passing through the rotating center, on the optical table. The projection of each incident beam path OBi can be determined by the rightangled triangular OABi with 兩ABi兩 = 兩OA兩tan i 共i = 1 , 2兲. The two apertures were placed on the paths of the incident beams with their centers at the same height of the rotating center. Finally, we aligned the two beams to pass through the corresponding aperture center and the FAD center.
册
2 2 2
2
冋
1
1 ,
1 ,
冑3 冑3 冑3
册
关0.966,0.183,0.183兴 关0.183,0.966,0.183兴 关0.183,0.183,0.966兴
ជ 0 = 1.3E0关circular polarization兴 E ជ 1 = 1.396Es关0.036+ i0.255, −0.101− i0.645, −0.092 E − i0.706兴 ជ 2 = 1.396Es关0.092+ i0.706, 0.036− i0.255, −0.101 E − i0.645兴 ជ 3 = 1.396Es关−0.101− i0.645, −0.092− i0.706, 0.036 E + i0.255兴
We roughly estimated the accuracy of the experimental beam angles in comparison to the theoretical design. It is reasonable to assume that the misalignment of incident beams 兩⌬s兩 ⬍ 1 mm. Thus, the deviation of the beam angle was estimated as 兩⌬i兩 = arctan共兩⌬s兩 / D兲 ⬵ 0.06° for D = 1 m, where D is the distance between the aperture and sample stage. A larger value of D would lead to higher accuracy of the incident angle. The current control of the incident beam alignment is sufficiently accurate for fabrication of SU-8 diamond-like structures with relatively small lattice constants. As seen in Fig. 2, further study of the robustness of the optical setup against incident angle deviation will be necessary for fabrication of larger size lattices and/or in photoresist with larger shrinkage.
ACKNOWLEDGMENT This work is supported by the Office of Naval Research (ONR) Grant No. N00014-05-0303.
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