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Method to fabricate orthogonal crossed gratings based on a dual Lloyd's mirror interferometer
Optics Communications 360 (2016) 68–72
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Method to fabricate orthogonal crossed gratings based on a dual Lloyd's mirror interferometer Hengyan Zhou, Lijiang Zeng n State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Tsinghua University, Beijing 100084, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 August 2015 Received in revised form 26 September 2015 Accepted 4 October 2015
We propose a dual Lloyd's mirror interferometer with two mutually perpendicular mirrors to fabricate orthogonal crossed gratings through a single exposure. Theoretical analysis shows that in this interferometer the angle between two main periodic directions of the crossed grating is solely determined by the normal directions of the two Lloyd's mirrors and the substrate. To verify this theoretical prediction, four groups of crossed gratings were fabricated. The measurement results agree well with the theoretical relationship. & 2015 Elsevier B.V. All rights reserved.
Keywords: Crossed grating Lloyd's mirror interferometer Orthogonality error Microstructure fabrication Holographic interferometry
1. Introduction A planar encoder can be used to measure two-dimensional (2D) displacements with nanometer resolution [1,2]. Because of the compact structure, the planar encoder is immune to the environmental disturbance and less vulnerable to Abbe errors, thus having a broad range of application prospect in the field of semiconductor manufacturing [3] and precision machining [4]. The key component in a planar encoder is a crossed grating, i.e., a grating that is periodic in two directions. Crossed gratings have also been used for evaluating the 2D contouring errors of machine tools [5], or calibrating the lateral scales of many kinds of microscopes [6,7] such as the scanning electron microscope (SEM) and the atomic force microscope (AFM). The angle between the two periodic directions is an important technical parameter for a crossed grating. For example, if this angle deviates from 90°, cross-talk error will be caused during the measurement in a planar encoder [8]. Therefore, it is necessary to study the factors influencing the angle between the two periodic directions during the fabrication process. Interference lithography is a traditional method for fabricating crossed gratings. Recently, both two-beam interference technique [9,10] and multiple-beam interference technique [11–14] have been investigated theoretically and experimentally. In the twobeam interference technique, a crossed grating can be made by rotating the substrate between two sequential exposures [15,16]; n
Corresponding author. E-mail address: [email protected] (L. Zeng).
http://dx.doi.org/10.1016/j.optcom.2015.10.017 0030-4018/& 2015 Elsevier B.V. All rights reserved.
while the multiple-beam interference technique brings convenience because a crossed grating can be produced through a single exposure [14]. The Lloyd's mirror interferometer system has distinct advantages such as simple configuration, fast alignment and high flexibility in adjusting grating period, and thus is a typical setup for the multiple-beam interference technique. Several different configurations of multi-beam Lloyd's mirror interferometers were proposed recently for fabricating two-dimensional periodic structures with a single exposure. Li et al. [17] presented a two-axis Lloyd's mirror interferometer to fabricate crossed gratings, in which the angles between the substrate and two Lloyd's mirrors are both larger than 90°. Boor et al. [18] created hexagonal hole/dot arrays by a modified Lloyd's mirror interferometer where the two Lloyd's mirrors are at an angle of 120° and both perpendicular to the substrate. Vala et al. [19] proposed a corner reflector-like interferometer formed by two Lloyd's mirrors and a substrate to fabricate periodic plasmonic arrays with rectangular symmetry. However, the above researches did not focus on the relationship between the system configuration and the angle of two periodic directions for the crossed grating. In this paper, we present a dual Lloyd's mirror interferometer system with two mutually perpendicular mirrors to fabricate orthogonal crossed gratings. This system allows fabricating a crossed grating with a single exposure. The factors influencing the angle between the two periodic directions of the crossed grating are analyzed in detail and a theoretical formula is given that shows the angle of two periodic directions is solely determined by the normal directions of the two Lloyd's mirrors and the substrate.
H. Zhou, L. Zeng / Optics Communications 360 (2016) 68–72
Experimental results agree well with the theoretical prediction. Our formula is applicable to not only our system but also the systems of Refs. [17–19].
69
bˆ 2
κˆ 2 bˆ 1
2. Exposure system The dual Lloyd's mirror interferometer is depicted in Fig. 1(a). A linearly polarized laser beam is divided into two beams by the polarization beam splitter PBS1 and one of them is directed by the mirror M0. The two beams are cleaned up by the spatial filters SF1 and SF2, collimated by the lenses L1 and L2, limited to have rectangular sectional profiles by the optical diaphragms D1 and D2, reflected by the mirrors M1 and M2, and finally form the collimated exposure beams I1 and I2. The half-wave plate WP0 is used for adjusting the intensity ratio between I1 and I2, while the halfwave plates WP1 and WP2 are used for adjusting the beams' polarization directions. The Lloyd's mirrors R1, R2, and the substrate S are perpendicular to each other. The path WPi-SFi-Li-DiMi-(Ri/S) forms a Lloyd's mirror interferometer, where i¼1,2. The white dash–dot line and dotted line represent the optical axes of the two paths, and they go through the intersection lines of the planar surfaces of Ri and S. Half of beam I1 meets S directly, while the other half is reflected by R1 before reaching S, as shown in Fig. 1(b). The two parts interfere with each other and produce a set of line fringes [red lines in Fig. 1(b)] on the surface of S. Similarly, beam I2 is projected onto R2 and S, forming another set of line fringes on S (approximately perpendicular to the first set, not shown in the figure). Suppose there is no interference between I1 and I2 in Fig. 1. The photoresist-coated substrate S is exposed by the two sets of line fringes, and then a crossed grating is generated as shown in Fig. 2 (a), where the gray dots form the lattice of the crossed grating; the red lines and green lines represent two sets of grating grooves ^ ^ produced by beams I1 and I2 respectively; b1 and b2 are unit vectors along the two periodic directions whose included angle is α. Fig. 2(b) is the corresponding reciprocal lattice, where κ^1 and κ^2 are unit vectors along the grating vectors K1 and K2 respectively, and their included angle ω is supplementary to the angle α,
Fig. 2. The real space lattice (a) and reciprocal space lattice (b) of a crossed grating.
namely
ω ¼180° α.
3. Theoretical analysis Theoretical calculation for the value of angle α is performed as ^, n ^ as the unit normal vectors of R1, ^ , and n follows. We define n 1 2 R2, and S, respectively, as shown in Fig. 1(b). The beam Ii is considered as a perfect plane wave and the surface of Ri is considered as an ideal plane. The wave vectors of beam Ii and its reflected beam after Ri are respectively denoted by ki and kir (i¼1,2). The relationship between them is
^ )n ^ k ir = k i − 2(k i⋅n i i.
PBS1
WP0 M0
WP1
L1
D1
SF1
M1
WP2 SF2 L2
I2
D2
M2
R2
R1 S
R2
R1
nˆ 2
nˆ 1
k1r nˆ
I1
I1
k1
S Fig. 1. Optical setup for fabricating crossed gratings. (a) The dual Lloyd's mirror interferometer. (b) Layout of the Lloyd's mirrors and the substrate.
(1)
Denote Δki as the difference between ki and kir. The equiphase plane of the interference field in space produced by Ii and its reflected beam is
^ )n ^ Δk i⋅r = 2(k i⋅n i i⋅r = φ = const,
(2)
where r is the position vector of an arbitrary point and φ is the phase at that point. Δki is perpendicular to the equiphase plane, and its projection on the substrate surface is the grating vector, namely
^ × Δk × n ^ = 2(k ⋅n ^ ^ ^ ^ ^ Ki = n i i i)[n i − (n i⋅n)n]. Therefore, the angle
cos α = − cos ω =
Laser
κˆ 1
(3)
α is determined by
−K1⋅K2 = K1 K2
^ ⋅n ^ ^ ^ ^ ^ (n 1 )(n2⋅n) − n1⋅n2 . ^ ⋅n ^)2 1 − (n ^ ⋅n ^)2 1 − (n 1
(4)
2
The above equation implies that the angle α is independent of the incident wave vector ki and solely determined by the normal directions of the two Lloyd's mirrors and the substrate. In planar encoders, to decrease cross-talk errors during the measurement, orthogonal crossed gratings (i.e., α ¼ 90°) are needed, for which the value of (α 90°) is defined as the orthogonality ^ ⋅n ^=0 error and denoted by θ. Eq. (4) demonstrates that when n i ^ ⋅n ^ = 0, orthogonal crossed gratings can be ob(i¼1, or 2) and n 1
2
tained. For simplicity three variables u, v, and w are defined as ^ ⋅n ^ ^ ⋅n ^ = cos β , and w = n ^ ⋅n ^ = cos β , v = n u=n 1 2 = cos β0 , where 2 1 2 1 β0, β1, and β2 are the intersection angles between n^ , n^ , and n^ . For u « 1 and v « 1, Eq. (4) can be approximated as
⎛ 1 ⎞⎛ 1 ⎞ cos α = − sin θ ≈ (uv − w )⎜ 1 + u2⎟⎜ 1 + v2⎟. ⎝ 2 ⎠⎝ 2 ⎠
1
2
(5)
The above equation demonstrates that w has a first-order effect on sin θ, while the effects of uv, u2, and v2 are in second-order. If the accuracy of the orthogonality error θ is required to be 0.2″, those second-order terms can be negligible when |u| o 10 3 and | v| o 10 3, namely |90° βi| o206″ (i ¼1, 2). We refer to this condition as the attitude condition of the substrate S. When S meets the attitude condition, Eq. (5) can be approximated as sin θ Ew¼ cos β0, namely
70
θ ≈ 90∘ − β0 ,
H. Zhou, L. Zeng / Optics Communications 360 (2016) 68–72
(6)
which means the orthogonality error θ is complementary in algebraic sense to the angle between R1 and R2. Orthogonal crossed gratings can be obtained when R1 and R2 are perpendicular.
4. Experimental results We experimentally tested the above theoretical prediction. The exposure source was a He–Cd laser with 441.6 nm wavelength. The focal lengths of the lenses L1 and L2 were both 816 mm. During the experiment, two key adjusting steps were needed. One was to adjust and measure the angle between the two Lloyd's mirrors by an autocollimator; the other was to adjust the substrate to meet the attitude condition. First, an autocollimator (TA US 300-57 made by TRIOPTICS GmbH, 70.25″ measuring accuracy) was used to measure the angle between the two Lloyd's mirrors. As the combination of R1 and R2 can be regarded as a right-angle prism whose internal material was air, we chose the software's 90° Prism Error mode to measure the angle between R1 and R2. The measuring beam from the autocollimator illuminated R1 and R2 simultaneously, as Fig. 3 (a) shows. The upper and lower halves of the measuring beam were reflected and formed two crosshairs in the software's camera window, depicted in red and blue respectively in Fig. 3(b). The relative position between the two crosshairs was read by the software and it equaled two times of |90° β0|. If the two crosshairs cross or overlap, their relative position cannot be read correctly. So this autocollimator's measuring range in 90° Prism Error mode was from 600″ to 1505″, and we will verify Eq. (6) in this range. It should be noted that if different parts of the autocollimator's measuring beam were used, the measured results might be different because of the beam's wavefront aberration. For example, we chose the software's Mirror Tilt Angle mode to measure the tilt angle of a mirror M (with 0.05 λ reflection wavefront), as shown in Fig. 3(c). Using a beam block B to shield the upper or lower half of the measuring beam respectively, we found the measuring results had a difference of about 2″. This phenomenon indicated that the wavefront aberration of the measuring beam would result in measurement errors when different parts of the measuring beam were used. The measuring accuracy for |90° β0| in the case of Fig. 3(a) should be estimated in the order of 2″. Second, the substrate was adjusted to meet the attitude condition. For the system shown in Fig. 1(a), only when S is perpen^, n ^ , and k are coplanar, the exposure beam Ii dicular to Ri, and n i i can return antiparallel to the original direction after being reflected by Ri and S successively, and finally go back into the pinhole of SFi. By adjusting the substrate, we can make the distance
between the returned beam spot and the pinhole's center smaller than 0.5 mm. Given the focal length of Li, the angle βi can thus be ensured in the range of |90° βi| o63″ (i¼ 1, 2), which meets the attitude condition. The adjustment for the substrate was accomplished through a motorized six-axis aligner (8095 Motorized SixAxis Aligner made by Newport Corporation, 0.2 μrad angular resolution, not shown in the figure) under the substrate which can tilt it precisely. Before exposure, a substrate S0 without photoresist ^ was layer was put into the system and its normal direction n adjusted by the above method. A He–Ne laser beam with 632.8 nm wavelength fell onto the substrate S0 and the reflected beam spot's ^ direction of S0. Then we put position P was used to record the n the photoresist-coated substrate S to be exposed into the system ^ direction according to the spot position P. The and adjusted its n adjusting accuracy through the spot position can reach 50″, higher than 63″, so the attitude condition can still be met for S. The 632.8 nm wavelength was chosen because the photoresist was insensitive to it. After the above two adjusting steps, we made nine crossed gratings on the quartz substrates coated with a 80 nm-thick chromium film and a 175 nm-thick photoresist layer. The period of the crossed gratings was 0.57 μm and the size was 45 mm 45 mm. Fig. 4 shows the AFM image of one fabricated crossed grating. Each of the nine crossed gratings was fabricated under different β 0 values, and they were divided into four groups (two pieces in each group except the second group which includes three pieces) according to the β0 value. We utilized an interferometry method of high accuracy proposed by Feng et al. [8] to measure the orthogonality error of the crossed gratings. Every grating was measured six times and the average was taken as the final value of the orthogonality error θ, shown as the blue circles in Fig. 5. The circles in each group overlap because gratings of each group had nearly equal orthogonality error values (with no more than 10″ difference). The relation between θ and (90° β 0) was fitted with linear relationship. The fitted result shown as the blue line in Fig. 5 was θ ¼ 1.000006 (90° β 0 ) 1.70″, and the correlation coefficient was R2 ¼0.99999987. The residual errors, i.e., the differences between the measured orthogonality errors and the calculated orthogonality errors according to the fitted result, were smaller than 0.6″ (pink crosses in Fig. 5). Such a result matched well with Eq. (6), except the additional item 1.70″ which was probably introduced by the wavefront aberration of measuring beam from the autocollimator. An autocollimator with a better beam wavefront would solve this problem. Limited by the measuring range of the autocollimator, we cannot fabricate orthogonal crossed gratings in higher accuracy. However, if a goniometer with high precision is available, smaller orthogonality error can be achieved.
Fig. 3. Measurement method with the autocollimator. (a) Measurement for |90° β0|. The angle β0 is supposed to be near 90°, so the reflected beam is almost antiparallel to the incident beam. (b) Two crosshairs in the software's camera window. Their relative position can be read by the software. (c) Measurement for the mirror tilt using different parts of the measuring beam.
H. Zhou, L. Zeng / Optics Communications 360 (2016) 68–72
200 nm 100 4 µm 3
4
A
71
B
3
3
2
2
1
1
1
A
4
B
0.8 0.6 0.4 0.2
2 0
0 1
1 2
3
4 µm
0
Fig. 4. AFM image of the fabricated crossed grating.
1
2
3
4
0
1
2
3
0
4
Fig. 6. Theoretically calculated intensity distribution of the interference patterns when two Lloyd's mirrors are at an angle of (a) 90° and (b) 85°. The two figures are plotted with the same color scale and normalization. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2 1
M1
2 1
3 M2
M1
3
2
1
4 5 3 M2
M2 M1
S
S
S
Fig. 7. Three systems consisting of two Lloyd's mirrors. (a) Two-axis Lloyd's mirror interferometer in Ref. [17]. (b) Three-beam Lloyd's mirror interferometer in Ref. [18]. (c) Corner reflector-like interferometer in Ref. [19].
Fig. 5. Experimental results drawn with two ordinates. The left blue ordinate and right pink ordinate represent the orthogonality error and residual error, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
5. Discussion In our system, if the polarization directions of beam Ii or the angle between the two Lloyd's mirrors are not appropriate, interference will occur between I1 and I2. The finally formed interference pattern may be complicated or even non-periodic. In order to ensure the periodicity, we limit our system to satisfy two conditions: the polarization direction of beam Ii is adjusted to be TE ^, n ^ , and k are coplanar. Under the above two conpolarized; n i i ditions, the holographically recorded pattern will always be periodic; however, rigorously speaking, a pattern recorded with two orthogonal Lloyd's mirrors will be drastically different from that recorded with two non-orthogonal Lloyd's mirrors, in not only the crossing angle but also the two periodic directions and the two periods. Fig. 6(a) shows the theoretically calculated interference pattern when the two Lloyd's mirrors are perfectly perpendicular (β0 ¼90°), in which case beams I1 and I2 do not interfere, and the minimum periodic unit is the dash–dot rectangle. Fig. 6(b) shows the theoretically calculated pattern when the two Lloyds' mirrors are not perpendicular (β0 ¼ 85°). Due to the interference of I1 and I2, there is a modulation of intensity maxima compared to case (a), so the minimum periodic unit is the dotted parallelogram. The intensity ratio between points B and A is (1 |cos β0|)/(1 þ|cos β0|). If the ratio is higher than 80%, namely |90° β0| o6.4°, the intensity difference between points A and B can be ignored and the solid parallelogram can be regarded as the minimum periodic unit whose inner angle α can still be calculated by Eq. (4). Therefore, in
our system |90° β0| o6.4° is regarded as the applicable condition for Eq. (4). Moreover, it should be noted that Eq. (4) is applicable to not only our system but also many other systems consisting of two Lloyd's mirrors, including systems of Refs. [17–19] as shown in Fig. 7. For the system in Ref. [17] [Fig. 7(a)], the angles between the substrate S and the Lloyd's mirrors M1 and M2 are both larger than 90°. The incident beam is divided into three sub-beams after polarization modulation. Sub-beam 1 is directly projected on S, and sub-beams 2 and 3 reach S after being reflected by M1 and M2, respectively. Due to the different polarization directions (shown as the red double-headed arrows), sub-beam 1 will interfere with sub-beams 2 and 3, and form two grating vectors; while subbeams 2 and 3 do not interfere with each other. Eq. (4) can be used for calculating the angle between the two periodic directions. In fact, Eq. (4) can be transformed into another form that is more suitable for Ref. [17]
cos α =
^ ×n ^)⋅(n ^ ×n ^) −(n 1 2 . ^ ⋅n ^ 2 1 − (n ^ ⋅n ^2 1 − (n 1 ) 2 )
(7)
The above equation implies that if the intersection lines of the substrate surface and two Lloyd's mirrors' surfaces are perpendicular to each other, the angle α will be 90° and orthogonal crossed gratings are produced. For the system in Ref. [18] [Fig. 7(b)], the two Lloyd's mirrors M1 and M2 are at an angle of 120°, and both perpendicular to the substrate S. The incident beam is divided into three sub-beams as well: sub-beam 1 is directly projected on S; sub-beams 2 and 3 reach S after being reflected by M1 and M2, respectively. Different from the case in Ref. [17], the three subbeams interfere with one another and form three grating vectors: K1 and K2 are grating vectors formed by the interference of subbeam 1 with sub-beams 2 and 3, respectively, while K3 is the
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H. Zhou, L. Zeng / Optics Communications 360 (2016) 68–72
grating vector formed by sub-beams 2 and 3. Their relation is K3 ¼K2 K1 according to Eq. (3), meaning that K3 does not alter the periodic distribution of the pattern formed by K1 and K2. Therefore the angle between two periodic directions of the crossed grating can be calculated by Eq. (4) and the result is 120°, matching with the hexagonal pattern presented in Ref. [18]. In the system of Ref. [19] [Fig. 7(c)], the incident beam is divided into five sub-beams: sub-beam 1 is directly projected onto the substrate S; sub-beams 2 and 3 reach S after being reflected by M1 and M2, respectively; sub-beams 4 and 5 are reflected by both M1 and M2 before reaching S. When the two Lloyd's mirrors are strictly perpendicular, sub-beams 4 and 5 will have the same wave vector after being reflected by both Lloyd's mirrors, but have different projection areas on the surface of S and different polarization directions. Hence there will exist two sets of four-beam (beams 1, 2, 3, 4 and beams 1, 2, 3, 5) interference patterns. Take the first set (beams 1, 2, 3, 4) for example. Four beams interfere with one another and form six grating vectors. Among them K1 and K2 are formed by the interference of sub-beam 1 with sub-beams 2 and 3, respectively. Similar to the case of Ref. [18], all the other four grating vectors can be expressed as the combination of K1 and K2. So Eq. (4) is still viable for the calculation of the angle between two periodic directions. However, if the two Lloyd's mirrors deviate from perpendicular, the other four grating vectors will have more complicated expressions and result in non-periodic interference pattern; besides, sub-beams 4 and 5 will have different wave vectors after being reflected by both Lloyd's mirrors, and multiple types of multiple-beam (3-beam, 4-beam, and/or 5-beam) interference will occur simultaneously. In that case, Eq. (4) is inapplicable. In addition, the systems of Ref. [17] and Ref. [19] can be used to fabricate orthogonal crossed gratings. Compared to them, our system has some advantages. In Ref. [17] the two exposure beams for the same dimension have different incident angles, so the fabricated crossed gratings have asymmetrical groove profiles and asymmetrical diffraction efficiencies for positive and negative diffraction orders in each dimension. Our system avoids such a problem because the substrate is perpendicular to each Lloyd's mirror and the incident angles of the two exposure beams for the same dimension are identical. In Ref. [19], as described above, there exist two sets of different four-beam interference patterns even if the two Lloyd's mirrors are perfectly perpendicular, so it is difficult to get orthogonal crossed gratings with large regular area. For our system much larger regular size can be obtained because only a single type of interference pattern exists.
6. Conclusion We have designed and set up a dual Lloyd's mirror interferometer as the exposure system, through which a crossed grating can be fabricated after a single exposure. Theoretical analysis shows the angle between the two periodic directions is solely determined by the normal directions of the two Lloyd's mirrors and the substrate. The theoretical formula is universal to many systems consisting of two Lloyd's mirrors. For orthogonal crossed gratings, the orthogonality error is complementary in algebraic sense to the angle between the two Lloyd's mirrors when
the substrate meets the attitude condition. Experimental results demonstrate the theoretical prediction and the residual errors are quite small. If a goniometer with high precision is used to measure the angle between the two Lloyd's mirrors, crossed gratings with small orthogonality error can be obtained.
Acknowledgments The authors thank Lifeng Li of our research group for sharing his thoughts on the theoretical aspects of this paper. This work was supported by the National Natural Science Foundation of China under Project no. 51427805.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Method to fabricate orthogonal crossed gratings based on a dual Lloyd's mirror interferometer Hengyan Zhou, Lijiang Zeng n State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Tsinghua University, Beijing 100084, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 August 2015 Received in revised form 26 September 2015 Accepted 4 October 2015
We propose a dual Lloyd's mirror interferometer with two mutually perpendicular mirrors to fabricate orthogonal crossed gratings through a single exposure. Theoretical analysis shows that in this interferometer the angle between two main periodic directions of the crossed grating is solely determined by the normal directions of the two Lloyd's mirrors and the substrate. To verify this theoretical prediction, four groups of crossed gratings were fabricated. The measurement results agree well with the theoretical relationship. & 2015 Elsevier B.V. All rights reserved.
Keywords: Crossed grating Lloyd's mirror interferometer Orthogonality error Microstructure fabrication Holographic interferometry
1. Introduction A planar encoder can be used to measure two-dimensional (2D) displacements with nanometer resolution [1,2]. Because of the compact structure, the planar encoder is immune to the environmental disturbance and less vulnerable to Abbe errors, thus having a broad range of application prospect in the field of semiconductor manufacturing [3] and precision machining [4]. The key component in a planar encoder is a crossed grating, i.e., a grating that is periodic in two directions. Crossed gratings have also been used for evaluating the 2D contouring errors of machine tools [5], or calibrating the lateral scales of many kinds of microscopes [6,7] such as the scanning electron microscope (SEM) and the atomic force microscope (AFM). The angle between the two periodic directions is an important technical parameter for a crossed grating. For example, if this angle deviates from 90°, cross-talk error will be caused during the measurement in a planar encoder [8]. Therefore, it is necessary to study the factors influencing the angle between the two periodic directions during the fabrication process. Interference lithography is a traditional method for fabricating crossed gratings. Recently, both two-beam interference technique [9,10] and multiple-beam interference technique [11–14] have been investigated theoretically and experimentally. In the twobeam interference technique, a crossed grating can be made by rotating the substrate between two sequential exposures [15,16]; n
Corresponding author. E-mail address: [email protected] (L. Zeng).
http://dx.doi.org/10.1016/j.optcom.2015.10.017 0030-4018/& 2015 Elsevier B.V. All rights reserved.
while the multiple-beam interference technique brings convenience because a crossed grating can be produced through a single exposure [14]. The Lloyd's mirror interferometer system has distinct advantages such as simple configuration, fast alignment and high flexibility in adjusting grating period, and thus is a typical setup for the multiple-beam interference technique. Several different configurations of multi-beam Lloyd's mirror interferometers were proposed recently for fabricating two-dimensional periodic structures with a single exposure. Li et al. [17] presented a two-axis Lloyd's mirror interferometer to fabricate crossed gratings, in which the angles between the substrate and two Lloyd's mirrors are both larger than 90°. Boor et al. [18] created hexagonal hole/dot arrays by a modified Lloyd's mirror interferometer where the two Lloyd's mirrors are at an angle of 120° and both perpendicular to the substrate. Vala et al. [19] proposed a corner reflector-like interferometer formed by two Lloyd's mirrors and a substrate to fabricate periodic plasmonic arrays with rectangular symmetry. However, the above researches did not focus on the relationship between the system configuration and the angle of two periodic directions for the crossed grating. In this paper, we present a dual Lloyd's mirror interferometer system with two mutually perpendicular mirrors to fabricate orthogonal crossed gratings. This system allows fabricating a crossed grating with a single exposure. The factors influencing the angle between the two periodic directions of the crossed grating are analyzed in detail and a theoretical formula is given that shows the angle of two periodic directions is solely determined by the normal directions of the two Lloyd's mirrors and the substrate.
H. Zhou, L. Zeng / Optics Communications 360 (2016) 68–72
Experimental results agree well with the theoretical prediction. Our formula is applicable to not only our system but also the systems of Refs. [17–19].
69
bˆ 2
κˆ 2 bˆ 1
2. Exposure system The dual Lloyd's mirror interferometer is depicted in Fig. 1(a). A linearly polarized laser beam is divided into two beams by the polarization beam splitter PBS1 and one of them is directed by the mirror M0. The two beams are cleaned up by the spatial filters SF1 and SF2, collimated by the lenses L1 and L2, limited to have rectangular sectional profiles by the optical diaphragms D1 and D2, reflected by the mirrors M1 and M2, and finally form the collimated exposure beams I1 and I2. The half-wave plate WP0 is used for adjusting the intensity ratio between I1 and I2, while the halfwave plates WP1 and WP2 are used for adjusting the beams' polarization directions. The Lloyd's mirrors R1, R2, and the substrate S are perpendicular to each other. The path WPi-SFi-Li-DiMi-(Ri/S) forms a Lloyd's mirror interferometer, where i¼1,2. The white dash–dot line and dotted line represent the optical axes of the two paths, and they go through the intersection lines of the planar surfaces of Ri and S. Half of beam I1 meets S directly, while the other half is reflected by R1 before reaching S, as shown in Fig. 1(b). The two parts interfere with each other and produce a set of line fringes [red lines in Fig. 1(b)] on the surface of S. Similarly, beam I2 is projected onto R2 and S, forming another set of line fringes on S (approximately perpendicular to the first set, not shown in the figure). Suppose there is no interference between I1 and I2 in Fig. 1. The photoresist-coated substrate S is exposed by the two sets of line fringes, and then a crossed grating is generated as shown in Fig. 2 (a), where the gray dots form the lattice of the crossed grating; the red lines and green lines represent two sets of grating grooves ^ ^ produced by beams I1 and I2 respectively; b1 and b2 are unit vectors along the two periodic directions whose included angle is α. Fig. 2(b) is the corresponding reciprocal lattice, where κ^1 and κ^2 are unit vectors along the grating vectors K1 and K2 respectively, and their included angle ω is supplementary to the angle α,
Fig. 2. The real space lattice (a) and reciprocal space lattice (b) of a crossed grating.
namely
ω ¼180° α.
3. Theoretical analysis Theoretical calculation for the value of angle α is performed as ^, n ^ as the unit normal vectors of R1, ^ , and n follows. We define n 1 2 R2, and S, respectively, as shown in Fig. 1(b). The beam Ii is considered as a perfect plane wave and the surface of Ri is considered as an ideal plane. The wave vectors of beam Ii and its reflected beam after Ri are respectively denoted by ki and kir (i¼1,2). The relationship between them is
^ )n ^ k ir = k i − 2(k i⋅n i i.
PBS1
WP0 M0
WP1
L1
D1
SF1
M1
WP2 SF2 L2
I2
D2
M2
R2
R1 S
R2
R1
nˆ 2
nˆ 1
k1r nˆ
I1
I1
k1
S Fig. 1. Optical setup for fabricating crossed gratings. (a) The dual Lloyd's mirror interferometer. (b) Layout of the Lloyd's mirrors and the substrate.
(1)
Denote Δki as the difference between ki and kir. The equiphase plane of the interference field in space produced by Ii and its reflected beam is
^ )n ^ Δk i⋅r = 2(k i⋅n i i⋅r = φ = const,
(2)
where r is the position vector of an arbitrary point and φ is the phase at that point. Δki is perpendicular to the equiphase plane, and its projection on the substrate surface is the grating vector, namely
^ × Δk × n ^ = 2(k ⋅n ^ ^ ^ ^ ^ Ki = n i i i)[n i − (n i⋅n)n]. Therefore, the angle
cos α = − cos ω =
Laser
κˆ 1
(3)
α is determined by
−K1⋅K2 = K1 K2
^ ⋅n ^ ^ ^ ^ ^ (n 1 )(n2⋅n) − n1⋅n2 . ^ ⋅n ^)2 1 − (n ^ ⋅n ^)2 1 − (n 1
(4)
2
The above equation implies that the angle α is independent of the incident wave vector ki and solely determined by the normal directions of the two Lloyd's mirrors and the substrate. In planar encoders, to decrease cross-talk errors during the measurement, orthogonal crossed gratings (i.e., α ¼ 90°) are needed, for which the value of (α 90°) is defined as the orthogonality ^ ⋅n ^=0 error and denoted by θ. Eq. (4) demonstrates that when n i ^ ⋅n ^ = 0, orthogonal crossed gratings can be ob(i¼1, or 2) and n 1
2
tained. For simplicity three variables u, v, and w are defined as ^ ⋅n ^ ^ ⋅n ^ = cos β , and w = n ^ ⋅n ^ = cos β , v = n u=n 1 2 = cos β0 , where 2 1 2 1 β0, β1, and β2 are the intersection angles between n^ , n^ , and n^ . For u « 1 and v « 1, Eq. (4) can be approximated as
⎛ 1 ⎞⎛ 1 ⎞ cos α = − sin θ ≈ (uv − w )⎜ 1 + u2⎟⎜ 1 + v2⎟. ⎝ 2 ⎠⎝ 2 ⎠
1
2
(5)
The above equation demonstrates that w has a first-order effect on sin θ, while the effects of uv, u2, and v2 are in second-order. If the accuracy of the orthogonality error θ is required to be 0.2″, those second-order terms can be negligible when |u| o 10 3 and | v| o 10 3, namely |90° βi| o206″ (i ¼1, 2). We refer to this condition as the attitude condition of the substrate S. When S meets the attitude condition, Eq. (5) can be approximated as sin θ Ew¼ cos β0, namely
70
θ ≈ 90∘ − β0 ,
H. Zhou, L. Zeng / Optics Communications 360 (2016) 68–72
(6)
which means the orthogonality error θ is complementary in algebraic sense to the angle between R1 and R2. Orthogonal crossed gratings can be obtained when R1 and R2 are perpendicular.
4. Experimental results We experimentally tested the above theoretical prediction. The exposure source was a He–Cd laser with 441.6 nm wavelength. The focal lengths of the lenses L1 and L2 were both 816 mm. During the experiment, two key adjusting steps were needed. One was to adjust and measure the angle between the two Lloyd's mirrors by an autocollimator; the other was to adjust the substrate to meet the attitude condition. First, an autocollimator (TA US 300-57 made by TRIOPTICS GmbH, 70.25″ measuring accuracy) was used to measure the angle between the two Lloyd's mirrors. As the combination of R1 and R2 can be regarded as a right-angle prism whose internal material was air, we chose the software's 90° Prism Error mode to measure the angle between R1 and R2. The measuring beam from the autocollimator illuminated R1 and R2 simultaneously, as Fig. 3 (a) shows. The upper and lower halves of the measuring beam were reflected and formed two crosshairs in the software's camera window, depicted in red and blue respectively in Fig. 3(b). The relative position between the two crosshairs was read by the software and it equaled two times of |90° β0|. If the two crosshairs cross or overlap, their relative position cannot be read correctly. So this autocollimator's measuring range in 90° Prism Error mode was from 600″ to 1505″, and we will verify Eq. (6) in this range. It should be noted that if different parts of the autocollimator's measuring beam were used, the measured results might be different because of the beam's wavefront aberration. For example, we chose the software's Mirror Tilt Angle mode to measure the tilt angle of a mirror M (with 0.05 λ reflection wavefront), as shown in Fig. 3(c). Using a beam block B to shield the upper or lower half of the measuring beam respectively, we found the measuring results had a difference of about 2″. This phenomenon indicated that the wavefront aberration of the measuring beam would result in measurement errors when different parts of the measuring beam were used. The measuring accuracy for |90° β0| in the case of Fig. 3(a) should be estimated in the order of 2″. Second, the substrate was adjusted to meet the attitude condition. For the system shown in Fig. 1(a), only when S is perpen^, n ^ , and k are coplanar, the exposure beam Ii dicular to Ri, and n i i can return antiparallel to the original direction after being reflected by Ri and S successively, and finally go back into the pinhole of SFi. By adjusting the substrate, we can make the distance
between the returned beam spot and the pinhole's center smaller than 0.5 mm. Given the focal length of Li, the angle βi can thus be ensured in the range of |90° βi| o63″ (i¼ 1, 2), which meets the attitude condition. The adjustment for the substrate was accomplished through a motorized six-axis aligner (8095 Motorized SixAxis Aligner made by Newport Corporation, 0.2 μrad angular resolution, not shown in the figure) under the substrate which can tilt it precisely. Before exposure, a substrate S0 without photoresist ^ was layer was put into the system and its normal direction n adjusted by the above method. A He–Ne laser beam with 632.8 nm wavelength fell onto the substrate S0 and the reflected beam spot's ^ direction of S0. Then we put position P was used to record the n the photoresist-coated substrate S to be exposed into the system ^ direction according to the spot position P. The and adjusted its n adjusting accuracy through the spot position can reach 50″, higher than 63″, so the attitude condition can still be met for S. The 632.8 nm wavelength was chosen because the photoresist was insensitive to it. After the above two adjusting steps, we made nine crossed gratings on the quartz substrates coated with a 80 nm-thick chromium film and a 175 nm-thick photoresist layer. The period of the crossed gratings was 0.57 μm and the size was 45 mm 45 mm. Fig. 4 shows the AFM image of one fabricated crossed grating. Each of the nine crossed gratings was fabricated under different β 0 values, and they were divided into four groups (two pieces in each group except the second group which includes three pieces) according to the β0 value. We utilized an interferometry method of high accuracy proposed by Feng et al. [8] to measure the orthogonality error of the crossed gratings. Every grating was measured six times and the average was taken as the final value of the orthogonality error θ, shown as the blue circles in Fig. 5. The circles in each group overlap because gratings of each group had nearly equal orthogonality error values (with no more than 10″ difference). The relation between θ and (90° β 0) was fitted with linear relationship. The fitted result shown as the blue line in Fig. 5 was θ ¼ 1.000006 (90° β 0 ) 1.70″, and the correlation coefficient was R2 ¼0.99999987. The residual errors, i.e., the differences between the measured orthogonality errors and the calculated orthogonality errors according to the fitted result, were smaller than 0.6″ (pink crosses in Fig. 5). Such a result matched well with Eq. (6), except the additional item 1.70″ which was probably introduced by the wavefront aberration of measuring beam from the autocollimator. An autocollimator with a better beam wavefront would solve this problem. Limited by the measuring range of the autocollimator, we cannot fabricate orthogonal crossed gratings in higher accuracy. However, if a goniometer with high precision is available, smaller orthogonality error can be achieved.
Fig. 3. Measurement method with the autocollimator. (a) Measurement for |90° β0|. The angle β0 is supposed to be near 90°, so the reflected beam is almost antiparallel to the incident beam. (b) Two crosshairs in the software's camera window. Their relative position can be read by the software. (c) Measurement for the mirror tilt using different parts of the measuring beam.
H. Zhou, L. Zeng / Optics Communications 360 (2016) 68–72
200 nm 100 4 µm 3
4
A
71
B
3
3
2
2
1
1
1
A
4
B
0.8 0.6 0.4 0.2
2 0
0 1
1 2
3
4 µm
0
Fig. 4. AFM image of the fabricated crossed grating.
1
2
3
4
0
1
2
3
0
4
Fig. 6. Theoretically calculated intensity distribution of the interference patterns when two Lloyd's mirrors are at an angle of (a) 90° and (b) 85°. The two figures are plotted with the same color scale and normalization. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2 1
M1
2 1
3 M2
M1
3
2
1
4 5 3 M2
M2 M1
S
S
S
Fig. 7. Three systems consisting of two Lloyd's mirrors. (a) Two-axis Lloyd's mirror interferometer in Ref. [17]. (b) Three-beam Lloyd's mirror interferometer in Ref. [18]. (c) Corner reflector-like interferometer in Ref. [19].
Fig. 5. Experimental results drawn with two ordinates. The left blue ordinate and right pink ordinate represent the orthogonality error and residual error, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
5. Discussion In our system, if the polarization directions of beam Ii or the angle between the two Lloyd's mirrors are not appropriate, interference will occur between I1 and I2. The finally formed interference pattern may be complicated or even non-periodic. In order to ensure the periodicity, we limit our system to satisfy two conditions: the polarization direction of beam Ii is adjusted to be TE ^, n ^ , and k are coplanar. Under the above two conpolarized; n i i ditions, the holographically recorded pattern will always be periodic; however, rigorously speaking, a pattern recorded with two orthogonal Lloyd's mirrors will be drastically different from that recorded with two non-orthogonal Lloyd's mirrors, in not only the crossing angle but also the two periodic directions and the two periods. Fig. 6(a) shows the theoretically calculated interference pattern when the two Lloyd's mirrors are perfectly perpendicular (β0 ¼90°), in which case beams I1 and I2 do not interfere, and the minimum periodic unit is the dash–dot rectangle. Fig. 6(b) shows the theoretically calculated pattern when the two Lloyds' mirrors are not perpendicular (β0 ¼ 85°). Due to the interference of I1 and I2, there is a modulation of intensity maxima compared to case (a), so the minimum periodic unit is the dotted parallelogram. The intensity ratio between points B and A is (1 |cos β0|)/(1 þ|cos β0|). If the ratio is higher than 80%, namely |90° β0| o6.4°, the intensity difference between points A and B can be ignored and the solid parallelogram can be regarded as the minimum periodic unit whose inner angle α can still be calculated by Eq. (4). Therefore, in
our system |90° β0| o6.4° is regarded as the applicable condition for Eq. (4). Moreover, it should be noted that Eq. (4) is applicable to not only our system but also many other systems consisting of two Lloyd's mirrors, including systems of Refs. [17–19] as shown in Fig. 7. For the system in Ref. [17] [Fig. 7(a)], the angles between the substrate S and the Lloyd's mirrors M1 and M2 are both larger than 90°. The incident beam is divided into three sub-beams after polarization modulation. Sub-beam 1 is directly projected on S, and sub-beams 2 and 3 reach S after being reflected by M1 and M2, respectively. Due to the different polarization directions (shown as the red double-headed arrows), sub-beam 1 will interfere with sub-beams 2 and 3, and form two grating vectors; while subbeams 2 and 3 do not interfere with each other. Eq. (4) can be used for calculating the angle between the two periodic directions. In fact, Eq. (4) can be transformed into another form that is more suitable for Ref. [17]
cos α =
^ ×n ^)⋅(n ^ ×n ^) −(n 1 2 . ^ ⋅n ^ 2 1 − (n ^ ⋅n ^2 1 − (n 1 ) 2 )
(7)
The above equation implies that if the intersection lines of the substrate surface and two Lloyd's mirrors' surfaces are perpendicular to each other, the angle α will be 90° and orthogonal crossed gratings are produced. For the system in Ref. [18] [Fig. 7(b)], the two Lloyd's mirrors M1 and M2 are at an angle of 120°, and both perpendicular to the substrate S. The incident beam is divided into three sub-beams as well: sub-beam 1 is directly projected on S; sub-beams 2 and 3 reach S after being reflected by M1 and M2, respectively. Different from the case in Ref. [17], the three subbeams interfere with one another and form three grating vectors: K1 and K2 are grating vectors formed by the interference of subbeam 1 with sub-beams 2 and 3, respectively, while K3 is the
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H. Zhou, L. Zeng / Optics Communications 360 (2016) 68–72
grating vector formed by sub-beams 2 and 3. Their relation is K3 ¼K2 K1 according to Eq. (3), meaning that K3 does not alter the periodic distribution of the pattern formed by K1 and K2. Therefore the angle between two periodic directions of the crossed grating can be calculated by Eq. (4) and the result is 120°, matching with the hexagonal pattern presented in Ref. [18]. In the system of Ref. [19] [Fig. 7(c)], the incident beam is divided into five sub-beams: sub-beam 1 is directly projected onto the substrate S; sub-beams 2 and 3 reach S after being reflected by M1 and M2, respectively; sub-beams 4 and 5 are reflected by both M1 and M2 before reaching S. When the two Lloyd's mirrors are strictly perpendicular, sub-beams 4 and 5 will have the same wave vector after being reflected by both Lloyd's mirrors, but have different projection areas on the surface of S and different polarization directions. Hence there will exist two sets of four-beam (beams 1, 2, 3, 4 and beams 1, 2, 3, 5) interference patterns. Take the first set (beams 1, 2, 3, 4) for example. Four beams interfere with one another and form six grating vectors. Among them K1 and K2 are formed by the interference of sub-beam 1 with sub-beams 2 and 3, respectively. Similar to the case of Ref. [18], all the other four grating vectors can be expressed as the combination of K1 and K2. So Eq. (4) is still viable for the calculation of the angle between two periodic directions. However, if the two Lloyd's mirrors deviate from perpendicular, the other four grating vectors will have more complicated expressions and result in non-periodic interference pattern; besides, sub-beams 4 and 5 will have different wave vectors after being reflected by both Lloyd's mirrors, and multiple types of multiple-beam (3-beam, 4-beam, and/or 5-beam) interference will occur simultaneously. In that case, Eq. (4) is inapplicable. In addition, the systems of Ref. [17] and Ref. [19] can be used to fabricate orthogonal crossed gratings. Compared to them, our system has some advantages. In Ref. [17] the two exposure beams for the same dimension have different incident angles, so the fabricated crossed gratings have asymmetrical groove profiles and asymmetrical diffraction efficiencies for positive and negative diffraction orders in each dimension. Our system avoids such a problem because the substrate is perpendicular to each Lloyd's mirror and the incident angles of the two exposure beams for the same dimension are identical. In Ref. [19], as described above, there exist two sets of different four-beam interference patterns even if the two Lloyd's mirrors are perfectly perpendicular, so it is difficult to get orthogonal crossed gratings with large regular area. For our system much larger regular size can be obtained because only a single type of interference pattern exists.
6. Conclusion We have designed and set up a dual Lloyd's mirror interferometer as the exposure system, through which a crossed grating can be fabricated after a single exposure. Theoretical analysis shows the angle between the two periodic directions is solely determined by the normal directions of the two Lloyd's mirrors and the substrate. The theoretical formula is universal to many systems consisting of two Lloyd's mirrors. For orthogonal crossed gratings, the orthogonality error is complementary in algebraic sense to the angle between the two Lloyd's mirrors when
the substrate meets the attitude condition. Experimental results demonstrate the theoretical prediction and the residual errors are quite small. If a goniometer with high precision is used to measure the angle between the two Lloyd's mirrors, crossed gratings with small orthogonality error can be obtained.
Acknowledgments The authors thank Lifeng Li of our research group for sharing his thoughts on the theoretical aspects of this paper. This work was supported by the National Natural Science Foundation of China under Project no. 51427805.
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