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Littrow-type self-aligned laser encoder with high tolerance using double diffractions
Optics Communications 297 (2013) 89–97
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Littrow-type self-aligned laser encoder with high tolerance using double diffractions Chyan-Chyi Wu a, Cheng-Chih Hsu b,n, Ju-Yi Lee c, Yan-Zou Chen a,d, Jhih-Sheng Yang a a
Advanced Photonics Laboratory, Department of Mechanical and Electromechanical Engineering, Tamkang University, Taiwan Department of Photonics Engineering, Yuan-Ze University, Chungli 320, Taoyuan, Taiwan c Institute of Optomechanics, National Central University, Taiwan, R.O.C d Institute of Mechanical Engineering, National Taiwan University, Taiwan b
a r t i c l e i n f o
abstract
Article history: Received 4 September 2012 Received in revised form 15 January 2013 Accepted 17 January 2013 Available online 12 February 2013
This study proposes a Littrow-type self-aligned laser encoder system (LisaLENS) with double diffractions. The LisaLENS has simple optical paths, facilitating alignment and manufacture, and fewer optical components than current laser encoders. This study analyzes the theoretical measurement resolution of this system and verifies the feasibilities of its long- and short-range measurements through experiments. Experimental results show that the LisaLENS has a resolution of 5.5 7 1.4 nm. Thus, this design has wide applications in various fields, especially for monitoring the displacement and vibration of precision-motorized stages. Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved.
Keywords: Displacement Grating Long range Littrow
1. Introduction Laser encoders are not subjected to environmental disturbances because they transfer the measurement scale from the laser wavelength to the grating period. Laser encoders have played an important role in precise positioning systems used by semiconductor and liquid crystal display (LCD) manufacturers. Based on rapid advances in nanotechnology, the 25-nm node on 450 mm wafers will likely become commonplace in 2014 [1]. Two important factors determine if a laser encoder has wide applications. One is the runout tolerance between the optical head and the grating scale (GS), which is called the head-to-scale tolerance. Laser encoders inevitably suffer a serious runout problem, especially when they are applied to high through-put or high-speed manufacture of the coming node. The appearance of runout can induce a displacement measurement error, or the interference signals may suddenly vanish. The other is the tolerance of the grating period, which comes from the grating manufacture and the thermal variation when a laser encoder is operating. A tight tolerance of the grating period represents the manufacturing difficulty and high cost. The literature contains two ways to raise the head-to-scale tolerance of laser encoders. In our previous work [2], we adopted
n
Corresponding author. Tel.: þ886 3 4638800; fax: þ 886 3 4514281. E-mail addresses: [email protected], [email protected] (C.-C. Hsu).
a 1 x telescope configuration in a laser encoder to give it a high head-to-scale tolerance. In another previous work [3], we combined a 1 x telescope configuration and a perpendicular optical configuration in our laser encoder to give it both a high head-toscale tolerance and a high manufacture tolerance. The 1 x telescope can effectively handle in-plane and out-of-plane optical paths, so it is a good choice for raising head-to-scale tolerance. Kao et al. [4] presented a laser encoder with a grating in the Littrow configuration. That encoder had a maximum measurement error of 53 nm and repeatability within 720 nm in the 100 mm measurement range. Cheng and Fan proposed a laser encoder with a Littrow configuration [5]. Cheng’s encoder attained repeatability within 15 nm in a 15-mm travel range. Fan et al. [6] used two sets of Littrow configurations with two laser diode sources to construct a planar encoder for a planar stage displacement measurement. The Littrow configuration can retro-reflect the laser beam at the specific angle of incidence. However, the Littrow configuration for a laser encoder must be accompanied by specially arranged optics in order to have high head-to-scale tolerance. The works by Kao et al., Cheng et al., and Fan et al. used optics that can handle only in-plane optical paths. Thus, when the runout of roll or yaw happens, their laser encoder cannot correct the out-of-plane optical paths. To our knowledge, no studies have focused on raising the grating manufacture tolerance through the optical configuration of a laser encoder. The grating period tolerance generally is near 1% of the grating period [7]; that is, the grating manufacture
0030-4018/$ - see front matter Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.01.048
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tolerance is 16 nm for the grating scale with a grating period of 1.6 mm. This tolerance specification leads to a strict environmental condition for the grating manufacture. Also, expensive ultraprecision equipment is usually required for the grating manufacture, one example being the NanoRuler from MIT [8]. This study proposes a Littrow-type self-aligned laser encoder system (LisaLENS) with the least optics. The LisaLENS has an optical configuration with a corner cube (CC) and a beam expander (BE), so the LisaLENS has a high tolerance on both the head-to-scale runout and the grating period. It has simple optical paths, facilitating easy alignment or manufacture, so it is called a self-aligned laser encoder. The experiments in this study verify the feasibilities of long- and short-range measurements. Experimental results show that the LisaLENS has a resolution of 5.571.4 nm. Thus, the LisaLENS is suitable for long- and shortrange measurements from the nanometer to the millimeter scale.
2. Theory 2.1. Measurement principle Fig. 1 shows a schematic diagram of the LisaLENS. This study analyzes both paths of the þ2nd- and 2nd-order diffracted beams to explain the displacement measurement principle of the LisaLENS. The laser beam from a polarized laser source (LS) is expanded by a beam expander (BE). A polarizing beam splitter (PBS) divides the polarized light from the BE into two linearly polarized beams, called p-light, transmitted at the PBS, and s-light, reflected at the PBS. For the optical Arm 1, the p-light first passes through the quarter-wave plate (Q1) and becomes a circularly polarized beam. Then the circularly polarized beam is bent by the mirror (M1), incidents onto the grating scale (GS), and is diffracted by the GS. The 2nd-order diffracted beam goes back along the incident path (the dash path in Fig. 1). The 2nd-order diffracted beam is s-polarized after the second time it passes through the Q1. The s-polarized beam is reflected sequentially at the PBS, at the corner cube (CC) twice, and then at the PBS again, shown as the dash path in Fig. 1. After the s-light passes through the Q1 for the third time, it becomes a circularly polarized light. The circularly polarized beam incidents onto the GS again and is again diffracted as the 2nd-order diffracted beam. The second time diffracted light becomes p-light after passing through the Q1 for the fourth time. Thereafter, the p-light passes through the PBS and is directed to photodetectors (the PD1 and PD2 in Fig. 1). For the optical Arm 2, the s-light from the BE is reflected at the PBS, is
bent by the mirror M2, and goes along the solid path. After being diffracted into the þ2nd-order diffracted beam by the GS two times and passing through the Q2 four times, the s-light recovers to s-light and is reflected at the PBS. In summary, the CC can ensure that the p- and s-light directed to the Q3 have parallel optical paths (see Section 2.2 for details). Thereafter, the p-light and s-light traverse into a circular polarization interferometer, which consists of a quarter waveplate Q3, a beam splitter BS, and two polarizers P1 and P2. The interference fringe is captured by photodetectors PD1 and PD2, respectively. The operational principle of the LisaLENS is based on the Doppler effect of the moving GS. Because the LisaLENS adopts a Littrow-type optical path, the diffracted beam theoretically returns along the incident direction. The grating equation [9] indicates that the incident angle and the diffractive angle are equal, and can be written as follows: 9yL 9 ¼ 9yi 9 ¼ sin1
l
r
,
ð1Þ
where yi is the incident angle, yL is the diffractive angle for the Littrow-type optical path on the GS, l is the wavelength of the LS, and r is the grating period. Because of the Doppler effect of the moving GS, after double diffractions the corresponding angular frequency shifts of Arm 1 and Arm 2 (Fig. 1), Do1 and Do2, respectively, are as follows [10]:
Do1 ¼
Do2 ¼ þ
8pu
r
,
8pu
r
ð2Þ
,
ð3Þ
where u is the moving velocity of GS in the þ x-direction. From (2) and (3), one can realize that double diffractions can raise the measurement sensitivity. For the optical Arm 1, the laser beam follows the path PBS-Q1-M1-GS-M1-Q1-PBS-CCPBS-Q1-M1-GS-M1-Q1-PBS-Q3-BS. For the optical Arm 2, the laser beam follows the path PBS-Q2-M2-GSM2-Q2-PBS-CC-PBS-Q2-M2-GS-M2-Q2-PBS-Q3BS. After departing from the quarter waveplate Q3, the electric fields ! ! of these two optical waves from Arm 1 and Arm 2, E 1 and E 2 , can be written in the following form: ( ) 1 ! 1 E 1 ¼ pffiffiffi E10 ejðot þ Do1 tfl1 Þ , ð4Þ j 2 ! 1 E 2 ¼ pffiffiffi E20 ejðot þ Do2 tfl2 Þ 2
j 1
,
ð5Þ
! where E10 and E20 are the amplitudes of the electric fields E 1 and ! E 2 , respectively, o is the angular frequency of the laser source LS, fl1 and fl2 are the phase shifts generated pffiffiffiffiffiffiffi by the optical paths along Arm 1 and Arm 2, respectively, j¼ 1, and t is time. Eqs. (4) and ! ! (5) show that the electric fields E 1 and E 2 are right and left ! ! circularly polarized, respectively. In this case, E 1 and E 2 interfere with each other and pass through the beam splitter (BS). Finally, the photodetectors PD1 and PD2 measure the interference fields. The ! resulting electric fields entering photodetectors PD1 and PD2, E D1 ! and E D2 , respectively, can be represented as follows: ( ) ! E10 ejðot þ Do1 tfl1 Þ þ jE20 ejðot þ Do2 tfl2 Þ 1 E D1 ¼ pffiffiffi , ð6Þ 0 2 2
Fig. 1. Schematics of the LisaLENS and the experiment setup.
! 1 þj E D2 ¼ pffiffiffi 4 2
(
E10 ejðot þ Do1 tfl1 Þ þ E20 ejðot þ Do2 tfl2 Þ E10 ejðot þ Do1 tfl1 Þ þ E20 ejðot þ Do2 tfl2 Þ
) ,
ð7Þ
C.-C. Wu et al. / Optics Communications 297 (2013) 89–97
The intensities ID1 and ID2, measured by photodetectors PD1 and PD2, respectively, can be written as follows: o 1n 2 E10 þ E220 þ 2E10 E20 sin ð2DoÞtðfl2 fl1 Þ , ð8Þ ID1 ¼ 8 ID2 ¼
o 1n 2 E10 þ E220 þ 2E10 E20 cos ð2DoÞtðfl2 fl1 Þ , 8
ð9Þ
where Do2 ¼ Do1 ¼ Do. The proper arrangement on the optical path can lead to E10 ¼E20. Assume fl1 ¼ fl2 when there are no environmental disturbances. Thus, ID1 and ID2 can be reduced as follows: ID1 p1þ sin½ð2DoÞt,
ð10Þ
ID2 p1þ cos½ð2DoÞt:
ð11Þ
The phase change in ID1 and ID2 signals, F, comes from the Doppler frequency shift because of the GS movement. For an arbitrary small duration dt, the GS has a small displacement dx¼udt, and the corresponding phase changes dF ¼2Dodt. Eqs. (2) and (3) show the relationship between the phase change in ID1 and ID2, because of the displacement of GS dx, as follows: dF ¼
16p
r
dx,
ð12Þ
Consider that the coordinate system coincides with three ridges of a corner cube with a ridge length of a, as shown in Fig. 3. On the input/output plane, an incident beam enters the corner cube at a point A, is reflected sequentially at points B, C, D, and finally returns to a point A0 . In Fig. 3, the corresponding unit normal ! ! vectors for x y, y z, and x z planes are n 1 ¼ e^ 3 , n 2 ¼ e^ 1 , ! and n 3 ¼ e^ 2 respectively. Here e^ 1 , e^ 2 , and e^ 2 are the unit vectors of the x -, y -, and z-axes respectively. Assume that the unit ! direction vectors of optical paths AB, BC, CD, and DA0 are v 1 , ! ! ! v 2 , v 3 and v 4 respectively. By a simple vector analysis, we can obtain the relation: ! ! n iþ1 ¼ ! n i 2ð! n i U! n i Þ n i , i ¼ 1,2,3: ð14Þ Note that ‘‘U’’ denotes the inner product. Assume that ! v 1 ¼ a1 e^ 1 þ a2 e^ 2 þ a3 e^ 3 , where a1, a2, and a3 are arbitrary real ! numbers. From (14), we can have n 2 ¼ a1 e^ 1 þ a2 e^ 2 a3 e^ 3 . In ! the same way, we can easily have v 3 ¼ a1 e^ 1 þ a2 e^ 2 a3 e^ 3 , and ! ! ! v 4 ¼ a1 e^ 1 a2 e^ 2 a3 e^ 3 . As v 1 ¼ v 4 , we conclude that the incident and return beams are certainly parallel to each other for a corner cube. ! For an incident beam with an incident wavevector k in as shown in Fig. 4(a), the components of the diffracted wavevector !0 for a three-dimensional grating diffraction, k m (see Fig. 4(b)), can be written as follows:
or
r dF, dx ¼ 16p
0
ð13Þ
Based on (13) the grating displacement, dx can be determined by measuring the phase difference dF between the two sinusoidal signals if the grating period r is known. A period displacement produces a phase change of 16p; that is, the GS displaces r=8 for a cycle of interference signals. 2.2. Raising head-to-scale tolerance with the corner cube Fig. 2 shows the schematics of four types of runout for the laser encoder: pitch; roll; yaw; and stand-off. Physically, the appearance of runout may spoil the interference condition of the optical path for the laser encoder such that visibility is degraded, unexpected phase shift presents, or the interference signal suddenly vanishes. For the LisaLENS, a corner cube (the CC) is adopted to keep the two interference beams traveling along the same direction. Thus the head-to-scale tolerance can be effectively raised. A corner cube is commonly made of glass with three reflective planes perpendicular to each other, and an input/output plane.
Fig. 2. Schematics of four types of runout for the laser encoder: pitch; roll; yaw; and stand-off.
91
kx0 ,m ¼ 0
2p
l
sin yi Ucos f þml=r ,
2p
sin f, l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 kz0 ,m ¼ 1sin2 fðsin yi Ucos f þml=rÞ2 , l
ky0 ,m ¼
2p
ð15Þ
where m is the diffraction order, k0 x0 ,m, k0 y0 ,m, and k0 z0 ,m are x0 -, y0 !0 and z0 -components of a wavevector k m of the m-th order diffracted beam, yi is the angle of incidence in the x0 –z0 plane, ! and f is the included angle between k in and the x0 –z0 plane. Note that the sign convention for yi and f obeys the right-handed rule. Let us further study the runout effect on the LisaLENS. For convenience, consider a runoutless configuration of the LisaLENS in Fig. 5. The points z0i (i¼1, 2, 3, 4) are the corresponding incident points on the GS when there is no runout. z0 0i are the image points of the incidence points z0i on the GS for Arm 1 and Arm 2, where i¼1, 2, 3, 4. Applying the sine theorem in triangles A1z01B1, A2z02B2, A3z03B3 and A4z04B4 in Fig. 5, we can have the following geometrical relations: 0
Ai z0i ¼ Ai z0i ¼
A i Bi , sin 2yM þ cos 2yM
i ¼ 1, 2, 3, 4:
ð16Þ
Fig. 3. The coordinate system coincides with three ridges of a corner cube with a ridge length of a.
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Fig. 5. Schematics of the incidence points and their image points for runout analysis. For a runoutless case, z0 0i are the image points of the incidence points z0i on the GS for Arm 1 and Arm 2, where i¼ 1–4. The point 0 is the origin of the x–y–z coordinate system.
Fig. 4. (a) Schematics of a three-dimensional grating diffraction in x–y–z coordinate system. c represents the runout of yaw. (b) Schematics of a diffracted beam under three-dimensional grating diffraction in an x–y–z coordinate system.
where Ai are the incidence points on the M1 and the M2, Bi are the intersection points between the extended light rays and the grating surface, and z0 i are the image points of the incidence points zi on the GS under runout, where i¼ 1, 2, 3, 4. With the help of Eq. (16), we can analyze the runout tolerance using the image points of the incidence points on the GS. Thus, we will adopt a simplified model to do the first-order tolerance analysis on the head-to-scale runout of the LisaLENS in the following sections (see Figs. 6 and 7).
2.2.1. Stand-off or roll Firstly, f ¼0 and the x-axis coincides with the x0 -axis for the runout of stand-off or roll. From (15), we always have k0 x, 2 ¼ k0 x, þ 2, k0 y, 2 ¼k0 y, þ 2, and then k0 z, 2 ¼k0 z, þ 2 in Arm 1 and Arm 2 for each diffraction. As the designed incident points on the GS for the Arm 1 and Arm 2 are symmetric to the PBS polarization
Fig. 6. Schematics of the runout of pitch. The incident points on the GS for Arm 1 are z1 and z2. The incident points on the GS for Arm 2 are z3 and z4. The points z0 3 and z0 4 are the images of points z3 and z4 with respect to the M1, the points z0 1 and z0 2 are the images of points z1 and z2 with respect to the M2, and the points z00 3 and z00 4 are the images of points of the image points z0 3 and z0 4 with respect to the PBS polarization film plane (the y–z plane). Here we omit M1 and M2 for convenience.
film plane, all of the corresponding paths of Arm 1 and Arm 2 are also symmetric to the polarization film plane of the PBS. Thus the two interference beams always travel along the same optical path as the originally designed path. That is, the interference signals will essentially not be spoiled under the runout of stand-off or roll. The runout specification of stand-off or roll is constrained only by the field of view of the photodetector.
2.2.2. Pitch Secondly, consider the runout of pitch, as shown in Fig. 6. The point o is the origin of the x–y–z coordinate system. L21 is the distance of the point z0 1 to the PBS, L01 is the distance of the point z0 01 to the PBS, L11 is the distance of the point z0 3 to the PBS, L12 is the distance of the point z00 4 to the PBS, L02 is the distance of the
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where p is the runout of pitch. The transverse separation of interference beams can reduce the visibility of the interference fringe and the interference signal intensities. Assume that the beam diameter incident onto the GS is D. Let a ratio b ¼ Drunout =D represent the relative transverse separation of two interference beams, where runout is pitch. Note that 0 r b o1 for two-beam interference. In general, the smaller the b is, the better the interference signals. Accordingly, we can estimate the runout tolerance of pitch as follows: pf2L01 þ6Lcc þ2L02 g r bD:
ð18Þ
If the Lcc ¼25 mm, L01 ¼ 42 mm, and L02 ¼34 mm for a prototype of the LisaLENS, the estimated runout tolerance of pitch with b ¼ 0.1 and D ¼10 mm is about 0.211 or 751 arc-sec. This tolerance value is about 37 times greater than the value in our previous work (20 arc-sec with b ¼0.5 and D ¼2 mm) [2]. Note that the pitch runout of a laser encoder generally results from the yaw runout of a stage. As the optical paths of Arm 1 and Arm 2 remain symmetric to the runoutless path when pitch runout happens (see Fig. 6), the optical path difference (OPD) between these two arms will be zero within the CC. Similarly within the PBS, the optical paths of Arm 1 and Arm 2 are also symmetric to the runoutless optical path, so there is no OPD between these two arms in the PBS. We can align the center of the incident points on the GS of Arm 1 at the origin point o. Similarly, we can also align the center of the incident points on the GS of Arm 2 at the origin point o. In this way, the OPD between Arm 1 and Arm 2 can be effectively compensated when the pitch runout happens. Practically, the common pitch runout of the stage carrying the GS is smaller than 10 4 rad [11]. When the deviation of the center alignment of incident points at the origin point o is within 1 mm, we can safely have an OPD of about 0.1 nm. The OPD may produce an additional phase shift and then a displacement error of about 30 pm for the LisaLENS. A better alignment of the center of incident points will further slash the OPD effect from pitch runout. 2.2.3. Yaw Thirdly, consider the runout of yaw with its schematics, as shown in Fig. 4(b). From (15) and the coordinate transformation, we can have k0 x, þ 2 ¼ k0 x, 2, k0 y, þ 2 ¼ k0 y, 2, and k0 z, þ 2 ¼k0 z, 2 for the diffracted beams of Arm 1 and Arm 2. Accordingly, we can write down the absolute diffraction angles of the þ2nd (for Arm 2) and 2nd (for Arm 1) order diffracted beams in the x–z plane, 9q9, as follows: rsinyL 2lcosc 9q9 ¼ tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ð19Þ r2 cos2 yL 4lðlrcoscsinyL Þ
Fig. 7. Schematics of the runout of yaw: (a) optical path projection in x–z plane; (b) optical path projection out of x–z plane. y01, y02, y2, y3, and y4 are the incident points on the GS. l01, l02, and l22 are the distances from the PBS to y01, y02, and y3 respectively.
point z02 to the PBS, and L22 is the distance of the point z0 2 to the PBS. Lcc is the diameter of the CC. Dpitch is the transverse separation of two parallel interference beams. Note that we apply the first-order analysis to the runout of pitch so that the refraction effects on the Q1, Q2, and PBS can be neglected. We can calculate the transverse separation of two parallel interference beams, Dpitch, as follows:
Dpitch ffi pfL11 þ3Lcc þL12 gArm 1 þ pfL21 þ3Lcc þ L22 gArm 2 ffi pf2L01 þ6Lcc þ 2L02 g,
ð17Þ
where c is the runout of yaw (see Fig. 4). Similarly, the absolute diffraction angle of the þ 2nd- and 2nd-order diffracted beams, 9f9, can be written as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 2l2 þ 2l2 cosð2cÞ 1 : 9f9 ¼ cos ð20Þ r From Fig. 7, we can see that the diffraction angle 9q9 makes the interference beams displace along the z-axis, and the diffraction angle 9f9 makes the interference beams separated in the þy- and y-directions. Even though these two interference beams displace or separate, they stay parallel to each other under the runout of yaw. We can calculate the displacement and separation of the two interference beams, dyaw, and Dyaw respectively (see Fig. 7(a) and (b)), using the first-order analysis as follows: dyaw ffi 9yL -q9fl01 þ 3Lcc þ l22 g,
ð21Þ
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and
Dyaw ffi 9f9f2l01 þ6Lcc þ 2l22 g:
ð22Þ
Using (21) and (22), we can estimate the runout tolerance of yaw to constrain both dyaw and Dyaw to smaller than bD. If the Lcc ¼25 mm, l01 ¼42 mm, and l22 ¼34 mm for a prototype of the LisaLENS, the estimated runout tolerance of yaw with b ¼0.1 and D¼10 mm is about 0.251 (or 0.0044 rad). This tolerance value is about one-twelfth of the value in our previous work (31 with b ¼0.5 and D¼ 2 mm) [2]. Note that the yaw runout of a laser encoder generally results from the pitch runout of a stage. Generally, the pitch specification for a common low-cost linear stage is 710 4 rad [11]. Thus the yaw tolerance of the LisaLENS is practically enough. As the LisaLENS adopts a diode laser as the laser source, the path difference between two interference beams can be ensured within a few microns. Also, the optical paths of Arm 1 and Arm 2 remain symmetric to the z-axis when yaw runout happens. Thus, even though the optical paths of Arm 1 and Arm 2 change under yaw runout, the optical path difference (OPD) between the two interference beams will be zero theoretically. 2.3. Raising grating manufacture tolerance with beam expander For the LisaLENS, assume that the grating manufacture error for each grating period is ei. The laser spot has a diameter of D. The number of p the ffiffiffi grating period in such a laser spot, N, can be estimated as N ¼ 2D=r. The resultant measurement error for N grating lines, e^ , can be written as:
ei e^ ¼ pffiffiffiffi : N
Fig. 8. The fabrication process of the grating scale using the laser writer.
ð23Þ
Let the measurement range for the LisaLENS be R. The number of cycles of the interference signals for the whole range R can be estimated by ½8R=r, where [ ] is the Gaussian sign. Accordingly, the total displacement error which comes from the grating manufacture error, e, can be written as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uX re2 ð24Þ e ¼ t pffiffiffii , 2D i where i¼ 1 to ½8R=r. From (24), the larger the laser spot diameter is, the larger the grating manufacture tolerance can be for a given accuracy assessment for the grating period. Thus we can adopt a beam expander configuration to raise the grating manufacture tolerance. Fig. 9. The schematic layout of the home-built laser writer. M1, M2, and M3 are mirrors; NPBS: non-polarization beam splitter; CCD: charge coupled device.
3. Fabrication of grating scale The GS is a key component in a laser encoder because it serves as the measurement scale. The proposed device used a homemade laser writer to write the line pattern of the GS. Fig. 8 shows the fabrication process of the GS using the laser writer. Fig. 9 shows the schematic layout of the homemade laser writer. The laser writer adopted a 50 mW He–Cd laser with a 442 nm wavelength (KimmonTM IK5451R-E) as the lithography light source. An acousto-optical deflector (AOD) (IsometTM LS55 V with IsometTM D323B analog driver) made the laser beam locally scan the substrate surface. An acousto-optical intensity (AOI) modulator (IsometTM 1205C with IsometTM 232A-1 analog driver) controlled the laser beam intensity for lithography. An acousto-optical modulator (AOM) (IsometTM 1250C with IsometTM 225A-1 digital driver) served as a high-frequency switch for lithography. An objective (OlympusTM) with a long working distance was mounted in a computercontrolled autofocus system (model: LTAF 8000, Teletrac Inc.). The GS substrate was quartz glass with a thermal expansion of 5 10 7/1C. The substrate was held by a vacuum chunk carried
by an x–y air-bearing stage (U-Star Tech). The air-bearing stage was driven by x- and y-axis linear motors (KollmorgenTM Platinum DDL). To enable the accurate positioning of the x–y stage, x- and y-axis linear encoders (HeidenhainTM LIP401) sensed the corresponding displacements as the feedback signals. The travel ranges of the air-bearing stage in the x- and y-axes were 200 mm, achieving a positioning accuracy of 0.1 mm. The maximum velocity was 50 mm/s. The maximum acceleration was 2000 mm/s2. The AOD, AOI, AOM, and x–y air-bearing stage were controlled using an industrial computer through a turbo programmable multi-axis control card (Delta Tau Data System Inc.). The grating line pattern was inputted in bitmap format. The manufactured grating topography was measured using an atomic force microscope (AFM), and Fig. 10 shows the measurement results. The grating period was measured at nine different locations on the grating surface [12], and statistically the grating period value was 1.62470.034 mm. The grating was deposited gold film at 40 nm thick to have better reflection. The diffraction efficiency for the 2nd diffracted beam was about 8%.
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95
Fig. 10. AFM photograph of the manufactured grating scale.
4. Experiments and results All of the experiments in this study were performed at an atmospheric temperature, pressure, and relative humidity of 2272 1C, 101.32570.7 kPa, and (45710)% RH, respectively. The vibration amplitude was smaller than 0.25 mm for low frequency excitation from 0.1 Hz to 30 Hz, and the acceleration was smaller than 0.001 g for the excitation frequency from 30 Hz to 200 Hz. The displacement provided by a linear stage was measured to verify the feasibility of the proposed method. Fig. 1 shows the experimental configuration. The light source consisted of a linearly polarized laser diode with a wavelength at 650 nm (model: L650P007, Thorlabs). The GS was mounted on the precision linear stage to diffract the laser beams from the laser source. Two grating scales were used for experiments. For short-range experiments, we used the grating scale described in Section 3. For long-range experiments, we adopted a commercial grating scale with a grating period of 1.6 mm (manufactured by Taiwan Mask Cooperation). The commercial grating period tolerance was guaranteed within 2 nm. The experiments in this study adopted two types of linear stages for long- and short-range displacements. One was a motorized stage (model: SGSP26-200(X), SIGMA KOKI Inc.), for long-range experiments, and the other was a stage driven by a multilayer piezo stack actuator (model: PL055.30, Physik Instrumente), for short-range experiments. The displacement signals ID1 and ID2 were sent to a high-speed multichannel data acquisition card (model: PCI-6143, National Instruments Corporation) with an analog input resolution of 16 bits. The phase change dF, which codes the grating displacement information, was measured by PC-based software programmed by LabVIEWTM software (version: 8.5, National Instruments Corporation). The direct current terms of ID1 and ID2 were mapped into a precalibrated Lissajous circle so that the direct current terms of ID1 and ID2 could be removed. Accordingly, we used an arctangent algorithm to calculate the phase shift because of GS displacement. The experimental results of the long-range measurements were compared with the commercial laser dynamic calibration system (model: HP5529A, HP Inc.). The experimental results of the short-range measurement were referenced to the input voltage waveform, since we have no instrument which has a resolution high enough to observe displacements smaller than 10 nm. 4.1. Long-range displacement According to the user’s manual, the linear stage has a minimum achievable movement of less than 50 nm. The moving velocity of the stage is approximately 1.42 mm/s. In the first experiment, we moved the motorized stage at 20-, 40-, 60-, 80-, and 100-mm ranges based on the reading of the HP5529A, and at
Fig. 11. The 100-mm range measurement results of the LisaLENS and the HP5529A.
Fig. 12. Repeatability test results in long-range measurements for the LisaLENS.
the same time we used the LisaLENS to measure the stage displacement. Each range measurement was repeated five times. Fig. 11 shows the 100-mm range measurement results of the LisaLENS and the HP5529A. The dashed line in this figure was deliberately biased by about 10 s on the time axis to differentiate the LisaLENS and HP5529A data. The final displacement reading for the LisaLENS was 100.001030 713 nm. In order to examine the repeatability performance of the LisaLENS, we compared 20-, 40-, 60-, 80- and 100-mm range measurement results with that of the HP5529A. Fig. 12 shows the repeatability test results in long-range measurements for the LisaLENS. Note that the mean displacement of five times of measurements for each range is approximately proportional to the measurement range. Thus, the measurement error for such long-range measurements may result from the cosine error. The cosine error may exist because the measurement axis does not exactly coincide with the LisaLENS axis. The measurement discrepancy can be further improved by optimizing the optomechanical design and reducing alignment errors. The grating period variation caused by manufacture tolerance might contribute to these discrepancies. Section 4.1 provides further discussion on this topic. These results demonstrate that the LisaLENS can measure long-range displacements.
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tolerance of the grating period can be conservatively estimated to be approximately 40 nm (dr) using error assessment theory. The results in Figs. 11 and 12 show that the average displacement error is 616 nm for long-range experiments. However, the contribution of the grating period variation to the displacement error is less than 20 nm, revealing a difference of more than an order of magnitude. Clearly, the error caused by the grating period variation can be ignored for the long-range displacement experiments in this study. Similarly, the contribution of the grating period variation to the displacement error is less than 0.1 nm for the short-range experiments within 10 nm. Thus, the error from the grating period tolerance is negligible. 5.2. System parameters
Fig. 13. LisaLENS measurement results of 10-nm range displacement under a periodic input voltage and a triangular waveform.
4.2. Short-range displacement The short-range experiments in this study adopted the configuration depicted in Fig. 1. However, the stage-driving mechanism was changed to a multilayer piezo stack actuator (model: PL055, Physik Instrumente). The relationship between the input voltages and the nanoscale displacements of the piezo stack actuating stage was predetermined using the HP5529A, the accuracy of which was in the range of a few micrometers. To verify the LisaLENS measurement capability on the nanometer scale, we drove the piezoelectric stack actuator using periodic input voltages with triangular and sinusoidal waveforms. Fig. 13 shows the LisaLENS measurement results for 10-nm range displacement under the periodic input voltage with a triangular waveform. These short-range test results indicate that the LisaLENS has the capability of measuring nanometer displacements.
5. Discussion 5.1. Grating period tolerance Eq. (13) shows the total differentiation of dx:
dðdxÞ ¼
dF r dr þ dðdFÞ: 16p 16p
ð25Þ
5.2.1. Stability Immunity to environmental disturbances can dramatically influence the wide applications of the proposed laser encoder. To verify the stability of the LisaLENS, which reflects its immunity to environmental disturbance, we immobilized the stage carrying the GS for half an hour. The driving servomotor was turned off electrically. We used the LisaLENS and the HP5529A to monitor the displacement of the stage carrying the GS. Fig. 14 shows the system stability test results for the LisaLENS and the HP5529A for 1800 s. Results show that the LisaLENS data had substantially smaller variations over time than the HP5529A data. The LisaLENS data truly reflected the state of the still stage. Fig. 14 shows the superior immunity of the LisaLENS to environmental disturbances. This figure also shows why an optical interferometer is an improper choice for nanoscale displacement measurements in engineering environments. The system drift of the LisaLENS is substantially smaller than that of the HP5529A in such environmental conditions. 5.2.2. Sensitivity The sensitivity of an instrument is typically calculated as the ratio of the transducer output signal intensity to the measured physical quantity. However, it is difficult to apply this concept to laser encoders or optical interferometers. Consider the quantities resulting from the processing of the transducer/LisaLENS0 output signals; that is, dF; dx is the measured physical quantity. Eq. (13) shows the measurement sensitivity for the LisaLENS: dF 16p ¼ : dx r
ð27Þ
where the total differentiation d(dx) represents the displacement measurement error. Eq. (25) shows two main error sources for the proposed system: the grating period variation dr and the phase shift error d(dF). In this system, a smaller grating period r leads to a smaller phase noise influence. The manufacture tolerance of the grating period directly influences the accuracy of the displacement measurement. The period variation, however, comes from random manufacture error. Consider the period variation and assume that the phase shift error is zero. Assume that the interference signals are only in arbitrary cycles (2p) or arbitrary r=8 displacement. The corresponding displacement error can be represented as:
dðdxÞ ¼
dr 8
in a cycle:
ð26Þ
The laser spot diameter on the GS is approximately 2 mm and covers approximately 6250 grating lines. For the 1 mm travel range of the GS, the interference signals run approximately 5000 cycles. With an equal accuracy consideration and 50 nm displacement accuracy requirement for the 1 mm travel range, the manufacture
Fig. 14. The system stability test results for the LisaLENS and the HP5529A for 1800 s.
C.-C. Wu et al. / Optics Communications 297 (2013) 89–97
A grating with a period of 1.6 mm leads to dF=dx ¼ 1:81=nm:
97
6. Conclusion ð28Þ
Eq. (28) represents the measurement sensitivity for the LisaLENS. 5.2.3. Resolution For a 16-bit analog to digit conversion (ADC) resolution, (28) produces a theoretical displacement measurement resolution smaller than 0.6 pm for the LisaLENS. However, the measured phase suffers from various high- and low-frequency noises. The high-frequency phase noises may come from the laser source, photodetectors, and operational amplifier electronics, whereas the low-frequency phase noise may result from slow variation of the optical path difference, such as foundation vibration and atmospheric disturbances. Consider the experiment shown in Fig. 14. Because the stage is held still (dF ¼0), the effect of the grating period variation can be dropped from (25). That is, the result in Fig. 14 represents only the phase noises. The measurement time for all of the experiments in this study is within 180 s. For each time interval of 180 s, we can have the corresponding maximum and minimum displacement data. The estimated resolution can be represented using the difference between the maximum and minimum displacement data. Thus we can have the estimated resolution statistically. A review of every 180 s time interval in Fig. 14 shows an estimated measurement resolution of approximately 5.5 71.4 nm. If the effective bit number for ADC is considered, we can obtain the equivalent effective bit number of approximately 4 bits. 5.2.4. Head-to-scale tolerance In Section 2.2, we choose b ¼0.1 in all runout tolerance estimations for the LisaLENS. This choice can ensure that the runout effect on the measurement is small enough. Actually, as the two interference beams of the LisaLENS are essentially parallel to each other, we can arrange two focus lenses for each of the PD1 and PD2 respectively. The active areas of the PD1 and PD2 are aligned at the focal distances of each of the lenses. The optical axis of each lens is aligned parallel to the two interference beams such that the two interference beams always overlap at the same positions on the PD1 and PD2. In this way, the runout effect on the interference signals can be reduced to zero, and the head-to-scale tolerance of the LisaLENS can be further raised. 5.2.5. Limit on stage moving velocity For experiments in this paper, a high performance field programmable gate array (FPGA) was developed to real-time implement the arctangent phase decoding method to 20 MHz. The real-time decoded displacement information was sent back using a high-speed PCI-DIO card installed on a PC computer. Based on our hardware setup, the stage moving velocity is limited to smaller than 3 mm/s. A moving velocity larger than 3 mm/s will induce phase unwrapping error.
This study presents the LisaLENS, which uses two direct return diffracted beams to interfere with each other. It adopts an optical configuration with a corner cube and a beam expander to ensure high tolerance on the head-to-scale runout and the grating period. The GS displacement can be determined by the relationship between the measured optical phase change and the grating period. Experimental results demonstrate that the LisaLENS is able to measure both long- and short-range displacements. The theoretical prediction of the measurement resolution is approximately 0.6 pm. If only high-frequency noises are considered, the measurement resolution of the system is estimated to be 5.571.4 nm. The LisaLENS also offers the advantages of high sensitivity and a long measurement range, making it useful for monitoring the displacement and vibration of a precision motorized stage in various applications.
Acknowledgments The authors would like to acknowledge the support of the National Science Council (NSC) 98–2218-E-032-009-, 99–2628-E032-001-, and 100–2628-E-032-001-). We also thank Mr. M.H. Wen (E-Pin Optical Industry Company, Taiwan) and Professor C.K. Lee (National Taiwan University) for providing us with access to equipment.
References [1] /http://www.itrs.netS. [2] C.-K. Lee, C.-C. Wu, S.-J. Chen, L.-B. Yu, Y.-C. Chang, Y.-F. Wang, J.-Y. Chen, J.W.-J. Wu, Applied Optics 43 (2004) 5754. [3] C.-C. Wu, W.-J. Wu, Z.-S. Pan, C.-K. Lee, Applied Optics 46 (2007) 3169. [4] C.F. Kao, S.H. Lu, H.M. Shen, K.C. Fan, Japanese Journal of Applied Physics 47 (2008) 1833. [5] F. Cheng, K.-C. Fan, Applied Optics 50 (2011) 4550. [6] K. Fan, Y. Zhang, J. Miao, F. Cheng, Error compensation of grating interferometer due to angular error of linear stage, in: Proceedings of the IEEE International Conference on Advanced Intelligent Mechatronics, 2012, p. 428. [7] M.L. Schattenburg, C. Chen, P.N. Everett, J. Ferrera, P. Konkola, H.I. Smith, Journal of Vacuum Science and Technology B: Microelectronics and Nanometer Structures 17 (1999) 2692. [8] C.G. Chen, P.T. Konkola, R.K. Heilmann, C. Joo, M.L. Schattenburg, Nanometeraccurate grating fabrication with scanning beam interference lithography, in: Proceedings of the Conference on Nano- and Micro-technology: Materials, Processes, Packaging, and Systems, Part of AU02 SPIE0 s International Symposium on Smart Materials, Nano- and Micro-Smart Systems, SPIE, Melbourne, Australia, 2002. [9] F.L. Pedrotti, L.M. Pedrotti, L.S. Pedrotti, Introduction to Optics, 3rd ed., Prentice-Hall International, London, 2006. [10] L.E. Drain, The Laser Doppler Technique, John Wiley, New York, 1980. [11] /http://www.newport.comS. [12] I. Misumi, S. Gonda, T. Kurosawa, K. Takamasu, Measurement Science and Technology 14 (2003) 463.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Littrow-type self-aligned laser encoder with high tolerance using double diffractions Chyan-Chyi Wu a, Cheng-Chih Hsu b,n, Ju-Yi Lee c, Yan-Zou Chen a,d, Jhih-Sheng Yang a a
Advanced Photonics Laboratory, Department of Mechanical and Electromechanical Engineering, Tamkang University, Taiwan Department of Photonics Engineering, Yuan-Ze University, Chungli 320, Taoyuan, Taiwan c Institute of Optomechanics, National Central University, Taiwan, R.O.C d Institute of Mechanical Engineering, National Taiwan University, Taiwan b
a r t i c l e i n f o
abstract
Article history: Received 4 September 2012 Received in revised form 15 January 2013 Accepted 17 January 2013 Available online 12 February 2013
This study proposes a Littrow-type self-aligned laser encoder system (LisaLENS) with double diffractions. The LisaLENS has simple optical paths, facilitating alignment and manufacture, and fewer optical components than current laser encoders. This study analyzes the theoretical measurement resolution of this system and verifies the feasibilities of its long- and short-range measurements through experiments. Experimental results show that the LisaLENS has a resolution of 5.5 7 1.4 nm. Thus, this design has wide applications in various fields, especially for monitoring the displacement and vibration of precision-motorized stages. Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved.
Keywords: Displacement Grating Long range Littrow
1. Introduction Laser encoders are not subjected to environmental disturbances because they transfer the measurement scale from the laser wavelength to the grating period. Laser encoders have played an important role in precise positioning systems used by semiconductor and liquid crystal display (LCD) manufacturers. Based on rapid advances in nanotechnology, the 25-nm node on 450 mm wafers will likely become commonplace in 2014 [1]. Two important factors determine if a laser encoder has wide applications. One is the runout tolerance between the optical head and the grating scale (GS), which is called the head-to-scale tolerance. Laser encoders inevitably suffer a serious runout problem, especially when they are applied to high through-put or high-speed manufacture of the coming node. The appearance of runout can induce a displacement measurement error, or the interference signals may suddenly vanish. The other is the tolerance of the grating period, which comes from the grating manufacture and the thermal variation when a laser encoder is operating. A tight tolerance of the grating period represents the manufacturing difficulty and high cost. The literature contains two ways to raise the head-to-scale tolerance of laser encoders. In our previous work [2], we adopted
n
Corresponding author. Tel.: þ886 3 4638800; fax: þ 886 3 4514281. E-mail addresses: [email protected], [email protected] (C.-C. Hsu).
a 1 x telescope configuration in a laser encoder to give it a high head-to-scale tolerance. In another previous work [3], we combined a 1 x telescope configuration and a perpendicular optical configuration in our laser encoder to give it both a high head-toscale tolerance and a high manufacture tolerance. The 1 x telescope can effectively handle in-plane and out-of-plane optical paths, so it is a good choice for raising head-to-scale tolerance. Kao et al. [4] presented a laser encoder with a grating in the Littrow configuration. That encoder had a maximum measurement error of 53 nm and repeatability within 720 nm in the 100 mm measurement range. Cheng and Fan proposed a laser encoder with a Littrow configuration [5]. Cheng’s encoder attained repeatability within 15 nm in a 15-mm travel range. Fan et al. [6] used two sets of Littrow configurations with two laser diode sources to construct a planar encoder for a planar stage displacement measurement. The Littrow configuration can retro-reflect the laser beam at the specific angle of incidence. However, the Littrow configuration for a laser encoder must be accompanied by specially arranged optics in order to have high head-to-scale tolerance. The works by Kao et al., Cheng et al., and Fan et al. used optics that can handle only in-plane optical paths. Thus, when the runout of roll or yaw happens, their laser encoder cannot correct the out-of-plane optical paths. To our knowledge, no studies have focused on raising the grating manufacture tolerance through the optical configuration of a laser encoder. The grating period tolerance generally is near 1% of the grating period [7]; that is, the grating manufacture
0030-4018/$ - see front matter Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.01.048
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tolerance is 16 nm for the grating scale with a grating period of 1.6 mm. This tolerance specification leads to a strict environmental condition for the grating manufacture. Also, expensive ultraprecision equipment is usually required for the grating manufacture, one example being the NanoRuler from MIT [8]. This study proposes a Littrow-type self-aligned laser encoder system (LisaLENS) with the least optics. The LisaLENS has an optical configuration with a corner cube (CC) and a beam expander (BE), so the LisaLENS has a high tolerance on both the head-to-scale runout and the grating period. It has simple optical paths, facilitating easy alignment or manufacture, so it is called a self-aligned laser encoder. The experiments in this study verify the feasibilities of long- and short-range measurements. Experimental results show that the LisaLENS has a resolution of 5.571.4 nm. Thus, the LisaLENS is suitable for long- and shortrange measurements from the nanometer to the millimeter scale.
2. Theory 2.1. Measurement principle Fig. 1 shows a schematic diagram of the LisaLENS. This study analyzes both paths of the þ2nd- and 2nd-order diffracted beams to explain the displacement measurement principle of the LisaLENS. The laser beam from a polarized laser source (LS) is expanded by a beam expander (BE). A polarizing beam splitter (PBS) divides the polarized light from the BE into two linearly polarized beams, called p-light, transmitted at the PBS, and s-light, reflected at the PBS. For the optical Arm 1, the p-light first passes through the quarter-wave plate (Q1) and becomes a circularly polarized beam. Then the circularly polarized beam is bent by the mirror (M1), incidents onto the grating scale (GS), and is diffracted by the GS. The 2nd-order diffracted beam goes back along the incident path (the dash path in Fig. 1). The 2nd-order diffracted beam is s-polarized after the second time it passes through the Q1. The s-polarized beam is reflected sequentially at the PBS, at the corner cube (CC) twice, and then at the PBS again, shown as the dash path in Fig. 1. After the s-light passes through the Q1 for the third time, it becomes a circularly polarized light. The circularly polarized beam incidents onto the GS again and is again diffracted as the 2nd-order diffracted beam. The second time diffracted light becomes p-light after passing through the Q1 for the fourth time. Thereafter, the p-light passes through the PBS and is directed to photodetectors (the PD1 and PD2 in Fig. 1). For the optical Arm 2, the s-light from the BE is reflected at the PBS, is
bent by the mirror M2, and goes along the solid path. After being diffracted into the þ2nd-order diffracted beam by the GS two times and passing through the Q2 four times, the s-light recovers to s-light and is reflected at the PBS. In summary, the CC can ensure that the p- and s-light directed to the Q3 have parallel optical paths (see Section 2.2 for details). Thereafter, the p-light and s-light traverse into a circular polarization interferometer, which consists of a quarter waveplate Q3, a beam splitter BS, and two polarizers P1 and P2. The interference fringe is captured by photodetectors PD1 and PD2, respectively. The operational principle of the LisaLENS is based on the Doppler effect of the moving GS. Because the LisaLENS adopts a Littrow-type optical path, the diffracted beam theoretically returns along the incident direction. The grating equation [9] indicates that the incident angle and the diffractive angle are equal, and can be written as follows: 9yL 9 ¼ 9yi 9 ¼ sin1
l
r
,
ð1Þ
where yi is the incident angle, yL is the diffractive angle for the Littrow-type optical path on the GS, l is the wavelength of the LS, and r is the grating period. Because of the Doppler effect of the moving GS, after double diffractions the corresponding angular frequency shifts of Arm 1 and Arm 2 (Fig. 1), Do1 and Do2, respectively, are as follows [10]:
Do1 ¼
Do2 ¼ þ
8pu
r
,
8pu
r
ð2Þ
,
ð3Þ
where u is the moving velocity of GS in the þ x-direction. From (2) and (3), one can realize that double diffractions can raise the measurement sensitivity. For the optical Arm 1, the laser beam follows the path PBS-Q1-M1-GS-M1-Q1-PBS-CCPBS-Q1-M1-GS-M1-Q1-PBS-Q3-BS. For the optical Arm 2, the laser beam follows the path PBS-Q2-M2-GSM2-Q2-PBS-CC-PBS-Q2-M2-GS-M2-Q2-PBS-Q3BS. After departing from the quarter waveplate Q3, the electric fields ! ! of these two optical waves from Arm 1 and Arm 2, E 1 and E 2 , can be written in the following form: ( ) 1 ! 1 E 1 ¼ pffiffiffi E10 ejðot þ Do1 tfl1 Þ , ð4Þ j 2 ! 1 E 2 ¼ pffiffiffi E20 ejðot þ Do2 tfl2 Þ 2
j 1
,
ð5Þ
! where E10 and E20 are the amplitudes of the electric fields E 1 and ! E 2 , respectively, o is the angular frequency of the laser source LS, fl1 and fl2 are the phase shifts generated pffiffiffiffiffiffiffi by the optical paths along Arm 1 and Arm 2, respectively, j¼ 1, and t is time. Eqs. (4) and ! ! (5) show that the electric fields E 1 and E 2 are right and left ! ! circularly polarized, respectively. In this case, E 1 and E 2 interfere with each other and pass through the beam splitter (BS). Finally, the photodetectors PD1 and PD2 measure the interference fields. The ! resulting electric fields entering photodetectors PD1 and PD2, E D1 ! and E D2 , respectively, can be represented as follows: ( ) ! E10 ejðot þ Do1 tfl1 Þ þ jE20 ejðot þ Do2 tfl2 Þ 1 E D1 ¼ pffiffiffi , ð6Þ 0 2 2
Fig. 1. Schematics of the LisaLENS and the experiment setup.
! 1 þj E D2 ¼ pffiffiffi 4 2
(
E10 ejðot þ Do1 tfl1 Þ þ E20 ejðot þ Do2 tfl2 Þ E10 ejðot þ Do1 tfl1 Þ þ E20 ejðot þ Do2 tfl2 Þ
) ,
ð7Þ
C.-C. Wu et al. / Optics Communications 297 (2013) 89–97
The intensities ID1 and ID2, measured by photodetectors PD1 and PD2, respectively, can be written as follows: o 1n 2 E10 þ E220 þ 2E10 E20 sin ð2DoÞtðfl2 fl1 Þ , ð8Þ ID1 ¼ 8 ID2 ¼
o 1n 2 E10 þ E220 þ 2E10 E20 cos ð2DoÞtðfl2 fl1 Þ , 8
ð9Þ
where Do2 ¼ Do1 ¼ Do. The proper arrangement on the optical path can lead to E10 ¼E20. Assume fl1 ¼ fl2 when there are no environmental disturbances. Thus, ID1 and ID2 can be reduced as follows: ID1 p1þ sin½ð2DoÞt,
ð10Þ
ID2 p1þ cos½ð2DoÞt:
ð11Þ
The phase change in ID1 and ID2 signals, F, comes from the Doppler frequency shift because of the GS movement. For an arbitrary small duration dt, the GS has a small displacement dx¼udt, and the corresponding phase changes dF ¼2Dodt. Eqs. (2) and (3) show the relationship between the phase change in ID1 and ID2, because of the displacement of GS dx, as follows: dF ¼
16p
r
dx,
ð12Þ
Consider that the coordinate system coincides with three ridges of a corner cube with a ridge length of a, as shown in Fig. 3. On the input/output plane, an incident beam enters the corner cube at a point A, is reflected sequentially at points B, C, D, and finally returns to a point A0 . In Fig. 3, the corresponding unit normal ! ! vectors for x y, y z, and x z planes are n 1 ¼ e^ 3 , n 2 ¼ e^ 1 , ! and n 3 ¼ e^ 2 respectively. Here e^ 1 , e^ 2 , and e^ 2 are the unit vectors of the x -, y -, and z-axes respectively. Assume that the unit ! direction vectors of optical paths AB, BC, CD, and DA0 are v 1 , ! ! ! v 2 , v 3 and v 4 respectively. By a simple vector analysis, we can obtain the relation: ! ! n iþ1 ¼ ! n i 2ð! n i U! n i Þ n i , i ¼ 1,2,3: ð14Þ Note that ‘‘U’’ denotes the inner product. Assume that ! v 1 ¼ a1 e^ 1 þ a2 e^ 2 þ a3 e^ 3 , where a1, a2, and a3 are arbitrary real ! numbers. From (14), we can have n 2 ¼ a1 e^ 1 þ a2 e^ 2 a3 e^ 3 . In ! the same way, we can easily have v 3 ¼ a1 e^ 1 þ a2 e^ 2 a3 e^ 3 , and ! ! ! v 4 ¼ a1 e^ 1 a2 e^ 2 a3 e^ 3 . As v 1 ¼ v 4 , we conclude that the incident and return beams are certainly parallel to each other for a corner cube. ! For an incident beam with an incident wavevector k in as shown in Fig. 4(a), the components of the diffracted wavevector !0 for a three-dimensional grating diffraction, k m (see Fig. 4(b)), can be written as follows:
or
r dF, dx ¼ 16p
0
ð13Þ
Based on (13) the grating displacement, dx can be determined by measuring the phase difference dF between the two sinusoidal signals if the grating period r is known. A period displacement produces a phase change of 16p; that is, the GS displaces r=8 for a cycle of interference signals. 2.2. Raising head-to-scale tolerance with the corner cube Fig. 2 shows the schematics of four types of runout for the laser encoder: pitch; roll; yaw; and stand-off. Physically, the appearance of runout may spoil the interference condition of the optical path for the laser encoder such that visibility is degraded, unexpected phase shift presents, or the interference signal suddenly vanishes. For the LisaLENS, a corner cube (the CC) is adopted to keep the two interference beams traveling along the same direction. Thus the head-to-scale tolerance can be effectively raised. A corner cube is commonly made of glass with three reflective planes perpendicular to each other, and an input/output plane.
Fig. 2. Schematics of four types of runout for the laser encoder: pitch; roll; yaw; and stand-off.
91
kx0 ,m ¼ 0
2p
l
sin yi Ucos f þml=r ,
2p
sin f, l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 kz0 ,m ¼ 1sin2 fðsin yi Ucos f þml=rÞ2 , l
ky0 ,m ¼
2p
ð15Þ
where m is the diffraction order, k0 x0 ,m, k0 y0 ,m, and k0 z0 ,m are x0 -, y0 !0 and z0 -components of a wavevector k m of the m-th order diffracted beam, yi is the angle of incidence in the x0 –z0 plane, ! and f is the included angle between k in and the x0 –z0 plane. Note that the sign convention for yi and f obeys the right-handed rule. Let us further study the runout effect on the LisaLENS. For convenience, consider a runoutless configuration of the LisaLENS in Fig. 5. The points z0i (i¼1, 2, 3, 4) are the corresponding incident points on the GS when there is no runout. z0 0i are the image points of the incidence points z0i on the GS for Arm 1 and Arm 2, where i¼1, 2, 3, 4. Applying the sine theorem in triangles A1z01B1, A2z02B2, A3z03B3 and A4z04B4 in Fig. 5, we can have the following geometrical relations: 0
Ai z0i ¼ Ai z0i ¼
A i Bi , sin 2yM þ cos 2yM
i ¼ 1, 2, 3, 4:
ð16Þ
Fig. 3. The coordinate system coincides with three ridges of a corner cube with a ridge length of a.
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Fig. 5. Schematics of the incidence points and their image points for runout analysis. For a runoutless case, z0 0i are the image points of the incidence points z0i on the GS for Arm 1 and Arm 2, where i¼ 1–4. The point 0 is the origin of the x–y–z coordinate system.
Fig. 4. (a) Schematics of a three-dimensional grating diffraction in x–y–z coordinate system. c represents the runout of yaw. (b) Schematics of a diffracted beam under three-dimensional grating diffraction in an x–y–z coordinate system.
where Ai are the incidence points on the M1 and the M2, Bi are the intersection points between the extended light rays and the grating surface, and z0 i are the image points of the incidence points zi on the GS under runout, where i¼ 1, 2, 3, 4. With the help of Eq. (16), we can analyze the runout tolerance using the image points of the incidence points on the GS. Thus, we will adopt a simplified model to do the first-order tolerance analysis on the head-to-scale runout of the LisaLENS in the following sections (see Figs. 6 and 7).
2.2.1. Stand-off or roll Firstly, f ¼0 and the x-axis coincides with the x0 -axis for the runout of stand-off or roll. From (15), we always have k0 x, 2 ¼ k0 x, þ 2, k0 y, 2 ¼k0 y, þ 2, and then k0 z, 2 ¼k0 z, þ 2 in Arm 1 and Arm 2 for each diffraction. As the designed incident points on the GS for the Arm 1 and Arm 2 are symmetric to the PBS polarization
Fig. 6. Schematics of the runout of pitch. The incident points on the GS for Arm 1 are z1 and z2. The incident points on the GS for Arm 2 are z3 and z4. The points z0 3 and z0 4 are the images of points z3 and z4 with respect to the M1, the points z0 1 and z0 2 are the images of points z1 and z2 with respect to the M2, and the points z00 3 and z00 4 are the images of points of the image points z0 3 and z0 4 with respect to the PBS polarization film plane (the y–z plane). Here we omit M1 and M2 for convenience.
film plane, all of the corresponding paths of Arm 1 and Arm 2 are also symmetric to the polarization film plane of the PBS. Thus the two interference beams always travel along the same optical path as the originally designed path. That is, the interference signals will essentially not be spoiled under the runout of stand-off or roll. The runout specification of stand-off or roll is constrained only by the field of view of the photodetector.
2.2.2. Pitch Secondly, consider the runout of pitch, as shown in Fig. 6. The point o is the origin of the x–y–z coordinate system. L21 is the distance of the point z0 1 to the PBS, L01 is the distance of the point z0 01 to the PBS, L11 is the distance of the point z0 3 to the PBS, L12 is the distance of the point z00 4 to the PBS, L02 is the distance of the
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where p is the runout of pitch. The transverse separation of interference beams can reduce the visibility of the interference fringe and the interference signal intensities. Assume that the beam diameter incident onto the GS is D. Let a ratio b ¼ Drunout =D represent the relative transverse separation of two interference beams, where runout is pitch. Note that 0 r b o1 for two-beam interference. In general, the smaller the b is, the better the interference signals. Accordingly, we can estimate the runout tolerance of pitch as follows: pf2L01 þ6Lcc þ2L02 g r bD:
ð18Þ
If the Lcc ¼25 mm, L01 ¼ 42 mm, and L02 ¼34 mm for a prototype of the LisaLENS, the estimated runout tolerance of pitch with b ¼ 0.1 and D ¼10 mm is about 0.211 or 751 arc-sec. This tolerance value is about 37 times greater than the value in our previous work (20 arc-sec with b ¼0.5 and D ¼2 mm) [2]. Note that the pitch runout of a laser encoder generally results from the yaw runout of a stage. As the optical paths of Arm 1 and Arm 2 remain symmetric to the runoutless path when pitch runout happens (see Fig. 6), the optical path difference (OPD) between these two arms will be zero within the CC. Similarly within the PBS, the optical paths of Arm 1 and Arm 2 are also symmetric to the runoutless optical path, so there is no OPD between these two arms in the PBS. We can align the center of the incident points on the GS of Arm 1 at the origin point o. Similarly, we can also align the center of the incident points on the GS of Arm 2 at the origin point o. In this way, the OPD between Arm 1 and Arm 2 can be effectively compensated when the pitch runout happens. Practically, the common pitch runout of the stage carrying the GS is smaller than 10 4 rad [11]. When the deviation of the center alignment of incident points at the origin point o is within 1 mm, we can safely have an OPD of about 0.1 nm. The OPD may produce an additional phase shift and then a displacement error of about 30 pm for the LisaLENS. A better alignment of the center of incident points will further slash the OPD effect from pitch runout. 2.2.3. Yaw Thirdly, consider the runout of yaw with its schematics, as shown in Fig. 4(b). From (15) and the coordinate transformation, we can have k0 x, þ 2 ¼ k0 x, 2, k0 y, þ 2 ¼ k0 y, 2, and k0 z, þ 2 ¼k0 z, 2 for the diffracted beams of Arm 1 and Arm 2. Accordingly, we can write down the absolute diffraction angles of the þ2nd (for Arm 2) and 2nd (for Arm 1) order diffracted beams in the x–z plane, 9q9, as follows: rsinyL 2lcosc 9q9 ¼ tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ð19Þ r2 cos2 yL 4lðlrcoscsinyL Þ
Fig. 7. Schematics of the runout of yaw: (a) optical path projection in x–z plane; (b) optical path projection out of x–z plane. y01, y02, y2, y3, and y4 are the incident points on the GS. l01, l02, and l22 are the distances from the PBS to y01, y02, and y3 respectively.
point z02 to the PBS, and L22 is the distance of the point z0 2 to the PBS. Lcc is the diameter of the CC. Dpitch is the transverse separation of two parallel interference beams. Note that we apply the first-order analysis to the runout of pitch so that the refraction effects on the Q1, Q2, and PBS can be neglected. We can calculate the transverse separation of two parallel interference beams, Dpitch, as follows:
Dpitch ffi pfL11 þ3Lcc þL12 gArm 1 þ pfL21 þ3Lcc þ L22 gArm 2 ffi pf2L01 þ6Lcc þ 2L02 g,
ð17Þ
where c is the runout of yaw (see Fig. 4). Similarly, the absolute diffraction angle of the þ 2nd- and 2nd-order diffracted beams, 9f9, can be written as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 2l2 þ 2l2 cosð2cÞ 1 : 9f9 ¼ cos ð20Þ r From Fig. 7, we can see that the diffraction angle 9q9 makes the interference beams displace along the z-axis, and the diffraction angle 9f9 makes the interference beams separated in the þy- and y-directions. Even though these two interference beams displace or separate, they stay parallel to each other under the runout of yaw. We can calculate the displacement and separation of the two interference beams, dyaw, and Dyaw respectively (see Fig. 7(a) and (b)), using the first-order analysis as follows: dyaw ffi 9yL -q9fl01 þ 3Lcc þ l22 g,
ð21Þ
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and
Dyaw ffi 9f9f2l01 þ6Lcc þ 2l22 g:
ð22Þ
Using (21) and (22), we can estimate the runout tolerance of yaw to constrain both dyaw and Dyaw to smaller than bD. If the Lcc ¼25 mm, l01 ¼42 mm, and l22 ¼34 mm for a prototype of the LisaLENS, the estimated runout tolerance of yaw with b ¼0.1 and D¼10 mm is about 0.251 (or 0.0044 rad). This tolerance value is about one-twelfth of the value in our previous work (31 with b ¼0.5 and D¼ 2 mm) [2]. Note that the yaw runout of a laser encoder generally results from the pitch runout of a stage. Generally, the pitch specification for a common low-cost linear stage is 710 4 rad [11]. Thus the yaw tolerance of the LisaLENS is practically enough. As the LisaLENS adopts a diode laser as the laser source, the path difference between two interference beams can be ensured within a few microns. Also, the optical paths of Arm 1 and Arm 2 remain symmetric to the z-axis when yaw runout happens. Thus, even though the optical paths of Arm 1 and Arm 2 change under yaw runout, the optical path difference (OPD) between the two interference beams will be zero theoretically. 2.3. Raising grating manufacture tolerance with beam expander For the LisaLENS, assume that the grating manufacture error for each grating period is ei. The laser spot has a diameter of D. The number of p the ffiffiffi grating period in such a laser spot, N, can be estimated as N ¼ 2D=r. The resultant measurement error for N grating lines, e^ , can be written as:
ei e^ ¼ pffiffiffiffi : N
Fig. 8. The fabrication process of the grating scale using the laser writer.
ð23Þ
Let the measurement range for the LisaLENS be R. The number of cycles of the interference signals for the whole range R can be estimated by ½8R=r, where [ ] is the Gaussian sign. Accordingly, the total displacement error which comes from the grating manufacture error, e, can be written as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uX re2 ð24Þ e ¼ t pffiffiffii , 2D i where i¼ 1 to ½8R=r. From (24), the larger the laser spot diameter is, the larger the grating manufacture tolerance can be for a given accuracy assessment for the grating period. Thus we can adopt a beam expander configuration to raise the grating manufacture tolerance. Fig. 9. The schematic layout of the home-built laser writer. M1, M2, and M3 are mirrors; NPBS: non-polarization beam splitter; CCD: charge coupled device.
3. Fabrication of grating scale The GS is a key component in a laser encoder because it serves as the measurement scale. The proposed device used a homemade laser writer to write the line pattern of the GS. Fig. 8 shows the fabrication process of the GS using the laser writer. Fig. 9 shows the schematic layout of the homemade laser writer. The laser writer adopted a 50 mW He–Cd laser with a 442 nm wavelength (KimmonTM IK5451R-E) as the lithography light source. An acousto-optical deflector (AOD) (IsometTM LS55 V with IsometTM D323B analog driver) made the laser beam locally scan the substrate surface. An acousto-optical intensity (AOI) modulator (IsometTM 1205C with IsometTM 232A-1 analog driver) controlled the laser beam intensity for lithography. An acousto-optical modulator (AOM) (IsometTM 1250C with IsometTM 225A-1 digital driver) served as a high-frequency switch for lithography. An objective (OlympusTM) with a long working distance was mounted in a computercontrolled autofocus system (model: LTAF 8000, Teletrac Inc.). The GS substrate was quartz glass with a thermal expansion of 5 10 7/1C. The substrate was held by a vacuum chunk carried
by an x–y air-bearing stage (U-Star Tech). The air-bearing stage was driven by x- and y-axis linear motors (KollmorgenTM Platinum DDL). To enable the accurate positioning of the x–y stage, x- and y-axis linear encoders (HeidenhainTM LIP401) sensed the corresponding displacements as the feedback signals. The travel ranges of the air-bearing stage in the x- and y-axes were 200 mm, achieving a positioning accuracy of 0.1 mm. The maximum velocity was 50 mm/s. The maximum acceleration was 2000 mm/s2. The AOD, AOI, AOM, and x–y air-bearing stage were controlled using an industrial computer through a turbo programmable multi-axis control card (Delta Tau Data System Inc.). The grating line pattern was inputted in bitmap format. The manufactured grating topography was measured using an atomic force microscope (AFM), and Fig. 10 shows the measurement results. The grating period was measured at nine different locations on the grating surface [12], and statistically the grating period value was 1.62470.034 mm. The grating was deposited gold film at 40 nm thick to have better reflection. The diffraction efficiency for the 2nd diffracted beam was about 8%.
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Fig. 10. AFM photograph of the manufactured grating scale.
4. Experiments and results All of the experiments in this study were performed at an atmospheric temperature, pressure, and relative humidity of 2272 1C, 101.32570.7 kPa, and (45710)% RH, respectively. The vibration amplitude was smaller than 0.25 mm for low frequency excitation from 0.1 Hz to 30 Hz, and the acceleration was smaller than 0.001 g for the excitation frequency from 30 Hz to 200 Hz. The displacement provided by a linear stage was measured to verify the feasibility of the proposed method. Fig. 1 shows the experimental configuration. The light source consisted of a linearly polarized laser diode with a wavelength at 650 nm (model: L650P007, Thorlabs). The GS was mounted on the precision linear stage to diffract the laser beams from the laser source. Two grating scales were used for experiments. For short-range experiments, we used the grating scale described in Section 3. For long-range experiments, we adopted a commercial grating scale with a grating period of 1.6 mm (manufactured by Taiwan Mask Cooperation). The commercial grating period tolerance was guaranteed within 2 nm. The experiments in this study adopted two types of linear stages for long- and short-range displacements. One was a motorized stage (model: SGSP26-200(X), SIGMA KOKI Inc.), for long-range experiments, and the other was a stage driven by a multilayer piezo stack actuator (model: PL055.30, Physik Instrumente), for short-range experiments. The displacement signals ID1 and ID2 were sent to a high-speed multichannel data acquisition card (model: PCI-6143, National Instruments Corporation) with an analog input resolution of 16 bits. The phase change dF, which codes the grating displacement information, was measured by PC-based software programmed by LabVIEWTM software (version: 8.5, National Instruments Corporation). The direct current terms of ID1 and ID2 were mapped into a precalibrated Lissajous circle so that the direct current terms of ID1 and ID2 could be removed. Accordingly, we used an arctangent algorithm to calculate the phase shift because of GS displacement. The experimental results of the long-range measurements were compared with the commercial laser dynamic calibration system (model: HP5529A, HP Inc.). The experimental results of the short-range measurement were referenced to the input voltage waveform, since we have no instrument which has a resolution high enough to observe displacements smaller than 10 nm. 4.1. Long-range displacement According to the user’s manual, the linear stage has a minimum achievable movement of less than 50 nm. The moving velocity of the stage is approximately 1.42 mm/s. In the first experiment, we moved the motorized stage at 20-, 40-, 60-, 80-, and 100-mm ranges based on the reading of the HP5529A, and at
Fig. 11. The 100-mm range measurement results of the LisaLENS and the HP5529A.
Fig. 12. Repeatability test results in long-range measurements for the LisaLENS.
the same time we used the LisaLENS to measure the stage displacement. Each range measurement was repeated five times. Fig. 11 shows the 100-mm range measurement results of the LisaLENS and the HP5529A. The dashed line in this figure was deliberately biased by about 10 s on the time axis to differentiate the LisaLENS and HP5529A data. The final displacement reading for the LisaLENS was 100.001030 713 nm. In order to examine the repeatability performance of the LisaLENS, we compared 20-, 40-, 60-, 80- and 100-mm range measurement results with that of the HP5529A. Fig. 12 shows the repeatability test results in long-range measurements for the LisaLENS. Note that the mean displacement of five times of measurements for each range is approximately proportional to the measurement range. Thus, the measurement error for such long-range measurements may result from the cosine error. The cosine error may exist because the measurement axis does not exactly coincide with the LisaLENS axis. The measurement discrepancy can be further improved by optimizing the optomechanical design and reducing alignment errors. The grating period variation caused by manufacture tolerance might contribute to these discrepancies. Section 4.1 provides further discussion on this topic. These results demonstrate that the LisaLENS can measure long-range displacements.
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tolerance of the grating period can be conservatively estimated to be approximately 40 nm (dr) using error assessment theory. The results in Figs. 11 and 12 show that the average displacement error is 616 nm for long-range experiments. However, the contribution of the grating period variation to the displacement error is less than 20 nm, revealing a difference of more than an order of magnitude. Clearly, the error caused by the grating period variation can be ignored for the long-range displacement experiments in this study. Similarly, the contribution of the grating period variation to the displacement error is less than 0.1 nm for the short-range experiments within 10 nm. Thus, the error from the grating period tolerance is negligible. 5.2. System parameters
Fig. 13. LisaLENS measurement results of 10-nm range displacement under a periodic input voltage and a triangular waveform.
4.2. Short-range displacement The short-range experiments in this study adopted the configuration depicted in Fig. 1. However, the stage-driving mechanism was changed to a multilayer piezo stack actuator (model: PL055, Physik Instrumente). The relationship between the input voltages and the nanoscale displacements of the piezo stack actuating stage was predetermined using the HP5529A, the accuracy of which was in the range of a few micrometers. To verify the LisaLENS measurement capability on the nanometer scale, we drove the piezoelectric stack actuator using periodic input voltages with triangular and sinusoidal waveforms. Fig. 13 shows the LisaLENS measurement results for 10-nm range displacement under the periodic input voltage with a triangular waveform. These short-range test results indicate that the LisaLENS has the capability of measuring nanometer displacements.
5. Discussion 5.1. Grating period tolerance Eq. (13) shows the total differentiation of dx:
dðdxÞ ¼
dF r dr þ dðdFÞ: 16p 16p
ð25Þ
5.2.1. Stability Immunity to environmental disturbances can dramatically influence the wide applications of the proposed laser encoder. To verify the stability of the LisaLENS, which reflects its immunity to environmental disturbance, we immobilized the stage carrying the GS for half an hour. The driving servomotor was turned off electrically. We used the LisaLENS and the HP5529A to monitor the displacement of the stage carrying the GS. Fig. 14 shows the system stability test results for the LisaLENS and the HP5529A for 1800 s. Results show that the LisaLENS data had substantially smaller variations over time than the HP5529A data. The LisaLENS data truly reflected the state of the still stage. Fig. 14 shows the superior immunity of the LisaLENS to environmental disturbances. This figure also shows why an optical interferometer is an improper choice for nanoscale displacement measurements in engineering environments. The system drift of the LisaLENS is substantially smaller than that of the HP5529A in such environmental conditions. 5.2.2. Sensitivity The sensitivity of an instrument is typically calculated as the ratio of the transducer output signal intensity to the measured physical quantity. However, it is difficult to apply this concept to laser encoders or optical interferometers. Consider the quantities resulting from the processing of the transducer/LisaLENS0 output signals; that is, dF; dx is the measured physical quantity. Eq. (13) shows the measurement sensitivity for the LisaLENS: dF 16p ¼ : dx r
ð27Þ
where the total differentiation d(dx) represents the displacement measurement error. Eq. (25) shows two main error sources for the proposed system: the grating period variation dr and the phase shift error d(dF). In this system, a smaller grating period r leads to a smaller phase noise influence. The manufacture tolerance of the grating period directly influences the accuracy of the displacement measurement. The period variation, however, comes from random manufacture error. Consider the period variation and assume that the phase shift error is zero. Assume that the interference signals are only in arbitrary cycles (2p) or arbitrary r=8 displacement. The corresponding displacement error can be represented as:
dðdxÞ ¼
dr 8
in a cycle:
ð26Þ
The laser spot diameter on the GS is approximately 2 mm and covers approximately 6250 grating lines. For the 1 mm travel range of the GS, the interference signals run approximately 5000 cycles. With an equal accuracy consideration and 50 nm displacement accuracy requirement for the 1 mm travel range, the manufacture
Fig. 14. The system stability test results for the LisaLENS and the HP5529A for 1800 s.
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A grating with a period of 1.6 mm leads to dF=dx ¼ 1:81=nm:
97
6. Conclusion ð28Þ
Eq. (28) represents the measurement sensitivity for the LisaLENS. 5.2.3. Resolution For a 16-bit analog to digit conversion (ADC) resolution, (28) produces a theoretical displacement measurement resolution smaller than 0.6 pm for the LisaLENS. However, the measured phase suffers from various high- and low-frequency noises. The high-frequency phase noises may come from the laser source, photodetectors, and operational amplifier electronics, whereas the low-frequency phase noise may result from slow variation of the optical path difference, such as foundation vibration and atmospheric disturbances. Consider the experiment shown in Fig. 14. Because the stage is held still (dF ¼0), the effect of the grating period variation can be dropped from (25). That is, the result in Fig. 14 represents only the phase noises. The measurement time for all of the experiments in this study is within 180 s. For each time interval of 180 s, we can have the corresponding maximum and minimum displacement data. The estimated resolution can be represented using the difference between the maximum and minimum displacement data. Thus we can have the estimated resolution statistically. A review of every 180 s time interval in Fig. 14 shows an estimated measurement resolution of approximately 5.5 71.4 nm. If the effective bit number for ADC is considered, we can obtain the equivalent effective bit number of approximately 4 bits. 5.2.4. Head-to-scale tolerance In Section 2.2, we choose b ¼0.1 in all runout tolerance estimations for the LisaLENS. This choice can ensure that the runout effect on the measurement is small enough. Actually, as the two interference beams of the LisaLENS are essentially parallel to each other, we can arrange two focus lenses for each of the PD1 and PD2 respectively. The active areas of the PD1 and PD2 are aligned at the focal distances of each of the lenses. The optical axis of each lens is aligned parallel to the two interference beams such that the two interference beams always overlap at the same positions on the PD1 and PD2. In this way, the runout effect on the interference signals can be reduced to zero, and the head-to-scale tolerance of the LisaLENS can be further raised. 5.2.5. Limit on stage moving velocity For experiments in this paper, a high performance field programmable gate array (FPGA) was developed to real-time implement the arctangent phase decoding method to 20 MHz. The real-time decoded displacement information was sent back using a high-speed PCI-DIO card installed on a PC computer. Based on our hardware setup, the stage moving velocity is limited to smaller than 3 mm/s. A moving velocity larger than 3 mm/s will induce phase unwrapping error.
This study presents the LisaLENS, which uses two direct return diffracted beams to interfere with each other. It adopts an optical configuration with a corner cube and a beam expander to ensure high tolerance on the head-to-scale runout and the grating period. The GS displacement can be determined by the relationship between the measured optical phase change and the grating period. Experimental results demonstrate that the LisaLENS is able to measure both long- and short-range displacements. The theoretical prediction of the measurement resolution is approximately 0.6 pm. If only high-frequency noises are considered, the measurement resolution of the system is estimated to be 5.571.4 nm. The LisaLENS also offers the advantages of high sensitivity and a long measurement range, making it useful for monitoring the displacement and vibration of a precision motorized stage in various applications.
Acknowledgments The authors would like to acknowledge the support of the National Science Council (NSC) 98–2218-E-032-009-, 99–2628-E032-001-, and 100–2628-E-032-001-). We also thank Mr. M.H. Wen (E-Pin Optical Industry Company, Taiwan) and Professor C.K. Lee (National Taiwan University) for providing us with access to equipment.
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