OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 24 — Nov. 26, 2007
  • pp: 15759–15766
« Show journal navigation

A compact system for simultaneous measurement of linear and angular displacements of nano-stages

Jae Wan Kim, Chu-Shik Kang, Jong-Ahn Kim, Taebong Eom, Mijung Cho, and Hong Jin Kong  »View Author Affiliations


Optics Express, Vol. 15, Issue 24, pp. 15759-15766 (2007)
http://dx.doi.org/10.1364/OE.15.015759


View Full Text Article

Acrobat PDF (294 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We report on a novel compact interferometery system for measuring parasitic motions of a precision stage. It is a combination of a Michelson interferometer with an auto-collimator, of which full physical dimension is mere 70 mm×80 mm×35 mm (W×L×H) including optical components, photo-detectors, and electronic circuits. Since the beams, which measure displacement and angle, can be directed at the same position on the moving mirror, the system is applicable for testing small nano-stages where commercial interferometers are not able to be used. And thus, errors from nano-scale deformation of the moving mirror can be minimized. We find that the residual errors of linear and angular motion measurements are 2.5 nm in peak-to-peak and 0.2″, respectively.

© 2007 Optical Society of America

1. Introduction

Various applications in the fields of nanotechnology require a high performance precision stage system for imaging with scanning probe microscopes (SPMs), positioning wafers and masks, lithographing patterns, and so on. Since the critical dimension in these applications approaches scales of a few nanometers and possibly sub-nanometers in some cases, the required specification of stage system is getting more severe in accuracy besides resolution [1

1. R. Leach, J. Haycocks, K. Jackson, A. Lewis, S. Oldfield, and A. Yacot, “Advances in traceable nanometrology at the National Physical Laboratory,” Nanotechnology 12, R1–R6 (2001). [CrossRef]

,2

2. J.-A. Kim, J. W. Kim, B. C. Park, and T. B. Eom, “Measurement of microscope calibration standards in nanometrology using a metrological atomic force microscope,” Meas. Sci. Technol. 17, 1792–1800 (2006). [CrossRef]

]. The accuracy of stage system is affected by several factors including parasitic angular motions due to Abbe offset and these motions result in subsequent Abbe error which is one of most critical aspects in designing precision systems. Therefore, combined effects of the parasitic angular and linear motions on the stage need to be compensated to achieve the highest accuracy typically required in such applications [3

3. S. Gonda, T. Kurosawa, and Y. Tanimura, “Mechanical performances of a symmetrical, monolithic three-dimensional fine-motion stage for nanometrology,” Meas. Sci. Technol. 10, 986–993 (1999). [CrossRef]

5

5. W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Prec. Eng. 30, 96–103 (2006). [CrossRef]

].

Laser interferometer and auto-collimator are typical instruments for measuring linear and angular motions [6

6. A. Bergamin, G. Cavagnero, and G. Mana, “A displacement and angle interferometer with subatomic resolution,” Rev. Sci. Instrum. 64, 3076–3080 (1993). [CrossRef]

8

8. J, S. Oh, E. D. Bae, T. Keem, and S.-W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” International Journal of Machine Tools and Manufacture 46, 1748–1752 (2006). [CrossRef]

]. Laser interferometers is mostly employed for the linear displacement measurement, but also applied to the measurement of angular displacement through the modified configurations. Some of the configurations using multiple interferometers is patented and commercialized to measure the linear and angular displacement simultaneously (Agilent multi-axis interferometer 10737L). However the resolution of angular displacement measurement is determined by the distance between multiple measurement points and resolutions of interferometers. Therefore, in case of the precision systems which typically lack sufficient space for attaching multiple interferometers, it is difficult to achieve sub-arcsecond resolution due to the limited distance between the interferometers.

Auto-collimator is widely used for evaluating the parasitic angular motions of stage because it has high resolution and reliability with a simple structure [9

9. P. R. Yoder Jr., E. R. Schlesinger, and J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890–1895 (1975). [CrossRef] [PubMed]

,10

10. J. Yuan, X. Long, and K. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005). [CrossRef]

]. However, it is not usually applied in situ, and thus the application from the evaluation result is only limited to rough estimation of the magnitude of parasitic angular motions. For this reason, they can not be estimated and compensated properly in an actual state, which vary according to operating conditions such as load and moving speed [2

2. J.-A. Kim, J. W. Kim, B. C. Park, and T. B. Eom, “Measurement of microscope calibration standards in nanometrology using a metrological atomic force microscope,” Meas. Sci. Technol. 17, 1792–1800 (2006). [CrossRef]

,4

4. M. Holmes, R. Hocken, and D. Trumper, “The long-range scanning stage: a novel platform for scanned-probe microscopy,” Prec. Eng. 24, 191–209 (2000). [CrossRef]

,5

5. W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Prec. Eng. 30, 96–103 (2006). [CrossRef]

]. Therefore, simultaneous measurement using a laser interferometer and an auto-collimator is essential to compensate parasitic angular motion and achieve high accuracy of the stage system. In some cases, simple additions of these sensors were implemented successfully, but they usually require extra space for sensors, which increases the complexity of overall system.

Demand of a new device for convenient and complete characterization of precision stages led to development of the Michelson Interferometer (MI) Combined with an auto-collimator (MICA). Because the MI and auto-collimator share laser beam and are aligned on same optical axis, a compact design is available and the optical axis can be easily and precisely aligned orthogonally to the moving mirror. In this paper, the principle of operation is briefly described, and then the characteristics and the performance of MICA will be elaborated.

2. Principle of operation

MICA is a system to measure linear and angular displacement simultaneously. Its structure is a combination of a MI and an autocollimator. Although the overall structure of MI and autocollimator are different, MI and autocollimator have some common parts in the sense that optical beam from the light source is directed to a mirror and then reflected back. We could combine these two instruments by designing MICA such that the light reflected by a mirror is partially reflected at the beam splitter of MI. The reflected beam and the transmitted beam are used for MI and autocollimator, respectively.

Since the beams measuring linear and angular displacement are incident at the same point on the moving mirror, MICA has advantages in that it can be applied to linear and angular motion measurements for nano-stages whose physical dimension is too small for commercial laser interferometers to be used, and that possible measurement error arising from nano-scale deformation of the moving mirror, due to environmental effect, is minimized.

Fig. 1. Schematic diagram of basic structure of MICA. LS: light source; BS: beam splitter; QPD: quadratic photo diode; CL: collimating lens; PBS: polarizing beam splitter; M1 and M2: mirrors; Q1 and Q2: quarter wave plates: P: polarizer; L: lens; PD: photo diode.

A schematic diagram of the basic structure of MICA is given in Fig. 1. Light from a frequency stabilized laser whose linear polarization is at some angle with respect to the horizontal, is focused to behave like a point source. The diverging beam is then collimated by the lens CL and directed to the polarizing beam splitter PBS. The s-polarization component reflects at the PBS, and passes through a quarter-wave plate Q1 whose slow axis is at 45° to the horizontal. The beam is reflected by the reference mirror M1, and then again passes the Q1 resulting in a light with p-polarization. It passes through the PBS and directed to the polarizer P.

On the other hand, the p-polarization component of the collimated beam at PBS is transmitted and passes a quarter-wave plate Q2 whose slow axis is at 22.5° to the horizontal axis. After being reflected by the moving mirror M2, it passes the Q2 once again. The resulting polarization is linear with its direction at 45° to the horizontal axis. The s-polarization component is reflected at the PBS and directed to P, whereas the p-polarization component is transmitted through the PBS and reflected by the beam splitter BS to the position sensitive detector (for example, quadratic photo diode; QPD). This beam incident to QPD is used for the angle measurement, and the two orthogonal beams incident to P is mixed by P and their interference signal is used for the linear displacement.

The combination of an auto-collimator with a Michelson interferometer is possible by using the fact that when a linearly polarized light passes a quarter wave plate whose slow axis is at 22.5° to the horizontal, and then reflected back through the quarter-wave plate, the resulting polarization state of light is linear with its polarization vector rotated at 45° to the horizontal. This could easily be seen by using the Jones calculus. If we denote a quarter-wave plate with its slow axis at θ to the horizontal as Q(θ), it can be expressed as [11

11. G. R, Fowles, Introduction to Modern Optics (Dover Publications, 1989), Chap. 2.

]

Q(θ)=(cos2θ+isin2θ(1i)sinθcosθ(1i)sinθcosθsin2θ+icos2θ).
(1)

When this quarter-wave plate is seen by light propagating in opposite direction, it will be regarded as Q(-θ). Thus, when horizontally polarized light (1 0)T makes a round trip through this quarter-wave plate, the resulting polarization will be

Q(θ)RQ(θ)(10)=(sin2θcos2θsin2θ)
(2)

where R represents the reflection matrix, R=(1001). At θ=22.5°, the polarization becomes 12(11)T, which means that the beam contains both the p- and s-polarization states.

Small angular displacement is calculated by using the signals obtained at the QPD. The yaw (θ Y) or pitch angle (θ P) is given by

θY=rY2fandθP=rP2f,
(3)

where f is the focal length of the lens and r Y and r P are the distances along yaw and pitch directions measured by the QPD. If the signals at the four quadrants are denoted by V i(i=1,2,3,4), the yaw angle θ Y and the pitch angle θ P could be measured by the following relations,

θY=KYΘYandθP=KPΘP
(4)

where K Y and K P are the calibration constants for the yaw and pitch angle measurements, respectively, and ΘY, ΘP are defined respectively by

ΘY=V1V2+V3V4V1+V2+V3+V4
(5)
ΘP=V1+V2V3V4V1+V2+V3+V4.
(6)

The calibration constants should be determined by experiments.

3. Experiments

3.1 Design and fabrication of MICA

One of our key aims was to build it in compact size. This was achieved by using an optical fiber which made it possible to separate the light source from MICA.

In principle, MICA can either be used for homodyne laser interferometry or heterodyne laser interferometry. In case of heterodyne interferometry, however, when laser beam is propagated through an optical fiber, polarization mixing of the two orthogonal polarizations could occur, and there could be nonlinearity in the measurement results. Thus, we designed MICA for homodyne interferometry in our application.

For practical use of MICA, the basic configuration shown in Fig. 1 has been modified so that the error in linear displacement measurement arising from tilt of the moving mirror could be removed. This is achieved by using cube-corner prisms as shown in Fig. 2. In this double pass configuration, the laser beam exits in parallel with the input beam even if the moving mirror is tilted during movement. Thus, the linear displacement measurement is not affected by tilt of the moving mirror. Besides, the resolution is enhanced by factor 2.

Figure 2(a) shows the structure of the realized MICA, and its photograph is shown in Fig. 2(b). As shown in Fig. 2(b), physical size of MICA, which includes optical components, photo-detectors, and electronic circuits, is only 70 mm×80 mm×35 mm (W×L×H). The size of cube corner prisms and PBS is 1 inch and the surfaces are anti-reflection coated. The quarter-wave plates Q1 and Q2 of 10 mm diameter are aligned in manner that slow axes of Q1 and Q2 are at 45° and 22.5°, respectively, to the horizontal.

A frequency stabilized laser (Melles Griot) is coupled to a polarization maintaining single mode optical fiber (PMSMF), and used as the light source. The laser was used in power stabilized mode. The measured spectral and power stability of the laser were 5.7×10-9 and 0.16 %, respectively. The core size of the PMSMF is 4 µm and the cutoff wavelength is 570 nm. When coupling the laser to the PMSMF, a half-wave plate was used to match the polarization axis of laser with the fast axis of the fiber to minimize the cross coupling of polarization modes. The power fluctuation of laser measured during 6 hours at the exit of PMSMF was 0.23 % in rms, whereas that measured beyond the PBS was 0.62 % in rms. This level of fluctuation is small enough for the laser to be used for a homodyne interferometer.

Fig. 2. The (a) schematic diagram and (b) photograph of MICA. OI: optical isolator; BS1 and BS2: beam splitters; L and CL: lens; PBS1 and PBS2: polarizing beam splitter; W1 and W2: half-wave plate; Q1, Q2, and Q3: quarter-wave plates; CC1 and CC2: cube-corner prisms; M, M1, and M2: mirrors; PD1, PD2, and PD3: photo detectors; RP: right angle prism; PMSMF: polarization maintaining single mode fiber; QPD: quadratic photo detector; P: polarizer.

In order to maximize the contrast of the interference signals, the input polarization was adjusted by rotating the fiber end. By calculation, we set the input polarization such that the light is split at the PBS in the manner that 1/3 of the input intensity is reflected and 2/3 is transmitted. This angle corresponds to tan-1(1/√2)≅35.3°.

In general, four signals, ±sin and ±cos, are used to analyze the optical interference and to reduce the effect of power fluctuation of light. In our case, however, only three signals, ±sin and +cos, were used to reduce the size of MICA. These signals are measured by the photo-detectors, PD1, PD2, and PD3, and then transformed to two differential signals having phase difference of 90°. Offsets, amplitude difference, and phase shift deviation from 90° of these signals are set to zero by using an electronic circuit which was designed to reduce the nonlinearity error of the laser interferometer system. Resistances of resistors in this circuit are fixed so that the Lissajous figure of the sine and cosine signals is best fitted to a circle. The sine and cosine signals are then fed to the interpolation circuit (MicroEsystem Mercury encoder, M3500) and then to the motion board (NI PCI-7354), in which the conversion to length is made. We set interpolation of M3500 being 128 and the corresponding resolution is 1.2 nm.

Instead of using this route, we can also directly capture the sine and cosine signals with the data-acquisition (DAQ) board (NI PCI-6259) and perform the elliptical fitting in real time to minimize the nonlinearity of the interferometer. This measuring mode, however, is limited by the speed of the DAQ board, and is not applicable to high speed displacements.

The QPD is positioned such that when the moving mirror is vertical to the laser beam, the four segments of QPD are equally illuminated by the laser. The verticality of the mirror was confirmed by detecting laser light being transmitted back through the fiber to the laser. When a QPD is placed at the focal plane of the lens L, purely linear displacement of the moving mirror would not affect the signal of QPD. In our system, however, in order to enlarge the laser spot size at the QPD, QPD is placed about 150 µm away from the focal place of the lens. This configuration could, in general, induce error in angle measurement because position of the laser focal spot changes according to the position of the moving mirror. However, considering the displacement range of nano-stages which is normally about 100 µm, this error is evaluated to be less than a nanometer.

The PBS was tilted until the parasitic reflections from surfaces of PBS are imaged outside the QPD, in order to prevent error in angle measurement by these ghost signals.

3.2. Calibration and performance test of MICA

The residual nonlinearity of the interferometer system was checked by using the elliptical least squares fitting technique [12

12. C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996). [CrossRef]

]. Before the interpolation stage, the two interference signals were saved separately while translating the stage, and fitted to a general ellipse equation. The length values obtained from the fitted phases are used as the reference values, and the deviation of length values measured by the interferometer from the reference values are calculated. The results are plotted in Fig. 3 together with the deviation from the reference, of the length obtained by the raw signals without passing the nonlinearity compensation circuit. It can be seen from the graphs that the nonlinearity of about 20 nm (peak-to-peak) has been reduced to 2.5 nm.

Fig. 3. Residual nonlinearity of the laser interferometer

For more accurate measurements of low speed displacement, we can use alternate measurement mode in which real time elliptical fitting of the raw sine and cosine signals are performed. In this mode, the nonlinearity is reduced to the electronic noise level of 0.5 nm. For high speed measurements, we are planning to upgrade our system by applying the technique of intermittent compensation of nonlinearity error [13

13. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Removing nonlinearity of a homodyne interferomerer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43, 2443–2448 (2004). [CrossRef] [PubMed]

].

Fig. 4. Setup for calibration of autocollimator part of MICA. M: Mirror; CA and CB: cube-corner prism; S: spring, MM: micrometer; D: photo detector; O: pivot point of rotational arm.

The autocollimator part of MICA was calibrated by using a small angle generator (SAG) equipped with a laser angle interferometer [14

14. T. Eom, D. Chung, and J. Kim, “A small angle generator based on a laser angle interferometer,” International Journal of Precision Engineering and Manufacturing 8, 20–23 (2007).

]. The calibration setup is shown in Fig. 4. A motorized micrometer is used to finely rotate the arm of SAG, on which a plane mirror is mounted. This rotation of the mirror is detected by the MICA through the voltage changes on the QPD, and the yaw angle, θ Y, is measured by the laser interferometer by the relation θ Y=sin-1(d/L), where L is the distance between two cube-corner prisms (C A and C B), and d, which is measured by the laser interferometer, is the relative displacement of the cube-corner prisms. For small angle, ΘY will be linear to θ Y, and beyond some range, ΘY will converge to +1 or -1 depending on the tilt direction of the mirror.

Fig. 5. (a) Calibration of the auto-collimator part in MICA and (b) the residuals.

Figure 5(a) shows average of 5 repeatedly measured data of ΘY as a function of θ Y, which is the angle measured by the SAG. To evaluate the repeatability, we have taken the data during 2 days. We performed a least-squares fitting of ΘY to the polynomials of θ Y of degree 5, to find out the relation between θ Y and ΘY. The region from -30″ to 30″ shows linear relation between θ Y and ΘY. Figure 5(b) shows the residuals of 5 repeatedly measured data from the fitted curve. The cross shows the residuals of average data which were used for getting the fitted curve. The standard deviation of the residuals of the average is 0.004, which corresponds to angle of 0.2′. Similar calibration has been performed for the pitch angle measurement with the MICA rotated 90° around the optical axis of the interferometer.

Fig. 6. Test results of a nano-stage. (a) Parasitic motion and (b) residual of capacitance sensor

As an example of the application, parasitic motion of a linear nano-stage has been measured by using MICA. The nano-stage is home-made and is equipped with a capacitive sensor (PI D-100). The linear displacement is checked up to 18 µm, and the angular motions for yaw and pitch are measured simultaneously. The measurement results are plotted in Fig. 6. Signals from interferometer, cap sensor, and QPD are measured with 1 ms sampling time. Yaw angle is measured to be about 1 arcsecond whereas the pitch angle is measured to be less than 0.3 arcsecond, indicating higher stiffness of the stage for pitch motion. The cap sensor shows phase delay when the stage changes its moving direction.

Periodic signal component of about 0.2 arcsecond amplitude is included in the angle measurement data shown in Fig. 6(a). If we shorten the scan range for the measurement, the periodicity is more clearly observed. This periodic signal, which was not appearing in Fig. 5, is the interference noise arising from the interference between a beam reflected from a stationary surface and a beam reflected from the moving mirror. Although practically this interference noise can not be perfectly removed, it could be reduced if we minimize the number of reflecting surfaces by using uncoated cube-corner prisms and quarter-wave plates instead of the anti-reflection coated ones, and gluing them on the uncoated PBS surfaces with optical cement.

5. Conclusion

The MICA, which is a new, compact device for simultaneous measurement of linear displacement and angle, has been developed. Although it is a combination of a Michelson interferometer with an auto-collimator, it has compact size of 70 mm×80 mm×35 mm (W×L×H), including optical components, photo-detectors, and electronic circuits. Several configurations of interferometer can be adopted into the MICA and the size can be further reduced if we use single beam pass configuration. The residual nonlinear error and resolution of interferometer are 2.5 nm in peak-to-peak and 1.2 nm, respectively. The angular motion can be measured with accuracy of 0.2″ in range of ±30″.

Acknowledgement

This work was supported by the KRISS through the ‘Measurement Standards for Future Generation Needs and Emerging Technology’ project.

References and links

1.

R. Leach, J. Haycocks, K. Jackson, A. Lewis, S. Oldfield, and A. Yacot, “Advances in traceable nanometrology at the National Physical Laboratory,” Nanotechnology 12, R1–R6 (2001). [CrossRef]

2.

J.-A. Kim, J. W. Kim, B. C. Park, and T. B. Eom, “Measurement of microscope calibration standards in nanometrology using a metrological atomic force microscope,” Meas. Sci. Technol. 17, 1792–1800 (2006). [CrossRef]

3.

S. Gonda, T. Kurosawa, and Y. Tanimura, “Mechanical performances of a symmetrical, monolithic three-dimensional fine-motion stage for nanometrology,” Meas. Sci. Technol. 10, 986–993 (1999). [CrossRef]

4.

M. Holmes, R. Hocken, and D. Trumper, “The long-range scanning stage: a novel platform for scanned-probe microscopy,” Prec. Eng. 24, 191–209 (2000). [CrossRef]

5.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Prec. Eng. 30, 96–103 (2006). [CrossRef]

6.

A. Bergamin, G. Cavagnero, and G. Mana, “A displacement and angle interferometer with subatomic resolution,” Rev. Sci. Instrum. 64, 3076–3080 (1993). [CrossRef]

7.

G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646–1651 (1974). [CrossRef] [PubMed]

8.

J, S. Oh, E. D. Bae, T. Keem, and S.-W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” International Journal of Machine Tools and Manufacture 46, 1748–1752 (2006). [CrossRef]

9.

P. R. Yoder Jr., E. R. Schlesinger, and J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890–1895 (1975). [CrossRef] [PubMed]

10.

J. Yuan, X. Long, and K. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005). [CrossRef]

11.

G. R, Fowles, Introduction to Modern Optics (Dover Publications, 1989), Chap. 2.

12.

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996). [CrossRef]

13.

T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Removing nonlinearity of a homodyne interferomerer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43, 2443–2448 (2004). [CrossRef] [PubMed]

14.

T. Eom, D. Chung, and J. Kim, “A small angle generator based on a laser angle interferometer,” International Journal of Precision Engineering and Manufacturing 8, 20–23 (2007).

OCIS Codes
(120.1880) Instrumentation, measurement, and metrology : Detection
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: August 31, 2007
Revised Manuscript: November 4, 2007
Manuscript Accepted: November 8, 2007
Published: November 13, 2007

Citation
Jae-Wan Kim, Chu-Shik Kang, Jong-Ahn Kim, Taebong Eom, Mijung Cho, and Hong Jin Kong, "A compact system for simultaneous measurement of linear and angular displacements of nano-stages," Opt. Express 15, 15759-15766 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15759


Sort:  Year  |  Journal  |  Reset  

References

  1. R. Leach, J. Haycocks, K. Jackson, A. Lewis, S. Oldfield, and A. Yacot, "Advances in traceable nanometrology at the National Physical Laboratory," Nanotechnology 12, R1-R6 (2001). [CrossRef]
  2. J.-A. Kim, J. W. Kim, B. C. Park, and T. B. Eom, "Measurement of microscope calibration standards in nanometrology using a metrological atomic force microscope," Meas. Sci. Technol. 17, 1792-1800 (2006). [CrossRef]
  3. S. Gonda, T. Kurosawa and Y. Tanimura, "Mechanical performances of a symmetrical, monolithic three-dimensional fine-motion stage for nanometrology," Meas. Sci. Technol. 10, 986-993 (1999). [CrossRef]
  4. M. Holmes, R. Hocken, and D. Trumper, "The long-range scanning stage: a novel platform for scanned-probe microscopy," Precis. Eng. 24, 191-209 (2000). [CrossRef]
  5. W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, "Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage," Precis. Eng. 30, 96-103 (2006). [CrossRef]
  6. A. Bergamin, G. Cavagnero, and G. Mana, "A displacement and angle interferometer with subatomic resolution," Rev. Sci. Instrum. 64, 3076-3080 (1993). [CrossRef]
  7. G. D. Chapman, "Interferometric angular measurement," Appl. Opt. 13, 1646-1651 (1974). [CrossRef] [PubMed]
  8. J, S. Oh, E. D. Bae, T. Keem, and S.-W. Kim, "Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry," Int. J. Mach. Tools Manuf. 46, 1748-1752 (2006). [CrossRef]
  9. P. R. Yoder, Jr., E. R. Schlesinger, and J. L. Chickvary, "Active annular-beam laser autocollimator system," Appl. Opt. 14, 1890-1895 (1975). [CrossRef] [PubMed]
  10. J. Yuan, X. Long, and K. Yang, "Temperature-controlled autocollimator with ultrahigh angular measuring precision," Rev. Sci. Instrum. 76, 125106 (2005). [CrossRef]
  11. G. R. Fowles, Introduction to Modern Optics (Dover Publications, 1989), Chap. 2.
  12. C.-M. Wu, C.-S. Su, and G.-S. Peng, "Correction of nonlinearity in one-frequency optical interferometry," Meas. Sci. Technol. 7, 520-524 (1996). [CrossRef]
  13. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, "Removing nonlinearity of a homodyne interferomerer by adjusting the gains of its quadrature detector systems," Appl. Opt. 43, 2443-2448 (2004). [CrossRef] [PubMed]
  14. T. Eom, D. Chung, and J. Kim, "A small angle generator based on a laser angle interferometer," Int. J. Precis. Eng. Manuf. 8, 20-23 (2007).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited