A calibrator utilizing a low-coherent light source straightness interferometer and a compensation method is introduced for straightness measurements in this paper. Where the interference pattern, which is modulated by an envelope function, generated by the interferometer undergoes a shifting as the Wolaston prism of the interferometer experiences a lateral displacement, and the compensation method senses the displacement by driving the prism back to the position to restore the pattern. A setup, which is with a measurement sensitivity of 36.6° /μm, constructed for realizing the calibrator is demonstrated. The experimental results from the uses of the setup reveal that the setup is with a measurement resolution and stability of 0.019 and 0.08μm, respectively, validate the calibrator, and confirm the calibrator’s applicability of straightness measurements and advantage of extensible working distance.
© 2011 OSA
The instruments adopted for examining straightness errors  of linear translation stages can be divided into two categories: angular and lateral displacement types. The electronic levels [2–4] and autocollimators [5–7] belong to the angular displacement type. They separate the travel length of a stage to be detected into several spans, measure the inclinations of the spans, and determine the straightness by the use of the measured inclinations. The apparatuses using PSD (Position sensing detector) [8–10], QD (Quadrant detector) [11, 12], and heterodyne laser interferometers [13–17] are part of the lateral displacement type, they sense the lateral positions of the span nodes and determine the straightness by using the measured positions.
The interferometers comprise four important parts: a heterodyne laser source, a beam-divider, a beam-reflector, and two receivers. Of which one part of the laser is guided to one of the receivers to create a reference interference signal. The other part of the laser is divided into two divergent beams by the divider; the divergent beams propagate to the reflector and then return to their incoming paths; and the return beams are combined into one beam by the divider and finally directed to the other receiver to generate a measurement interference signal. As the divider travels to each of the span nodes, the lateral position can be extracted by comparing the phases of the measurement and the reference interference signals.
The interferometers have the advantages of high measurement resolution, sensitivity, and stability, they are therefore widely employed for stage performance examinations. The working distance, however, which is restrained by the angle between the divergent beams, is without extensibility. Since any action taken during the measurement for extending the working distance would interrupt the phase comparison, vary the paths of the return beams, and consequently ruin the testing with no recovery. The stitching method [18, 19] can be a solution of this issue. It measures the profiles of several partially overlapped sub-sections of the stage, compensates the offset and tilt values between adjacent sub-sections by using profile matching technique, and constructs the profile and then determines the straightness of the overall stage. Nevertheless, an error of a tilt compensation of an overlapping creates error to the subsequent measured profiles. And the farther away the subsequent measured profiles are, the larger the created error is.
A calibrator utilizing a low-coherent light source straightness interferometer and a compensation method is thus proposed. It possesses not only the advantages of the interferometers but also the ability of extending the working distance. In this paper, the straightness interferometer and compensation method are first introduced, a setup installed to accomplish the calibrator is then demonstrated, and the experiments and experimental results from the uses of the setup are finally presented and discussed.
2. The straightness interferometer and compensation method
The straightness interferometer: Fig. 1 (a) depicts the schematic diagram of the straightness interferometer, which is composed of a polarizer (P) with transmitting axis at 45°, beam-splitter (BS), Wollaston prism (WP), bi-wedge prism (BP), reflecting mirror (M), analyzer (A) with transmitting axis at 45°, and a CCD camera. As a light beam is incident into the interferometer, it is separated into two divergent beams of which the divergent angle is 2θ and the polarization states are linear and perpendicular by the Wollaston prism; the divergent beams travel through the bi-wedge prism and then make a reverse propagation at the mirror; and the reverse beams are combined by the Wollaston prism and finally guided to transmit through the analyzer to generate an interference pattern on the CCD camera.
Let the Jones matrices of the polarizer and analyzer be denoted as MP and MA, respectively, and the effective Jones matrix of the combination of the Wollaston prism, bi-wedge prism, and reflecting mirror be represented by Meff, the beam incident on the polarizer and that emerging from the analyzer have a relation of 
Hence, the intensity of the interference pattern, which is equal to , can be expressed as
This interference equation is only available for narrow-band source. For a broad-band source centered at λc, the equation has to be modified as 
In which, δ1is due to the lateral distance, x, between the Wollaston prism and bi-wedge prism, according to Ref ,
The above descriptions indicate that the interference pattern comprises a set of straight fringes; the zero-order fringe in the pattern is with the maximum intensity; and a small lateral displacement, Δx, of the Wollaston prism shifts all the fringes, including the zero-order one, away from the original location.
These imply, once the Wollaston prism is carried to experience a displacement along the z-axis by a translation stage, the small lateral displacement (i.e., as that demonstrated in Fig. 1(b)) of the prism due to the lateral position of the track of the translation stage can be determined by laterally moving the prism back until the zero-order fringe appears at the original location, and the inverse of the movement is the lateral displacement the prism has undergone.
The compensation method: The compensation method described below is employed for guiding the Wollaston prism back.
Figure 2 demonstrates three points, P, Q, and R, on the interference fringe pattern. Where point Q is located at the original location of the zero-order fringe; P and R are adjacent to Q; and is approximately perpendicular to the image fringes. A non-dimensional intensity S is thus defined as
Therefore, as a small lateral displacement of the Wollaston prism occurs, the compensation method senses the variation of the S value, commands the control system to immediately move the prism back according to the sensed variation and the slope of the S-curve, and retrieves lateral displacement of the prism. Note the judgment of whether the prism has been moved back is whether the intensity of point Q reaches its maximum or not. And the compensation method is consecutively executed during the moving of the Wollaston prism, the integration of the measured lateral displacements, i.e. , is thus equal to the lateral position, d, of the translation stage. The lateral position d is also revealed in Fig. 1(b).
3. The way of extending the working distance
As that demonstrated in Fig. 3 , the movement of the Wollaston prism is with a dead point. However, since a longitudinal displacement (i.e., displacement along the z-axis) as well as any other small displacements of the bi-wedge prism does not alter the propagation directions of the reverse beams, the measurement can be continued by moving the bi-wedge prism forward (i.e., move the prism from the place of the dash-line to that of the solid-line). Of course this movement would shift the fringes of the interference pattern, however a lateral movement (i.e., a movement along the x-axis) of the Wollaston prism is again able to drag the fringes back. The working distance of this interferometer is thus extensible.
4. The experimental setup of the calibrator
Figure 4 represents the setup constructed to realize the proposed calibrator, which is composed of a light source module, the straightness interferometer, and an image processing system.
The light source module is composed of a lamp, three lenses, and a pin hole (PH). It emits a broad-band collimated light beam with a central wavelength of λc = 550 nm to the optical setup of the straightness interferometer.
In the straightness interferometer, the Wollaston prism (Agilent 10774A from Agilent Technologies, Inc.) ejects two beams with a divergent angle of 2θ = 1.6° and it is assembled on a nano-stage (P-622.1 CD from Physik Instrumente GmbH & Co.) which is driven by a PZT actuator and with a capacitive sensor. The assembly of the Wollaston prism and nano-stage is abbreviated as the core-stage, it is in fact the probe of the calibrator and it is placed on the test object as the calibrator is adopted to examine the straightness of the object. In addition, the Wollaston prism is with a circular aperture, it makes the interference image be with a boundary as that shown in Fig. 2.
The image processing system contains a frame grabber, a stage controller/driver (E-625 from Physik Instrumente GmbH & Co.), and a personal computer. The frame grabber captures the interference patterns mapped on the CCD camera and sends these pattern images to the computer. The stage controller/driver amplifies the signals delivered from the computer to drive the nano-stage, and feeds the movements of the nano-stage back to the computer. And the computer can execute two programs: calibration and measurement.
The calibration program sends an analog signal to the nano-stage to bring the Wollaston prism to experience a lateral displacement scanning; the program then receives and stores the displacements of the prism and the images on the camera as the prism is scanning; and by using the stored displacements and images, the program determines the correlograms (intensity with respect to x) of the points P, Q, and R, which are further adopted to generate the S-curve and find the maximum intensity of the point Q. The measurement program determines the lateral positions at the span nodes of the stage to be examined by controlling the core-stage and using the compensation method; it then calculates the straightness error of the examined stage and exhibits the measurement result on the monitor. Here, P and R are away from Q by a distance of 4 pixels, the intensities of P, Q, and R of each image are determined by averaging the grey levels of 55 pixels around P, Q, and R, respectively. The area comprising the 55 pixels is defined as the average area.
Note the sensitivity of an instrument represents the ratio of the variation of the observed variable to the corresponding variation of the measured quantity. For the calibrator, the observed variable and the measured quantity are the phase of the interferometer and lateral displacement of the Wollaston prism, respectively. According to Eqs. (5) and (6), the sensitivity of the proposed calibrator can thus be written as . Substituting the described λc and θ into this equation reveals the sensitivity of the constructed calibrator, which is = 36.6°/μm.
5. Experiments and experimental results
The installed calibrator executed the calibration program first. As per the correlogram shown in Fig. 5 , the point Q had a maximum intensity of 234 gray level. As per the S-curve shown in Fig. 6 , the curve was with a slope of −0.188/μm.
The installed calibrator was then conducted to examine the stability of the installed setup and verify the validity and applicability of the proposed calibrator.
During the stability examination, the core-stage was in a stationary state and the calibrator performed the measurement for one hour. The measured displacements were due to the noise of environmental perturbations, and, as the result of the examination shown Fig. 7 , the displacements had a standard deviation of 0.019μm and a maximum variation of 0.08 μm. The former is regarded as the smallest change the calibrator can detect, the calibrator is therefore with a measurement resolution of 0.019μm. The latter is defined as the stability of the calibrator. Because the calibrator would detect a lateral displacement containing the component due to the noise, the calibrator may examine a lateral displacement with an error of the magnitude of the stability.
In the validity verification, a nano-stage (P-730.4 L from Physik Instrumente GmbH & Co.) with a PZT actuator and capacitive sensor was driven to alter its lateral position from −20 to 20 μm step by step, and the calibrator was conducted to measure the lateral positions at these steps. The alteration of the lateral position was repeated 10 times, the calibrator re-measured the positions as the alteration was repeated, and the measurement average at each step is revealed in Fig. 8 . Besides, the given lateral positions, the positions fed back from the examined nano-stage, and the differences between the measured and given positions are also exhibited in the same figure. As the small differences indicated in this figure, the given and measured positions are with good consistence, the validity of the proposed calibrator is thus confirmed.
And in the application verification, the installed calibrator was utilized to examine the horizontal straightness error of a translation stage driven by a stepping motor. The travel of the translation stage, along the direction of the z-axis, was divided into 20 1-cm-spans. The examination was performed with two measurements. The first of which commanded the translation stage to experience the travel 10 times, detected the lateral positions at the span nodes as the stage was traveling, evaluated the average and standard deviation of the measured positions at each node, and calculated the straightness error of the stage. Figure 9 represents the straightness error from the calculation. It is noticed that the measured lateral position comprises two components: one due to the straightness error of the stage, and the other one due to the parallelism of the light beam to the stage. Therefore, to remove the component of the second one, the calculation computed the least squares line of the curve linking the measurement averages and retrieved the straightness error by subtracting the least squares line from the curve. The second measurement was the same as the first one except, as the movement of the bi-wedge prism shown in Fig. 3, the bi-wedge prism was moved forward for 10 cm as the stage progressed to the node of 10 cm each time. The straightness error from the second measurement is also presented in Fig. 9. The results of the two measurements demonstrate the applicability of the calibrator. In addition, the consistency between the results indicates that the bi-wedge prism is allowed to be moved during the examination. In other words, the proposed calibrator is with the capability of extending the working distance.
The calibrator, shown in Fig. 4, was installed on an optical table during the described experiments. However, since the table was not placed in a standard lab for metrology and the optical elements were supported using commercial mounts, low-frequency vibrations, air turbulence, and temperature variation still drifted the elements and consequently decreased the stability of the calibrator. Isolating the environmental perturbations and using optical mounts with more mass and a low degree-of-freedom would increase the stability of the calibrator.
The average area for determining the intensity of each of the points P, Q, and R and the distance between the successive points of the three points were 55 pixels and 4 pixels, respectively. These were obtained according to two criteria: (1) whether the average area  can effectively smooth out the tremble (due to the perturbation from electronic noise, defects of optical components, stray light, and so on) in the correlograms and provide the correlograms with a high contrast zero-order fringe; and (2) whether the distance is able to give the S-curve a high slope. Thus, a series of trial and error is in general required for a newly installed calibrator. Note a larger average area can further depress the tremble but may decrease the contrast of the zero-order fringe, and a pattern with larger fringe spacing allows the use of a larger area and distance.
And, in the applicability verification, the calibrator was conducted to examine the horizontal straightness error of the translation stage. This may be insufficient since a vertical straightness error examination is in general demanded in a real stage inspection. Once the core-stage and bi-wedge prism are rotated, from the situation shown in Fig. 4, about the z-axis by 90 deg., the calibrator is capable of vertical straightness measurements.
In summary, this paper introduced a calibrator based on the use of a low-coherent light source straightness interferometer and compensation method, performed a setup for realizing the calibrator, and presented the experiments and experimental results from the applications of the setup. The setup was with high measurement sensitivity. The experimental results revealed that the setup was with high measurement resolution and stability, confirmed the validity and applicability of the proposed calibrator, and verified the advantage of extensible working distance of the calibrator.
The support of the National Science Council, Taiwan, under grant NSC 98-2221-E-027-009-MY2 is gratefully acknowledged.
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