Simultaneous multi-channel absolute position alignment by multi-order grating interferometry

Zhang Tao; Jiwen Cui; Jiubin Tan

doi:10.1364/OE.24.000802

1. Introduction

The ability to determine the absolute position of an object is one of the most basic requirements of modern sensing technology. Precision machines capable of nanometer-resolution positioning are essential for the efficient production of high-density microelectronic products [1, 2]. Progress in nanometer-resolution positioning systems has been motivated by the continual downscaling required of integrated circuit technology, which has created a strong need for ultra-high-precision absolute position alignment [3–5]. In an incremental displacement measurement system, a zero reference point is an important addition that can enable the measurement of absolute positions, origins of coordinate systems, and machine home positions [6, 7]. The alignment of reference positions in lithography alignment systems and the measurement of absolute positions in displacement measurement systems are very similar problems. Such measurements can be realized by detecting optimally aligned points or by obtaining the offsets between objects of interest and the zero reference point [8].

Previously, the geometric imaging method was often used. In this method, two geometric marks, such as crosses, are compared to perform a measurement. However, the accuracy of this method is limited and cannot meet the requirements of recently developed applications [9].

Zero reference codes (ZRCs) have been developed since 1986 [10] and consist of groups of unevenly spaced transparent and opaque slits. In [11], J. Sáez-Landete et al. proposed a new kind of binary code for precise positioning and alignment. However, ZRC-based methods are problematic as their designs and algorithms are highly complex.

In recent years, the moiré fringe method of performing absolute position alignment measurements has attracted increasing attention [12–17]. However, moiré fringe technology is mostly used in proximity lithography, and as the reference mark and measurement mark are located on the mask and wafer respectively, the gap between the mask and the wafer must be kept a very small distance. These features limit the applicability of moiré fringe technology to other situations.

Interferometric techniques adopting diffraction gratings are more frequently applied in absolute position alignment measurements [18]. Flanders proposed an interferometric measurement system that detects the intensities of two symmetric interfering beams diffracted by two gratings by using X-ray lithography [19]. Une et al. [20] presented a heterodyne interferometric method that employs two-frequency laser beams diffracted from gratings and associates their relative displacement with the phase change of a beat signal. Similar schemes are also utilized in alignment sensors for optical projection printing systems [21–23]. The grating is used as a measurement mark and to diffract light, after the initial light beams propagate to the diffraction grating, various orders of the resulting interference pattern are measured respectively to determine the critical intensities of the interfering beams. The phase variation of a particular diffracted order depends on the lateral shift of the grating, and this variation is registered in a detector. All of these methods take full advantage of the high sensitivity of diffraction-induced interferometry.

Performing an interference measurement involves determining the phase change of a signal. When the integer part of the interference is lost during shifting, only the fractional part of the interference is detected, resulting in a non-ambiguity range equal to the period of the measured interference signal. However, the relative phase detection accuracy is inversely proportional to this period. In general, shorter-period interference signals enable higher-accuracy measurements, while longer-period interference signals offer greater non-ambiguity ranges. Although using two gratings with slightly different periods can significantly expand the non-ambiguity range [21–23], a single grating requires less space and simplifies the system design, thus enabling greater flexibility in terms of applications. Thus, the tradeoff between measurement range and accuracy can be a limiting factor.

Combining the measurements of shorter-period and longer-period interference signals could be a possible means of achieving a reasonable non-ambiguity range while maintaining high accuracy. In this paper, we present a hybrid absolute position measurement method based on simultaneous measurement of interference signals with various periods. The key feature of this method is that the resulting measurements exhibit the non-ambiguity range of the longest-period interference signal and the accuracy of the shortest-period interference signal. The simultaneous measurements ensure the consistency and stability of the measurement results. Building on earlier work, we present herein a prototype multi-order grating interferometer that simultaneously measures multiple orders of interference signals [24]. In this system, multi-channel phase extraction method is applied to measure the phase information of various orders, and absolute position measurements are made via the phase extraction scheme along with parallel data processing of the multiple interference signals. In addition, the results of the experiment, which proved the validity of the theoretical method. The measurement performance of the proposed interferometer was evaluated to determine its potential use as an absolute position measurement system in ultra-high-precision machine applications, and the conclusions are presented herein.

2. Measurement principle

In our system, absolute position alignment is realized by using a symmetrical grating mark such as that shown in Fig. 1, where the center of the grating is defined as the fiducial position, and the offset between the fiducial position and the principal optical axis of the measurement system is defined as the absolute position offset x_mark. When a laser beam illuminates the grating, the beam is diffracted into discrete orders, and the measured interference signal results from the superposition of the diffracted beams.

Fig. 1 Measurement principle of proposed absolute position alignment method.

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The absolute position offset can be measured by scanning the grating to acquire a periodic interference signal. The intensity S(x) of the resulting signal can be calculated by using Eq. (1):

S (x) = A \cdot \cos (\frac{2 π}{P_{s i g n a l}} x + φ_{m a r k}) = A \cdot \cos (\frac{2 π}{P_{s i g n a l}} (x + x_{m a r k})) .

Where A is the amplitude of the resulting signal, P_signal is the period of the resulting signal, x is the displacement of the grating mark during scanning,

φ_{m a r k}

is the initial phase of the periodic interference signal which indicates the absolute position offset, as shown in Fig. 1.

The absolute position offset x_mark can thus be obtained using

x_{m a r k} = \frac{P_{s i g n a l}}{2 π} φ_{m a r k} .

Therefore, determining the absolute position offset becomes equivalent to measuring the optical phase.

However, in interferometry, phase values can only be detected modulo 2π, significantly limiting the range of grating periods that will yield unambiguous measurements. Using a grating with a large period, however, will result in lower phase detection accuracy. For applications related to engineering, it is essential to extend the non-ambiguity range while simultaneously ensuring high phase measurement accuracy.

In the proposed method, the absolute position alignment offset can be obtained by using the hybrid measurement concept presented in Fig. 2: the absolute position alignment is determined based on simultaneous measurement of the interferometric phases of the multiple discrete signal orders. Since the period of the interference signal is inversely proportional to the order number, lower-order signals yield larger unambiguous measurement ranges, while higher-order signals enable relatively higher phase detection accuracy. To combine these two characteristics, the alignment offset x_mark can be expressed in terms of P_high-order which is the period of the high-order signal. This approach effectively generates a reasonable non-ambiguity range while maintaining high accuracy:

Fig. 2 Diagram of combined coarse and fine absolute position alignment principle.

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x_{m a r k} = N \cdot P_{h i g h - o r d e r} + f \cdot P_{h i g h - o r d e r} .

In Eq. (3), the integer part N of the absolute position alignment offset is determined by the relatively imprecise lower-order signals. The second term provides information about the missing fractional part f of the alignment offset resulting from the relatively precise higher-order signals.

It should be stressed that, in actual engineering applications, due to environmental disturbances and the mechanical drift of the setup, the absolute position of the mark is not constant. In the proposed hybrid measurement method, the multiple signal orders must be measured simultaneously in order to ensure the consistency and stability of the absolute position measurement.

3. Absolute position alignment system

3.1 Multi-order grating interferometer module

A schematic diagram of the proposed system is shown in Fig. 3. In order to measure multiple orders of interference signals simultaneously to perform absolute position measurement, a multi-order grating interferometer based on a reversal shearing imaging system [24] is employed.

Fig. 3 Absolute position alignment system employed in this investigation.

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As schematically shown in Fig. 3, linearly P-polarized light passes through the illumination lens and is then emitted through the objective lens in the grating plane with a plane wavefront. The light diffracted by the grating is collimated by the objective lens. And the light passed through the quarter-wave plate I (QWP I) twice, causing it to become linearly S-polarized with respect to the polarizing beam splitter (PBS), which directs it to the reversal shearing interferometer.

The reversal shearing interferometer, which has a modified Mach–Zehnder design, generates two virtual images of the grating diffraction field that are rotated by 180° relative to each other. The polarization axis of the light is rotated by 45° as it passes through a half-wave plate (HWP). Subsequently, the light polarized 45° relative to the polarizing axis of the interferometer beam splitter is divided evenly into two beams. The S-polarized signal beam is reflected by a right-angle prism, passes through the QWP II twice as it travels along its path, and is then transmitted through the polarizing beam splitter of the interferometer. The resulting wavefront plane has 180° rotational shear. Similarly, the P-polarized signal beam is reflected by a reflection mirror, passes through the QWP III twice as it travels along its path, and is then reflected at the polarizing beam splitter of the interferometer. The two beams are recombined upon exiting the interferometer.

The combined beam is focused onto a photodetector by a tube lens that, together with the objective lens, forms a 4-f imaging system. In the imaging plane, two images of the light diffraction field are overlapped by the combination of the objective and tube lenses. In front of the tube lens, there is a polarizer of −45° polarization, which functions as the analyzer. Interference effects are, therefore, seen in the paths of the two beams emerging from the analyzer.

The diffraction light field U_ob(x) of the grating can be expressed by

U_{o b} (x) = R (x) \cdot U_{i l l} (x),

where

R (x)

is the diffraction coefficient of the reflection grating and

U_{i l l} (x)

is the field of the illumination beam with a plane wavefront, and x is the displacement of the grating-mark during scanning.

Because the optical system is configured as a 4-f imaging system, the field in the detection plane can be expressed as a Fourier series according to the Fourier optics principle - [24]:

U_{i m} (x) = \sum_{m = - M}^{M} A_{m} e^{- i \cdot m \cdot \frac{2 π}{P_{0}} (x + x_{m a r k})} U_{i l l} (x) .

Here,

U_{i m} (x)

is the field in the imaging plane, P₀ is the period of the grating, m is the order number of the grating diffraction field, A_m is the m^th-order diffraction efficiency of the grating, x_mark is the absolute position offset, and M is the highest order number measured.

According to the Fourier optics principle, only the finite frequency component contributes to the image field. The integration boundaries are considered by summing over a finite number of orders, where the order number m must fall within the numerical aperture NA of the objective lens:

- N A \leq m \cdot \frac{λ}{P_{0}} \leq N A .

The signal in the detected field is composed of two overlapping beams with orthogonal polarizations and the field emerging from the polarizer and can be expressed as

U_{i m}^{inter} (x) = \frac{1}{2} [U_{i m}^{} (x) - U_{i m}^{} (- x)] .

The detected signal is the integrated intensity of the whole detected field:

S (x) = \int_{D} U_{i m}^{int e r} (x, ξ_{0}) \cdot \bar{U_{i m}^{int e r} (x, ξ_{0})} d σ .

Where D is the detection area of the whole detected field,

σ

is the coordinate of the detected field. Then the signal can be calculated by substituting Eqs. (5) and (7) in Eq. (8):

S (x) = \frac{1}{2} I_{i l l} \sum_{m = 1}^{M} {A_{m} \bar{A_{m}} + A_{- m} \bar{A_{- m}} - (A_{m} \bar{A_{- m}} + A_{- m} \bar{A_{m}}) \cos [\frac{4 m π}{P_{0}} (x + x_{m a r k})]} .

Here, I_ill is the integrated intensity of the illumination profile. Then, Eq. (7) can be simplified to

S (x) \propto \sum_{m = 1}^{M} {R_{m} + T_{m} \cos [k_{m} (x + x_{m a r k})]}

where

R_{m} = A_{m} \bar{A_{m}} + A_{- m} \bar{A_{- m}}

,

T_{m} = - (A_{m} \bar{A_{- m}} + A_{- m} \bar{A_{m}})

, and

k_{m} = \frac{4 m π}{P_{0}}

.

According to Eqs. (9) and (10), the signal measured by the single detector is composed of multiple orders, and each order of the signal contains phase information that can be used to measure the alignment offset independently and simultaneously.

3.2 Multi-channel measurement phase information extraction method

To retrieve the phase information, a multi-channel simultaneous extraction method is performed, which is a type of quadrature detection and requires two reference signals that are related in quadrature (i.e., cosine and sine). The frequencies of the reference signals are exactly the same as that of the detected signal, and the resulting signal can be obtained from the mixed signal. This method of multi-channel phase information extraction method is depicted in Fig. 4.

Fig. 4 Diagram of phase information extraction method used for the measured signal.

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The interference signal is generated by scanning the grating-mark. It should be stressed that, the scanlength L must be integer multiple of the period P₀. And the scanning speed must be constant, so as to ensure the equidistant sampling. These two points ensure the detection result of a certain order will not be mixed with the other order signal in principle.

As shown in Fig. 5, the analog-to-digital converter takes N samples of the measured interference signal S(x) to yield the discrete signals S(n) (n = 0, ..., (N - 1)), as shown in Eq. (11):

S (n) = \sum_{m = 1}^{M} \sum_{n = 0}^{N - 1} {T_{m} + R_{m} \cos [k_{m} (n Δ + x_{m a r k})]} .

According to the Nyquist criterion, the sampling frequency must be larger than twice the frequency of the largest-order sampled signal.

Fig. 5 Diagram of scan sampling used for phase extraction detection.

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After the data have been sampled, the digitized signals are multiplied the quadrature discrete reference signals A_j(n) and B_j(n):

A_{j} (n) = \cos (k_{j} \cdot n Δ)

and

B_{j} (n) = \sin (k_{j} \cdot n Δ),

where j is the order of interference signal whose phase needs to be obtained. The resulting signals can be expressed as follows:

\begin{array}{l} R_{A j} (n) = S (n) \cdot A_{j} (n) \\ = \sum_{m = 0}^{M} \sum_{n = 0}^{N - 1} {R_{m} \cos (k_{j} \cdot n Δ) + T_{m} \cos [k_{m} (x_{m a r k} + n Δ)] \cdot \cos (k_{j} \cdot n Δ)} \\ = \frac{1}{2} \cos [\frac{1}{2} (N - 1) (k_{m} - k_{j}) Δ + k_{m} x_{m a r k}] \frac{\sin [(k_{m} - k_{j}) \frac{1}{2} N Δ]}{\sin [(k_{m} - k_{j}) \frac{1}{2} Δ]} \\ + \frac{1}{2} \cos [\frac{1}{2} (N - 1) (k_{m} + k_{j}) Δ + k_{m} x_{m a r k}] \frac{\sin [(k_{m} + k_{j}) \frac{1}{2} N Δ]}{\sin [(k_{m} + k_{j}) \frac{1}{2} Δ]} \end{array}

and

\begin{array}{l} R_{B j} (n) = S (n) \cdot B_{j} (n) \\ = \sum_{m = 0}^{M} \sum_{n = 0}^{N - 1} {R_{m} \sin (k_{j} \cdot n Δ) + T_{m} \cos [k_{m} (x_{m a r k} + n Δ)] \cdot \sin (k_{j} \cdot n Δ)} \\ = - \frac{1}{2} \sin [\frac{1}{2} (N - 1) (k_{m} - k_{j}) Δ + k_{m} x_{m a r k}] \frac{\sin [(k_{m} - k_{j}) \frac{1}{2} N Δ]}{\sin [(k_{m} - k_{j}) \frac{1}{2} Δ]} \\ + \frac{1}{2} \sin [\frac{1}{2} (N - 1) (k_{m} + k_{j}) Δ + k_{m} x_{m a r k}] \frac{\sin [(k_{m} + k_{j}) \frac{1}{2} N Δ]}{\sin [(k_{m} + k_{j}) \frac{1}{2} Δ]} \end{array}

Each order of signal is repeatedly modulated into both the sum and difference of its frequency and that of the reference signals. In case the scanlength L is integer multiple of the period P₀ of the reference signal.

L = N Δ = q \cdot P_{0},

here, q is integer, then Eq. (14) and (15) can simplify as follows:

R_{A j} (n) = \frac{1}{2} T_{j} \cos (k_{j} x_{m a r k}) \cdot N

and

R_{B j} (n) = - \frac{1}{2} T_{j} \sin (k_{j} x_{m a r k}) \cdot N .

The term retained depend only on the value of the optical phase

k_{j} x_{m a r k}

. Hence, the information of the j^th-order signal in the composite measured signal and the j^th-order absolute position offset

x_{m a r k}^{j^{t h} - o r d e r}

can be calculated by using the arctan functions

x_{m a r k}^{j^{t h} - o r d e r} = \frac{P_{0}}{j \cdot 4 π} arc \tan [- \frac{S (n) \cdot B_{j} (n)}{S (n) \cdot A_{j} (n)}] .

The phase information from multiple orders of signals can be used to determine the absolute position offset using the above method, and each order provides the measurement result independently.

Since the multiple orders of signals have various periods, the lower-order signals with longer periods can be used to estimate the absolute position offset for the subsequent higher-order signals. The phase information gathered at the absolute position offset x_mark by measuring different orders of signals can then be combined to recover x_mark with high precision. According to Eq. (3), its integer part N is determined by applying the floor function ⌊x⌋, which returns the largest integer ≤ x. Where $x_{m a r k}^{l o w - o r d e r}$ is the coarse measurement results made using a low-order signal, $x_{m a r k}^{h i g h - o r d e r}$ is the more precise measurement results made using a high-order signal. Together, the absolute position offset of the combined results can be expressed by:

x_{m a r k} = ⌊ \frac{x_{m a r k}^{l o w - o r d e r}}{P_{h i g h - o r d e r}} - \frac{x_{m a r k}^{h i g h - o r d e r}}{P_{h i g h - o r d e r}} + \frac{1}{2} ⌋ \cdot P_{h i g h - o r d e r} + x_{m a r k}^{h i g h - o r d e r},

Since the measurement uncertainties can lead to incorrect determination of the integer part N, mathematically, the corrected integer part N using this expression can provide a stable result against noise [25].

4. System performance test

4.1 Experimental setup

An experiment was performed to verify the effectiveness and feasibility of the proposed method. The absolute position measurement system was configured using the setup shown in section 2. A laser (JDSU 1145/P) with a wavelength of 633 nm was used as the light source. An amplified photodetector sensor (Thorlabs, amplified photodetector PDA36A) was employed to measure the integrated intensity of the interference field. An analog-to-digital conversion card (National Instruments PCI-6143) was applied to sample the reference and measured interference signals. In this experiment, a rectangular phase grating with a 50% duty cycle and a 16 μm period was adopted as the mark for the absolute position measurement. An objective lens with NA = 0.55 was used to collect the diffraction orders between ± 13th orders. The grating was mounted on a positioning stage (piezoelectric transducer, Physik Instrumente, model: PI-P-625.1CD). All of the devices were set on an active self-leveling isolation system (Thorlabs, PTR52514 and PTS602).

After the interference signals were detected, the multi-channel phase information extraction method introduced in section 3.2 was applied to process the sampled interference signals. The discrete reference signals A_j(n) and B_j(n) were generated purely numerically. Based on the derived phase information, the unknown absolute position offset x_mark was directly calculated by using Eq. (18).

To demonstrate the facility and performance of our proposed interferometer, a commercial incremental heterodyne interferometer (5517B, Agilent) was installed for comparison. The target was a mirror that was moved along with the positioning stage while position measurements were obtained using the incremental heterodyne interferometer. The whole system was enclosed within an insulating container to minimize air turbulence and temperature variation.

4.2 Pre-alignment

In this system, pre-alignment should be performed before conducting the absolute position alignment process proposed in this paper. In the setup employed in our investigation, the grating mark had a period of 16 μm, so the signals of order 1 and −1 could yield a non-ambiguity range of at most 8 μm when demodulated from the multi-order composite signal. Such a pre-value can be achieved by using a machine vision system to adjust the misalignment into the range necessary to perform the subsequent absolute position alignment.

During pre-alignment, the polarizer in front of the image plane should be removed, and the photodetector should be replaced with a CCD camera. As discussed in section 3.1, the field in the detection plane can be expressed as Eq. (5). By using the reversal interferometer, two diffraction patterns generated by the grating should be imaged on the CCD. The two overlapping images will shift in opposite directions as the grating is scanned. When the two images overlap completely, the pre-alignment process is complete. Feature extraction and image processing can confirm the non-ambiguity range of the subsequent absolute position alignment.

4.3 Output spectrum test

Firstly, qualitative analysis of the multi-order composite interference signal can be achieved by measuring the spectrum of the interferometer’s output response curve. To perform this test, the positioning stage should be moved at a constant speed (800μm/s) while the interference signal is obtained by the photodetector. As in the case of grating mark displacement, the output signal varies periodically. The Fourier-transformed frequency spectrum of the multi-order composite signal in Fig. 6 is clear, and signals up to the 13th order are observable simultaneously. These features are as expected, and can be explained by higher-order interference between the optical measurement system and the simultaneous generation of multiple orders of interference signals. Furthermore, the signal-to-noise ratio decreases as the order m increases; the signal-to-noise ratio of the 1st-order signal is about 35 dB, while that of the 13th-order signal is about 12 dB.

Fig. 6 Multi-order composite signal spectrum.

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4.4 Test of performances of various orders

To verify the performance of the proposed absolute position measurement system, it was compared experimentally with the incremental heterodyne interferometer mentioned earlier. By performing pre-alignment, the grating mark was adjusted to its nominal initial position, whose absolute position offset x_initial was measured using the proposed system. The grating mark and retro-reflector mounted on the positioning stage were moved to a certain position, and the absolute offset of this position x_position was measured. The position offset relative to the nominal initial position l_meas is then equal to x_position-x_initial and can be compared with the reference value l_ref obtained from the calibrated incremental heterodyne interferometer. In Fig. 7, the residuals l_meas-l_ref are depicted versus the interference signal order. These results were obtained by performing 15 measurements at measured position.

Fig. 7 Offset residuals measured at identical positions for different orders m.

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According to the discussion in section 2, the variation in the results should decrease with increasing signal order. As expected, the measured residuals shown in Fig. 7 significantly decrease with increasing m. However, the 11th- and 13th-order residuals are slightly greater than expected, while the residuals corresponding to the 9th-order signals exhibit the minimum standard deviation of 5.33 nm. This phenomenon can be explained by the fact that the interference signal intensity significantly decreases with increasing order, resulting in decreased signal-to-noise ratios for higher diffraction orders.

4.5 Comparison test for various measurement positions

A test was performed to compare the multi-order-based absolute positioning system and calibrated incremental heterodyne interferometer results at various measurement positions. The grating mark and retro-reflector were moved in 500 nm steps from the nominal position described in section 4.4. Ten positions were measured independently by each system for comparison.

Using the multi-order interferometry system and multi-channel phase information extraction method described above, the absolute position at each location was measured. The final result was calculated using Eq. (20), where $x_{m a r k}^{l o w - o r d e r}$ was measured by using the 1st-order signal, which yielded the largest non-ambiguity range, and the refined measurement parameter $x_{m a r k}^{h i g h - o r d e r}$ was obtained by using the 9th-order signal, which produced the optimum result.

In Fig. 8, the circle points and line indicate the results measured using the multi-order absolute position alignment system and calibrated incremental heterodyne interferometer, respectively. The results of the two systems agree well, yielding a highly linear relationship with a slope of 0.99703 and a 39.7 nm peak-to-valley residual error.

Fig. 8 Linear relationship between results of proposed system and HeNe interferometer.

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The residuals at each measurement position are shown in Fig. 9, and at each measurement position, the absolute position alignment was measured 15 times independently. Where the circle points and error bars indicate the mean values and standard deviations of the residuals, respectively. The mean value range from −10.12 nm to 9.89 nm, while the standard deviations vary from 6.51 nm to 11.48 nm.

Fig. 9 Residuals measured at various positions.

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5. Discussion

The experimental results above illustrate the ability of the proposed approach and system to measure the absolute position offset of a grating mark. Of course, the measurement accuracy reported herein for our approach and system were determined by the specific experimental conditions in this study. The noise in the measured signals can be attributed to the mechanical vibration of the scanning stage, electronic or thermal noise in the electronic circuit, and even environmental turbulence. Noise sources can be present anywhere from the illumination optics to the electronic signal processing system. Wherever a noise source exists, its influence on the alignment signal can always be written as a Fourier series expression, the error varies because the spectral components a_m and b_m vary:

N (x) = \sum_{m = 0}^{\infty} a_{m} \sin (k_{m} x) + b_{m} \cos (k_{m} x) .

An actual alignment signal can always be written as

S_{r e a l} (x) = S (x) + N (x) .

After substituting Eq. (21)–(22) into Eqs. (14)–(19) for the position offset, the following expression directly relating the error to the spectral Fourier component can be obtained:

E r r o r_{m} (x) = \frac{1}{k_{m}} arc \tan (- \frac{a_{m}}{T_{m} + b_{m}}) .

This expression directly relates the alignment position offset to the noise parameters. The variance of the noise parameters leads to the variance of the alignment position:

\begin{array}{l} V (E r r o r_{m} (x)) = {(\frac{\partial E r r o r_{m} (x)}{\partial a_{m}})}^{2} V (a_{m}) + {(\frac{\partial E r r o r_{m} (x)}{\partial b_{m}})}^{2} V (b_{m}) \\ + 2 (\frac{\partial E r r o r_{m} (x)}{\partial a_{m}}) (\frac{\partial E r r o r_{m} (x)}{\partial b_{m}}) cov (a_{m}, b_{m}) . \end{array}

The partial derivatives are evaluated at the average value of (a_m, b_m) = (0, 0), around which a linear approximation is valid. It seems very likely that a_m and b_m are completely uncorrelated, so their covariance can be assumed to be zero. The measurement uncertainty is the square root of the variance.

In case the amplitudes of the noise coefficients are much smaller than the signal amplitude, we can write

V (E r r o r_{m} (x)) \approx {(\frac{1}{k_{m} \cdot T_{m}})}^{2} V (a_{m})

and

U (E r r o r_{m} (x)) = \sqrt{V (E r r o r_{m} (x))} \approx \frac{P_{0}}{4 π m} \cdot \frac{U (a_{m})}{T_{m}} .

These expressions show that only an anti-symmetric noise coefficient at the same spatial frequency as the alignment signal leads to alignment error. This measurement uncertainty is inversely proportional to the alignment signal strength and to the diffraction order. The noise physically leads to a phase variation first, the position variation corresponding to a particular phase variation decreases with increasing order. As an example, the signal-to-noise ratio of the 9th-order component is about 15 dB shown in Fig. 6, and the measurement uncertainty was about 4.5 nm according to Eq. (26). Considering the influence of the calibrated incremental heterodyne interferometer measurement error, the residuals of the comparison results of the entire system would be little more than 10 nm, which is roughly consistent with the experimental results.

As calculated initially, the measurement results based on high-order interference signals were expected to be more accurate than those based on low-order interference signals, because the measurement uncertainty contribution to the phase determination scales with the period of the signal order. However, the expected superior performances of the highest-order (11th- and 13th-order) interference signals were obscured by noise, as mentioned above. The noise could be reduced by improving the experimental conditions and device performance, such as by choosing a laser source with lower drift, choosing a detector with lower noise, or keeping the environment and system more stable.

6. Conclusion

Simultaneous multi-channel absolute position alignment system is proposed in this paper, and its ability to determine the absolute position of a grating mark is evaluated. In the proposed technique, By employing a multi-order grating interferometer and a multi-channel phase extraction method the absolute position determined through a multi-channel scheme in which interference phases of various orders are obtained in parallel to enable stable and consistent measurement. A hybrid method combines several orders of measurement results, thus retaining the benefits of the large non-ambiguity ranges yielded by low-order signals and the precision provided by high-order signals, resulting in highly precise absolute position measurements. Comparison of the positions measured by the proposed system with those obtained using an incremental HeNe reference interferometer yielded residuals with mean values ranging from −10.12 nm to 9.89 nm and standard deviations varying from 6.51 nm to 11.48 nm. With the reported performance, we conclude that the proposed scheme is a potential candidate for the absolute position measurement method necessary for ultra-high-precision machine applications.

Acknowledgment

Thanks are given to National Natural Science Foundation of China (51575140) for the financial support.

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Simultaneous multi-channel absolute position alignment by multi-order grating interferometry

Abstract

1. Introduction

2. Measurement principle

3. Absolute position alignment system

3.1 Multi-order grating interferometer module

3.2 Multi-channel measurement phase information extraction method

4. System performance test

4.1 Experimental setup

4.2 Pre-alignment

4.3 Output spectrum test

4.4 Test of performances of various orders

4.5 Comparison test for various measurement positions

5. Discussion

6. Conclusion

Acknowledgment

References and links

Cited By

Figures (9)

Equations (26)

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