Fluid gauges comprising multiple differential pressure sensors

Abstract

The subject fluid gauges measure actual position of a workpiece relative to a target position. A gauge body that is positionable relative to the workpiece and that includes multiple differential-pressure (DP) sensors has a measurement channel and respective reference channels. Each DP sensor measures, over a respective individual dynamic pressure range, a differential pressure established by a respective fluid flow in the measurement channel relative to a fluid flow in a respective reference channel. The dynamic pressure ranges of the DP sensors substantially overlap each other. A controller is connected to and monitors the DP sensors. The controller is configured to select, for obtaining a differential pressure indicative of the position of the workpiece, a DP sensor sensing the smallest magnitude of DP.

Claims

1. A fluid gauge for measuring actual position of an object relative to a target position, comprising: a gauge body that is positionable relative to the object and that includes a measurement fluid passageway and multiple differential-pressure sensors which measure a difference between pressure of fluid in the measurement fluid passageway and pressure of fluid in a respective reference fluid passageways and which produce respective differential-pressure signals, the differential-pressure sensors measuring respective differential pressures established by respective fluid flows in the measurement fluid passageway relative to fluid flow in the respective reference fluid passageways having respective reference outlets situated at different gaps with respect to respective reference planes, each of the respective reference fluid passageways coupled to an associated flow restrictor, wherein each flow restrictor is associated with a different flow restriction and each of the respective reference fluid passageways are independent of each other; and a controller connected to and monitoring the differential-pressure sensors, the controller being configured to determine respective position estimates based on the differential-pressure signals, and to determine a measured position of the object based on determined weighted averages of the position estimates.

2. A fluid gauge for measuring an axial position of a workpiece, comprising: a fluid source coupled to supply a fluid; a measurement fluid passageway, one end of the measurement fluid passageway connected to the fluid source and another end of the measurement fluid passageway having a measurement outlet adjacent to the workpiece; a first reference fluid passageway, one end of the first reference fluid passageway connected to the fluid source and another end of the first reference fluid passageway having a first reference outlet situated at a first interval with respect to a first reference plane so that fluid from the fluid source flows to the first reference plane; a second reference fluid passageway, independent of the first fluid passageway, one end of the second reference fluid passageway connected to the fluid source and another end of the second reference fluid passageway having a second reference outlet situated at a second interval which differs from the first interval with respect to a second reference plane so that fluid from the fluid source flows to the second reference plane; a first differential-pressure sensor that detects a differential pressure between fluid passing through the measurement fluid passageway and fluid passing through the first reference fluid passageway; a second differential-pressure sensor that detects a differential pressure between fluid passing through the measurement fluid passageway and fluid passing through the second reference fluid passageway; and a controller configured to determine the axial position of the workpiece by using at least one of a first output from the first differential-pressure sensor and a second output from the second differential-pressure sensor.

3. The fluid gauge of claim 2, wherein the controller is configured to select one of the first output and the second output.

4. The fluid gauge of claim 2, wherein the first differential-pressure sensor has a first differential-pressure measurement range and the second differential-pressure sensor has a second differential-pressure measurement range partially overlapping the first differential-pressure measurement range.

5. The fluid gauge of claim 3, wherein the first output from the first differential-pressure sensor has a first magnitude and the second output from the second differential-pressure sensor has a second magnitude, and the controller determines the axial position based on the smaller of the first magnitude and the second magnitude.

6. The fluid gauge of claim 3, wherein the controller is configured to determine the axial position by using either the first output from the first differential-pressure sensor or the second output from the second differential-pressure sensor, whichever is closest to zero.

7. The fluid gauge of claim 4, wherein the controller is configured to determine the axial position based on whichever of the first output from the first differential-pressure sensor or the second output from the second differential-pressure sensor is closer zero to reduce an axial position measurement error.

8. The fluid gauge of claim 3, wherein the controller is configured to determine the axial position based on whichever of the first output from the first differential-pressure sensor or the second output from the second differential-pressure sensor is closer to zero to reduce an axial position measurement error.

9. The fluid gauge of claim 2, wherein the measurement outlet is provided at gap with respect to the workpiece.

10. A fluid gauge for measuring axial position of a workpiece, comprising: a fluid source coupled to supply a fluid; a measurement fluid passageway, one end of the measurement fluid passageway connected to the fluid source, and another end of the measurement fluid passageway having a measurement outlet adjacent to the workpiece; a first reference fluid passageway, one end of the first reference fluid passageway connected to the fluid source, and another end of the first reference fluid passageway connected to a first flow restrictor associated with a first flow restriction so that fluid flows from the fluid source to an output of the first flow restrictor; a second reference fluid passageway, independent of the first fluid passageway, one end of the second reference fluid passageway connected to the fluid source, and another end of the second reference fluid passageway connected to a second flow restrictor associated with a second flow restriction that is different from the first flow restriction so that fluid flows from the fluid source to an output of the second flow restrictor; a first differential-pressure sensor coupled to detect a differential pressure between the measurement fluid passageway and the first reference fluid passageway; a second differential-pressure sensor coupled to detect a differential pressure between the measurement fluid passageway and the second reference fluid passageway; and a controller configured to determine an axial position of the workpiece based on at least one of a first output from the first differential-pressure sensor and a second output from the second differential-pressure sensor.

11. The fluid gauge of claim 10, wherein the controller is configured to select one of the first output and the second output.

12. The fluid gauge of claim 10, wherein the first differential-pressure sensor has a first differential-pressure measurement range and the second differential-pressure sensor has a second differential-pressure measurement range which partially overlaps the first differential-pressure measurement range.

13. The fluid gauge of claim 12, wherein the first output has a first magnitude and the second output has a second magnitude, and the controller is configured to determine the axial position based on the smaller of the first magnitude and the second magnitude.

14. The fluid gauge of claim 12, wherein the controller is configured to determine the axial position based on whichever of the first output or the second output is associated with a differential pressure value that is closer to zero.

15. The fluid gauge of claim 12, wherein the controller is configured to determine the axial position based on whichever of the first output and the second output is associated with a smaller absolute value of differential pressure to reduce an axial position measurement error.

16. The fluid gauge of claim 12, wherein the controller is configured to determine the axial position based on whichever of the first output and the second output is associated with a differential pressure value that is closer to zero to reduce an axial position measurement error.

17. The fluid gauge of claim 10, wherein the measurement outlet is situated at a measurement gap with respect to the workpiece.

18. An exposure apparatus which exposes a workpiece, comprising the fluid gauge of claim 2 situated to measure an axial position of the workpiece.

19. The exposure apparatus of claim 18, further comprising a projection optical system which projects a pattern on the workpiece, wherein a measurement axis coincides with an optical axis of the projection optical system.

20. An exposure method for exposing a workpiece, comprising measuring an axial position of the workpiece by using the fluid gauge of claim 2; measuring an axial position of the workpiece by using an axial position measuring system; and calibrating the axial position measuring system by using an output from the fluid gauge.

21. A device manufacturing method, comprising: exposing a pattern onto the workpiece by using the exposure method of claim 20; developing the pattern which is transferred the workpiece; forming a mask layer having a shape corresponding to the pattern; and processing a surface of the workpiece through the mask layer.

22. An exposure apparatus which exposes a workpiece, comprising the fluid gauge of claim 10 situated to measure an axial position of the workpiece.

23. The exposure apparatus of claim 22, further comprising a projection optical system which projects a pattern on the workpiece, wherein a measurement axis coincides with an optical axis of the projection optical system.

24. An exposure method for exposing a workpiece, comprising measuring the axial position of the workpiece by using the fluid gauge of claim 10; measuring an axial position of the workpiece by using an axial position measuring system; and calibrating the axial position measuring system by using an output from the fluid gauge.

25. A device manufacturing method, comprising: exposing a pattern onto the workpiece by using the exposure method of claim 24; developing the pattern which is transferred the workpiece; forming a mask layer having a shape corresponding to the pattern; and processing a surface of the workpiece through the mask layer.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1A is a schematic diagram of a prior-art fluid gauge having three reference channels and one measurement channel, as discussed in U.S. Patent publication no. 2011/0157576.

(2) FIG. 1B is a plot of wafer height as a function of differential pressure, produced by a fluid gauge as discussed in U.S. Patent publication no. 2011/0157576.

(3) FIG. 2A is a schematic diagram of an embodiment of a fluid gauge comprising certain general features, including multiple differential-pressure (DP) sensors.

(4) FIG. 2B is a schematic diagram of the embodiment of FIG. 2A, but including more detail than shown in FIG. 2A.

(5) FIG. 3 is a plot of gap versus differential pressure for each of the respective DP sensors of an embodiment of a fluid gauge.

(6) FIG. 4 is a plot of height error as function of gap change (nm) produced by a fluid gauge comprising only one DP sensor.

(7) FIG. 5 is a plot of DP versus gap produced by an embodiment of a fluid gauge that includes three DP sensors.

(8) FIG. 6 is a plot of DP versus gap that is similar to FIG. 5 but also indicates A.

(9) FIG. 7 is a plot of an exemplary Gaussian distribution of DP.

(10) FIG. 8 is a plot of height error as a function of gap, including a plot of a simple average of the DP values produced by the DP sensors, and including a plot of height error when one simply switches from one DP sensor to another.

(11) FIG. 9 is a plot of relative error as a function of gap produced by an embodiment of the fluid gauge, in which the plot is of a weighted average.

(12) FIG. 10A is a flow-chart of an exemplary method for determining weighted-average coefficients.

(13) FIG. 10B is an algorithm diagram for the method shown in FIG. 10A.

(14) FIG. 10C is a flow-chart outlining a determination of height of a workpiece.

(15) FIG. 11 is a schematic diagram of a microlithographic exposure system, as a representative precision system, including features of the invention described herein.

(16) FIG. 12 is a flow-chart outlining a process for manufacturing a semiconductor device in accordance with the invention.

(17) FIG. 13 is a flow-chart of a portion of a device-manufacturing process in more detail.

DETAILED DESCRIPTION

(18) This disclosure is set forth in the context of representative embodiments that are not intended to be limiting in any way.

(19) The drawings are intended to illustrate the general manner of construction and are not necessarily to scale. In the detailed description and in the drawings themselves, specific illustrative examples are shown and described herein in detail. It will be understood, however, that the drawings and the detailed description are not intended to limit the invention to the particular forms disclosed, but are merely illustrative and intended to teach one of ordinary skill how to make and/or use the invention claimed herein.

(20) As used in this application and in the claims, the singular forms a, an, and the include the plural forms unless the context clearly dictates otherwise. Additionally, the term includes means comprises. Further, the term coupled encompasses mechanical as well as other practical ways of coupling or linking items together, and does not exclude the presence of intermediate elements between the coupled items.

(21) This disclosure is directed toward all novel and non-obvious features and aspects of the various disclosed embodiments, alone and in various combinations and sub-combinations with one another. The disclosed things and methods are not limited to any specific aspect or feature or combinations thereof, nor do the disclosed things and methods require that any one or more specific advantages be present or problems be solved.

(22) Although the operations of some of the disclosed methods are described in a particular, sequential order for convenient presentation, it should be understood that this manner of description encompasses rearrangement, unless a particular ordering is required by specific language set forth below. For example, operations described sequentially may in some cases be rearranged or performed concurrently. Moreover, for the sake of simplicity, the attached figures may not show the various ways in which the disclosed things and methods can be used in conjunction with other things and method. Additionally, the description sometimes uses terms like produce and provide to describe the disclosed methods. These terms are high-level abstractions of the actual operations that are performed. The actual operations that correspond to these terms will vary depending on the particular implementation and are readily discernible by one of ordinary skill in the art.

(23) Any mention herein of a controller or processor referred to in the singular will be understood to encompass use of multiple controllers or processors.

(24) In the following description, certain terms may be used such as up, down,, upper, lower, horizontal, vertical, left, right, and the like. These terms are used, where applicable, to provide some clarity of description when dealing with relative relationships. But, these terms are not intended to imply absolute relationships, positions, and/or orientations. For example, with respect to an object, an upper surface can become a lower surface simply by turning the object over. Nevertheless, it is still the same object.

(25) If a figure includes an orientation system that includes an x-axis, a y-axis that is orthogonal to the x-axis, and a z-axis that is orthogonal to both the x- and y-axes, it should be noted that any of these axes can also be referred to as the first, second, and/or third axes.

(26) In the embodiments described below, fluid gauges are provided that, compared to conventional fluid gauges (including conventional fluid gauges having multiple reference channels), are substantially less sensitive to noise and to variations in supply pressure and/or ambient pressure. These improvements are achieved by configuring the fluid gauges with multiple reference channels having their own respective reference pressure sensors, relative to which the fluid gauge can selectively obtain respective differential pressure (DP) measurements, and at least one measurement channel. Pairing of the measurement channel with a respective reference channel identifies a respective differential pressure (DP) sensor. The DP sensors' respective dynamic pressure ranges substantially overlap each other, thereby providing multiple measurements of the differential pressures and thus multiple estimates of the measurement gap while at least one DP sensor remains at or near a balanced condition (i.e., DP.sub.i=0). Alternatively or in addition, weighted averages of DP data from multiple DP sensors can be used to obtain gap measurements that are more accurate due to reduced sensitivity to variations in ambient pressure and/or fluid supply, and reduced sensitivity to sensor noise.

(27) A general depiction of a fluid gauge 50 according to various embodiments is in FIG. 2A. The depicted fluid gauge 50 is useful for measuring the position of a workpiece 52 along an axis 54 relative to a structure 56, such as an exposure system, process tool, or the like. The fluid gauge 50 comprises a gauge body 58 that is positionable relative to the workpiece 52 and that includes multiple DP sensors 60 (three are shown). Each DP sensor 60 measures, over a respective individual dynamic pressure range, a respective differential pressure established by fluid flow in a measurement gap between the gauge 50 and the workpiece 52, relative to a respective reference pressure established by a corresponding fluid flow in a respective reference gap. The respective DP ranges of the DP sensors 60, used to estimate the change in the measurement gap, substantially overlap each other and can be smaller than corresponding dynamic ranges produced by fluid gauges having fewer DP sensors per measurement channel, under the same operating conditions. A controller 62 is connected to the DP sensors 60 to receive respective DP signals from them. The controller 62 is also configured to select, for reference purposes, a particular DP sensor 60 whose DP value is closest to zero. Although three reference channels are utilized in this embodiment, any number greater than one may be used.

(28) As shown more specifically in FIG. 2B, this embodiment utilizes one measurement channel M and three reference channels R.sub.1, R.sub.2, R.sub.3 contained in a gauge body 58. Each reference channel has a respective gap RG.sub.1, RG.sub.2, RG.sub.3 and thus a respective measurement pressure. The reference channels R.sub.1, R.sub.2, R.sub.3 are arranged in parallel. Associated with the measurement channel M are a measurement gap MG and three respective differential pressures DP.sub.1, DP.sub.2, DP.sub.3 that correlate to the respective reference channels R.sub.1, R.sub.2, and R.sub.3. The differential pressures are utilized by the controller 62 as discussed above and elsewhere herein.

(29) Example operating conditions for this embodiment are shown in FIG. 3. In a conventional air gauge, one DP sensor (e.g., producing differential pressure DP.sub.2) would have to cover a differential-pressure range of approximately 4200 Pa for a gap ranging from 45-55 m (nominally 50 m; bold solid-line curve). In this embodiment, instead of employing only one DP sensor, three DP sensors are employed, which allows the range, used to estimate the change in the measurement gap, for each DP sensor to be correspondingly reduced while maintaining the substantial overlap of the three DP sensors.

(30) Associated with each channel M, R1, R2, R3 in this embodiment are substantially identical flow restrictors FR.sub.M, FR.sub.1, FR.sub.2, FR.sub.3. Also, in this embodiment, the conduitry of each channel has respective intrinsic resistance as indicated by the sawtooth symbols. In this embodiment the flow restrictors are different from each other. For example, the flow restrictors have different structures and restrict the fluid in different flow amounts. In an alternative configuration the flow restrictors are respective adjustable flow restrictors, or at least one of the flow restrictors is identical to another of the flow restrictors. In other alternative configurations, one or more of the reference gaps RG.sub.1, RG.sub.2, RG.sub.3 can be replaced with a respective controlled-bleed or adjustable flow restrictor. The gauge can be housed within a gauge body.

(31) Reference is now made to FIG. 3 showing three differential-pressure curves DP.sub.1, DP.sub.2, DP.sub.3 produced by an exemplary fluid gauge. Note that the differential-pressure curves have the same DP minima and the same DP maxima; hence, the curves substantially overlap each other. Meanwhile, the gap range covered by each sensor DP.sub.1, DP.sub.2, DP.sub.3 is reduced by approximately three. E.g., the reduced gap range for sensor DP.sub.2 is approximately 10/3=3.3 m. Thus, the range of differential pressure measurements, used by each DP sensor to determine a measurement-gap change, is reduced by approximately 3, compared to a gauge having a single DP sensor. E.g., the pressure-measurement range of DP.sub.2 is reduced by a factor of 3 from approximately 4200 Pa to approximately 1800 Pa. Note that sensor DP.sub.1 can now be used for measurements of gaps ranging from 45 to 48 m; sensor DP.sub.2 can now be used for measurement of gaps ranging from 48 to 52 m; and sensor DP.sub.3 can now be used for measurement of gaps ranging from 52 to 55 M. During use of the gauge, a particular DP sensor is selected that provides a respective DP measurement (DP.sub.i) that is at or near a balanced condition (DP.sub.i=0), which correspondingly reduces the sensitivity of the fluid gauge to changes in ambient pressure and/or supply pressure, compared to using one DP sensor. In FIG. 2B the reference gaps RG.sub.1, RG.sub.2, RG.sub.3 which are different from each other, can be replaced with respective variable flow restrictors, each adjusted to provide a respective rate of fluid bleed.

(32) Whenever an air gauge is in a balanced condition (where DP.sub.i0), the gauge is substantially insensitive to changes in either or both the supply pressure and ambient pressure. I.e., atmospheric pressure and/or supply pressure may vary, but without imparting a significant change to the DP.sub.i=0 condition. (This is closely analogous to the electrically balanced condition of a Wheatstone Bridge.) The insensitivity breaks down with increasing departure from the balanced condition, resulting in greater sensitivity of this fluid gauge to changes in supply pressure and/or ambient pressure. Hence, during operation of the fluid gauge, it is advantageous that the selected DP.sub.i be as small as possible. By using multiple reference channels arranged so that at least one (a selectable one) reference channel produces a small magnitude of DP (DP.sub.i0), a selected channel can be used to provide a measurement of the gap that is less sensitive to changes in supply pressure or atmospheric pressure. This selection can be done alone or in combination with weighting (see later below) to favor that channel.

(33) For small changes to supply pressure and/or to ambient pressure (.sub.Pa) and differential pressure (DP), the error in gap measurement (.sub.gap) is expressed by .sub.gap(.sub.Pa)(DP), where the factor depends on the pressure and flow conditions of the fluid gauge, and where the variable .sub.Pa refers to atmospheric pressure changes or supply pressure changes. Again, if DP0, then the gap measurement produced by the gauge is substantially insensitive to changes in supply pressure and/or ambient pressure.

(34) Dependence of air-gauge accuracy upon changes in ambient atmospheric pressure P.sub.atm can be estimated using a model from an air gauge environment-sensitivity evaluation, as well as flow data. Atmospheric pressure can change significantly day-to-day. For example, a P.sub.atm of 100 Pa from one day to the next is common and can be considerably larger. The corresponding gap-height error varies substantially linearly with corresponding changes in P.sub.atm and gap height relative to a gap in which DP0. See FIG. 4, which depicts a range of gap-height error when only one DP sensor is used (left-hand portion of the figure), compared to a range of error when three DP sensors are used (right-hand portion of the figure). Data are for two flow rates through the measurement channel of the gauge, namely 0.72 slm and 1.4 slm (wherein slm is an abbreviation of standard liter per minute). The graph in FIG. 4 depicts a change in gap range from 5000 to +5000 nm to approximately 1500 to +1500 nm, resulting from use of three DP sensors instead of only one to perform the gap measurement. Note that, while height error depends linearly on changes in P.sub.atm and height relative to gap where DP0, the plots pivot about the point at which zero gap change produces zero height error. In FIG. 4 the range of gap error obtained using three DP sensors is about 5 to +5 nm, compared to a range of error of about 15 to +15 nm using one DP sensor. Therefore, the range of error is reduced by using multiple DP sensors.

(35) In view of the above, measuring a gap using this air gauge can include monitoring ambient pressure and supply pressure, and making appropriate corrections to the gap determinations using these pressure data. Calibration can be made to a particular atmospheric-pressure condition and gap, from which height determinations are then made. This approach in principle obviates the need for multiple reference channels. However, the use of multiple reference channels can reduce the contribution of sensor noise. Also, the correction provided by monitoring the atmospheric and supply pressures will be smaller and therefore less demanding of the relevant sensor accuracies.

(36) For determination of gap, instead of (or in addition to) selecting the DP value closest to DP=0, the DP measurements can be combined together in a weighted average to reduce sensor noise while also minimizing sensitivity to changes in P.sub.supply or P.sub.atm. For example, for three reference channels, there are three corresponding determined estimates of the gap (h.sub.i), assuming substantial overlap:
h.sub.1=f.sub.1(DP.sub.1)
h.sub.2=f.sub.2(DP.sub.2)
h.sub.3=f.sub.3(DP.sub.3)
where ideally h=h.sub.1=h.sub.2=h.sub.3. The estimates h.sub.1 of height measurements are respective functions of the DP pressure; i.e., h.sub.i=f.sub.i(DP.sub.i). The functions f.sub.1, f.sub.2, and f.sub.3 are not necessarily equal to each other.

(37) A simple average of the three measurements:
h=(h.sub.1+h.sub.2+h.sub.3)/3=[f.sub.1(DP.sub.1)+f.sub.2(DP.sub.2)+f.sub.3(DP.sub.3)]/3
reduces sensor noise by a factor of ().sup.1/2, assuming they are uncorrelated (i.e., substantially random). See FIG. 5, in which the curves f.sub.1(DP.sub.1), f.sub.2(DP.sub.2), and f.sub.3(DP.sub.3) have substantial overlap. Note, for example, the particular gap (approximately 49 m) denoted by the vertical dashed line, and the three curves f.sub.1(DP.sub.1), f.sub.2(DP.sub.2), f.sub.3(DP.sub.3) in substantial overlap relative to that gap. Compare to FIG. 1B, in which there is substantially no overlap.

(38) The key here is to reduce sensitivity to changes in ambient pressure or supply pressure. Reducing sensitivity to sensor noise is a side advantage. This embodiment will, by its nature, improve both.

(39) The noise referred to here is largely intrinsic noise produced by the DP sensors (e.g., intrinsic noise produced by DP sensors). Intrinsic noise can be produced at any time the sensors are receiving electrical power, even when they are not currently being used to measure pressure. This and/or any other noise introduces errors into gap measurements performed using the DP sensors.

(40) Sensor noise is expected to be largely uncorrelated, so the above averaging should reduce its contribution to the measurement. In addition to sensor noise, acoustic noise within the measurement and reference channels can also degrade the height measurement. To the extent that acoustic noise within the reference channels is uncorrelated, the above averaging should reduce its effect on the height determination.

(41) Whenever DP0, changes in P.sub.supply or P.sub.atm will introduce a gap error h=(.sub.Pa)(DP). This leads to a total gap error (height error) of:

(42) $\begin{array}{c}h=\left(h_1+h_2+h_3\right)/3\\ ={}_{\mathrm{Pa}}\left[\left({\mathrm{DP}}_1+{\mathrm{DP}}_2+{\mathrm{DP}}_3\right)/3\right]\end{array}$
Thus, the height error h depends upon the average of DP.sub.1, DP.sub.2, and DP.sub.3. See FIG. 6. To a reasonable approximation, DP.sub.1DP.sub.2+ and DP.sub.3DP.sub.2, so the height error is approximately:

(43) $\begin{array}{c}h{}_{\mathrm{Pa}}\left[\left({\mathrm{DP}}_2++{\mathrm{DP}}_2+{\mathrm{DP}}_2-\right)/3\right]\\ ={}_{\mathrm{Pa}}{\mathrm{DP}}_2.\end{array}$
This is the error that would exist with a single sensor, so simply averaging data from multiple sensors may not appreciably change the height error (although the effects of sensor noise are reduced). However, the gap error still increases with DP.sub.2, so being able to select a sensor with the smallest magnitude of DP is still advantageous.

(44) According to a more general algorithm, the gap can be defined by:
h=(w.sub.1h.sub.1+w.sub.2h.sub.2+w.sub.3h.sub.3)=[w.sub.1(DP.sub.1)f.sub.1(DP.sub.1)+w.sub.2(DP.sub.2)f.sub.2(DP.sub.2)+w.sub.3(DP.sub.3)f.sub.3(DP.sub.3)],
where w.sub.1, w.sub.2, and w.sub.3 are respective normalized weighting coefficients and depend on the values DP.sub.i and w.sub.1+w.sub.2+w.sub.3=1. By way of example, w.sub.i(DP.sub.i) can have a Gaussian distribution g(DP.sub.i), such as that shown in FIG. 7, or other distribution. For proper normalization with Gaussian distributions:
w.sub.i(DP.sub.i)=g(DP.sub.i)/[g(DP.sub.1)+g(DP.sub.2)+g(DP.sub.3)]
The weights will select out the values of DP that are closest to zero, which reduces height error. Although some increase in sensor noise may occur, compared to a simple average, the result is nevertheless better than the result obtained using a single sensor.

(45) For example, consider a simple Gaussian for g(DP):
g(DP)=exp[(DP.sup.2/2.sup.2)].
For the gap error, let =(210.sup.4) nm/Pa.sup.2; .sub.Pa=100 Pa; and =600 Pa. Using the DP curves shown in FIG. 6, the height errors from individual DP sensors, and the error obtained from the weighted average are shown in FIG. 8. FIG. 8 includes three curves .sub.h1, .sub.h2, and .sub.h3 (delta h1, delta h2, delta h3, respectively) representing gap errors for three separate sensors. The light dashed-line curve is height error obtained by a single sensor but is also equal to a simple average of the three height errors. The sawtooth profile is similar to that shown in FIG. 1B, obtained by switching DP sensors as the gap changes. In contrast to any of the other profiles shown in FIG. 7, use of a weighted average height error (represented by the heavy line) with Gaussian weighting functions and the value of , above, provides a height error that is close to zero over most of the gap range (except at the extremes). Hence, use of the weighted average provides a height error that is substantially smaller than obtained using a single DP sensor, except at the extremes of the height range. It is noted that using additional DP sensors could extend the region in which the weighted average error is small. By making small adjustments to the g functions, the weighted average error can probably be made substantially zero over a range that is wider than the range shown in FIG. 6. In particular, selecting different values of for the three sensors may improve the error over the entire height range. Additionally, each weighting coefficient could have a different functional form if desired.

(46) Assuming that the noise of the three sensors is uncorrelated, and of the same magnitude, the total noise from the weighted sum of the sensor signals is proportional to N, wherein N(w.sub.1.sup.2+w.sub.2.sup.2+w.sub.3.sup.2).sup.1/2. If the weights were equal (e.g., w.sub.1=w.sub.2=w.sub.3=, corresponding to a simple average), then N=().sup.1/2, as shown previously. If only one sensor contributes significantly (e.g., w.sub.1=1 and w.sub.2=w.sub.3=0), then N=1. The result (FIG. 9) shows that significant noise reduction is possible using multiple sensors, weighted to minimize gap error.

(47) As noted elsewhere herein, one or more controllers or processors are programmed to determine the differential pressures and to perform calculations involving the differential pressures as described above. A flow-chart of an exemplary protocol for selecting weighted-average coefficients, as performed by the controller(s), is shown in FIG. 10A. A flow-chart of a corresponding embodiment of the algorithm used by the controller(s) in determining weighted height is provided in FIG. 10B.

(48) The first two algorithms described in FIG. 10B are straightforward. The third algorithm is more complicated. The weights can be determined by trial and error to reduce the height error, or they may be determined more formally to reduce the error to a minimum value. For example, suppose the functional forms for the weights are identical and depend upon one parameter Then:
W.sub.i(DP.sub.i)=w(,DP.sub.i)=w(,DP.sub.i(h))
The height error is then given by:
h=.sub.Pa[w(,DP.sub.1(h))DP.sub.1(h)+w(,DP.sub.2(h))DP.sub.2(h)+w(,DP.sub.3(h)]
thereby representing the height error as a function of height h. The value of , which minimizes the height error over the range of height h, can then be determined using standard minimization techniques, such as, but not restricted to, least squares, Simplex, or Levenberg-Marquardt. These techniques can be extended to more general cases where is different for each weight, or different functional forms are used.

(49) FIG. 10A is a flow-chart of an exemplary method for determining weighted-average coefficients. In step 201, height is adjusted to obtain a relationship of DP versus height. In step 202, the DP reference-channel flow restrictors are set so that a DP=0 condition occurs within the expected height-measurement range of the gauge. For example, DP.sub.1, DP.sub.2, and DP.sub.3 are set so that:

(50) DP.sub.2=0 at approximately the center of the height-measurement range;

(51) DP.sub.1=0 somewhere between the minimum height and the center of the height-measurement range; and

(52) DP.sub.3=0 somewhere between the maximum height and the center of the height-measurement range.

(53) As one example, if the height range is given by h.sub.0h:

(54) Adjust reference channel 1 so that DP.sub.1=0 at a height of approximately h.sub.00.5h;

(55) Adjust reference channel 2 so that DP.sub.2=0 at a height of approximately h.sub.0; and

(56) Adjust reference channel 3 so that DP.sub.3=0 at a height of approximately h.sub.0+0.5h.

(57) In step 203, a determination is made of the relation between height and pressure for each reference channel and sensor from measurements. For example:

(58) h.sub.1=(DP.sub.1)height estimate for channel 1 h.sub.2=f.sub.2(DP.sub.2)height estimate for channel 2 h.sub.3=f.sub.3(DP.sub.3)height estimate for channel 3
In step 204, weights are selected. Referring to the example shown in FIG. 10B, h.sub.1 is determined from DP.sub.1 (h.sub.2 and h.sub.3 are determined similarly from DP.sub.2 and DP.sub.3, respectively). Respective weights w.sub.1, w.sub.2, w.sub.3 are determined from h.sub.1, h.sub.2, h.sub.3, respectively, according to a desired algorithm. Three exemplary algorithms are shown in FIG. 10B: 1. Determine average: w.sub.1=w.sub.2=w.sub.3= 2. Set the DP sensor closest to zero (e.g., if i=m, then w(i=m)1 and w(im)=0) 3. Select w.sub.1, w.sub.2, and w.sub.3 to minimize height error over the expected range of heights
In step 205, the measurements are combined using the weights to obtain an improved height estimate:
h.sub.est=w.sub.1h.sub.1+w.sub.2h.sub.2+w.sub.3h.sub.3=w.sub.1f.sub.1(DP.sub.1)+w.sub.2f.sub.2(DP.sub.2)+w.sub.3f.sub.3(DP.sub.3),
wherein w.sub.1, w.sub.2, and w.sub.3 are functions of DP.sub.1, DP.sub.2, and DP.sub.3, respectively.

(59) Referring further to FIG. 10B, the depicted algorithms detail the calibration of DP.sub.1, but the algorithms are extendable to DP.sub.2 and DP.sub.3. In block 210, the relation DP.sub.1(h) is determined based on a series of DP measurements DP.sub.1 (212) and corresponding height measurements 214. The height measurements h are used by the controller(s) 216 for actuating, as required, a workpiece height actuator 218. Upon determining DP.sub.1(h), the reference flow restrictor is adjusted (220) to make DP.sub.1=0 at a desired height h. In block 222, the relation DP.sub.1(h) is inverted to get h.sub.1=f.sub.1(DP.sub.1), from which the corresponding weight w.sub.1 is determined (224). The weights w.sub.2 and w.sub.3 are determined in a similar manner, based on corresponding determinations of h.sub.2 and h.sub.3.

(60) FIG. 10C is a diagram showing an embodiment of a method for determining height of a workpiece from three DP values, namely DP.sub.1, DP.sub.2, and DP.sub.3. From these DP values, unweighted heights h.sub.1=f.sub.1(DP.sub.1), h.sub.2 f.sub.2(DP.sub.2), and h.sub.3=f.sub.3(DP.sub.3) are determined. Respective weights w.sub.1, w.sub.2, w.sub.3 are added to each height, and the weighted heights are summed to yield h=w.sub.1h.sub.1+w.sub.2h.sub.2+w.sub.3h.sub.3. Again, w.sub.1, w.sub.2, w.sub.3 are selected to reduce height error over the expected range of heights, using one of the algorithms shown previously.

(61) Included in this disclosure are any of various precision systems comprising a stage or the like that holds a workpiece or other item useful in a manufacture, relative to an axis, and that determines location of the stage at high accuracy and precision using devices and methods as described above. An example of a precision system is a microlithography system or exposure tool used for manufacturing semiconductor devices. A schematic depiction of an exemplary microlithography system 110, comprising features as described above, is provided in FIG. 11. The system 110 includes a system frame 112, an illumination system 114, an imaging-optical system 116, a reticle-stage assembly 118, a substrate-stage assembly 120, a positioning system 122, and a system-controller 124. The configuration of the components of the system 110 is particularly useful for transferring a pattern (not shown) of an integrated circuit from a reticle 126 onto a substrate 128 (e.g., a semiconductor wafer 128). The system 110 mounts to a mounting base 130, e.g., the ground, a base, or floor or other supporting structure. The system also includes a fluid-gauge measurement system 122A that measures the position of the substrate 128 (as an exemplary workpiece) along an axis (e.g., the z-axis or optical axis) with improved accuracy and precision. In certain embodiment the fluid-gauge measurement system 122A is configured so that environmental conditions near the workpiece and/or a photoresist-coated surface of the substrate 128 do not adversely influence the accuracy of the fluid gauge 122A.

(62) Since the fluid-gauge measurement system 122A utilizes multiple references, the dynamic range of the fluid-gauge 122A is relatively large. Consequently, the substrate 128 can be positioned with improved accuracy, and the microlithography system 110 can be used to manufacture electronic devices having higher circuit densities.

(63) There are various types of microlithographic systems. For example, the depicted system 110 can be used as a scanning type photolithography system. Alternatively, the exposure system 110 can be a step-and-repeat type microlithography system. However, the use of the exposure system 110 is not limited to a photolithography system for semiconductor manufacturing. The exposure system 110 can be used as, for example, an LCD photolithography system that exposes a liquid-crystal display device pattern onto a rectangular glass plate, or as a photolithography system for manufacturing thin-film magnetic heads.

(64) The system frame 112 is rigid and supports the components of the exposure system 110. The system frame 112 shown in FIG. 10 supports the reticle-stage assembly 118, the optical assembly 116, the wafer-stage assembly 20, and the illumination system 114 above the mounting base 130.

(65) The illumination system 114 includes an illumination source 132 and an illumination-optical assembly 134. The illumination source 132 emits a beam (irradiation) of light energy. The illumination-optical assembly 134 guides the beam of light energy from the illumination source 132 to the optical assembly 116. The illumination source 132 can be a mercury-lamp g-line source (436 nm), a mercury-lamp i-line source (365 nm), a KrF excimer laser (248 nm), an ArF excimer laser (193 nm), an F.sub.2 laser (157 nm), an EUV source (13.5 nm), or an x-ray source. Alternatively, the illumination source 132 can generate a charged particle beam such as an electron beam.

(66) The optical assembly 116 projects and/or focuses light leaving the reticle 126 to the substrate 128. Depending upon the configuration of the exposure system, the optical assembly 116 can magnify or reduce the image illuminated on the reticle 126.

(67) The reticle-stage assembly 118 holds and positions the reticle 126 relative to the optical assembly 116 and to the substrate 128. In FIG. 11 the reticle-stage assembly 118 includes a reticle stage 118A that retains the reticle 126, and a reticle-stage mover assembly 118B that positions the reticle stage 118A and the reticle 126. The reticle-stage mover assembly 118B can be configured to move the reticle 126 along the x-, y-, and z-axes, and about the x-, y-, and z-axes.

(68) Somewhat similarly, the substrate-stage assembly 120 holds and positions the substrate 128 relative to the projected image of the illuminated portions of the reticle 126. In FIG. 11 the substrate-stage assembly 210 includes a substrate stage 120A that retains the substrate 128, and a substrate-stage mover assembly 120B that positions the substrate stage 120A and the substrate 128. The substrate-stage mover assembly can be configured to move the substrate 128 along the x-, y-, and/or z-axes, and about the x-, y-, and z-axes.

(69) The positioning system 122 monitors movement of the reticle 126 and the substrate 128 relative to the optical assembly 116 or other reference. With this information the apparatus-control system 124 can control the reticle-stage assembly 118 to precisely position the reticle 126 and the substrate-stage assembly 120 to precisely position the substrate 128. For example, the positioning system 122 can utilize multiple laser interferometers, encoders, autofocus systems, and/or other measuring devices.

(70) In FIG. 11 the positioning system 122 includes: (i) a reticle-measurement system 122B (that monitors the position of the reticle stage 118B and the reticle 126), (ii) a substrate-measurement system 122B that monitors the position of the substrate stage 120A along the x- and y-axes, and about the z-axis, and (iii) the fluid gauge 122A that monitors the position of the substrate 128 relative to the optical assembly 116 along an optical axis 116A (e.g., the z-axis).

(71) Additionally, in certain embodiments the positioning system 122 can include an autofocus system 122D that monitors the position of the substrate 128 relative to the optical assembly 116 along the z-axis (the optical axis 116A), about the x-axis, and about the y-axis. A suitable autofocus system 122D is a slit-type system that directs multiple slit images of light at a glancing angle of incidence at the substrate 128 and measures the light reflected from the substrate 128. A further discussion of a slit-type autofocus system 122D is set forth in U.S. Pat. No. 4,650,983, which is incorporated herein by reference as far as permitted by law.

(72) In one non-exclusive embodiment, a fluid gauge 122A is used in conjunction with an autofocus system 122D to calibrate the autofocus system 122D prior to processing (e.g., transferring images to) the substrate 128 to improve the accuracy of the autofocus system 122D.

(73) The apparatus-control system 124 is connected to the reticle-stage assembly 118, the substrate-stage assembly 120, and the positioning system 122. The apparatus-control system 124 receives information from the positioning system 122 and controls the stage assemblies 118, 120 to precisely position the reticle 126 and the substrate 128. The apparatus-control system 124 can includes one or more processors and circuits.

(74) An exemplary process for manufacturing semiconductor devices, including an exposure step, is shown in FIG. 12. In step 901 the device's function and performance characteristics are designed. Next, in step 902, a mask (reticle) having a desired pattern is designed according to the previous designing step, and in a parallel step 903 a wafer is made from a suitable semiconductor material. The mask pattern designed in step 902 is exposed onto the wafer from step 903 in step 904 by a microlithography system described herein in accordance with the present invention. In step 905 the semiconductor device is assembled (including the dicing process, bonding process, and packaging process. Finally, the device is inspected in step 906.

(75) FIG. 13 is a flowchart of the above-mentioned step 904 in the case of fabricating semiconductor devices. In FIG. 13, in step 911(oxidation step), the wafer surface is oxidized. In step 912 (CVD step), an insulation film is formed on the wafer surface. In step 913 (electrode-formation step), electrodes are formed on the wafer by vapor deposition. In step 914 (ion-implantation step), ions are implanted in the wafer. The above-mentioned steps 911-914 constitute the preprocessing steps for wafers during wafer processing, and selection is made at each step according to processing requirements.

(76) At each stage of wafer-processing, when the above-mentioned preprocessing steps have been completed, the following post-processing steps are implemented. During post-processing, first, in step 915 (photoresist-formation step), photoresist is applied to a wafer. Next, in step 916 (exposure step), the above-mentioned exposure device is used to transfer the circuit pattern of a mask (reticle) to a wafer. Then, in step 917 (developing step), the exposed wafer is developed, and in step 918 (etching step), parts other than residual photoresist (exposed material surface) are removed by etching. In step 919 (photoresist-removal step), unnecessary photoresist remaining after etching is removed. Multiple circuit patterns are formed by repetition of these pre-processing and post-processing steps.

(77) It will be understood that gauges as disclosed herein are merely illustrative of the currently preferred embodiments, and that no limitations re intended to impact the details of construction or design herein shown, other than as described. In certain embodiments as described above, the fluid gauge is configured to monitor the position of a wafer or other lithographic workpiece relative to an optical assembly and used in conjunction with an auto-focus system that monitors position of a lithographic substrate relative to the optical assembly. However, use of the gauge is not limited to monitoring the position of a lithographic substrate. For example, the fluid gauge can be configured to monitor the position of a reticle relative to the optical assembly and used in conjunction with the auto-focus system that monitors the position of the reticle relative to the optical assembly.

(78) Whereas the invention has been described in connection with representative embodiments, it will be understood that it is not limited to those embodiments. On the contrary, it is intended to encompass all alternatives, modifications, and equivalents as may be included within the spirit and scope of the invention as defined by the appended claims.