**3.2 Error analysis of length calibration systems**

Making measurements with nanoscale precision poses several major difficulties. Environmental fluctuations such as vibration or temperature change have a large effect at the nanoscale. For example, any external change to the large machines used in manufacturing microelectronics components will affect the creation of nanoscale features and their crucially important alignment to each other. The ability to measure these influences, and thereafter to minimize them, is therefore vital.

The error budget is an accuracy model of the machine in its environment expressed in terms of cause-and-effect relationships. It may involve random quantities such as seismic vibrations or deterministic quantities such as deflections due to gravity or weights of moving axes and payloads. The error budget helps identifying where to focus resources to improve the accuracy of an existing machine or one under development. It provides useful information for the specification of subsystem precision requirements to achieve an overall balance at the levels of difficulty.

The technical basis for the error budget rests on two assumptions:


A practical difficulty arises because we generally cannot quantify the errors in complete detail especially at the design stage. Although an error may vary spatially and temporally, usually the only estimate will be a bounding envelope and perhaps an approximate frequency of variation.

In general, machine tools errors can be divided into two categories: systematic and random errors. Systematic errors can be described by deterministic mathematical and engineering models. Random errors are difficult to model and to compensate. In the calibration of line standards, several essential sources need to be identified and eliminated or corrected. The following error map, see Fig. 5, presented as cause and effect diagram, gives the sources of uncertainty classified by type and their origin (Jakštas, 2006).

as measuring tapes or rods. Many materials are used including steel, Invar, glass, glassceramics, silicon, and fused silica. Cross sectional shape can be rectangular, "H," modified "U" (flat bottom), or a modified "X" (Tresca). At present, the line scale interferometer is limited to graduation widths ranging from submicrometer to 100 mm, and spacings ranging from less than 1 mm up to 1025 mm. Spacings are generally measured from center to center

Some devices that are not strictly linear scales are measured in the line scale interferometer. These include end standards in a size range (250 mm to 1000 mm) that can present

Two dimensional patterns are measured by treating each row and column of graduations as an independent scale and, when possible, an estimate of orthogonality can be made by

Making measurements with nanoscale precision poses several major difficulties. Environmental fluctuations such as vibration or temperature change have a large effect at the nanoscale. For example, any external change to the large machines used in manufacturing microelectronics components will affect the creation of nanoscale features and their crucially important alignment to each other. The ability to measure these

The error budget is an accuracy model of the machine in its environment expressed in terms of cause-and-effect relationships. It may involve random quantities such as seismic vibrations or deterministic quantities such as deflections due to gravity or weights of moving axes and payloads. The error budget helps identifying where to focus resources to improve the accuracy of an existing machine or one under development. It provides useful information for the specification of subsystem precision requirements to achieve an overall

the total error in a given direction is the sum of all individual error components in that

 individual error components have physical causes that can be identified and quantified. A practical difficulty arises because we generally cannot quantify the errors in complete detail especially at the design stage. Although an error may vary spatially and temporally, usually the only estimate will be a bounding envelope and perhaps an approximate

In general, machine tools errors can be divided into two categories: systematic and random errors. Systematic errors can be described by deterministic mathematical and engineering models. Random errors are difficult to model and to compensate. In the calibration of line standards, several essential sources need to be identified and eliminated or corrected. The following error map, see Fig. 5, presented as cause and effect diagram, gives the sources of

of the graduations, but can also be measured from edge to edge.

measurement problems with laser interferometry.

**3.2 Error analysis of length calibration systems** 

influences, and thereafter to minimize them, is therefore vital.

The technical basis for the error budget rests on two assumptions:

uncertainty classified by type and their origin (Jakštas, 2006).

measuring the diagonals.

balance at the levels of difficulty.

direction, and

frequency of variation.

Fig. 5. Example of error model for line scale calibration system

Among the key factors that affect the accuracy of length calibration systems are geometric deviations and thermal effects on the comparator components and the scale. Mechanical limitations of calibration systems are featured by the whole complex of a mechanical system including error compensation circuits.

Thermally induced errors in measurement are an accuracy-limiting factor in the length metrology. Temperature, barometric pressure and humidity influence the refractive index of air and, thus, the wavelength of light; temperature also affects the length of the scale being measured. The philosophy of precise dimensional measurements is, therefore, to approximate thermal equilibrium near the reference temperature 20 oC for measurement condition. In order to calibrate especially long line scales some prerequisites have to be fulfilled as far as temperature influence is concerned:


Temperature problems are a widely assessed error source in precision measuring machines. While analyzing high precision calibration systems it is essential to evaluate an average volume temperature of some parts of a mechanical comparator as well as the temperature of the scale. Under real calibration conditions, temperature measurement is possible only at certain points. The response time of temperature sensors is also rather long (from several seconds to several minutes). Therefore, fast temperature changes cannot be detected, and, consequently, the measurement uncertainty increases. The thermally induced errors in the long-stroke measuring machine are more significant because of their size and complexity. Due to high requirements for geometrical stability of calibration system, the temperature deformations caused by changes of several hundredth parts of Kelvin must be considered. The cause-and-effect relationships can be calculated in considerable detail using modern finite element (FE) analysis and empirical heat transfer formulas, but doing so requires considerable knowledge about the design and environment, see chapter 3.3.

The results of temperature measurements performed in the comparator laboratory (shown in Fig. 6) have revealed that maximum temperature deviations from the mean amount to about ±0.25 K (during the whole period of system operation) and the main reasons that induce temperature gradients are laser source, computation and control unit, and thermally non-isolated part of the ground floor. The measurement of air temperature during the calibration procedure due to the introduced dynamic mode indicated much smaller temperature variances, and calculated standard deviation was 0.034 K. For precision calibration of the line scales not only the closeness of laboratory temperature to nominal 20 oC temperature is of crucial importance, but also temperature stability (deviations) during calibration procedure. Therefore, a period of temperature stabilization that can last for 10-14 hours is necessary in order to accomplish high precision calibration procedures.

Fig. 6. Temperature measurements in the laboratory of length calibration system

#### **3.3 Modeling**

Error-related problems specific to length measurements are caused primarily by geometrical and thermal deviations of the comparator components and the scale. One of critical tasks in

certain points. The response time of temperature sensors is also rather long (from several seconds to several minutes). Therefore, fast temperature changes cannot be detected, and, consequently, the measurement uncertainty increases. The thermally induced errors in the long-stroke measuring machine are more significant because of their size and complexity. Due to high requirements for geometrical stability of calibration system, the temperature deformations caused by changes of several hundredth parts of Kelvin must be considered. The cause-and-effect relationships can be calculated in considerable detail using modern finite element (FE) analysis and empirical heat transfer formulas, but doing so requires

The results of temperature measurements performed in the comparator laboratory (shown in Fig. 6) have revealed that maximum temperature deviations from the mean amount to about ±0.25 K (during the whole period of system operation) and the main reasons that induce temperature gradients are laser source, computation and control unit, and thermally non-isolated part of the ground floor. The measurement of air temperature during the calibration procedure due to the introduced dynamic mode indicated much smaller temperature variances, and calculated standard deviation was 0.034 K. For precision calibration of the line scales not only the closeness of laboratory temperature to nominal 20 oC temperature is of crucial importance, but also temperature stability (deviations) during calibration procedure. Therefore, a period of temperature stabilization that can last for 10-14 hours is necessary in order to accomplish high precision calibration procedures.

considerable knowledge about the design and environment, see chapter 3.3.

Fig. 6. Temperature measurements in the laboratory of length calibration system

Error-related problems specific to length measurements are caused primarily by geometrical and thermal deviations of the comparator components and the scale. One of critical tasks in

**3.3 Modeling** 

the dynamic calibration of the scales is microscope image acquisition, i.e. bringing the specimen into focus before taking an image and measuring any feature. Due to imperfections of the stage, i.e. inaccuracy of the motion of the scanning mechanism and vibrations, the microscope slide is not perfectly perpendicular in regard to the optical axis of the imaging system. Measuring even a slightly vibrating structure with any degree of accuracy is prone to an error with an optical microscope. Any deviation in the distance of microscope lens with respect to the scale - for example, when the surface is in motion during the data acquisition process - introduces measurement errors. The magnitude of the resulting error can range from a few nm to several μm depending on the magnitude of such disturbances and the measurement setup. The most common error associated with small vibrations is the error in detection of the line scale graduation.

Structures of precision length measuring machines are often too sophisticated to be modelled precisely by applying simple methods. Therefore complex models as well as their analysis tools are needed in order to perform qualitative and quantitative description and analysis of determinants of the precision length calibration process.

In this work finite element method (FEM) techniques were used for behavior simulation of the comparator carriage, CCD microscope and calibrated line standard itself under the influence of dynamic and thermal factors, like variations of environmental temperature, vibrations in the structure caused by seismic excitation.

The state-of-the-art FE technique was applied in order to evaluate the possible influence of dynamic and thermal factors upon the inaccuracies of measurement. Two basic physical phenomena were of interest:


All necessary aspects of the dynamic behavior of the comparator can be investigated by employing small displacement elastic structural models as (Jakštas et al., 2006)

$$\left\{ \left[ M \right] \middle| \left\{ \ddot{\boldsymbol{L}} \right\} + \left[ \boldsymbol{C} \right] \middle| \left\{ \dot{\boldsymbol{L}} \right\} + \left[ \boldsymbol{K} \right] \middle| \left\{ \boldsymbol{L} \right\} \right\} = \left\{ F(t) + Q(t) \right\} \tag{2}$$

[*K*], [*M*], [*C*] are stiffness, mass and damping matrices of the structure; {*F*} is nodal vector of external excitation forces; *Q t* is nodal force vector caused by the temperature propagation effect; *U* , *U* , *U* , are nodal displacement, velocity and acceleration vectors.

FE models enable us to simulate all 3D displace-ment or vibration patterns of the structure. Vertical vibrations of the microscope may lead to defocusing. Vibrations in the direction of motion may cause detection errors in determining positions of graduation lines.

In practice the excitation vectors *F t* caused by external dynamic effects or by moving parts of the structure are not known explicitly, but often are subjected to external excitation propagating through the base and supports of the comparator structure. A spectrum analysis is one in which results of a modal analysis are used with a known spectrum to calculate displacements and stresses in the structure modeled. The model is capable of predicting the system's behavior under thermal load and enables us to investigate the thermo-mechanical processes in the system, by taking into account both static and dynamic disturbances and parameter deviations.

The dynamic response of the comparator as a result of seismic excitation was investigated by employing small displacement elastic structural models (Jakštas, 2006; Jakštas et al. 2006). Modeling of seismic excitations in the comparator structure has shown that maximum displacements are expected at the bottom plane of the microscope objective and can amount more than 100 nm.

Modal analysis of the spatial carriage and microscope deviations induced by seismic excitations as well as those caused by operation of the carriage drive vibrations has shown that dynamic factors may contribute significantly to the calibration uncertainty budget. To minimize, in particular, the vibrations of the measurement reflector the construction of the carriage structure was optimized and drive-originated were vibrations reduced.

Extensive investigations were accomplished to both reduce the dynamically induced deviations originated by the dynamic excitations of the mechanical structure and optimize the comparator design. The precision measurements were performed in order to evaluate the impact of small vibration on performance of the line scale calibration process. The experimental results revealed, in particular, that the sample standard deviation of the driveinduced relative displacements between the moving reflector of the interferometer and the measurement point of the microscope may reach 0.662 μm (at calibration speed 3 mm/s); they were considerably reduced by optimization the carriage structure and elimination the undesirable modes of vibration.

Measurement results depicted in Fig. 7 show an improvement of systems stability using the optimized microscope carriage structure (blue line). The sample standard deviation of the drive-induced relative displacements of measurement mirror was reduced from 0.178 μm down to 0.054 μm (Kaušinis et al., 2009).

Fig. 7. Comparison of vibrations of the interferometer measurement mirror

thermo-mechanical processes in the system, by taking into account both static and dynamic

The dynamic response of the comparator as a result of seismic excitation was investigated by employing small displacement elastic structural models (Jakštas, 2006; Jakštas et al. 2006). Modeling of seismic excitations in the comparator structure has shown that maximum displacements are expected at the bottom plane of the microscope objective and can amount

Modal analysis of the spatial carriage and microscope deviations induced by seismic excitations as well as those caused by operation of the carriage drive vibrations has shown that dynamic factors may contribute significantly to the calibration uncertainty budget. To minimize, in particular, the vibrations of the measurement reflector the construction of the

Extensive investigations were accomplished to both reduce the dynamically induced deviations originated by the dynamic excitations of the mechanical structure and optimize the comparator design. The precision measurements were performed in order to evaluate the impact of small vibration on performance of the line scale calibration process. The experimental results revealed, in particular, that the sample standard deviation of the driveinduced relative displacements between the moving reflector of the interferometer and the measurement point of the microscope may reach 0.662 μm (at calibration speed 3 mm/s); they were considerably reduced by optimization the carriage structure and elimination the

Measurement results depicted in Fig. 7 show an improvement of systems stability using the optimized microscope carriage structure (blue line). The sample standard deviation of the drive-induced relative displacements of measurement mirror was reduced from 0.178 μm

Fig. 7. Comparison of vibrations of the interferometer measurement mirror

carriage structure was optimized and drive-originated were vibrations reduced.

disturbances and parameter deviations.

more than 100 nm.

undesirable modes of vibration.

down to 0.054 μm (Kaušinis et al., 2009).

Temperature problems are a widely assessed error source in precision measuring machines. While analyzing high precision calibration systems it is necessary to evaluate an average volume temperature of some parts of a mechanical comparator as well as the temperature of the scale. Under real calibration conditions, temperature measurement is possible only at certain points. The response time of temperature sensors is also rather long (from several seconds to several minutes). Therefore, fast temperature changes cannot be detected, and, consequently, the measurement uncertainty increases. The thermally induced errors in the long-stroke measuring machine are even more significant because of their size and complexity. Due to high requirements for geometrical stability of calibration system, the temperature deformations caused by changes of several hundredth parts of Kelvin must be considered.

The cause-and-effect relationships can be calculated in considerable detail using modern FE analysis and empirical heat transfer formulas, but doing so requires considerable knowledge about the design and environment.

The impact of temperature on the mechanical deformation of the line scale can be simulated in several ways:


One of precarious temperature disturbances is the heat spread out by the CCD camera of the measuring microscope. As the steady-state temperature under the operating conditions is known, the thermal expansion process can be modeled by using the FE simulation, and the temperature values can be found at all points of the microscope structure. Having the temperature values obtained, the displacements due to thermal expansion can be calculated at any point of the structure.

The equation of the structure heat balance reads as follows:

$$\mathbb{E}\left[\mathbf{C}\right]\big|\dot{T}\big) + \left[K\_{Th}\big|\big|T\big) = \left\{S\_{\Rightarrow}\right\}\tag{3}$$

where *C* - matrix of thermal capacity, *KTh* - matrix of thermal conductivity, *S* - nodal vector of heat sources of the element determined by the heat exchange over the surface of the body.

The solution presents the nodal temperature values, which are further used as loads in the problem of thermal expansion of the structure as:

$$\begin{bmatrix} \mathbf{K} \end{bmatrix} \begin{Bmatrix} \mathbf{U} \end{Bmatrix} = \begin{Bmatrix} \mathbf{Q} \end{Bmatrix} \tag{4}$$

where *K* – stiffness matrix of the element; *Q* – vector of nodal forces determined by temperature loads.

The FE computational model of the structure was set up, in which the temperatures of the structure and the ambient air could be calculated. The model is based on the coupling of the following physical phenomena:


In the computational model phenomena 1-4 have been described by means of ANSYS (FLOTRAN) element FLUID142. The element can be used under two different conditions:


The model is capable to predict the system's behavior under thermal load and enables us to investigate thermo-mechanical processes in the system and facilitates finding proper structural solutions to reduce the impact of thermal load on the calibration accuracy.

Displacements in the structure caused by the calculated temperatures field are depicted in Fig. 8.

Fig. 8. Displacements in the structure caused by the calculated temperatures field; CCD camera is fixed to a side of the microscope: (a) deformation of the construction; (b) vectors of nodal displacements; (c) deformation of the structure

Vector *Q* in equation (1) can be easily determined if the temperature field inside of the structure is known. If only the surrounding transient temperature field is known, the temperatures inside of the structure can be obtained by solving the thermal conductivity problem

5. Formation of deformations in comparator structure due to the non-homogenous

In the computational model phenomena 1-4 have been described by means of ANSYS (FLOTRAN) element FLUID142. The element can be used under two different conditions: - liquid (gas) dynamics described by the continuity equation, the advection-diffusion

The model is capable to predict the system's behavior under thermal load and enables us to investigate thermo-mechanical processes in the system and facilitates finding proper

Displacements in the structure caused by the calculated temperatures field are depicted in

(a) (b) (c)

Vector *Q* in equation (1) can be easily determined if the temperature field inside of the structure is known. If only the surrounding transient temperature field is known, the temperatures inside of the structure can be obtained by solving the thermal conductivity

Fig. 8. Displacements in the structure caused by the calculated temperatures field; CCD camera is fixed to a side of the microscope: (a) deformation of the construction; (b) vectors of

nodal displacements; (c) deformation of the structure


structural solutions to reduce the impact of thermal load on the calibration accuracy.

1. Heat transfer by the ambient air due to its thermal conductivity;

4. Heat transfer by comparator structure due to its thermal conductivity;

2. Convective heat transfer (due to the motion of the air); 3. Heat exchange between the air and comparator structure;

thermal field generated in it.

Fig. 8.

problem

equation and the ideal gas state equation;

$$
\begin{bmatrix} \mathbf{C}\_{\text{th}} \\ \end{bmatrix} \begin{Bmatrix} \mathbf{T} \end{Bmatrix} + \begin{bmatrix} \mathbf{K}\_{\text{th}} \\ \end{bmatrix} \begin{Bmatrix} T & \\ \end{Bmatrix} = \begin{Bmatrix} \mathbf{S}\_{\text{ev}} \\ \end{Bmatrix} + \begin{Bmatrix} \mathbf{Q}\_{\text{th}} \\ \end{Bmatrix} \tag{5}
$$

where *Cth* **,** *Kth* - thermal capacity and conductivity matrices of the structure, *S* nodal vector of heat sources determined by heat convection across the surface of the structure.

In order to determine distribution of the temperature fields around the CCD camera, temperature sensors were arranged, and temperature distribution measured at CCD cut-off, warming up, and steady operating conditions.

Calibration error caused by a thermal CCD camera impact under steady-state calibration conditions is of a random character and in real-time it cannot be compensated by mathematical methods. Within the experiments conducted an estimate of the variance of this error amounts to +/- 0.23 µm at the 95% probability level (Barauskas et al., 2011).

### **3.4 Line scale calibrations**

High-precision measurements of line scales basically apply two main modes of calibration: static and dynamic. Currently, static line detection systems are predominantly used in metrology institutes and calibration laboratories worldwide. The static method is potentially more accurate but somewhat slow, whereas the dynamic method offers taking advantages of scale calibration in terms of speed, accuracy and throughput. It also allows the construction simplification, because high-precision settling of the moving scale or microscope is unnecessary, and the measurement process is less influenced by ambient environmental conditions. On the other hand, the dynamic method encounters difficulties induced by measurement speed fluctuations, time delays, noise and vibrations especially during the graduation line detection. Mechanical limitations for the dynamic mode are featured by the whole complex of mechanical system including error compensation circuits.

As the dynamic calibration process is to be examined in real time, the experiments of the line scale calibration with both a slit and moving CCD microscope have been carried out in specific operating modes, and the dependence of the accuracy of dynamic calibration vs. speed has been studied.

The calibration experiments were performed that intended to document current capabilities to carry out line scale calibrations on high quality graduated scales made of low thermal expansion substrates. The line scale standard made of the glass ceramic Zerodur was available for calibration purpose from PTB. The dimensions of the scale are 230 mm in length, 25 mm in width and 14 mm in height. The graduation represents a total length of 200 mm and consists of line structures with 1 mm length and 2.5 μm width. The line structures are reflecting on transparent substrates. The measurand that was determined on the line scale standard is the deviations from the nominal lengths for 1 mm lines (1 mm pitch). Fig. 9 shows the deviations from the nominal positions for the weighted mean, calculated on the basis of the set of 6 independent measurement runs taken at the microscope carriage speed 3mm/s. The environmental chamber and scale temperatures were held within ±0.05 °C during the measurements. The positions of the line are corrected for the influence of the temperature deviation from 20 °C and pressure deviations from 1013.25 hPa.

Fig. 9. Calibration results on Zerodur scale, 200 mm graduation, 1 mm step.

It has been demonstrated that the capabilities of a newly developed comparator are close to the calibration capabilities of analogous long scale calibration systems in the other countries, available from BIPM key comparison database, see Fig. 10, and still can be improved first of all by embedding an automatic line focusing system and tightening tolerances of ambient conditions in the laboratory.

Fig. 10. Comparison of long line scale calibration capabilities
