Issue #1/2022

MEASUREMENT OF DISPLACEMENT USING A MOVABLE HOLOGRAPHIC DIFFRACTION GRATING IN AN INTERFERENCE FIELD

**B.G.Turukhano, N.Turukhano, I.A.Turukhano**MEASUREMENT OF DISPLACEMENT USING A MOVABLE HOLOGRAPHIC DIFFRACTION GRATING IN AN INTERFERENCE FIELD

10.22184/1993-8578.2022.15.1.46.56

INTRODUCTION

LHE and AHE are based on combination of the HDG with interference field and can be used for determination of nanoscale displacement magnitude. The authors have developed and realized technology of synthesis of linear nano HDG with accuracy ±0.4 µm/m and length over one meter and radial HDG with accuracy ±0.2 angular second, as well as developed a number of linear and angular holographic measuring systems based on these sensors.

The phase distribution of the investigated linear interference field (IF) along the OX axis can be represented as follows:

Φ(x) = 2kx + Δf(x), (1)

here k = 2π/λsin/θ/2 = π/d0 is the wave number, d0 = λ/sinθ/2 is the line period of the interference field formed by the two plane waves S10 и S20.

DESCRIPTION OF THE EXPERIMENT

In this work we investigate a possibility of determining the magnitude of displacement by means of a linear holographic diffraction grating (Ризм) acting together with an interference field. During their interaction the interference moiré and obturation bands (MD and OB) are formed depending on the angle between the bars of the IF and the measuring holographic diffraction grating (Ризм), as well as on the equality or difference of their periods.

Let us show that the set problem of determining the magnitude of displacement and size of a product or object can be solved by means of an orderly moving Ризм in the IF. The essence of the method consists in the fact that the information about phase distribution of IF is taken in the system of points rigidly connected with the moving lattice [1]. In this case, the phase difference between any points will be uniquely determined by the phase distribution of Ризм dashes only. The phase difference introduced by the interferometer itself will be a constant, additively included in each act of measurement.

It is required to find the phase distribution of the interference fringes in the pattern of interaction of the interference field with a holographic diffraction grating. Let us consider interactions of the interference field of a IF with a holographic diffraction grating Ризм. The method of phase distribution along the measured axis consists in determination of the phase distribution along the chosen moving direction including determination of all boundary conditions which allow complete phase matching of the phase distribution values at displacement.

Let us represent Ризм itself using two quasi-planar waves [2–3]. Two quasiplanar monochromatic waves with complex amplitudes are:

S1G (x1,y1) – exp i

[k1G r1 +Ψ1G (x1,y1)], (2)

S2G (x1,y1) – exp i [k2G r1 +Ψ2G (x1,y1)] (3)

and they form IF in the X1OY1 plane where Ризм with the amplitude transmission:

T(x1,y1) α cos[kG r1 + ΨG (x1,y1)], (4)

here kG = k1G – k2G is the wave vector of the grating, and the function that characterises the phase distribution of the bars is Ризм:

ΨG(x,y) = Ψ1G(x1,y1) – Ψ2G(x1,y1) (5)

In this case the grating period is:

dG = π/KG =λ/2sinθ1/2, (6)

where θ1 is the angle between wave vectors k1G and k2G, and KG = |KG| means the grating wave number.

Let us place the grating in the XOY plane in the XYZ coordinate system, which is rotated with respect to the XYZ coordinate system by angles υ, γ, φ respectively, and illuminate it with two quasi-plane monochromatic waves with complex amplitudes:

S1 (x,y) – exp i [k1 r +Ψ1 (x,y)],

S2 (x,y) – exp i [k2 r +Ψ2 (x,y)], (7)

where k1 = k1G, k2 = k2G.

Waves S1(x,y) and S2(x,y) form the investigated IFs with a phase distribution in the XOY plane:

Ψ (x,y) = Ψ2 (x,y) – Ψ1 (x,y) (8)

Let us introduce a parameter δ characterising the mismatch of the IF period with respect to the Ризм period, so that the wave number of the field is:

k = |k2 – k1| = kG (1- δ); |δ|«1, (9)

where δ is the mismatch between the IF period and the grating period.

Interaction of the IF with the grating in the XOY plane results in a pattern of interference fringes (If). The joint solution of equations (4) and (6), taking into account the transition formulas from one system to another (r1»r), leads to the following expression for the IfF intensity distribution:

I(x,y) = {1 + cos{kG /cosγ [(xcosφ +

+ ysinφ) – (1 – δ)xcosγ]} +

+ ΨG (x1,y1) – Ψ (x,y)}, (10)

Since I(x,y) changes little if the angle υ changes over a wide range, the terms containing υ in (9) are omitted.

From (9), we can write the IfF phase distribution Φ (x,y) as

Φ(x,y) = kG/cosγ [(xcosφ + ysinφ) –

– (1 – δ)xcosγ] + [ΨG (x1,y1) – Ψ (x,y)]. (11)

Thus, we obtained an expression that allows us to describe the IfF pattern in the most general case, the interaction of an arbitrarily oriented nonideal HDG with a period-mismatched IF. It should be noted that since the mismatch parameter was chosen to be sufficiently small (the case most often realized in practice), expression (10) is valid not only in the XOY plane but also at sufficiently large distances in the direction of propagation of each of the interfering waves.

Let us now proceed to analyse the phase distribution of the IfF and rewrite (10) as

Φ(x,y) = Φр (x,y) +

+ [ΨG (x1,y1) – Ψ (x,y)], (12)

where

Φ(x,y)р = kG/cosγ [(xcosφ +

+ ysinφ) – (1 – δ)xcosγ]. (13)

It is easy to see that Φ(x,y)р describes the interaction of an ideal HDG with an ideal IF formed by two plane waves, while the second term of equation (12), enclosed in square brackets, describes the deviations of the phase distributions of the IF and HDG from the ideal ones.

Let us focus on the analysis of Φ(x,y)р. The equation describing the configuration and location of MBs in the case of IFs and HDGs has the form:

kG/cosγ [(xcosφ + ysinφ) –

– (1 – δ)xcosγ] = 2πn, (14)

where n is an integer. It is easy to see that equation (13) describes a series of equidistant lines with a period:

dM = dG cosγ/[(cosφ –

– cosγ + δ cosγ)2 + sin2φ]0.5 (15)

tgα = – 1/ cosγ cosφ

[cosγ (1 – δ) + cosφ] (16)

The most interesting in practice is the case of φ«1; δ«1; γ«1. In this case, neglecting the values of the second order of smallness we have:

dM = dG /[φ2 + δ2]0.5 (17)

α ≈ arctg(φ/2 + (γ2/2 – δ))/ φ) (18)

Formula (17) shows that the angle γ enters (17) as a value of the second order of smallness as compared to φ and δ; moreover, when δ = γ2/2, the value of α depends on only one variable α = α(φ). Thus, by selecting the mismatch parameter δ we can compensate for the HDG slope angle. Therefore, from now on, we will consider only two variables: the angle φ of the slope of the HDG bars with respect to the fringes of the IF and the mismatch parameter δ with respect to the period.

According to expression (11) for Φ(x,y) at δ«1 and φ«1, ΨG(x1,y1) ≈ Ψ(x,y) and, excluding from consideration γ, we can write:

Φ(x,y) = Φр (x,y) +

+ [ΨG (x1,y1)- Ψ (x,y)], (19)

where

Φ(x,y)р ≈ kG[(xcosφ + ysinφ) – (1 – δ)x].

Expression (18) describes the MB phase distribution quite accurately, in all practical cases.

Thus, we have determined the phase distribution of MF (18) from the interaction between the IF and Ризм located in the IF aperture.

RESULTS AND DISCUSSION

This picture will form the basis for operation of a holographic LHE linear displacement sensor depending on the displacement of Рmeas. in an interferometer IF or in an IF formed behind the same grating illuminated by a coherent emission source.

The clipped light generated by the radiation source rigidly connected to the sensor carriage after reflection from the mirror, falls on the measuring and display gratings (the order of the gratings can be any, depending on the design of the movable linear displacement detector, MLDD). Photodetectors are installed in the MB field behind the gratings.

At the beginning we adjust the grating fringes parallel to each other by means of a frame containing means for turning the Ринд fringes relative to the Ринд fringes and obtaining moiré conjugation of MB, which are perpendicular to the direction of the grinding fringes in the first approximation. It is desirable to obtain MBs as wide as possible (Fig.1).

The device is further tuned to produce an appropriate nonius conjugation. This can be done by changing divergence of the light source (Fig.2).

Fig.2 gives an optical diagram for obtaining nonius fringes. In the OX direction is the Ризм recorded in a two-beam interferometer with an angle θ/2 between its arms. The diffraction grating is illuminated by two beams S1 and S2 at equal angles (θ/2 + δ).

Nonius fringes (Fig.3) are similar to moiré fringes, but they are parallel to the grinding fringes.

When determining the linear dimension of an object, Ринд is displaced relative to the measurement dimension or Ринд is displaced relative to the indicator dimension (which is equivalent).

Although it contains the nods required for the initial setting of the relative positions of the two gratings, Ризм and Ринд, it cannot eliminate the measurement errors introduced by the movement of one of the gratings relative to the other fixed grating by inaccurate guides of the linear encoder itself or the external device to which its moving grating is fastened.

When moving along inaccurate guides, the originally set period of moiré fringes is not retained, as the frame with the indicator grating fixed in it will repeat roughness of the guide, which will necessarily lead to a change in the angle of its bars relative to the Ризм fringes and, consequently, to a variable MB period, which in turn leads to an error in the readout of the movement.

In this regard, HDGs use nonius or obturator fringes rather than moiré strips (Fig.3). In addition, it is possible to design a device which will substantially reduce the error introduced by the guide which has non-linearity in the direction of travel.

In order to eliminate variation of the MB period during movement, the glass guide of the HDG itself can be used, for example, as a guide along which the Ринд moves. For this purpose, the glass guide is rigidly connected to the end of the measuring tape (Fig.4) [4]. The glass guide of the HDG itself, rigidly connected to the face of the measuring tape, is used in this device as a guide, along which the reading head moves. This guide must be at least as long as the linear dimension of the object being measured. As the reader head moves along the glass guide, the support bearings move along the base surface of the glass guide and the spring-loaded bearing, respectively, along its reverse surface (Fig.5). The length of this guide is not less than the linear dimension of the object or the measured displacement value.

If the base surface of the glass guide is of good quality, the movable grating will move exactly along the chosen (constructively) axis of the measuring diffraction grating and perpendicular to its bars. This means that the IBs formed behind the gratings retain their period and slope and will not affect the accuracy of measurements, which will only depend on the quality of the measuring array and the quality of the manufacture of the sensor itself. Thus, we have an opportunity to fully realize the accuracy characteristics of the HDG (Fig.6).

Domestic industry serially produces special glass (Borsky glass on tin melt) that has the required precision characteristics for the glass guide and thus for obtaining nanocharacteristics of HDGs. This will make it possible to carry out sufficiently cheap and efficient serial production of domestic high-precision and high-resolution HDGs above 0.1 µm/m.

The working principle of the HDG for displacement measurement is as follows. When the reader head is moved, whilst measuring the linear dimensions of an object, the indicator grating is shifted with respect to the diffraction measuring grating 2 located on the substrate of the measuring tape ruler. The light beam generated by the light source rigidly coupled to the reader head is collimated by the collimator and passes through the indicator and measurement gratings. A matrix of photodetectors is installed in the field of interference fringes behind the grating, which converts the intensity distribution of the interference fringes into electrical signals. When the reader head is shifted simultaneously with the object during determination of its linear dimension, the indicator grating is shifted relative to the measuring grating and variable electrical signals shifted by 90° in phase are generated at the outputs of the matrix photodetectors. These signals are then fed to the electronics unit, where a comparator is used to generate counting pulses that determine the linear size of the object. The bearing rigidly connected to the indicator grating 6, moving on the base surface of the measuring ruler substrate and the spring-loaded bearing of the same group moving on the reverse surface of the substrate allow to keep a constant gap between the gratings Ризм and Ринд, thus ensuring the probe operability during the whole range of object linear dimension measurement regardless of the quality of the guide device to which the moving part of the HDG is fixed. Therefore this device cannot be used for high resolution displacement measurement, let alone nanoscale resolution.

The output of the photodetector generates alternating electrical signals phase-shifted by 90°. These signals are then fed into the electronics unit, where a comparator generates counting pulses which determine the linear size of the object or the amount of movement. Equipping this device with means for turning the bars against each other in order to obtain the required MB period or means for changing the convergence of the beams allows of increasing the measurement accuracy, including reducing the error due to better fixation of device elements, resulting in less variation of device parameters and an increase in their contrast, as well as reducing the distortion of the form of IF, in whose field the photodetector is installed.

The main technical characteristics of HDG are given in Table 2. The range of linear motion transformations vary from 10–2÷10–3 μm to 1 m and more. The limit of acceptable error value at normal temperature value of 20 °С and its deviation depending on the accuracy class of the transducers must not exceed the values given in Table 1. The values of limiting errors indicated in Table 1 include all kinds of systematic errors, inherent to transducers of a particular type, and their random components, and L is the length of linear movement, metres.

The binding of the communication node to the external device is done on a case-by-case basis depending on the design features of the external device, but in such a way that this node has direct communication capabilities with the LHE reading head.

Peer review info

Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal's website and SELibrary eLIBRARY.RU.

Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

LHE and AHE are based on combination of the HDG with interference field and can be used for determination of nanoscale displacement magnitude. The authors have developed and realized technology of synthesis of linear nano HDG with accuracy ±0.4 µm/m and length over one meter and radial HDG with accuracy ±0.2 angular second, as well as developed a number of linear and angular holographic measuring systems based on these sensors.

The phase distribution of the investigated linear interference field (IF) along the OX axis can be represented as follows:

Φ(x) = 2kx + Δf(x), (1)

here k = 2π/λsin/θ/2 = π/d0 is the wave number, d0 = λ/sinθ/2 is the line period of the interference field formed by the two plane waves S10 и S20.

DESCRIPTION OF THE EXPERIMENT

In this work we investigate a possibility of determining the magnitude of displacement by means of a linear holographic diffraction grating (Ризм) acting together with an interference field. During their interaction the interference moiré and obturation bands (MD and OB) are formed depending on the angle between the bars of the IF and the measuring holographic diffraction grating (Ризм), as well as on the equality or difference of their periods.

Let us show that the set problem of determining the magnitude of displacement and size of a product or object can be solved by means of an orderly moving Ризм in the IF. The essence of the method consists in the fact that the information about phase distribution of IF is taken in the system of points rigidly connected with the moving lattice [1]. In this case, the phase difference between any points will be uniquely determined by the phase distribution of Ризм dashes only. The phase difference introduced by the interferometer itself will be a constant, additively included in each act of measurement.

It is required to find the phase distribution of the interference fringes in the pattern of interaction of the interference field with a holographic diffraction grating. Let us consider interactions of the interference field of a IF with a holographic diffraction grating Ризм. The method of phase distribution along the measured axis consists in determination of the phase distribution along the chosen moving direction including determination of all boundary conditions which allow complete phase matching of the phase distribution values at displacement.

Let us represent Ризм itself using two quasi-planar waves [2–3]. Two quasiplanar monochromatic waves with complex amplitudes are:

S1G (x1,y1) – exp i

[k1G r1 +Ψ1G (x1,y1)], (2)

S2G (x1,y1) – exp i [k2G r1 +Ψ2G (x1,y1)] (3)

and they form IF in the X1OY1 plane where Ризм with the amplitude transmission:

T(x1,y1) α cos[kG r1 + ΨG (x1,y1)], (4)

here kG = k1G – k2G is the wave vector of the grating, and the function that characterises the phase distribution of the bars is Ризм:

ΨG(x,y) = Ψ1G(x1,y1) – Ψ2G(x1,y1) (5)

In this case the grating period is:

dG = π/KG =λ/2sinθ1/2, (6)

where θ1 is the angle between wave vectors k1G and k2G, and KG = |KG| means the grating wave number.

Let us place the grating in the XOY plane in the XYZ coordinate system, which is rotated with respect to the XYZ coordinate system by angles υ, γ, φ respectively, and illuminate it with two quasi-plane monochromatic waves with complex amplitudes:

S1 (x,y) – exp i [k1 r +Ψ1 (x,y)],

S2 (x,y) – exp i [k2 r +Ψ2 (x,y)], (7)

where k1 = k1G, k2 = k2G.

Waves S1(x,y) and S2(x,y) form the investigated IFs with a phase distribution in the XOY plane:

Ψ (x,y) = Ψ2 (x,y) – Ψ1 (x,y) (8)

Let us introduce a parameter δ characterising the mismatch of the IF period with respect to the Ризм period, so that the wave number of the field is:

k = |k2 – k1| = kG (1- δ); |δ|«1, (9)

where δ is the mismatch between the IF period and the grating period.

Interaction of the IF with the grating in the XOY plane results in a pattern of interference fringes (If). The joint solution of equations (4) and (6), taking into account the transition formulas from one system to another (r1»r), leads to the following expression for the IfF intensity distribution:

I(x,y) = {1 + cos{kG /cosγ [(xcosφ +

+ ysinφ) – (1 – δ)xcosγ]} +

+ ΨG (x1,y1) – Ψ (x,y)}, (10)

Since I(x,y) changes little if the angle υ changes over a wide range, the terms containing υ in (9) are omitted.

From (9), we can write the IfF phase distribution Φ (x,y) as

Φ(x,y) = kG/cosγ [(xcosφ + ysinφ) –

– (1 – δ)xcosγ] + [ΨG (x1,y1) – Ψ (x,y)]. (11)

Thus, we obtained an expression that allows us to describe the IfF pattern in the most general case, the interaction of an arbitrarily oriented nonideal HDG with a period-mismatched IF. It should be noted that since the mismatch parameter was chosen to be sufficiently small (the case most often realized in practice), expression (10) is valid not only in the XOY plane but also at sufficiently large distances in the direction of propagation of each of the interfering waves.

Let us now proceed to analyse the phase distribution of the IfF and rewrite (10) as

Φ(x,y) = Φр (x,y) +

+ [ΨG (x1,y1) – Ψ (x,y)], (12)

where

Φ(x,y)р = kG/cosγ [(xcosφ +

+ ysinφ) – (1 – δ)xcosγ]. (13)

It is easy to see that Φ(x,y)р describes the interaction of an ideal HDG with an ideal IF formed by two plane waves, while the second term of equation (12), enclosed in square brackets, describes the deviations of the phase distributions of the IF and HDG from the ideal ones.

Let us focus on the analysis of Φ(x,y)р. The equation describing the configuration and location of MBs in the case of IFs and HDGs has the form:

kG/cosγ [(xcosφ + ysinφ) –

– (1 – δ)xcosγ] = 2πn, (14)

where n is an integer. It is easy to see that equation (13) describes a series of equidistant lines with a period:

dM = dG cosγ/[(cosφ –

– cosγ + δ cosγ)2 + sin2φ]0.5 (15)

tgα = – 1/ cosγ cosφ

[cosγ (1 – δ) + cosφ] (16)

The most interesting in practice is the case of φ«1; δ«1; γ«1. In this case, neglecting the values of the second order of smallness we have:

dM = dG /[φ2 + δ2]0.5 (17)

α ≈ arctg(φ/2 + (γ2/2 – δ))/ φ) (18)

Formula (17) shows that the angle γ enters (17) as a value of the second order of smallness as compared to φ and δ; moreover, when δ = γ2/2, the value of α depends on only one variable α = α(φ). Thus, by selecting the mismatch parameter δ we can compensate for the HDG slope angle. Therefore, from now on, we will consider only two variables: the angle φ of the slope of the HDG bars with respect to the fringes of the IF and the mismatch parameter δ with respect to the period.

According to expression (11) for Φ(x,y) at δ«1 and φ«1, ΨG(x1,y1) ≈ Ψ(x,y) and, excluding from consideration γ, we can write:

Φ(x,y) = Φр (x,y) +

+ [ΨG (x1,y1)- Ψ (x,y)], (19)

where

Φ(x,y)р ≈ kG[(xcosφ + ysinφ) – (1 – δ)x].

Expression (18) describes the MB phase distribution quite accurately, in all practical cases.

Thus, we have determined the phase distribution of MF (18) from the interaction between the IF and Ризм located in the IF aperture.

RESULTS AND DISCUSSION

This picture will form the basis for operation of a holographic LHE linear displacement sensor depending on the displacement of Рmeas. in an interferometer IF or in an IF formed behind the same grating illuminated by a coherent emission source.

The clipped light generated by the radiation source rigidly connected to the sensor carriage after reflection from the mirror, falls on the measuring and display gratings (the order of the gratings can be any, depending on the design of the movable linear displacement detector, MLDD). Photodetectors are installed in the MB field behind the gratings.

At the beginning we adjust the grating fringes parallel to each other by means of a frame containing means for turning the Ринд fringes relative to the Ринд fringes and obtaining moiré conjugation of MB, which are perpendicular to the direction of the grinding fringes in the first approximation. It is desirable to obtain MBs as wide as possible (Fig.1).

The device is further tuned to produce an appropriate nonius conjugation. This can be done by changing divergence of the light source (Fig.2).

Fig.2 gives an optical diagram for obtaining nonius fringes. In the OX direction is the Ризм recorded in a two-beam interferometer with an angle θ/2 between its arms. The diffraction grating is illuminated by two beams S1 and S2 at equal angles (θ/2 + δ).

Nonius fringes (Fig.3) are similar to moiré fringes, but they are parallel to the grinding fringes.

When determining the linear dimension of an object, Ринд is displaced relative to the measurement dimension or Ринд is displaced relative to the indicator dimension (which is equivalent).

Although it contains the nods required for the initial setting of the relative positions of the two gratings, Ризм and Ринд, it cannot eliminate the measurement errors introduced by the movement of one of the gratings relative to the other fixed grating by inaccurate guides of the linear encoder itself or the external device to which its moving grating is fastened.

When moving along inaccurate guides, the originally set period of moiré fringes is not retained, as the frame with the indicator grating fixed in it will repeat roughness of the guide, which will necessarily lead to a change in the angle of its bars relative to the Ризм fringes and, consequently, to a variable MB period, which in turn leads to an error in the readout of the movement.

In this regard, HDGs use nonius or obturator fringes rather than moiré strips (Fig.3). In addition, it is possible to design a device which will substantially reduce the error introduced by the guide which has non-linearity in the direction of travel.

In order to eliminate variation of the MB period during movement, the glass guide of the HDG itself can be used, for example, as a guide along which the Ринд moves. For this purpose, the glass guide is rigidly connected to the end of the measuring tape (Fig.4) [4]. The glass guide of the HDG itself, rigidly connected to the face of the measuring tape, is used in this device as a guide, along which the reading head moves. This guide must be at least as long as the linear dimension of the object being measured. As the reader head moves along the glass guide, the support bearings move along the base surface of the glass guide and the spring-loaded bearing, respectively, along its reverse surface (Fig.5). The length of this guide is not less than the linear dimension of the object or the measured displacement value.

If the base surface of the glass guide is of good quality, the movable grating will move exactly along the chosen (constructively) axis of the measuring diffraction grating and perpendicular to its bars. This means that the IBs formed behind the gratings retain their period and slope and will not affect the accuracy of measurements, which will only depend on the quality of the measuring array and the quality of the manufacture of the sensor itself. Thus, we have an opportunity to fully realize the accuracy characteristics of the HDG (Fig.6).

Domestic industry serially produces special glass (Borsky glass on tin melt) that has the required precision characteristics for the glass guide and thus for obtaining nanocharacteristics of HDGs. This will make it possible to carry out sufficiently cheap and efficient serial production of domestic high-precision and high-resolution HDGs above 0.1 µm/m.

The working principle of the HDG for displacement measurement is as follows. When the reader head is moved, whilst measuring the linear dimensions of an object, the indicator grating is shifted with respect to the diffraction measuring grating 2 located on the substrate of the measuring tape ruler. The light beam generated by the light source rigidly coupled to the reader head is collimated by the collimator and passes through the indicator and measurement gratings. A matrix of photodetectors is installed in the field of interference fringes behind the grating, which converts the intensity distribution of the interference fringes into electrical signals. When the reader head is shifted simultaneously with the object during determination of its linear dimension, the indicator grating is shifted relative to the measuring grating and variable electrical signals shifted by 90° in phase are generated at the outputs of the matrix photodetectors. These signals are then fed to the electronics unit, where a comparator is used to generate counting pulses that determine the linear size of the object. The bearing rigidly connected to the indicator grating 6, moving on the base surface of the measuring ruler substrate and the spring-loaded bearing of the same group moving on the reverse surface of the substrate allow to keep a constant gap between the gratings Ризм and Ринд, thus ensuring the probe operability during the whole range of object linear dimension measurement regardless of the quality of the guide device to which the moving part of the HDG is fixed. Therefore this device cannot be used for high resolution displacement measurement, let alone nanoscale resolution.

The output of the photodetector generates alternating electrical signals phase-shifted by 90°. These signals are then fed into the electronics unit, where a comparator generates counting pulses which determine the linear size of the object or the amount of movement. Equipping this device with means for turning the bars against each other in order to obtain the required MB period or means for changing the convergence of the beams allows of increasing the measurement accuracy, including reducing the error due to better fixation of device elements, resulting in less variation of device parameters and an increase in their contrast, as well as reducing the distortion of the form of IF, in whose field the photodetector is installed.

The main technical characteristics of HDG are given in Table 2. The range of linear motion transformations vary from 10–2÷10–3 μm to 1 m and more. The limit of acceptable error value at normal temperature value of 20 °С and its deviation depending on the accuracy class of the transducers must not exceed the values given in Table 1. The values of limiting errors indicated in Table 1 include all kinds of systematic errors, inherent to transducers of a particular type, and their random components, and L is the length of linear movement, metres.

The binding of the communication node to the external device is done on a case-by-case basis depending on the design features of the external device, but in such a way that this node has direct communication capabilities with the LHE reading head.

Peer review info

Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal's website and SELibrary eLIBRARY.RU.

Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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