rus
Issue #1/2023
B.G.Turukhano, N.Turukhano, Yu.M.Lavrov, O.G.Ermolenko, S.N.Khanov
LINEAR NANOMEASURING HOLOGRAPHIC SENSOR
LINEAR NANOMEASURING HOLOGRAPHIC SENSOR
DOI: https://doi.org/10.22184/1993-8578.2023.16.1.58.68
Linear nanomeasuring holographic encoder (LNHS) refers to measuring technology, more exactly precisely, to the field of highly accurate measurements of lengths and displacements by linear holographic sensors based on holographic diffraction gratings (HDG), and can be used in mechanical engineering for precision equipment, including machine tools, optical-mechanical and aerospace industry for measuring end standards, etc. Application of such sensors improves accuracy and resolution of LNHS when measuring linear dimensions in the entire measured range of displacements up to a meter or more, regardless of the quality of the guides of the system. Hence, there is an expansion of the range of processed or studied objects while maintaining high accuracy and resolution of the measuring system.
Linear nanomeasuring holographic encoder (LNHS) refers to measuring technology, more exactly precisely, to the field of highly accurate measurements of lengths and displacements by linear holographic sensors based on holographic diffraction gratings (HDG), and can be used in mechanical engineering for precision equipment, including machine tools, optical-mechanical and aerospace industry for measuring end standards, etc. Application of such sensors improves accuracy and resolution of LNHS when measuring linear dimensions in the entire measured range of displacements up to a meter or more, regardless of the quality of the guides of the system. Hence, there is an expansion of the range of processed or studied objects while maintaining high accuracy and resolution of the measuring system.
Теги: holographic diffraction grating vertical nanolength holographic encoder голографическая дифракционная решетка нанодлиномер голографический
INTRODUCTION
Linear displacement sensors are very versatile in design and widely used in all fields of science and technology. Sensors based on holographic diffraction gratings (HDGs) are characterized by the greatest accuracy, as they allow of receiving stable signals in the nano-displacement range, which do not require special processing (as is done in the case of rifled diffraction gratings), and their illumination with coherent light from a red semiconductor laser additionally improves signal quality [1]. Another significant advantage of HDGs is that only three orders of diffraction are formed behind the gratings: +1, -1 and 0.
In Russia, a LNHS up to 1,000 mm in length has been developed and produced, which operates on the basis of holographic diffraction gratings with 1 micron pitch [2]. Currently, a linear displacement sensor [3] is known for measuring linear dimensions of objects.
There are various ways of using a guide: as an external device on which a reading head, to which the encoder is attached, travels to measure linear displacement, or as a stand-alone guide of the sensor itself on which its reading head moves. Length of the self-contained guide must not be shorter than the linear dimension of the object to be measured.
However, in order to maintain the encoder accuracy when measuring, it is necessary that the HDGind moves along some axis strictly perpendicular to the HDGizm strokes. This factor places certain condition on the guide rail accuracy along which the encoder head with the HDGind. moves. For example, if the emcoder accuracy is approximately 1 micron per metre and we want to measure an object with the same accuracy, then, naturally, the quality of the guide used to move a reading head should be not worse than said value. As a rule, such guides are not produced, especially for longer lengths (up to 1 metre), and even more than one metre, as it is very difficult, time-consuming and expensive to manufacture them. Different external units have different accuracy guides, which makes it virtually impossible to use precision sensors, especially those based on high-frequency HDGs (up to 1,000 lines/mm) in devices, machine tools, instruments, etc. with poor quality guides. At the same time, in all devices where LNHEs are used, no measures have been taken to eliminate moiré fringes (Fig.1) arising due to poor-quality guides, which distort the true length or displacement value. Known is an LNHS described in work [2]. In its development the authors set a problem of excluding the influence of the external device guides and provide a self-contained high quality guide during measurements It contains two glass substrates mounted along a motion line. One of the substrates contains the HDGizm, the other – the self-contained guide of the sensor itself. The glued section is shaped like a T-bar (Fig.2). This design provides necessary rigidity. The glass substrates are made of Borsky glass, which has high flatness characteristics. The HDGizm is glued to the guide base with one of its faces. The carriage is the necessary mechanism for reading coordinates from the HDGizm. The carriage accommodates the HDRizm readout. The LNHS case houses the HDGizm with the guide rail and the carriage containing the illumination system, the HDGind and two matrices, each containing two photodetectors arranged one in each diffraction order. In this case the autonomous guide has less influence on the result of the displacement measurement than in the case of an external guide or an autonomous guide obtained by machining. In each matrix photodetectors are arranged in a straight line perpendicular to the axis of the diffraction beams and grating strokes. Such straight line enables the obturator fringes to be counted.
The phase distribution of the interference fringes in the interaction pattern of the interference field (IF) with holographic diffraction grating along the OX axis can be represented as
Φ(x) = 2kx + Δf(x), (1)
where k=2π/λsin/θ/2 = π/d0 is the wave number, d0 = λ/sin/θ/2 is the line period of the interference field formed by two plane waves S10 and S20. In this paper we study a possibility of determining magnitude of displacement by means of a measuring holographic diffraction grating (HDGizm) acting together with the interference field from an interferometer or IF formed behind the indicator grating. Their interaction produces interference moiré and obturator fringes (Fig.3), depending on equality or difference of their periods. The arrows in Fig.3 coincide with the direction of movement of the LNMHE reading head while measuring the object length.
Let us show that the set problem can be solved by orderly moving the HDRizm. The essence of the method is that information about the phase distribution is taken in a system of points rigidly connected with the moving grating. In this case the phase difference change between the points will be uniquely determined by phase distribution of the interference field only. The additional phase difference introduced by the interferometer or indicator grating itself will be a constant, additively included in every measurement.
The method for determining phase distribution along the measured axis is reduced to determining phase distribution along the chosen direction of displacement, including determination of all the boundary conditions that allow complete phase agreement of phase distributions values during displacement. Let us consider interactions of the interference field with HDGizm in the case of moiré fringes formation in the case of quasiplanar waves. Let two quasiplanar monochromatic waves with complex amplitudes:
S1G(x1,y1) – exp(i)[k1G r1 +Ψ1G (x1,y1)], (2)
S1G(x1,y1) – exp(i)[k2G r1 +Ψ2G (x1,y1)], (3)
form in the X1OY1 plane, in which the grating with amplitude transmittance is recorded
T(x1,y1) = αcos[kGr1+ΨG(x1,y1)], (4)
where kG=k1G – k2G is the wave vector of the grade, and
ΨG (x,y) = Ψ1G (x1,y1) – Ψ2G (x1,y1), (5)
is a function that describes phase distribution of the grade strokes.
In this case the grade period is
dG = π/KG =λ/2sinθ1/2, (6)
where θ1 is the angle between the wave vectors k1G and k2G, KG = |KG| is the wave number of the grating. Let us place the grating in the XOY plane in the XYZ coordinate system, which is rotated relatively to the XYZ coordinate system by angles υγφ respectively, and illuminate it with two quasiplane 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 in the XOY plane the investigated IF with phase distribution
Ψ(x,y) = Ψ2(x,y) – Ψ1(x,y), (8)
Then we introduce the parameter δ characterising mismatch of the IF period with respect to the HDGizm period so that the wave number of the field:
k = |k2 – k1| = kG(1–δ); |δ|"1, (9)
As a result of interaction of IF with the grating in the XOY plane, a pattern of interference obturation and moiré fringes is formed. The joint solution of equations (5) and (7) taking into account the standard transition formulas from one system to another (r1"r) leads to the following expression for the moiré fringe 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 (10) are omitted.
From (10) we can write phase distribution of the interference moiré fringes Φ(x,y) as
Φ(x,y) = kG/cosγ[(xcosφ + ysinφ) –
– (1 – δ)xcosγ] + [ΨG (x1,y1) – Ψ (x,y)]. (11)
Thus, we obtain an expression that allows us to describe the pattern of interference moiré fringes in the most general case, interaction of an arbitrarily oriented non-ideal HDGizm with a period-mismatched IF. Note that since the mismatch parameter was chosen to be small enough (the case most frequently encountered in practice), expression (11) is valid not only in the XOY plane but also at sufficiently large distances in direction of propagation of each of the interfering waves. Let us now proceed to analyze the phase distribution of the interference moiré fringes. Let us rewrite (11) 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].
It is easy to see that Φ(x,y)р describes interaction of an ideal HDGizm with an ideal IF formed by two plane waves, while the second term of equation (11), enclosed in square brackets, describes deviations of the phase distributions of the IF and HDGizm from the ideal ones. Let us focus on the analysis of Φ(x,y)р. The equation describing the configuration and location of moiré fringes in the case of IF and HDGizm is as follows:
kG/cosγ[(xcosφ + ysinφ) – (1 – δ)xcosγ] = 2πn, (13)
where n is integer. Equation (13) describes a series of equidistant lines with a period:
dM = dG cosγ/(cosφ – cosγ + δcosγ)2 + sin2φ, (14)
tgα = – 1/ cosγcosφ[cosγ(1 – δ) + cosφ]. (15)
Most interesting in practice is the case of φ"1 and γ"1. In this case, neglecting the values of the second order of smallness, we have dM = dG/φ2 + δ2:
α ≈ arctg(φ/2 + γ2/2 – δ/ φ). (16)
Equation (16) shows that the angle γ enters 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 fitting the mismatch parameter δ, we can compensate for the angle of slope of HDGizm. Therefore, from now on, we will consider only two variables: φ, the slope angle of the HDGizm bars with respect to the fringes of the IF and the mismatch parameter δ, the period mismatch. According to expression (11), for Φ(x,y) at δ "1 and φ "1, ΨG(x1,y1) ≈ Ψ(x,y) and, excluding γ from consideration, we can write:
Φ(x,y) = Φр (x,y) + [ΨG (x1,y1) – Ψ (x,y)], (17)
where Φ(x,y)р ≈ kG [(xcosφ + ysinφ) – (1 – δ)x].
Expression (17) quite accurately describes phase distribution of interference moiré fringes (in all practically important cases). Similarly, phase distribution of interference obturation fringes obtained due to the frequency difference between the IF and HDGizm can be found in a similar way. These patterns (Fig.1 and Fig.3) will make the basis for operation of the holographic linear displacement sensor. A collimated beam of radiation, generated by a radiation source rigidly connected to the reading head, falls on the grating HDGind and HDGizm. When the reading head is moved during object linear measurement, the HDGind moves relative to the HDGizm.
In the field of interference fringes generated behind the gratings, the photodetector arrays are placed on a line perpendicular to the grating strokes to record interference obturation fringes (Fig.3). The arrows show directions of the fringes (nonius) displacement. When one of the gratings moves in the same direction the photodetectors convert intensity distribution of the fringes into electrical signals. When the reading head moves simultaneously with the object during determination of its linear size, the HDGind moves relative to the HDGizm and variable electrical signals, shifted in phase by 90 degrees, are generated at the outputs of the matrix photodetectors.
These signals are then fed into the electronic unit, where a comparator generates counting pulses which determine linear dimension of the object or the magnitude of its movement. The bearings (of the first bearing group), which are rigidly connected to the indicator grating, move along the base surface of the measuring grating substrate and the spring-loaded bearing of the same group, which moves on the reverse surface of this substrate, allows of maintaining a constant gap between the gratings, thus ensuring operability of the sensor throughout the measurement of the linear size of the object regardless of the guiding device quality, to which the moving part of the HDE is fixed.
Disadvantages of the prototype device
In developing the above-described HDG device, the objective was to reduce influence of the external device guides and to create, in the device design, its own (autonomous) guide, more accurate and independent of the external device guide; due to it the measurement results would be less dependent on quality of the external device (machine, device, etc), and to some extent, would depend only on quality of its own guide. The unit described above uses a self-contained, in-built HDG in the sensor itself. However, the size of any real guide has an effect on the result of displacement or length measurement and this effect exists despite the fact that quality of the stand-alone guide is much higher than that of the external device guide. The proposed "Borsky" glass as a self-contained guide, itself, due to its manufacturing technology, has high flatness over long lengths and it does not require additional and complex machining. Nevertheless, moiré fringes appear due to existence of real (not perfect) guides. With emergence and solution of new modern tasks in science and technology, the demands to accuracy and resolution of HDGs are constantly increasing, so this issue needs a special solution. The authors of this paper propose to eliminate the influence of an autonomous guide on the displacement value when measuring the length of an individual article. The solution is not to improve quality of the guide by machining, which is expensive, extremely difficult and impractical, particularly for longer lengths of up to a metre or more, but by implementing real-time digital compensation for the error introduced by the guide by movement or length of the item being verified or manufactured. The HDG device (Fig.4) contains a measuring diffraction grating with a substrate, an indicator grating and a reading head, an emission source, a collimator, and a matrix of photodetectors. The HDG also contains a glass autonomous guide that has a base and a back surface, and is rigidly attached to the end face of the HDGizm substrate (Fig.2). The length of the self-contained guide is not less than the linear size of the object to be measured. The photodetectors of the matrix (Fig.5) are mounted in two after the diffraction gratings in each of its two diffraction orders, each with the axis parallel to the beam axis of the radiation source.
Each pair of photodetectors is mounted symmetrically about the beam axis of each of the diffraction orders and along the line perpendicular to their axes. In addition, one photodetector is added to each pair of photodetectors, located in the laser beam, on the same line as one of the existing ones and at the same distance from it, equal to the distance from the first two and perpendicularly to the line connecting the first two photodetectors from the same side.
The reading head is rigidly connected to the moving part of the external device. Inaccuracy of the guides, along which one of the gratings moves, affects the value of the moiré fringe period and hence the value of movement itself, which is determined by them, so that not this value but the value of the obturation fringes is used for length measurement, since they are independent (invariant) of the guides. In order to eliminate the error introduced by this disadvantage it is necessary to use perfectly flat guides, which is practically impossible in practice. It is therefore proposed that this error is eliminated in this unit by subtracting it from the total value of displacement. Thus, in view of the fact that the HDG measures linear displacement by means of obturator strips, where Δоб is the true value of displacement introduced by the obturator strips, and δнапр.об. is the amount of displacement error introduced by the obturation strips through the use of inaccurate guides:
Δперем.об. = Δоб + δнапр.об. (18)
Since the axis of the bearings, serving to move one grating along the guide relative to the other is perpendicular to the movement direction, and the grating can only tilt in the direction of movement, at this HDG design the value of δнапр.об. = 0 and Δперем.об. = Δоб., which means that the obturator strips of this sensor are independent (invariant) of the quality of its own guide. On the other hand, the displacement value obtained by recording the moiré fringes Δперем.м., within the framework of the new sensor based on photodetectors 7 and 8 (Fig.5) is as follows:
Δперем.м = Δм + δнапр.м., (19)
where Δм is the magnitude of the displacement itself, in case it is determined by means of moiré fringes, and δнапр.м. is the additional magnitude of displacement introduced by the use of inaccurate guides. In the proposed HDG device it is possible to determine the value of δнапр.м. using the sensors based on photodetectors 5, 7 and 6, 8. In this case, the true value of displacement can be found as follows:
Δперем. = Δоб. – δнапр.м., (20)
where
δнапр.м. = (Δперем.м. – Δперем.об.). (21)
HDG device (Fig.4) functions as follows: semiconductor laser 1 (Fig.5) emits a coherent beam directed at a collimator forming a slightly divergent beam. The divergent beam allows operation with both "obturator" and "moiré" bands [3]. This beam is directed onto HDG 3 and HDG 4 at a Bragg angle. In the case of HDRs with 1000 strokes/mm, it is an angle of about 20 degrees. After diffraction onto HDGizm grating 3, the light falls onto HDGind grating 4. In this particular case the length of the GDGizm is 1,000 mm. Gratings 3 and 4 are parallel to each other. A collimator is necessary to fine-tune the interference fringes period. In case of a divergent laser beam, the period of strokes of the first diffraction grating 3 increases as this grating gets farther away from the second grating 4. Thus, the frequency of the projected bars of the first grating changes due to diffraction in the plane of the second grating. The difference between the periods of the measuring and indicating gratings determines the period of the obturation bands formed behind the gratings. These bands, hitting the photodetectors 5 and 6, create sinusoidal signals (Fig.6). Photodetectors 5 and 6 are mounted on the invariant line, which is perpendicular to the grating bars. In case of a perfectly flat guide, the magnitude of HDG displacement determined by the obturator fringes will correspond to the true displacement magnitude. Select the width of the interference fringes so that two photodetectors from each photodetector matrix, produce two signals shifted by 90 degrees. A shift of 90 degrees is necessary to determine direction of carriage movement (reversal) and for further interpolation of the signals, which greatly increases resolution of the sensor. Photodetectors 7 and 8 are mounted on a line perpendicular to the invariant line and parallel to the strokes. A running sine wave is produced when the HDGind moves.
It is most convenient to control the 90 degree shift during adjustment using the Lissajous figure method (Fig.7). It is necessary to obtain a Lissajous figure in the form of a circle from both photodetector pair 5 and 7 and from photodetector pair 6–8. In that case the device would read both obturatorial, by means of a pair of photodetectors 5 and 6, and moiré stripes by means of photodetectors 5–7 and 6–8. At the same movement along the ideal guideline their magnitudes will be equal, i.e. Δм. = Δоб.. With a non-ideal guideline their difference is equal to δнапр.м.:
Δперем.м. – Δоб. = δнапр.м. (22)
In the device proposed by the authors, δнапр.м. has no influence on the measured value of the displacement length of photodetectors 5 and 6 along the invariance line. In this case, also independently of each other the displacements of obturation and moiré fringes are read by means of two sensors, respectively one sensor is based on photodetectors 5 and 6, and another is based on photodetectors 7 and 8. Considering equation (21) at each displacement point and in real time with the help of appropriate digital software, we can find the length of the true displacement Δперем by subtracting the contribution δнапр.м. from the value Δоб. by determining it with moiré and obturator strips according to expression (20).
CONCLUSIONS
Linear nano-measuring holographic sensor based on holographic diffraction gratings, according to invention [4], has high metrological characteristics, as it is able to eliminate influence of an autonomous guide on the magnitude of movement. Such sensor completely replaces any analogues of similar linear motion sensors.
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 SEL eLibrary 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. ■
Linear displacement sensors are very versatile in design and widely used in all fields of science and technology. Sensors based on holographic diffraction gratings (HDGs) are characterized by the greatest accuracy, as they allow of receiving stable signals in the nano-displacement range, which do not require special processing (as is done in the case of rifled diffraction gratings), and their illumination with coherent light from a red semiconductor laser additionally improves signal quality [1]. Another significant advantage of HDGs is that only three orders of diffraction are formed behind the gratings: +1, -1 and 0.
In Russia, a LNHS up to 1,000 mm in length has been developed and produced, which operates on the basis of holographic diffraction gratings with 1 micron pitch [2]. Currently, a linear displacement sensor [3] is known for measuring linear dimensions of objects.
There are various ways of using a guide: as an external device on which a reading head, to which the encoder is attached, travels to measure linear displacement, or as a stand-alone guide of the sensor itself on which its reading head moves. Length of the self-contained guide must not be shorter than the linear dimension of the object to be measured.
However, in order to maintain the encoder accuracy when measuring, it is necessary that the HDGind moves along some axis strictly perpendicular to the HDGizm strokes. This factor places certain condition on the guide rail accuracy along which the encoder head with the HDGind. moves. For example, if the emcoder accuracy is approximately 1 micron per metre and we want to measure an object with the same accuracy, then, naturally, the quality of the guide used to move a reading head should be not worse than said value. As a rule, such guides are not produced, especially for longer lengths (up to 1 metre), and even more than one metre, as it is very difficult, time-consuming and expensive to manufacture them. Different external units have different accuracy guides, which makes it virtually impossible to use precision sensors, especially those based on high-frequency HDGs (up to 1,000 lines/mm) in devices, machine tools, instruments, etc. with poor quality guides. At the same time, in all devices where LNHEs are used, no measures have been taken to eliminate moiré fringes (Fig.1) arising due to poor-quality guides, which distort the true length or displacement value. Known is an LNHS described in work [2]. In its development the authors set a problem of excluding the influence of the external device guides and provide a self-contained high quality guide during measurements It contains two glass substrates mounted along a motion line. One of the substrates contains the HDGizm, the other – the self-contained guide of the sensor itself. The glued section is shaped like a T-bar (Fig.2). This design provides necessary rigidity. The glass substrates are made of Borsky glass, which has high flatness characteristics. The HDGizm is glued to the guide base with one of its faces. The carriage is the necessary mechanism for reading coordinates from the HDGizm. The carriage accommodates the HDRizm readout. The LNHS case houses the HDGizm with the guide rail and the carriage containing the illumination system, the HDGind and two matrices, each containing two photodetectors arranged one in each diffraction order. In this case the autonomous guide has less influence on the result of the displacement measurement than in the case of an external guide or an autonomous guide obtained by machining. In each matrix photodetectors are arranged in a straight line perpendicular to the axis of the diffraction beams and grating strokes. Such straight line enables the obturator fringes to be counted.
The phase distribution of the interference fringes in the interaction pattern of the interference field (IF) with holographic diffraction grating along the OX axis can be represented as
Φ(x) = 2kx + Δf(x), (1)
where k=2π/λsin/θ/2 = π/d0 is the wave number, d0 = λ/sin/θ/2 is the line period of the interference field formed by two plane waves S10 and S20. In this paper we study a possibility of determining magnitude of displacement by means of a measuring holographic diffraction grating (HDGizm) acting together with the interference field from an interferometer or IF formed behind the indicator grating. Their interaction produces interference moiré and obturator fringes (Fig.3), depending on equality or difference of their periods. The arrows in Fig.3 coincide with the direction of movement of the LNMHE reading head while measuring the object length.
Let us show that the set problem can be solved by orderly moving the HDRizm. The essence of the method is that information about the phase distribution is taken in a system of points rigidly connected with the moving grating. In this case the phase difference change between the points will be uniquely determined by phase distribution of the interference field only. The additional phase difference introduced by the interferometer or indicator grating itself will be a constant, additively included in every measurement.
The method for determining phase distribution along the measured axis is reduced to determining phase distribution along the chosen direction of displacement, including determination of all the boundary conditions that allow complete phase agreement of phase distributions values during displacement. Let us consider interactions of the interference field with HDGizm in the case of moiré fringes formation in the case of quasiplanar waves. Let two quasiplanar monochromatic waves with complex amplitudes:
S1G(x1,y1) – exp(i)[k1G r1 +Ψ1G (x1,y1)], (2)
S1G(x1,y1) – exp(i)[k2G r1 +Ψ2G (x1,y1)], (3)
form in the X1OY1 plane, in which the grating with amplitude transmittance is recorded
T(x1,y1) = αcos[kGr1+ΨG(x1,y1)], (4)
where kG=k1G – k2G is the wave vector of the grade, and
ΨG (x,y) = Ψ1G (x1,y1) – Ψ2G (x1,y1), (5)
is a function that describes phase distribution of the grade strokes.
In this case the grade period is
dG = π/KG =λ/2sinθ1/2, (6)
where θ1 is the angle between the wave vectors k1G and k2G, KG = |KG| is the wave number of the grating. Let us place the grating in the XOY plane in the XYZ coordinate system, which is rotated relatively to the XYZ coordinate system by angles υγφ respectively, and illuminate it with two quasiplane 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 in the XOY plane the investigated IF with phase distribution
Ψ(x,y) = Ψ2(x,y) – Ψ1(x,y), (8)
Then we introduce the parameter δ characterising mismatch of the IF period with respect to the HDGizm period so that the wave number of the field:
k = |k2 – k1| = kG(1–δ); |δ|"1, (9)
As a result of interaction of IF with the grating in the XOY plane, a pattern of interference obturation and moiré fringes is formed. The joint solution of equations (5) and (7) taking into account the standard transition formulas from one system to another (r1"r) leads to the following expression for the moiré fringe 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 (10) are omitted.
From (10) we can write phase distribution of the interference moiré fringes Φ(x,y) as
Φ(x,y) = kG/cosγ[(xcosφ + ysinφ) –
– (1 – δ)xcosγ] + [ΨG (x1,y1) – Ψ (x,y)]. (11)
Thus, we obtain an expression that allows us to describe the pattern of interference moiré fringes in the most general case, interaction of an arbitrarily oriented non-ideal HDGizm with a period-mismatched IF. Note that since the mismatch parameter was chosen to be small enough (the case most frequently encountered in practice), expression (11) is valid not only in the XOY plane but also at sufficiently large distances in direction of propagation of each of the interfering waves. Let us now proceed to analyze the phase distribution of the interference moiré fringes. Let us rewrite (11) 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].
It is easy to see that Φ(x,y)р describes interaction of an ideal HDGizm with an ideal IF formed by two plane waves, while the second term of equation (11), enclosed in square brackets, describes deviations of the phase distributions of the IF and HDGizm from the ideal ones. Let us focus on the analysis of Φ(x,y)р. The equation describing the configuration and location of moiré fringes in the case of IF and HDGizm is as follows:
kG/cosγ[(xcosφ + ysinφ) – (1 – δ)xcosγ] = 2πn, (13)
where n is integer. Equation (13) describes a series of equidistant lines with a period:
dM = dG cosγ/(cosφ – cosγ + δcosγ)2 + sin2φ, (14)
tgα = – 1/ cosγcosφ[cosγ(1 – δ) + cosφ]. (15)
Most interesting in practice is the case of φ"1 and γ"1. In this case, neglecting the values of the second order of smallness, we have dM = dG/φ2 + δ2:
α ≈ arctg(φ/2 + γ2/2 – δ/ φ). (16)
Equation (16) shows that the angle γ enters 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 fitting the mismatch parameter δ, we can compensate for the angle of slope of HDGizm. Therefore, from now on, we will consider only two variables: φ, the slope angle of the HDGizm bars with respect to the fringes of the IF and the mismatch parameter δ, the period mismatch. According to expression (11), for Φ(x,y) at δ "1 and φ "1, ΨG(x1,y1) ≈ Ψ(x,y) and, excluding γ from consideration, we can write:
Φ(x,y) = Φр (x,y) + [ΨG (x1,y1) – Ψ (x,y)], (17)
where Φ(x,y)р ≈ kG [(xcosφ + ysinφ) – (1 – δ)x].
Expression (17) quite accurately describes phase distribution of interference moiré fringes (in all practically important cases). Similarly, phase distribution of interference obturation fringes obtained due to the frequency difference between the IF and HDGizm can be found in a similar way. These patterns (Fig.1 and Fig.3) will make the basis for operation of the holographic linear displacement sensor. A collimated beam of radiation, generated by a radiation source rigidly connected to the reading head, falls on the grating HDGind and HDGizm. When the reading head is moved during object linear measurement, the HDGind moves relative to the HDGizm.
In the field of interference fringes generated behind the gratings, the photodetector arrays are placed on a line perpendicular to the grating strokes to record interference obturation fringes (Fig.3). The arrows show directions of the fringes (nonius) displacement. When one of the gratings moves in the same direction the photodetectors convert intensity distribution of the fringes into electrical signals. When the reading head moves simultaneously with the object during determination of its linear size, the HDGind moves relative to the HDGizm and variable electrical signals, shifted in phase by 90 degrees, are generated at the outputs of the matrix photodetectors.
These signals are then fed into the electronic unit, where a comparator generates counting pulses which determine linear dimension of the object or the magnitude of its movement. The bearings (of the first bearing group), which are rigidly connected to the indicator grating, move along the base surface of the measuring grating substrate and the spring-loaded bearing of the same group, which moves on the reverse surface of this substrate, allows of maintaining a constant gap between the gratings, thus ensuring operability of the sensor throughout the measurement of the linear size of the object regardless of the guiding device quality, to which the moving part of the HDE is fixed.
Disadvantages of the prototype device
In developing the above-described HDG device, the objective was to reduce influence of the external device guides and to create, in the device design, its own (autonomous) guide, more accurate and independent of the external device guide; due to it the measurement results would be less dependent on quality of the external device (machine, device, etc), and to some extent, would depend only on quality of its own guide. The unit described above uses a self-contained, in-built HDG in the sensor itself. However, the size of any real guide has an effect on the result of displacement or length measurement and this effect exists despite the fact that quality of the stand-alone guide is much higher than that of the external device guide. The proposed "Borsky" glass as a self-contained guide, itself, due to its manufacturing technology, has high flatness over long lengths and it does not require additional and complex machining. Nevertheless, moiré fringes appear due to existence of real (not perfect) guides. With emergence and solution of new modern tasks in science and technology, the demands to accuracy and resolution of HDGs are constantly increasing, so this issue needs a special solution. The authors of this paper propose to eliminate the influence of an autonomous guide on the displacement value when measuring the length of an individual article. The solution is not to improve quality of the guide by machining, which is expensive, extremely difficult and impractical, particularly for longer lengths of up to a metre or more, but by implementing real-time digital compensation for the error introduced by the guide by movement or length of the item being verified or manufactured. The HDG device (Fig.4) contains a measuring diffraction grating with a substrate, an indicator grating and a reading head, an emission source, a collimator, and a matrix of photodetectors. The HDG also contains a glass autonomous guide that has a base and a back surface, and is rigidly attached to the end face of the HDGizm substrate (Fig.2). The length of the self-contained guide is not less than the linear size of the object to be measured. The photodetectors of the matrix (Fig.5) are mounted in two after the diffraction gratings in each of its two diffraction orders, each with the axis parallel to the beam axis of the radiation source.
Each pair of photodetectors is mounted symmetrically about the beam axis of each of the diffraction orders and along the line perpendicular to their axes. In addition, one photodetector is added to each pair of photodetectors, located in the laser beam, on the same line as one of the existing ones and at the same distance from it, equal to the distance from the first two and perpendicularly to the line connecting the first two photodetectors from the same side.
The reading head is rigidly connected to the moving part of the external device. Inaccuracy of the guides, along which one of the gratings moves, affects the value of the moiré fringe period and hence the value of movement itself, which is determined by them, so that not this value but the value of the obturation fringes is used for length measurement, since they are independent (invariant) of the guides. In order to eliminate the error introduced by this disadvantage it is necessary to use perfectly flat guides, which is practically impossible in practice. It is therefore proposed that this error is eliminated in this unit by subtracting it from the total value of displacement. Thus, in view of the fact that the HDG measures linear displacement by means of obturator strips, where Δоб is the true value of displacement introduced by the obturator strips, and δнапр.об. is the amount of displacement error introduced by the obturation strips through the use of inaccurate guides:
Δперем.об. = Δоб + δнапр.об. (18)
Since the axis of the bearings, serving to move one grating along the guide relative to the other is perpendicular to the movement direction, and the grating can only tilt in the direction of movement, at this HDG design the value of δнапр.об. = 0 and Δперем.об. = Δоб., which means that the obturator strips of this sensor are independent (invariant) of the quality of its own guide. On the other hand, the displacement value obtained by recording the moiré fringes Δперем.м., within the framework of the new sensor based on photodetectors 7 and 8 (Fig.5) is as follows:
Δперем.м = Δм + δнапр.м., (19)
where Δм is the magnitude of the displacement itself, in case it is determined by means of moiré fringes, and δнапр.м. is the additional magnitude of displacement introduced by the use of inaccurate guides. In the proposed HDG device it is possible to determine the value of δнапр.м. using the sensors based on photodetectors 5, 7 and 6, 8. In this case, the true value of displacement can be found as follows:
Δперем. = Δоб. – δнапр.м., (20)
where
δнапр.м. = (Δперем.м. – Δперем.об.). (21)
HDG device (Fig.4) functions as follows: semiconductor laser 1 (Fig.5) emits a coherent beam directed at a collimator forming a slightly divergent beam. The divergent beam allows operation with both "obturator" and "moiré" bands [3]. This beam is directed onto HDG 3 and HDG 4 at a Bragg angle. In the case of HDRs with 1000 strokes/mm, it is an angle of about 20 degrees. After diffraction onto HDGizm grating 3, the light falls onto HDGind grating 4. In this particular case the length of the GDGizm is 1,000 mm. Gratings 3 and 4 are parallel to each other. A collimator is necessary to fine-tune the interference fringes period. In case of a divergent laser beam, the period of strokes of the first diffraction grating 3 increases as this grating gets farther away from the second grating 4. Thus, the frequency of the projected bars of the first grating changes due to diffraction in the plane of the second grating. The difference between the periods of the measuring and indicating gratings determines the period of the obturation bands formed behind the gratings. These bands, hitting the photodetectors 5 and 6, create sinusoidal signals (Fig.6). Photodetectors 5 and 6 are mounted on the invariant line, which is perpendicular to the grating bars. In case of a perfectly flat guide, the magnitude of HDG displacement determined by the obturator fringes will correspond to the true displacement magnitude. Select the width of the interference fringes so that two photodetectors from each photodetector matrix, produce two signals shifted by 90 degrees. A shift of 90 degrees is necessary to determine direction of carriage movement (reversal) and for further interpolation of the signals, which greatly increases resolution of the sensor. Photodetectors 7 and 8 are mounted on a line perpendicular to the invariant line and parallel to the strokes. A running sine wave is produced when the HDGind moves.
It is most convenient to control the 90 degree shift during adjustment using the Lissajous figure method (Fig.7). It is necessary to obtain a Lissajous figure in the form of a circle from both photodetector pair 5 and 7 and from photodetector pair 6–8. In that case the device would read both obturatorial, by means of a pair of photodetectors 5 and 6, and moiré stripes by means of photodetectors 5–7 and 6–8. At the same movement along the ideal guideline their magnitudes will be equal, i.e. Δм. = Δоб.. With a non-ideal guideline their difference is equal to δнапр.м.:
Δперем.м. – Δоб. = δнапр.м. (22)
In the device proposed by the authors, δнапр.м. has no influence on the measured value of the displacement length of photodetectors 5 and 6 along the invariance line. In this case, also independently of each other the displacements of obturation and moiré fringes are read by means of two sensors, respectively one sensor is based on photodetectors 5 and 6, and another is based on photodetectors 7 and 8. Considering equation (21) at each displacement point and in real time with the help of appropriate digital software, we can find the length of the true displacement Δперем by subtracting the contribution δнапр.м. from the value Δоб. by determining it with moiré and obturator strips according to expression (20).
CONCLUSIONS
Linear nano-measuring holographic sensor based on holographic diffraction gratings, according to invention [4], has high metrological characteristics, as it is able to eliminate influence of an autonomous guide on the magnitude of movement. Such sensor completely replaces any analogues of similar linear motion sensors.
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 SEL eLibrary 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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