rus
Issue #5/2021
B.G.Turukhano, N.Turukhano, I.A.Turukhano
Recording holographic diffraction grids using a pulse laser
Recording holographic diffraction grids using a pulse laser
DOI: 10.22184/1993-8578.2021.14.5.312.319
This paper deals with a device for holographic diffraction gratings (HDG) recording using a pulse laser, which makes it possible to increase the accuracy of grating and its diffraction efficiency. For this purpose use was made of the interferometer for recording gratings and an optically connected second source of coherent radiation located at the input. Such configuration eliminates the need to track corrections during illumination due to a short exposure period. A synchronization unit was introduced into the optoelectronic circuit which output was connected to the second source of pulsed coherent radiation, and the input – to the AC voltage source of the optoelectronic circuit.
This paper deals with a device for holographic diffraction gratings (HDG) recording using a pulse laser, which makes it possible to increase the accuracy of grating and its diffraction efficiency. For this purpose use was made of the interferometer for recording gratings and an optically connected second source of coherent radiation located at the input. Such configuration eliminates the need to track corrections during illumination due to a short exposure period. A synchronization unit was introduced into the optoelectronic circuit which output was connected to the second source of pulsed coherent radiation, and the input – to the AC voltage source of the optoelectronic circuit.
Теги: diffraction efficiency holographic diffraction grids interferometer optoelectronic circuit pulse laser source of pulsed coherent radiation synchronization unit блок синхронизации голографические дифракционные решетки дифракционная эффективность импульсный лазер интерферометр источник импульсного когерентного излучения оптоэлектронная схема
Recording holographic diffraction grids using a pulse laser
INTRODUCTION
Holographic diffraction grids (HDG) are used in different measuring systems, such as linear and angular displacement sensors, length meters and multi-coordinate measuring systems, and make it possible to obtain higher accuracy and resolution compared with the systems based on traditional optics.
Nowadays, the ever increasing requirements are applied to the modern measuring systems. Moreover, the characteristics of metalworking equipment do not fully respond to the growing demands, so, it is imperative to improve quality of the machine parts manufacture. The state of the art and product quality in machine-tool industry depends much on the related branches that produce component parts, including optical and measuring devices. In this paper, we consider problems of metrological accuracy of the measuring systems equipped with high precision HDG-based sensors.
ADJUSTMENT OF AN INTERFEROMETER FOR THE HDG RECORDING
While recording using an interferometer it should first be adjusted with a continuous laser and, after this, recording is carried out with a pulse laser.
An angle between the interferometer arms is chosen depending on the necessary HDF frequency. Phase modulation of light beams is used to detect phase shifts [1]. The phase difference value in the interference field (IF) Δφ1k and Δφ2k are determined by photodetectors placed along the line parallel to wave front shift (ζ). When a quasilinear wave of an interferometer arm falls upon a grid, it changes the IF phase in such a way that its phase distribution is the precise copy of the falling wave phase distribution. This phase distribution also transforms in the amplitude distribution in the interference field too. So, a quasilinear wave S of a complex amplitude is formed in the interferometer output:
S (x, y) = S0 (x, y) exp [if (x, y)], (1)
here S0 (x, y) – amplitude distribution, f (x, y) – phase distribution of a wave.
The device for HD recording using a pulse laser is shown in Fig.1. The laser ray S (1) preserves its coherence and interference with the second ray, thereby forming the interference pattern.
This interferometer forms the complex amplitude in an IF output:
Sx вых (x, y) = a S (x, y) +
+ b S (x + ζ, y). (2)
IF intensity is defined by the formula:
I (x, y, t) = |exp i [f (x, y) + arg a(t)] +
+ exp i [f (x+ ζ, y) arg b(t)]|2 =
= 2{1 + cos [f (x+ ζ, y) = (3)
= f (x, y) + c(t)]},
here x, y – coordinates in outer substrate plane, ζ – shift along OX axis, a and b (2) – complex functions dependent on real optics used in the interferometer. In the dynamic mode of the interferometer operation a and b are time functions.
IF intensity is described by the formula:
I (x, y, t) = |[exp i[f (x, y) +
+ arg a(t)] + exp i [f (x+ ζ, y) +
+arg b(t)]|2 = 2{1 + (4)
+ cos [f (x+ ζ, y) – f (x, y) + c(t)]},
here c(t) = arg b(t) – arg а(t).
If the function c(t) in (3) changes according to the linear law, the intensity changes according to the harmonic law at each point of IF with a frequency of ν = ω/2π.
The phase difference between the signals in two points of IF (x0, y0) and (x, y) can be written as:
Δφx (x, y; x0, y0) =
= [f (x+ ζ, y) – f (x, y)] – (5)
– [f (x0 + ζ, y0) – f (x0, y0)].
It is clear from (5), that the function Δφx (x, y; x0, y0) is invariant to transformation of the function f (x, y) described as:
f (x, y) = f (x0, y0) +
+ с (x) + g (y) + d, (6)
here с and d – arbitrary constants,
g (y) – arbitrary function.
The recurrent expression (4) can be reduced to the relation for the values of f function (x, y) in a number of points located with the period ζ along X axis:
(7)
here n = 1, 2….Nx, Nx = , Dx – field aperture along OX axis.
Let’s make a definition of function f (x0,y) (5). In order to do this, we make a shift of the wave front along OY axis, perpendicular to the initial shift direction. Similar to the above, the function f (x0, y) values in a series of points with a period µ, it is possible to write down:
(8)
here m = 1, 2….Ny, Ny = Dy/μ, Dy – field aperture along the OY axis.
Using the expressions (7) and (8), it is possible to write down the general expression for the values of f (x, y) function in the point net with the periods ζ and µ:
(9)
where С1 = [f (x0 + ζ, y0) – f (x0, y0)] / ζ, С2 = [f (x0 + y0 + µ) – f (x0, y0)] / µ, d = f (x0, y0), and Δφx and Δφy (7) include information about the wave front curvature and are measured during the experiment. The constants С1, С2 and d can be determined only from boundary conditions which form the support plane of comparison. In doing so, it seems the most correct to use such boundary conditions where there is a coincidence of the initial and finite bands of the studied IF.
The IF intensity distribution can be formulated as follows:
I (x, y, t) = 2 {cos [f(x + ζ, y) –
– f (x, y) + Ω (x, y, t)]}, (10)
here Ω (x, y, t) = arg a (x, y, t) – arg b (x, y, t), Ω (x, y, t) = Ω (x, y) + c (t), Ω (x, y) – the member describing phase distortions that appeared due to the interferometer.
ERRORS IN DETECTION OF THE IF PHASE DISTRIBUTION
The systematic and random errors occur while detecting the IF phase distribution. The first ones connected with choosing the method, and the second ones occurring during the experiment.
The phase error ΔΩ of the proposed method is associated with the errors caused by aberrations of the optical system used [2–3]:
(11)
(12)
here h and n – thickness and refraction indices, accordingly. If the optics quality is such that ΔΩ (x)/dx ≤ 1λ/cm, and dx ≤ 0,1mm, and Δα ≤ 5о , we have |ΔΩ(x) | max<λ/100.
It is also possible to estimate the magnitude of dispersion Δn, (9) characterizing the non-uniformity of the IF bands distribution:
(13)
Since the values Δφ1k and Δφ2k (10) are measured and included in the same way in both arms, they have the same dispersion σn and, taking into account the standard error and according to the results of 10 measurements (m = 10) the values of the phase difference Δφi between the same points, we finally obtain:
(14)
As in our case S ≤ 2π/300, N = 6, σN/2 ≤ 5.10–3 µm, the total error (11) does not exceed 3.10–2 µm.
To assess the interference field quality, we introduce a function:
Q = {Δn}max – {Δn}min, (15)
that is the integral characteristics of the aberration of telescopic systems (TS) and characterizes the maximum deviation of the IF band distribution function from the perfectly uniform distribution (Fig.1).
HDG RECORDING USING A PULSE LASER
The selected correction values are further supported by the constants using a direct voltage unit 20 to the illumination of the first section of the HDG. After that, the pulse source of coherent radiation 2 is triggered from the control and switching unit by a synchropulse from block 19 for the illumination of the recording layer. The duration of the light pulse source of coherent radiation is about 10–8 s, which is much smaller than the duration of the illumination carried out in a continuous laser.
The synchropulse to launch a pulse source of coherent radiation can be formed in the synchronization unit 19 (Fig.1), which is constructed according to the well-known principles. The synchronization unit contains a differentiating circuit for the formation of a negative pulse for the decay of the saw-shaped pulse. A positive trigger generator based on a Schmitt trigger contains an adjusting delay circuit per a vibrator, which includes a delay controller based on a variable resistor, and also contains a differentiating circuit of the rear front of the signal from the delay circuit and the launch moment forming scheme. Therefore, no external and internal factors like vibration and deformation of mechanical nodes may not impact, for such a short time, the image of the interference field copied onto the synthesized grid, and that is why they are administered before illumination of the correction, and retain their value with high accuracy during the illumination itself. The synchropulse from the variable voltage generator 21 serves as a "reference" pulse to determine a position of the moiré band in space. The required values of Δφ can be obtained using the switching and control unit 22, which commutes the desired photodetector which signals are fed to the phazometer 25 via selective amplifiers 23 and 24.
Afterwards, the pulse source of coherent radiation is triggered to illuminate the lattice. Due to a short exposure time of the correction value, Δφ retains its value.
After illumination, it is possible to move the carriage and record the second section of the HDG. As a result of non-parallel strokes movement of the recorded first section, they changed their position relative to the lines of the interference field, which will be copied in the second area. For this purpose, appropriate corrections are introduced. After entering the corrections, an illumination of the second grid section is performed by the next start of the pulse source of coherent radiation without tracking corrections during the illumination. The cycle is repeated until the grid of necessary length is synthesized.
EXPERIMENTAL RESULTS
Aberrations of the real lens systems used in the interferometer are associated with inaccuracy of their processing, cause asymmetric distortion of wave fronts and cannot be eliminated by their correct location along the beams.
The Zeidel aberrations: coma, distortion, astigmatism and curvature of the field depend on the orientation of the TS [4] in the interferometer.
Spherical aberrations, for the identical telescopic systems do not affect the Q value because of their similarity in both arms.
Aberrations of the real TS related to their processing may not be eliminated.
Application of the proposed method made it possible to obtain a uniform distribution of the interference bands (IB) no worse than λ/100 and record the holographic grids used in the holographic systems of linear movements [5] with high accuracy and resolution up to 1 nm.
The proposed device allows of increasing accuracy of the synthesized grid and its diffraction efficiency due to:
CONCLUSIONS
The proposed method is suitable for studying a degree of periodicity of the interference bands of the holographic 2-radiation interferometer.
As a result of the studies, as can be seen from the above graph (Fig.2) of the uniform distribution of the bands, it is possible to determine the phase distribution of the interference bands and adjust the interferometer in such a way that the uniformity of the band distribution will not be worse than 2 nm on a plot of 100 mm.
Therefore, it can be concluded that the main errors are associated with inaccuracy of processing the optical elements of the interferometer.
In the case of small aberrations of optical elements, the value of normalized intensity in the center of the wave beam aperture is practically independent of the aberration nature, and differs from the ideal case by the magnitude, proportional to the range of the wave front.
Increased metrological accuracy of the HDG will allow to develop high-precision sensors on their base and bring the modern measuring systems to a higher level. ■
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.
INTRODUCTION
Holographic diffraction grids (HDG) are used in different measuring systems, such as linear and angular displacement sensors, length meters and multi-coordinate measuring systems, and make it possible to obtain higher accuracy and resolution compared with the systems based on traditional optics.
Nowadays, the ever increasing requirements are applied to the modern measuring systems. Moreover, the characteristics of metalworking equipment do not fully respond to the growing demands, so, it is imperative to improve quality of the machine parts manufacture. The state of the art and product quality in machine-tool industry depends much on the related branches that produce component parts, including optical and measuring devices. In this paper, we consider problems of metrological accuracy of the measuring systems equipped with high precision HDG-based sensors.
ADJUSTMENT OF AN INTERFEROMETER FOR THE HDG RECORDING
While recording using an interferometer it should first be adjusted with a continuous laser and, after this, recording is carried out with a pulse laser.
An angle between the interferometer arms is chosen depending on the necessary HDF frequency. Phase modulation of light beams is used to detect phase shifts [1]. The phase difference value in the interference field (IF) Δφ1k and Δφ2k are determined by photodetectors placed along the line parallel to wave front shift (ζ). When a quasilinear wave of an interferometer arm falls upon a grid, it changes the IF phase in such a way that its phase distribution is the precise copy of the falling wave phase distribution. This phase distribution also transforms in the amplitude distribution in the interference field too. So, a quasilinear wave S of a complex amplitude is formed in the interferometer output:
S (x, y) = S0 (x, y) exp [if (x, y)], (1)
here S0 (x, y) – amplitude distribution, f (x, y) – phase distribution of a wave.
The device for HD recording using a pulse laser is shown in Fig.1. The laser ray S (1) preserves its coherence and interference with the second ray, thereby forming the interference pattern.
This interferometer forms the complex amplitude in an IF output:
Sx вых (x, y) = a S (x, y) +
+ b S (x + ζ, y). (2)
IF intensity is defined by the formula:
I (x, y, t) = |exp i [f (x, y) + arg a(t)] +
+ exp i [f (x+ ζ, y) arg b(t)]|2 =
= 2{1 + cos [f (x+ ζ, y) = (3)
= f (x, y) + c(t)]},
here x, y – coordinates in outer substrate plane, ζ – shift along OX axis, a and b (2) – complex functions dependent on real optics used in the interferometer. In the dynamic mode of the interferometer operation a and b are time functions.
IF intensity is described by the formula:
I (x, y, t) = |[exp i[f (x, y) +
+ arg a(t)] + exp i [f (x+ ζ, y) +
+arg b(t)]|2 = 2{1 + (4)
+ cos [f (x+ ζ, y) – f (x, y) + c(t)]},
here c(t) = arg b(t) – arg а(t).
If the function c(t) in (3) changes according to the linear law, the intensity changes according to the harmonic law at each point of IF with a frequency of ν = ω/2π.
The phase difference between the signals in two points of IF (x0, y0) and (x, y) can be written as:
Δφx (x, y; x0, y0) =
= [f (x+ ζ, y) – f (x, y)] – (5)
– [f (x0 + ζ, y0) – f (x0, y0)].
It is clear from (5), that the function Δφx (x, y; x0, y0) is invariant to transformation of the function f (x, y) described as:
f (x, y) = f (x0, y0) +
+ с (x) + g (y) + d, (6)
here с and d – arbitrary constants,
g (y) – arbitrary function.
The recurrent expression (4) can be reduced to the relation for the values of f function (x, y) in a number of points located with the period ζ along X axis:
(7)
here n = 1, 2….Nx, Nx = , Dx – field aperture along OX axis.
Let’s make a definition of function f (x0,y) (5). In order to do this, we make a shift of the wave front along OY axis, perpendicular to the initial shift direction. Similar to the above, the function f (x0, y) values in a series of points with a period µ, it is possible to write down:
(8)
here m = 1, 2….Ny, Ny = Dy/μ, Dy – field aperture along the OY axis.
Using the expressions (7) and (8), it is possible to write down the general expression for the values of f (x, y) function in the point net with the periods ζ and µ:
(9)
where С1 = [f (x0 + ζ, y0) – f (x0, y0)] / ζ, С2 = [f (x0 + y0 + µ) – f (x0, y0)] / µ, d = f (x0, y0), and Δφx and Δφy (7) include information about the wave front curvature and are measured during the experiment. The constants С1, С2 and d can be determined only from boundary conditions which form the support plane of comparison. In doing so, it seems the most correct to use such boundary conditions where there is a coincidence of the initial and finite bands of the studied IF.
The IF intensity distribution can be formulated as follows:
I (x, y, t) = 2 {cos [f(x + ζ, y) –
– f (x, y) + Ω (x, y, t)]}, (10)
here Ω (x, y, t) = arg a (x, y, t) – arg b (x, y, t), Ω (x, y, t) = Ω (x, y) + c (t), Ω (x, y) – the member describing phase distortions that appeared due to the interferometer.
ERRORS IN DETECTION OF THE IF PHASE DISTRIBUTION
The systematic and random errors occur while detecting the IF phase distribution. The first ones connected with choosing the method, and the second ones occurring during the experiment.
The phase error ΔΩ of the proposed method is associated with the errors caused by aberrations of the optical system used [2–3]:
(11)
(12)
here h and n – thickness and refraction indices, accordingly. If the optics quality is such that ΔΩ (x)/dx ≤ 1λ/cm, and dx ≤ 0,1mm, and Δα ≤ 5о , we have |ΔΩ(x) | max<λ/100.
It is also possible to estimate the magnitude of dispersion Δn, (9) characterizing the non-uniformity of the IF bands distribution:
(13)
Since the values Δφ1k and Δφ2k (10) are measured and included in the same way in both arms, they have the same dispersion σn and, taking into account the standard error and according to the results of 10 measurements (m = 10) the values of the phase difference Δφi between the same points, we finally obtain:
(14)
As in our case S ≤ 2π/300, N = 6, σN/2 ≤ 5.10–3 µm, the total error (11) does not exceed 3.10–2 µm.
To assess the interference field quality, we introduce a function:
Q = {Δn}max – {Δn}min, (15)
that is the integral characteristics of the aberration of telescopic systems (TS) and characterizes the maximum deviation of the IF band distribution function from the perfectly uniform distribution (Fig.1).
HDG RECORDING USING A PULSE LASER
The selected correction values are further supported by the constants using a direct voltage unit 20 to the illumination of the first section of the HDG. After that, the pulse source of coherent radiation 2 is triggered from the control and switching unit by a synchropulse from block 19 for the illumination of the recording layer. The duration of the light pulse source of coherent radiation is about 10–8 s, which is much smaller than the duration of the illumination carried out in a continuous laser.
The synchropulse to launch a pulse source of coherent radiation can be formed in the synchronization unit 19 (Fig.1), which is constructed according to the well-known principles. The synchronization unit contains a differentiating circuit for the formation of a negative pulse for the decay of the saw-shaped pulse. A positive trigger generator based on a Schmitt trigger contains an adjusting delay circuit per a vibrator, which includes a delay controller based on a variable resistor, and also contains a differentiating circuit of the rear front of the signal from the delay circuit and the launch moment forming scheme. Therefore, no external and internal factors like vibration and deformation of mechanical nodes may not impact, for such a short time, the image of the interference field copied onto the synthesized grid, and that is why they are administered before illumination of the correction, and retain their value with high accuracy during the illumination itself. The synchropulse from the variable voltage generator 21 serves as a "reference" pulse to determine a position of the moiré band in space. The required values of Δφ can be obtained using the switching and control unit 22, which commutes the desired photodetector which signals are fed to the phazometer 25 via selective amplifiers 23 and 24.
Afterwards, the pulse source of coherent radiation is triggered to illuminate the lattice. Due to a short exposure time of the correction value, Δφ retains its value.
After illumination, it is possible to move the carriage and record the second section of the HDG. As a result of non-parallel strokes movement of the recorded first section, they changed their position relative to the lines of the interference field, which will be copied in the second area. For this purpose, appropriate corrections are introduced. After entering the corrections, an illumination of the second grid section is performed by the next start of the pulse source of coherent radiation without tracking corrections during the illumination. The cycle is repeated until the grid of necessary length is synthesized.
EXPERIMENTAL RESULTS
Aberrations of the real lens systems used in the interferometer are associated with inaccuracy of their processing, cause asymmetric distortion of wave fronts and cannot be eliminated by their correct location along the beams.
The Zeidel aberrations: coma, distortion, astigmatism and curvature of the field depend on the orientation of the TS [4] in the interferometer.
Spherical aberrations, for the identical telescopic systems do not affect the Q value because of their similarity in both arms.
Aberrations of the real TS related to their processing may not be eliminated.
Application of the proposed method made it possible to obtain a uniform distribution of the interference bands (IB) no worse than λ/100 and record the holographic grids used in the holographic systems of linear movements [5] with high accuracy and resolution up to 1 nm.
The proposed device allows of increasing accuracy of the synthesized grid and its diffraction efficiency due to:
- the lack of need for adjustment of corrections during the illumination due to the smallness of the value of the illumination time;
- the lack of "dead time" associated with turning off the modulation before illumination. which is necessary for the mirror to adopt a stationary position, during which the corrections may not be controlled and restored;
- the lack of the process of turning on and off the modulation with all ensuing negative consequences;
- for the first time the linear automated systems of a particularly high accuracy were also created: linear sensors up to 1 meter with an accuracy of ± 0.3 μm, as well as holographic length (HL) with lengths: 30, 100, 200, accuracy 0.1 / 0.2 / 0.3 μm, with a resolution of up to 10 Nm and above and at a measurement speed up to 500 mm/s.
CONCLUSIONS
The proposed method is suitable for studying a degree of periodicity of the interference bands of the holographic 2-radiation interferometer.
As a result of the studies, as can be seen from the above graph (Fig.2) of the uniform distribution of the bands, it is possible to determine the phase distribution of the interference bands and adjust the interferometer in such a way that the uniformity of the band distribution will not be worse than 2 nm on a plot of 100 mm.
Therefore, it can be concluded that the main errors are associated with inaccuracy of processing the optical elements of the interferometer.
In the case of small aberrations of optical elements, the value of normalized intensity in the center of the wave beam aperture is practically independent of the aberration nature, and differs from the ideal case by the magnitude, proportional to the range of the wave front.
Increased metrological accuracy of the HDG will allow to develop high-precision sensors on their base and bring the modern measuring systems to a higher level. ■
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.
Readers feedback
