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Abstract
Replacing the bulky traditional optical elements in the optical system with a holographic optical element (HOE) is conducive to the functional integration and volume miniaturization. However, when the HOE is used in the infrared system, the mismatch between the recording wavelength and the working wavelength will lead to the reduction of diffraction efficiency and the introduction of aberration, which will seriously affect the performance of the optical system. This paper proposes a design and fabrication method of multifunctional infrared HOEs that can be used in laser Doppler velocimeter (LDV), which can reduce the effect of wavelength mismatch on HOE performance while integrating the functions of the optical system. The restriction relationship and selection method of parameters in typical LDV are summarized; the decrease of diffraction efficiency due to the mismatch between recording and working wavelengths is compensated by designing the angle of signal and reference wave of the HOE; and the aberration caused by wavelength mismatch is compensated by cylindrical lens. The optical experiment shows that the HOE can produce two groups of fringes with opposite gradient, which proves the feasibility of the proposed method. Moreover, this method has a certain degree of universality, and it is expected to design and fabricate HOEs for any working wavelength in the near-infrared band.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser Doppler velocimeter (LDV) is an instrument that uses laser Doppler effect to measure the motion parameters of fluid or solid. Because of the advantages of non-contact [1], high spatial resolution [2–6] and fast dynamic response [7–11], laser velocimetry is of great significance in measurement, especially in rotation measurement [12–15], flow field studies [16–20] and optical biosensors [21,22]. When LDV is used as an experimental tool for fluid measurement, it is important to obtain multiple fluid parameters at the same time. The dual wavelength LDV is a typical representative of multi parameter measurement [23,24]. It can not only measure the velocity of the tracer particles in the direction perpendicular to the optical axis, but also measure the velocity and position of the particles along the optical axis, which is conducive to obtaining more comprehensive and rich information about the fluid in a single measurement. This LDV mainly improves the traditional system in the generation of fringes and signal processing. But due to the use of traditional optical elements to arrange the optical path, it still has problems such as large volume and difficult to install.
Although there are some attempts to miniaturize the light source [25,26] and transmission unit [27–29], little attention has been paid to the miniaturization of measurement unit from the perspective of functional integration. Holographic optical element (HOE) has the advantages of small size, light weight, low cost and reusability [30]. Replacing traditional optical elements with HOE is a promising solution for the integration of measurement unit functions. But due to the mismatch between the recording wavelength and the working wavelength, which leads to the problems of diffraction efficiency degradation and aberration, only a few LDV schemes have applied HOE, and many of them stay at the stage of idea without further design and verification [31].
Some researches have been devoted to solving the problems caused by wavelength mismatch, such as optimizing the location of recording point source [32], recursive design method based on auxiliary hologram [33], iterative design method based on ray tracing [34], holographic wavefront printing [35], etc. However, the above methods have the problems of large calculation in the design process and complex optical setups for recording, and these methods are only optimized for holographic lenses. Due to the requirement of LDV integration, the dependence of signal processing on the parameters of optical elements and the functional requirements of non-imaging, multiple functions need to be integrated in a single HOE and the design and analysis methods of the multifunctional HOE are different from holographic lenses. Therefore, it is of practical significance to design and fabricate multifunctional infrared HOE to improve the integration of LDV.
This paper proposes a design and fabrication method of HOE which can be used in compact LDV. The relationship between the parameters of HOE and those of other parts of LDV is analyzed. Kogelnik coupled wave theory is used to analyze the decrease of diffraction efficiency caused by wavelength mismatch, and angle shift is used to compensate for wavelength shift. Moreover, the cause of aberration is analyzed by ray tracing method, and the aberration is pre-compensated in the recording light path. This work can be seen as an example of combining HOE with optical system to realize system function integration.
The arrangement of this paper is as follows: Section 2 describes the basic principles of selecting parameters and solving wavelength mismatch problems; Section 3 carries out the numerical simulation and optical experiments, and the results are analyzed. In the discussion, the important results of this paper were reviewed, the contributions of this paper were refined and the directions of future research were prospected.
2. Principle
2.1 Function integration and design method of HOE in LDV
2.1.1 Functional requirements of HOE
Figure 1 illustrates the functions undertaken by HOE in LDV. To explain the functions that the multi-function HOE needs to achieve, the principle of LDV is briefly introduced. A typical LDV includes a light source unit, a measurement unit, a transmission unit and a detection unit. For dual wavelength LDV, the light source unit contains two laser diodes of different wavelengths, and the light emitted by the diode is combined through the dichroic mirror or fiber coupler. The combined laser enters the measurement unit through the transmission unit, which can be air or optical fiber. In the measurement unit, the laser is first collimated by the lens, and due to the dispersion of the converging lens, the lasers of different wavelengths will be focused at different axial positions. A grating is placed in the middle of the beam waists, so light with different wavelengths will have different curvature after splitting by the grating. Since the grating is also located at the front focal plane of the 4f system, the light split by the grating will intersect near the back focal plane of the 4f system to form a measurement volume (MV) with fringes, and because the beam waists are located on different sides of the grating, the interference fringes of different wavelengths in MV have opposite gradients, which will facilitate the measurement of velocity in the z direction. When LDV is working, the particles or particle flow to be measured will pass through MV, the backscattered light will be collected by the second lens of 4f system, the collected light will enter the detection unit through the transmission unit, the light of different wavelengths will be separated through the dichroic mirror and detected by different photodetectors, and the detected signal will be input to the signal processing circuit or computer for signal processing.
Fig. 1. The functions undertaken by HOE in LDV. The green, red and yellow lines represent light with short, long and mixed wavelengths, respectively. The difference between the traditional LDV and the HOE based LDV proposed in this paper is mainly reflected in the measurement unit. The yellow rectangle shows the common part of the two schemes. The traditional LDV (purple rectangle) needs to be focused by lens and split by grating before passing through the 4f system. The scheme proposed in this paper uses HOE (blue rectangle) to realize the above two functions simultaneously, and the reflective mode saves more space. LD: laser diode, DM: dichroic mirror, SF: spatial filter, M: mirror, PD: photo detector.
This LDV with convergent and divergent fringes in MV can realize simultaneous measurement of position and velocity. The principle is briefly described below, and the detailed process can be found in [36]. First, the two fringe spacing functions ${d_{1,2}}(z )$ are obtained through calibration. Next, the calibration function is defined as $q(z )= {d_1}(z )/{d_2}(z )$, and the calibration function is also a function of the coordinate z. Since the object velocity can be calculated by $v = d \cdot f$, $q(z )= {f_2}({v,z} )/{f_1}({v,z} )$ can be obtained, in which the doppler frequency ${f_{1,2}}({v,z} )$ can be obtained by processing the signal collected by the photodetector. In this way, for each group of ${f_{1,2}}({v,z} )$, a $q(z )$ can be calculated. According to the calibration function, the z coordinate of the position of the scattering object at this time can be obtained. Substituting the z coordinate into ${d_{1,2}}(z )$, the fringe spacing ${d_{1,2}}$ can be obtained. The transverse velocity v of the scattering object can be calculated from $v = {d_1}{f_1} = {d_2}{f_2}$, and if the chirp of doppler frequency is considered, the axial velocity ${v_z}$ can also be measured.
According to the previous description of the optical path and measurement principle of dual wavelength LDV, it can be seen that the calibration function $q(z )$ must be a monotone function, preferably a linear function, meaning that ${d_{1,2}}(z )$ is preferably a linear function with opposite slopes. As a result, the main function of the measurement unit is to generate two sets of fringes with opposite gradients of the fringe spacing. In the original LDV, this is achieved by a focusing lens, a grating and a 4f system. The focusing lens serves to focus the beam and separate the beam waist of different wavelengths, and the grating serves to split the beam. In this paper, HOE is used to replace the focusing lens and the grating to realize the integration of components and the miniaturization of the system. The blue rectangle in Fig. 1 shows the schematic diagram of the HOE based LDV, in which the HOE must have the functions of focusing, beam splitting and beam waist separation at the same time to generate two fan-shaped interference fringes with different gradients of the fringe spacing.
2.1.2 Design method of multifunctional HOE
Three beam interference is used to record HOE with the functions described above. For a certain working wavelength, an inclined plane wave is used as the reference light, and two identical Gaussian beams are used as the signal light. The included angle of the two signal lights is noted as $2\alpha $, and their bisectors are nearly perpendicular to the HOE surface. The vertical distance between the waist of the signal beam and the HOE surface is ${b_1}$. When the signal lights have real focus on the right of HOE, the sign of ${b_1}$ is positive, otherwise, the sign of ${b_1}$ is negative. The HOE fabricated according to the above principles will have the functions of focusing and beam splitting at the same time.
Because the parameters of the measurement unit are closely related to the measurement task, it is the most important part of LDV. The parameters of different parts such as HOE, 4f system and MV are interrelated. For the convenience of the following description, the definitions of each parameter are illustrated in Fig. 2 and the symbols are provided in Table 1.
Table 1. Parameters involved in the design process and their symbolic representation
The design and fabrication of HOE with the above functions face three difficulties: (1) Once the HOE is made, its parameters will be fixed. Therefore, the parameters of HOE should be carefully designed so that the parameters of MV such as stripe spacing and size can meet the requirements of the measurement task; (2) Since the avalanche photodiode commonly used in the detection module works in the near-infrared band, at least one operating wavelengths of the LDV are near-infrared light. However, holographic films used to make HOE are usually only sensitive to visible light, so it is necessary to compensate for the decrease in diffraction efficiency caused by wavelength offset in the design process; (3) Since reflective HOE is used and the two signal lights are Gaussian beams, it is necessary to compensate the aberration caused by the oblique incidence of the illumination wave into the HOE and the wavelength offset during fabrication process, otherwise it will lead to the distortion of the measured volume. In order to provide a complete solution to the difficulties described above, the flow chart of designing and fabricating HOE is summarized in Fig. 3, and the principle of each step will be described in detail below.
Fig. 3. Flowchart for designing and fabricating HOE. This flow chart starts from the lower left corner, and yellow, purple, green and blue represent design, fabrication, verification and process control respectively.
2.2 Wavelength matching of HOE
The mismatch between the recording wavelength and the working wavelength of HOE will lead to the decrease of diffraction efficiency, and because the signal light is not a plane wave, additional aberration will be introduced.
2.2.1 Improvement of diffraction efficiency
The recorded HOE can be regarded as a volume holographic grating (VHG), which can be analyzed by Kogelnik coupled wave theory. As shown in Fig. 4(a), when the incident light meets the Bragg condition, the wave vector of the reference light ${k_r}$, the wave vector of the signal light ${k_s}$ and the grating vector K will form a closed triangle. However, when the incident wavelength or incident angle deviates, the phase mismatch will be introduced because the Bragg condition is not satisfied.
Fig. 4. When the operating wavelength is different from the recording wavelength, (a) if the illumination wave still enters the HOE in the same direction as the reference light, the incident light deviates from the Bragg condition. Kogelnik assumes that ${k_c}$, ${k_d}$ and K still form a closed triangle, but the end point of ${k_d}$ is not on the K-vector circle. On the one hand, the diffraction efficiency will drop sharply, and on the other hand, the direction of the diffracted light needs to be consistent with the experiment to prove its correctness. (b) If the illumination light is shifted to a certain angle so that the Bragg condition is still satisfied, the diffracted light can still maintain a high diffraction efficiency.
For reflective HOE without absorption, the Bragg mismatch parameter $\xi $ can be calculated by the following formula
The coupling strength of the grating $\nu $ is
where ${\theta _r}$, ${\theta _s}$ and $\phi $ are the angle formed by ${k_r}$, ${k_s}$ and K respectively with the surface of HOE, ${n_0}$, $\Delta n$ and d are the refractive index, the modulation of refractive index and the thickness of the holographic material, respectively, $\lambda $ is the operating wavelength of HOE. With the help of the Bragg mismatch parameter and the coupling strength of the grating, the diffraction efficiency can be expressed asIt can be seen that the diffraction efficiency of HOE will be affected by angle and wavelength deviation at the same time. When the deviation of wavelength is not too large, it can be compensated by the angle deviation of the incident light. As shown in Fig. 4(b), when the incident light with another wavelength meets the Bragg condition, the wave vector of the incident light ${k_c}$, the wave vector of the diffracted light ${k_d}$ and the grating vector K will also form a closed triangle. This method can greatly improve the diffraction efficiency decline caused by wavelength mismatch.
2.2.2 Aberration correction
To simplify the analysis, only the interference between the reference light and one signal light is considered, and another signal light can be analyzed by the same way. In the experiment, the Gaussian beam is used as the signal light, but when the beam waist is small, it can be approximately analyzed by ray tracing. Figure 5(a) shows the recording and reconstruction of HOE using the K-vector, where green and red represent the wave vector used to record and reconstruct HOE respectively, black represents the grating vector, and the plane $z = 0$ represents the surface of the holographic material. When the curvature radius of the signal light is different, the intersection point of the diffracted wave vector is also different.
Fig. 5. (a) K vector is used to analyze the recording and reconstruction of HOE. To show the geometric relationship of wave vector during recording and reconstruction more clearly, the angle relationship in the figure is accurate, while the length of wave vector is different from the actual size. (b) Oblique incidence results in different curvature radii of signal light in planes $xoz$ and $yoz$. Beam 1 is one of the signal beams, and beam 2 is the equivalent beam with the same curvature radius of beam 1 in the yoz plane.
To make the wave vector of the reconstructed light approximately perpendicular to the surface of the HOE, the signal light during recording needs to be inclined to the surface of the HOE. Next, the reason why the signal light with oblique incidence has different curvature radii in planes $xoz$ and $yoz$ is explained and the difference of curvature radii is calculated.
When the size of HOE is small, the difference between the curvature radii of the signal light can be calculated by geometric method. As illustrated in Fig. 5(b), ${\theta _s}$ is the angle between the signal light and the z-axis, ${l_y}$ and ${l_x}$ are the curvature radii of the signal light in planes $yoz$ and $xoz$, which can be approximated by ${b_{1 - y}}$ and ${b_{1 - x}}$ due to the small difference between them, and ${x_{HOE}}$ and ${y_{HOE}}$ are the size of HOE in the x and y directions, respectively. If the signal light is converted to the case where the optical axis is perpendicular to the HOE surface, its phase distribution on the HOE surface is
The difference between the signal light curvature radii of different planes can be expressed as
After $\Delta {b_1}$ is obtained, the astigmatism of diffracted light can be simulated by the ray tracing of K-vector, and this calculation will provide a basis for astigmatism correction.
3. Parameter design
The goal of designing HOE parameters is to make the MV suitable for specific measurement tasks. Since the input light is modulated by the HOE and 4f systems, and then the measured volume is formed, the calculation method of each part parameters in the above optical path and the relationship between them need to be understood in order to design the parameters of HOE. Therefore, the parameter calculation method of each part is introduced and deduced separately, and the constraint relationship between the parameters is studied. Finally, the parameters of HOE are designed based on the above results.
3.1 Measurement volume
Take the intersection point of two beams as the origin, the bisector of the beam is the z axis, and the plane of the beam is the $yoz$ plane to establish a three-dimensional right-hand Cartesian coordinate system. Then the calculation problem involved in MV can be summarized as: when the distance from the beam waist to $xoy$ plane ${b_1}$ and the angle between the two beams $2\alpha $ are known, calculate other parameters of MV. The variation of fringe spacing with z coordinate can be calculated by [37]:
3.2 4f system
The parameters of the 4f system only involve the focal lengths of the two lenses. The key problem in this part is the transformation of the tilted Gaussian beam through the 4f system, that is, when the focal length of the lens and the distance from the beam waist to the front focal plane of the first lens are known, calculate the distance from the beam waist to the back focal plane of the second lens after the beam passes through the 4f system.
Establish the coordinate system as shown in Fig. 6, $({{x_0},{y_0}} )$ is the coordinate before passing through the optical system, while $({{x_1},{y_1}} )$ is the coordinate after passing through the optical system. ${z_d} ={-} {b_1}/cos\alpha $ is the distance from the beam waist to the coordinate origin (when the beam waist is on the right side of the origin, Z is negative, otherwise it is positive). The expression of the inclined Gaussian beam in the $xoy$ plane is
$k$ is wave vector and ${C_{00}}$ is a constant. Assume ${L_0}$ is the length of the optical system, A, B, C, and D are coefficients in the transmission matrix of the optical system, and q parameter is $q(z )= 1/\{{1/R(z )- i\lambda /[{\pi \omega (z )} ]} \}$. After applying Collins formula, the following expression can be obtained:
Due to the fact that light propagates in this system under paraxial approximation conditions and without diffraction constraints, the q parameter of the tilted Gaussian beam also satisfies the ABCD law, which means that the position of the beam waist after passing through the 4f system can be calculated through the q parameter and the transformation matrix of the lens. The included angle of the two beams after passing through the 4f system can be calculated by geometrical optics:
The transformation of 4f system is a bridge connecting the parameters of HOE and the parameters of measuring volume, the calculation above will provide an important basis for the design of HOE.
3.3 Design the parameters of HOE
In LDV, the Doppler signal is mainly obtained by collecting the scattered light of solid or liquid particles, so the scattering characteristics of particles are very important. In order to ensure the following property of the particles to measure the velocity of the fluid accurately, the particle size needs to be small enough, but too small particles will lead to weak scattering signal, which is not conducive to collection. Therefore, the commonly used particle with diameters from several to dozens of microns, and the practical particle size in LDV is close to or slightly larger than the wavelength [38]. In this paper, it is assumed that the measurement task and other units except the measurement unit are determined, that is, the particle size and working wavelength are known. Then the design task is transformed into the determination of HOE and 4f system parameters to make the fringe spacing and the size of MV meet the measurement requirements.
The selection of ${b_1}$ is the key to HOE design, but since ${b_1}$ is also closely related to other parameters, the selection of parameters is a process of constant attempts and repeated adjustments. The idea of selecting HOE parameters in this paper is similar to the optimization process of optical system by optical design software, and Fig. 7 illustrates the flow chart for designing HOE parameters. The typical working process of optical design software is to first determine the initial structure and merit function, and then optimize the parameters to make the merit function as small as possible. The parameters of HOE are much less than those of optical system, so this process can be completed by manual calculation. In this design task, the parameters of the measured volume are equivalent to the design targets in the optical design software, while the parameters of HOE and 4f need to give a set of initial parameters with reference to a typical LDV. According to the calculation method given above, the parameters of the measured volume can be calculated from the parameters of HOE and 4f systems, and the difference between this result and the design target is the merit function. The goal of the design task is to make the value of the merit function as small as possible.
Fig. 7. Flowchart for designing the parameters of HOE. When the value of the merit function is small enough, the current parameter is the result.
The design method in the optical design software usually adopts the damped least square method, and this process needs to be completed manually in the selection of HOE parameters, which requires understanding the influence direction and degree of different HOE parameters on the measured volume parameters. Therefore, under the condition of ${f_1} = {f_2}$, the influence of HOE parameters on the fringe spacing L and the change rate of the fringe spacing $\partial L(z )/\partial z$ is simulated, and the results are shown in Fig. 8.
Fig. 8. The influence of HOE parameters on measuring volume parameters. The columns from left to right show the influence of the included angle $\alpha $ of the signal light, the working wavelength $\lambda $, the radius of the beam waist ${\omega _0}$ and the position of the beam waist ${b_1}$ respectively. The upper and lower lines respectively show the influence of HOE parameters on the fringe spacing L and the change rate of fringe spacing $\partial L/\partial z$. The x-axis represents the position change along the z-direction, the y-axis represents independent variables such as $\alpha $, $\lambda $, ${\omega _0}$ and ${b_1}$, and the z-axis represents dependent variables such as L and $\partial L/\partial z$.
It can be seen from Fig. 8 that for the interference of MV before beam waist, the influence of wavelength and included angle of the signal light on parameters is independent of z. The longer the wavelength is, the greater the fringe spacing is, and the greater the absolute value of the fringe spacing change rate is. The influence of angle is opposite to that of wavelength. Contrary to the above, the influence of beam waist radius and position on parameters depends on z. When the beam waist is relatively small, the larger the beam waist is, the smaller the spacing that is originally greater than the spacing at the center is, and the larger the spacing that is originally smaller than the spacing at the center is. That is, the variation direction of $|{\partial L(z )/\partial z} |$ is opposite to that of the beam waist radius. The absolute value of the distance from the waist position to the HOE surface has complex effects on the fringes, this effect is monotonous when $|{{b_1}} |$ is small, and the influence is opposite to that of beam waist radius. Because $|{{b_1}} |$ has a nonlinear effect on fringe parameters, it is usually determined first.
According to the APD and particle radius, the operating wavelengths are selected as 639 nm and 785 nm, and the recording wavelength is 639 nm. However, considering that the near-infrared fringes are difficult to be captured by CCD, 532 nm and 639 nm are used as the working wavelengths and 532 nm is used as recording wavelength to facilitate the verification experiment. This change is only to illustrate the feasibility of HOE design and fabrication method. In the actual system, red light and near-infrared light are still used as working wavelengths. In order to ensure the uniformity in the experiment, the diameter of the light spot must be at least 2.5 mm, that is, the minimum $|{{b_1}} |$ can be 2 cm. For better uniformity, $|{{b_1}} |$ is selected as 5 cm in this paper, so the diameter of light spot on HOE is about 7 mm, which can ensure the quality of diffracted light. For the included angle, whether from the perspective of wavelength mismatch compensation or fringe spacing expansion, it is expected that the angle should be as small as possible, so half of the angle is selected as 2 °. If three beams are used for recording, the two signal beams will be affected by the optical elements in the other optical path because they are too close. To overcome this problem, two signal beams are used to record with the reference beam respectively, and then combined into a HOE with complete functions. In the experiment, the waist radius of the beam will vary from 50um to 150um due to the different focal length of the lens used, and the average value of 100um is used in the simulation. The fringes in the measured volume are simulated with the above parameters, which are shown in Fig. 9. To make the gradient of fringes spacing more obvious, the proportion of pixels is adjusted appropriately.
4. Simulation and experiment
In this section, the HOE for compact LDV is simulated and experimentally verified. The diffraction efficiency decrease and aberration caused by wavelength mismatch are compensated respectively. And then the gradient of fringe spacing in MV is used to verify that the fabricated HOE can meet the requirements.
4.1 Compensation of diffraction efficiency
As illustrated in Fig. 10(a), the K-vector circle is drawn in the polar coordinate system. The green circle represents 532 nm for recording and the red circle represents 639 nm for reconstruction. The line 0-180 represents the plane where the HOE is located. The acute angle between the K-vector of the reference wave and the HOE surface is 35.41°, and the acute angle between the angular bisector of the signal waves and the HOE surface is 69.92°. The length and tilt angle of the grating vector in the two vector triangles are the same, and the two ends are located on the vector circle of the reconstructed wave after translation. During reconstruction, the acute angle between the K-vector of the illumination wave and the HOE surface is 55.49°, and the diffracted light exits perpendicular to the HOE, which represents the direction of the angular bisector of the two diffracted waves. It is worth noting that the above angles are all in the holographic material, but the refractive index of the holographic material is quite different from that of the air. It is necessary to consider the refraction of light at the interface. The actual recording optical path is shown in Fig. 10(b). The acute angle between the reference wave and the prism surface is 75.43°, and the acute angle between the angular bisector of the signal wave in the air and the HOE surface is 58.77°.
Fig. 10. The decrease of diffraction efficiency caused by wavelength mismatch is compensated by shifting the angle of incident light. (a) K-vector circle is used to analyze the angle of recording light. (b) The schematic diagram for recording HOE is shown, in which the prism is used to make the angle of the reference light in the holographic material meet the design requirements. (c) and (d) show the variation of diffraction efficiency with the change of wavelength and incident angle of illumination light.
Figure 10(c) and (d) show the variation of the diffraction efficiency with the change of wavelength and incident angle of illumination light. It can be seen that if the wavelength and angle are consistent with those recorded, the theoretical diffraction efficiency is 100%. When the reconstructed wavelength becomes 639 nm, if the angle remains the same, the diffraction efficiency will be only 4.06%, but if the angle shifts to 55.49°, the diffraction efficiency can reach 99.94%. During the experiment, it is necessary to shift the two signal beams to both sides of the angular bisector, and the diffracted light will also shift. If the angle of shift is not large, the closed vector triangle can still be formed during reconstruction, which has little impact on the diffraction efficiency. After fabrication, the efficiency of the two diffracted beams is 78% and 80% respectively, and the diffraction efficiency in the experiment is lower than the theoretical value mainly because the contrast of the stripes in the holographic material decreases due to the vibration of the air during the recording process.
4.2 Compensation of astigmatism
Since the reconstructed wavelength is inconsistent with the recorded wavelength and the signal light is not a parallel light, a variety of aberrations such as spherical aberration, coma, and astigmatism can be seen in the diffracted light. These aberrations may be caused by holographic lens itself or the fact that the illumination light is not parallel, the incident angle of the recorded or reconstructed light is inconsistent with the designed angle, and the refraction of the light at the interface between the air and the holographic material.
Among these aberrations, the spherical aberration and coma are only caused by holographic lenses, and due to the small size of HOE, these aberrations can be ignored. But astigmatism is not only caused by holographic lenses, but also by off axis incidence, and will cause serious distortion of the overall contour of the measured volume. Therefore, the following method is used to simulate the astigmatism and compensate it in the experiment according to the simulation results.
As shown in Fig. 11, since the signal light is incident obliquely, when the radius of curvature of the $xoz$ plane is 5.00 cm, the equivalent curvature radius of the $yoz$ plane is 7.08 cm. If the difference between the curvature radii of these two planes is defined as the astigmatism, the signal light has an astigmatism of 2.08 cm, which will lead to astigmatism of diffracted light. The ray tracing diagram of HOE is illustrated in Fig. 12. It shows that when the radius of curvature of the signal light is 5.00 cm and 7.08 cm respectively, the curvature radius of the diffracted light is 4.19 cm and 5.79 cm respectively, that is, the diffracted light will have an astigmatism of 1.60 cm theoretically.
Fig. 12. The astigmatism caused by wavelength mismatch is simulated by ray tracing. (a) The green line represents the K-vector of the recording light with a wavelength of 532 nm, the red line represents the K-vector of the reconstructed light with a wavelength of 639 nm, and the black line represents the grating vector. (b) and (c) are magnified images of the area enclosed by the blue rectangle in (a) when the curvature radius of the signal light is 5.00 cm and 7.08 cm, respectively.
The astigmatism of the fabricated HOE is measured, and the results are intuitively shown in the upper row of Fig. 13. In the experiment, the diffracted light converges first in the $xoz$ plane and then in the $yoz$ plane, that is, as the CCD gradually moves away from the HOE, the horizontal focus line is seen first, and then the vertical focus line is seen, which is consistent with the results of theoretical analysis. The difference between the curvature radii of the two planes is 1.50 cm, which is close to the simulation result. The error comes from the measurement of angle and distance, but the relative error of 6.25% is acceptable in the experiment. It can be inferred from the above experimental results that the ray tracing method can predict the amount of astigmatism, which can provide a basis for astigmatism compensation.
Fig. 13. Comparison of the shape of diffracted light before (upper row) and after (lower row) astigmatism correction. If the HOE surface is located at 0 mm, then before correction, the xoz plane is focused at 42.5 mm, while the diffracted beam of the yoz plane is focused at 57.5 mm. After correction, the diffracted beam is focused to a point at 52.5 mm.
According to the results of simulation and pre-experiment, a cylindrical lens is introduced into the optical path for fabricating HOE, so that the signal light has astigmatism opposite to that of the diffracted light, that is, the vertical focal line of the signal light is close to HOE, while the horizontal focal line is far from HOE. Then CCD is used to capture the diffracted light of HOE at different distances, and the results are shown in the lower line of Fig. 13. It can be seen that the rays converge at a point rather than two focal lines at different distances, indicating that the original astigmatism has been well corrected. The beam waist of the diffracted light is located at 52.5 mm, which is slightly different from the designed parameters. This is because the wavelength mismatch and astigmatism compensation will make the distance from the beam waist of the diffracted light to the HOE surface slightly different from that of the recording light. As long as the subsequent measurement is based on the calibrated parameters, the results will not be affected. However, if the goal that the results after fabricating are closer to the designed parameters needs to be achieved, the change rule of beam waist position under different wavelength mismatches can be obtained through experiments.
4.3 Interference fringes in measurement volume
Because the parameters of the fringes have a great influence on the measurement results, it is necessary to calibrate these parameters before using LDV to measure the fluid, rather than only using the results of theoretical calculation.
There are two main methods to calibrate the fringes. One is to make a metal wire with a diameter of several microns measure the volume at different distances and obtain the fringe spacing from the scattered signal [39]. Another calibration method is to capture the fringe in MV directly with CCD, and obtain the fringe parameters through image processing. The fringe parameters calibrated by the first method include the influence of signal collection and processing, and are suitable for use before measuring fluid. However, since the main content of this paper is the design and fabrication of HOE, it is important to prove that the HOE can achieve two sets of fringes with opposite gradient of spacing, and the actual value of the fringe spacing is consistent with the theoretical calculation result, so the second method is used to obtain the fringe parameters.
Since the pixel size of the CCD used in the experiment is 5um, it is impossible to detect that the fringe spacing is too small, and moiré fringes are easy to appear when the fringe spacing is close to the pixel size. Therefore, a 4f system with amplification effect is used in the verification experiment. Although the measured volume will be stretched, the trend of stripe spacing did not change, so it can still be used to verify the rationality of the designed HOE parameters. The focal length of lens 1 is 3.5 cm, and the focal length of lens 2 is 35 cm, when the working wavelength is 532 nm and 639 nm, the calculated fringe spacing at $xoy$ plane is 75$\mu m$ and 92$\mu m$, respectively. In the experiment, the fringes formed by different working wavelengths at different distances are shown in Fig. 14. It can be seen that the spacing of the fringes formed by 532 nm gradually increases, while the spacing of the fringes formed by 639 nm gradually decreases. The above results correspond to the cases that the beam waists are before and after MV, respectively, which are consistent with the theory.
Fig. 14. The captured fringe pattern of cross section in MV. The CCD acquires an image every 5 cm in the range of 15 cm to 70 cm.
The fringe spacing is obtained in the frequency domain by image processing. The fringe spacing of different wavelengths varies with the distance are plotted in Fig. 15, where the dots are the experimental data and the lines are the fitting results. It can be obtained that when the working wavelengths are 532 nm and 639 nm, the fringe spacings calculated according to the fitting function are 71.55$\mu m$ and 85.80$\mu m$ respectively, which are close to the theoretical result. The error mainly comes from two aspects: one is that the distance between CCD and HOE may not be measured accurately, and the other is that the fringe defect caused by the non-uniform beam may lead to inaccurate image processing results. The relative error of 4.6% and 6.7% is acceptable in the experiment, and the results show that the HOE for compact LDV can be manufactured using the design and fabrication methods proposed in this paper.
Fig. 15. Verification of the fringe spacing gradient of the HOE after fabrication. The green dot represents the experimental measurement value of the change of the fringe spacing along the z-axis in MV formed by the interference of the beam with the wavelength of 532 nm. The fitted green line indicates that the fringe spacing increases gradually with the increase of z. The red dot represents the experimental measurement value of the change of the fringe spacing along the z-axis in MV formed by the interference of the beam with the wavelength of 639 nm, and the fitted red line indicates that the fringe spacing decreases gradually with the increase of z.
5. Discussion
Previous research has made many attempts to integrate LDV, but mainly aimed at the miniaturization of light source unit, transmission unit and detection unit. In this paper, HOE is used to miniaturize the measurement module by integrate multiple functions in a single element, and the problems of diffraction efficiency degradation and aberration caused by the mismatch between recording and working wavelengths are solved. This paper analyzed and calculated the parameters of HOE with reference to the measurement task of typical LDV. After that, Kogelnik coupled wave theory was adopted to design the angle between the recording wave and the HOE surface when the working wavelength is 785 nm and the recording wavelength is 639 nm. The astigmatism introduced by wavelength mismatch is corrected by adding a cylindrical lens in the signal light path. Finally, the HOE designed and fabricated by this method can realize two sets of fringes with opposite gradients of the fringe spacing, which meets the functional requirements of LDV measurement unit for HOE.
The contribution of this paper is mainly reflected in two aspects. To improve the performance of LDV, HOE compresses the volume of LDV without affecting system functions, promoting the application of LDV in blood flow measurement, in situ shape measurement, etc. It not only extends the application range of LDV greatly, but also provides a feasible solution for space limited measurement. To design and fabricate HOE, this paper considers how to overcome the shortcomings while giving full play to the advantages of HOE in the system. The consideration of technical details in this process is helpful to expand the application scope of HOE and provide a possible path for the application of HOE in other systems.
There are still many studies that can be carried out on the basis of this paper. For example, to improve the uniformity of spot size when the beam waist is close to HOE is a major problem in the production of HOE, which can expand the optional range of HOE parameters. To better combine HOE and LDV, on the one hand, it can be tried to distinguish different fringes with light field parameters other than wavelength, on the other hand, HOE with adjustable parameters can be designed to enable LDV to adapt to different measurement tasks.
6. Conclusion
This paper presents a design and fabrication method of HOE which can integrate the functions of multiple traditional elements in the measurement unit of LDV system. The fabricated HOE has the functions of focusing, splitting and separating the beam waist at the same time, so it can save space by simplifying the optical path, and the optical path can be further folded due to the reflective structure of the HOE. Moreover, the advantages of HOE are fully utilized in LDV: on the one hand, the converging lens in the original system is large and heavy, while the thinness of HOE is conducive to reducing the weight of the system while miniaturizing; On the other hand, the traditional grating has the defects of long production cycle, high cost and easy to be damaged. HOE can overcome these defects, so it helps to reduce the production cost of LDV and improve its durability.
It is worth mentioning that the method proposed in this paper for designing and fabricating HOE has a certain degree of universality and can be used for reference in other researches. For the design of HOE, this paper analyzed the relationship between the system parameters of LDV in detail, which can also be used for the calculation and analysis of other LDVs. For the fabrication of HOE, the method proposed in this paper not only considered the feasibility of using the characteristics of HOE to improve the system performance, but also considered and solved many technical problems. For example, the mismatch between recording and operating wavelengths and the aberration correction are two main problems in the application of HOE. The solutions in this paper can provide inspirations and reference for solving such problems in other applications. In future, this method of designing and fabricating HOE is expected to integrate LDV, which can be applied to various applications of optical non-destructive measurement such as the measurement of flow field parameter in confined space.
Funding
National Natural Science Foundation of China (61975014, 62035003, U22A2079); Beijing Municipal Science & Technology Commission, Administrative Commission of Zhongguancun Science Park (Z211100004821012).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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