Refractive index measuring device and method based on Airy light beam

Abstract

The invention discloses a refractive index measuring device and method based on Airy light beams. The problem that an environment with an incident angle larger than 0 must be constructed for measuring the refractive index at present, and consequently a measuring device is complex is mainly solved. According to the refractive index measuring device and method based on the Airy light beams, an environment with the incident angle larger than 0 does not need to be constructed, images of the two Airy light beams with the vertical polarization direction can be obtained by rotating the polaroid in front of the CCD, and the refractive index of a sample can be accurately measured by measuring the distance between main spots of the two light beams. Therefore, the device and the measuring method are more direct, simpler and more convenient, and the Airy light beam used by the measuring device has no diffractivity, so that the light spot is smaller, and the precision is higher.

Description

Technical Field

[0001] This invention relates to the field of optical measurement, and more specifically to a refractive index measurement device and method based on an Airy beam. Background Technology

[0002] Measuring the refractive index of a substance is of great significance in scientific research and daily life, playing a crucial role in materials science, the food industry, environmental monitoring, and medical diagnostics. It is also one of the fundamental experimental topics in general physics. Methods for measuring refractive index mainly include Snell's law (such as the Abbe refractometer, beam deflection method, immersion method, and minimum deviation angle method), spectroscopic methods, interferometry, and polarization measurement methods. Among these, the Snell's law method is not only simple in principle and easy to operate, but also the most widely used, and the methods themselves are diverse. Regardless of the apparatus used, to apply Snell's law to measure refractive index, an environment with an incident angle greater than 0° must be constructed; this is precisely where the complexity or key to these apparatuses lies. Summary of the Invention

[0003] To overcome the shortcomings of the prior art, the present invention provides a refractive index measurement device and method based on Airy beam, which mainly solves the problem that current refractive index measurement requires the construction of an environment with an incident angle greater than 0, resulting in a relatively complex measurement device.

[0004] The technical solution of the present invention is as follows: A refractive index measurement device based on an Airy beam, comprising: Two-dimensional Airy beam emitting unit, used to generate two-dimensional Airy beams; The measurement unit includes a first beam splitter, a second 1 / 4λ wave plate, a container for storing the object to be measured, a second beam splitter, a polarizer, and a CCD, which are sequentially arranged on the optical path of the two-dimensional Airy beam. The two-dimensional Airy beam is incident perpendicularly on the surface of the object to be tested inside the container.

[0005] The two-dimensional Airy beam emitting unit includes a laser, a beam expander, a mirror, a spatial light modulator, a pinhole, a lens, and a first 1 / 4λ waveplate arranged sequentially.

[0006] It also includes a half-wave plate, which is disposed between the spatial light modulator and the laser.

[0007] The container has a cubic structure.

[0008] The lens is a Fourier lens.

[0009] The beam expander assembly includes a first lens and a second lens arranged coaxially.

[0010] The CCD has a pixel resolution of 5496*3672 and a pixel spacing of 2.4μm.

[0011] The laser is a helium-neon laser generator with a wavelength of 632.8 nm and a power of 10 mW.

[0012] A method for measuring refractive index based on an Airy beam includes the following steps: Step 1: Construct the refractive index measurement device based on the Airy beam as described above; Step 2: Pour the medium into the container; Step 3: Turn on the laser so that the beam is incident perpendicularly into the container. Rotate the polarizer until an image of two Airy beams with perpendicular polarization directions appears. Measure the offset d of the two-dimensional Airy beam at position z in the probe plane, and calculate the refractive index of the medium using the following formula.

[0013] in Let be the transmission distance of the Airy beam, i.e., the ccd position, k = 2π / λ be the wave number, L be the thickness of the medium, d be the offset of the two two-dimensional Airy beams at the z position of the probe plane, and x0 be the parameter in the optical field of the two-dimensional Airy beam, representing any lateral scale.

[0014] The beneficial effects of this invention are as follows: This invention provides a refractive index measurement device and method based on Airy beams. It eliminates the need for an environment with an incident angle greater than 0°. By rotating a polarizer in front of the CCD, images of two perpendicularly polarized Airy beams can be obtained. By measuring the offset of their parabolic trajectories, the refractive index of the sample can be accurately measured. Therefore, the device and measurement method are more direct and simpler. The Airy beam used in this measurement device is non-diffractive, resulting in a smaller spot size and higher accuracy. Attached Figure Description

[0015] Figure 1 (a) is a one-dimensional Airy beam; Figure 1 (b) is a two-dimensional Airy beam, with the dashed line representing the lateral offset pattern; Figure 1 (c) is the trajectory diagram of the Airy beam.

[0016] Figure 2 This represents the curved trajectory of the Airy beam as it passes through the medium; the blue dashed line represents the propagation trajectory in free space, and the red solid line represents the propagation trajectory of the beam through the medium. xd is the relative offset to the zd plane.

[0017] Figure 3 (a) shows the parabolic trajectories with different refractive indices, z1=0, L=0.05m; Figure 3 (b) The trajectory of the Airy beam at different locations, L=0.05m.

[0018] Figure 4 It expresses the change in the curved trajectory of the Airy beam as it passes through the medium; x d For z d Relative offset of the plane; BS1, BS2: beam splitters; G1: 1 / 4λ glass plate; PL1: polarizer.

[0019] Figure 5 This is a schematic diagram of the optical path; the Airy beam will be generated at position z in the initial plane.

[0020] Figure 6 (a) shows the sample stage and syringe; Figure 6 (b) is the sample to be measured.

[0021] Figure 7 Create a physical model of the system.

[0022] Figure 8 It is a single Airy beam phase hologram.

[0023] Figure 9 To capture a single Airy beam image at the focal plane using a CCD.

[0024] Figure 10 (a) Image of the position (x-polarization) of beam 1 after the addition of the sample, passing through the first beam splitter, 1 / 4λ wave plate, medium object, and second beam splitter. Figure 10 (b) A position image (y-polarization) of beam 2, which passes through the first beam splitter, the 1 / 4λ wave plate, and the medium object, is reflected by the second beam splitter and then passes through the medium object, the 1 / 4λ wave plate, and after being reflected by the first beam splitter, passes through the 1 / 4λ wave plate, the medium object, and the second beam splitter. Figure 10 (c) is an image where x-polarization and y-polarization coexist. Detailed Implementation

[0025] The invention will be further described below with reference to the accompanying drawings. A refractive index measuring device based on an Airy beam includes... Two-dimensional Airy beam emitting unit, used to generate two-dimensional Airy beams; The measurement unit includes a first beam splitter 24, a second 1 / 4λ wave plate 25, a container 29 for storing the object to be measured, a second beam splitter 26, a polarizer 27, and a CCD 28, which are sequentially arranged on the optical path of the two-dimensional Airy beam. The two-dimensional Airy beam is incident perpendicularly on the surface of the object to be tested inside the container.

[0026] The two-dimensional Airy beam emitting unit includes a laser 11, a beam expander 13, a reflector 14, a spatial light modulator 15, a pinhole 16, a lens 17, and a first 1 / 4λ waveplate 18 arranged sequentially.

[0027] It also includes a half-wave plate 12, which is disposed between the spatial light modulator and the laser.

[0028] The container has a cubic structure.

[0029] The lens is a Fourier lens.

[0030] The beam expander assembly includes a first lens and a second lens arranged coaxially.

[0031] The CCD has a pixel resolution of 5496*3672 and a pixel spacing of 2.4μm. The experiment used a CCD to measure the position of the light beam after it passed through the medium. The CCD model was Daheng MER-2000-19U3C, with a pixel resolution of 5496*3672 and a pixel spacing of 2.4μm.

[0032] The laser is a helium-neon laser generator with a wavelength of 632.8 nm and a power of 10 mW. The main instrument used to generate the Airy beam is a spatial light modulator (SLM). Specifically, MATLAB is used to extract the amplitude information M and phase information Φ of the light field U to be generated. Thus, the hologram used to generate this beam can be written as... Thus, if the extracted information comes from the Fourier transform of the light field... Therefore, SLM produces the light field after Fourier transform. At this point, a Fourier transform can be performed on the outgoing light field using the Fourier lens L3 to convert the light into a linear image. This is converted into the actual light field U.

[0033] A helium-neon laser generator with a wavelength of 632.8 nm and a power of 10 mW was used as the light source in the experiment. Because the SLM has polarization characteristics, a half-wave plate is needed to adjust the polarization of the emitted laser beam; the beam passes through the first lens L1 ( =50mm) and the second path L2 ( A beam expander system consisting of a 400mm beam can expand the beam by 8 times, completely covering the SLM screen and improving adjustment efficiency. After the beam is reflected by the SLM, multiple orders of diffracted light will appear. The first order diffracted light is selected through a pinhole and then passed through Fourier lens L3 to generate an Airy beam at the focal plane of L3.

[0034] The advantage of the experimental setup is that the parabolic trajectory Airy beam passes through the first beam splitter, the 1 / 4λ waveplate, and the second beam splitter in sequence, generating two Airy beams with perpendicular polarization directions and an optical path difference of 2L. Since these two Airy beams have different transmission distances in the sample and are perpendicular in polarization direction, the images of the two perpendicularly polarized Airy beams can be obtained by rotating the polarizer in front of the CCD. By measuring the offset of their parabolic trajectory, the refractive index of the sample can be accurately measured.

[0035] The specific experimental approach is as follows: A linearly polarized beam becomes a circularly polarized beam when passed through a 1 / 4λ waveplate, and the same applies to circularly polarized beams. Theoretically, the two beams to be measured are beam 1, which passes through the first beam splitter, the 1 / 4λ waveplate, the medium object, and the second beam splitter; and beam 2, which passes through the first beam splitter, the 1 / 4λ waveplate, the medium object, is reflected by the second beam splitter, passes through the medium object, the 1 / 4λ waveplate, is reflected by the first beam splitter, and then passes through the 1 / 4λ waveplate, the medium object, and the second beam splitter. Furthermore, the polarization directions of beams 1 and 2 are perpendicular. To ensure that the beams reaching the CCD are linearly polarized, a 1 / 4λ waveplate is added after lens L3 to first convert the linearly polarized light into circularly polarized light. Finally, adjusting the polarizer in front of the CCD allows us to obtain the images of beams 1 and 2 respectively.

[0036] This experiment can measure the refractive index of solids and liquids. The sample needs to be placed in a square container (5cm×5cm×5cm, 2mm thick, dimensions adjustable). A syringe is used to add the sample into the container. The container is fixed to a three-dimensional adjustment stage. The pitch angle of the container can be adjusted using the adjusting screws at the bottom of the stage. Figure 6 As shown in (a).

[0037] A method for measuring refractive index based on an Airy beam includes the following steps: Step 1: Construct the refractive index measurement device based on the Airy beam as described above; Step 2: Pour the medium into the container; Step 3: Turn on the laser so that the beam is incident perpendicularly into the container. Rotate the polarizer until an image of two Airy beams with perpendicular polarization directions appears. Measure the offset d of the two-dimensional Airy beam at position z in the probe plane, and calculate the refractive index of the medium using the following formula.

[0038] in Let be the transmission distance of the Airy beam, i.e., the ccd position, k = 2π / λ be the wave number, L be the thickness of the medium, d be the offset of the two two-dimensional Airy beams at the z position of the probe plane, and x0 be the parameter in the optical field of the two-dimensional Airy beam, representing any lateral scale.

[0039] The following sections will explain this application from various aspects, starting with its principles. An Airy beam is a curved trajectory beam, and its curved trajectory property can be used to improve ordinary optical experiments. In this experiment, we will, for the first time, utilize the curved trajectory property to measure the refractive index of a sample. Airy beams are divided into one-dimensional and two-dimensional Airy beams. The optical field expression of a one-dimensional Airy beam is as follows: (1) The light field expression for a two-dimensional Airy beam is: (2) Where Ai represents the Airy function, x0 and y0 represent the horizontal and vertical scaling factors, respectively, and a is the attenuation factor. The light field distributions of the two Airy beams are as follows: Figure 1 As shown in (a) and 1(b), a one-dimensional Airy beam consists of vertical lines, while a two-dimensional Airy beam consists of beams arranged at right angles. The brightest beam is called the main beam. During transmission, the main beam of the Airy beam will move laterally, thus producing a curved trajectory. The one-dimensional Airy beam is deflected in the x-direction, while the two-dimensional Airy beam is deflected in both the x and y directions.

[0040] Ordinary laser beams, such as Gaussian beams and Bessel beams, travel along a straight line in a homogeneous medium, but the Airy beam exhibits a curved trajectory. For example... Figure 1 As shown in (c), we can see that the propagation trajectory of the Airy beam is a curved line. Superficially, this seems to contradict Fermat's principle, but existing research shows that the overall beam center of the Airy beam still propagates along a straight line; only its main spot propagates along a parabolic trajectory. In a homogeneous medium with a refractive index of n, the expression for the parabolic trajectory of a one-dimensional Airy beam along the z-direction is: (3) In the above formula, x represents the horizontal position. Let k be the transmission distance of the Airy beam, k = 2π / λ be the wave number, and λ be the wavelength of the light.

[0041] When a beam of light with a straight trajectory is incident perpendicularly on a medium, the light rays are incident and exit perpendicularly. However, when the entire Airy beam is incident perpendicularly, the parabolic trajectory of its main spot changes, specifically as follows: Figure 2As shown in the figure, a medium with a refractive index of n is placed between z1 and z2. When the light ray reaches the z1 plane, the principal spot follows a parabolic trajectory: (4) Its in z The offset of plane 1 is x 1. The incident angle of the main spot is i 1: (5) Due to the incident angle of the main spot i 1 is very small and can be approximated as the slope of the parabolic trajectory at that point, that is: (6) When the angle is very small, Snell's law can also be approximated as follows: nθ 1 =i 1. Therefore, the angle of departure is... θ 1 is: (7) Similarly, based on the expression for a parabolic trajectory and an approximate Snell's law, it can be derived that... z 2. Locus of a plane x d1 Offset x 2. Angle of incidence i 2. Angle of emission θ The expression for 2 is: (8) (9) (10) (11) Finally, we can derive the expression for the trajectory of the light beam after it has completely passed through the medium as follows: Figure 2 (Solid red line in the middle) (12) in L = z 2- z 1 represents the thickness of the medium. The above formula can be simplified to the following expression: (13) From formula (3), the trajectory of the beam in free space is expressed as (i.e.) Figure 2 (middle blue dashed line) (14) Two trajectories on the detection plane z The difference in position is: (15) make t =1-1 / n The formula can be simplified to: (16) This expression is exactly a quadratic function of t, with the axis of symmetry being t = z / L > 1. However, in reality, the refractive index n of most samples is greater than 1, so the actual t < 1. Analysis of the function graph shows that as the refractive index n increases, t increases, and consequently, the offset xd also increases. Figure 3 As shown in (a). Secondly, from expression (17), it can also be seen that the offset xd increases linearly with the increase of the transmission distance z. Moreover, this offset is only related to the thickness of the medium and has nothing to do with the location of the medium (i.e., the specific value of z1), as shown in (a). Figure 3 As shown in (b). This means that in a specific experiment, we can place the medium at any position during the transmission process without affecting the final detection offset.

[0042] Based on this, we added a first beam splitter BS1 and a 1 / 4λ glass plate (i.e., the second 1 / 4λ glass plate) in front of the medium, and a second beam splitter BS2 and a polarizer PL1 behind the medium. The purpose was to generate two beams with perpendicular polarization. We denote the beam passing through the first beam splitter BS1, the 1 / 4λ glass plate, and the second beam splitter BS2 as beam 1; and the beam passing through the beam splitter BS1 and the 1 / 4λ glass plate, reflected by the second beam splitter BS2, passed through the 1 / 4λ glass plate, reflected by the first beam splitter BS1, and then passed through the 1 / 4λ glass plate and BS2 as beam 2. The polarization directions of these two beams are perpendicular to each other. When recording images with a CCD, simply rotating the polarizer PL1 will produce the beams. Figure 4 The image of two beams in the image simplifies the experimental procedure for measuring the refractive index of a sample and has advantages in terms of specific operation.

[0043] Since beam 2 travels 3L in the medium, and its total travel distance is 2L greater than that of beam 1, we can obtain the propagation formula for beam 2 based on formula (13): (17) In theory, for two one-dimensional Airy beams in the detection plane z The difference in position is: (18) For a two-dimensional Airy beam, since the beam is offset in both the x and y directions, and the offsets in both directions are the same when x0 = y0 in the expression, the final offset of the two-dimensional Airy beam after passing through the medium at a detection distance of z can be derived from formula (18): (19) Based on formulas (18) and (19), the refractive index of the medium can be further derived. n for: (20) In the above formula, Let be the transmission distance of the Airy beam, i.e., the ccd position, k = 2π / λ be the wave number, L = z2 - z1 be the thickness of the medium, d be the offset of the two two-dimensional Airy beams at position z on the probe plane, and x0 be the parameter in the optical field of the two-dimensional Airy beam, representing any lateral scale.

[0044] Devices for measuring the refractive index of objects based on the curved trajectory of Airy beams, such as... Figure 7 As shown. The optical components in the optical path are mainly divided into two parts: one is the generation of the two-dimensional Airy beam, including the spatial light modulator (SLM), laser, half-wave plate, lens, pinhole, etc.; the other is the measurement of the refractive index of the sample, including CCD, three-dimensional adjustment stage, square container, sample to be measured, first beam splitter, 4λ glass plate, polarizer, etc.

[0045] A helium-neon laser generator with a wavelength of 632.8 nm and a power of 10 mW was used as the light source in the experiment. Because the SLM has polarization characteristics, a half-wave plate is needed to adjust the polarization of the emitted laser beam; the beam passes through L1 ( =50mm) and L2 ( A beam expander system consisting of a 400mm beam can expand the beam by 8 times, completely covering the SLM screen and improving adjustment efficiency. After the beam is reflected by the SLM, multiple orders of diffracted light will appear. The first order diffracted light is selected through a pinhole and then passed through Fourier lens L3 to generate an Airy beam at the focal plane of L3.

[0046] The advantage of this experimental setup lies in the fact that the parabolic Airy beam passes sequentially through the first beam splitter, the quarter-λ waveplate, and the second beam splitter, generating two mutually perpendicular Airy beams with a path difference of 2L. Since these two Airy beams travel at different distances within the sample and are perpendicularly polarized, rotating the polarizer in front of the CCD allows for the acquisition of images of the two perpendicularly polarized Airy beams. By measuring the deviation of their parabolic trajectories, the refractive index of the sample can be accurately determined. The specific experimental approach is as follows: a linearly polarized beam becomes a circularly polarized beam when passing through the quarter-λ waveplate, and the same applies to circularly polarized beams. Theoretically, the two beams to be measured in the experiment are beam 1, which passes through the first beam splitter, a 1 / 4λ waveplate, a medium object, and a second beam splitter; and beam 2, which passes through the first beam splitter, a 1 / 4λ waveplate, and a medium object, is reflected by the second beam splitter, passes through the medium object, a 1 / 4λ waveplate, is reflected by the first beam splitter, and then passes through the 1 / 4λ waveplate, the medium object, and the second beam splitter. Furthermore, beams 1 and 2 are polarized perpendicularly. To ensure that the beams reaching the CCD are linearly polarized, a 1 / 4λ waveplate is added after lens L3 to first convert the linearly polarized light into circularly polarized light. Finally, by adjusting the polarizer in front of the CCD, images of beams 1 and 2 can be obtained respectively.

[0047] This experiment can measure the refractive index of solids and liquids. The sample needs to be placed in a square container (5cm × 5cm × 5cm, 2mm thick), and is added to the container using a syringe. The container is fixed to a three-dimensional adjustment stage. The pitch angle of the container can be adjusted using the adjusting screws at the bottom of the stage. Figure 6 As shown in (a).

[0048] The sample used in the experiment was water, such as Figure 6 As shown in (b).

[0049] The experiment used a CCD to measure the position of the beam after it passed through the medium. The CCD model was Daheng MER-2000-19U3C, with a pixel resolution of 5496*3672 and a pixel spacing of 2.4μm.

[0050] Experimental theoretical optical path diagram, such as Figure 5 As shown in the image. The final setup of the experiment is shown in the image. Figure 7 As shown.

[0051] First, adjust the pitch angle of the laser platform to ensure the emitted beam propagates horizontally forward, and then fix it in place. Next, add reflectors M1 and M2 to center the beam on the SLM screen. Then, add lenses L1 and L2 to form a beam expander system, expanding the laser beam size to completely cover the SLM screen. Ensure the two lenses are 450mm apart and that the center of the expanded beam spot coincides with the center of the SLM screen. Place a half-wave plate for subsequent polarization adjustment.

[0052] MATLAB can be used to perform a Fourier transform on the optical field of an Airy beam, obtaining its amplitude and phase information, and then a corresponding hologram can be generated, such as... Figure 10 As shown. The parameters selected for the experiment were x0=y0=0.006 mm, a=0.05, r d =0.5mm. Using the software accompanying the SLM, a corresponding phase hologram can be loaded onto the SLM. A first-order diffracted light can be selected using a pinhole, and a Fourier transform using a 400mm lens can generate the desired Airy beam at the lens's focal plane (z=0). The Airy beam detected by a CCD on the initial plane is shown below. Figure 9 As shown.

[0053] Example Table 1. Measurement of the refractive index of water using the Airy beam.

[0054] Δ can be obtained by total differential calculation according to formula (15). d and Δ n The formulas are respectively (twenty one) (twenty two) The relevant experimental parameter λ=632.8 nm , x 0 = 0.06 mm , L =0.055m, z =0.15 m After substituting, we can obtain (twenty three) The theoretical value of the refractive index of water can be found through querying. n Theoretical value = 1.33 (room temperature 25°C), therefore the relative error compared to the theoretical value is... (twenty four) Measurement of anhydrous ethanol (n) using a curved trajectory beam 理论 The results (=1.361) are shown in Table 2. The final measured result is... .

[0055] Table 2. Refractive index of anhydrous ethanol measured using an Airy beam.

[0056] The relative error between the measured refractive index of anhydrous ethanol and the theoretical value in this experiment is: (25) Measurement of glycerol (n) using a curved trajectory beam理论 The results (=1.474) are shown in Table 3. The final measured result is... .

[0058] Table 3. Airy beam measurement of the refractive index of glycerol

[0059] The relative error between the measured refractive index of glycerol in this experiment and the theoretical value is:

[0060] These results demonstrate the accuracy of formula (20), showing that measuring the refractive index of liquids using curved beams with different polarization is not only simple and effective, but also highly accurate.

[0061] The embodiments described with reference to the accompanying drawings are exemplary and intended to explain the invention, and should not be construed as limiting the invention. The embodiments should not be considered as limiting the invention, but any improvements made based on the spirit of the invention should be within the scope of protection of the invention.

Claims

1. A refractive index measuring device based on an Airy beam, characterized in that: include Two-dimensional Airy beam emitting unit, used to generate two-dimensional Airy beams; The measurement unit includes a first beam splitter (24), a second 1 / 4λ wave plate (25), a container (29) for storing the object to be measured, a second beam splitter (26), a polarizer (27), and a CCD (28) arranged sequentially on the optical path of the two-dimensional Airy beam. The two-dimensional Airy beam is incident perpendicularly on the surface of the object to be tested inside the container.

2. The refractive index measuring device based on an Airy beam according to claim 1, characterized in that: The two-dimensional Airy beam emitting unit includes a laser (11), a beam expander (13), a mirror (14), a spatial light modulator (15), a pinhole (16), a lens (17), and a first 1 / 4λ waveplate (18) arranged in sequence.

3. The refractive index measuring device based on an Airy beam according to claim 2, characterized in that: It also includes a half-wave plate (12), which is disposed between the spatial light modulator and the laser.

4. The refractive index measuring device based on an Airy beam according to claim 1, characterized in that: The container has a cubic structure.

5. The refractive index measuring device based on an Airy beam according to claim 1, characterized in that: The lens is a Fourier lens.

6. The refractive index measuring device based on an Airy beam according to claim 2, characterized in that: The beam expander assembly includes a first lens and a second lens arranged coaxially.

7. A refractive index measuring device based on an Airy beam according to any one of claims 1-6, characterized in that: The CCD has a pixel resolution of 5496*3672 and a pixel spacing of 2.4μm.

8. The refractive index measuring device based on an Airy beam according to claim 2, characterized in that: The laser is a helium-neon laser generator with a wavelength of 632.8 nm and a power of 10 mW.

9. A method for measuring the refractive index based on an Airy beam, characterized in that, Includes the following steps, Step 1: Construct a refractive index measurement device based on an Airy beam as described in any one of claims 1-8; Step 2: Pour the medium into the container; Step 3: Turn on the laser so that the beam is incident perpendicularly into the container. Rotate the polarizer until an image of two Airy beams with perpendicular polarization directions appears. Measure the offset d of the two-dimensional Airy beam at position z in the probe plane, and calculate the refractive index of the medium using the following formula. in Let be the transmission distance of the Airy beam, i.e., the ccd position, k = 2π / λ be the wave number, L be the thickness of the medium, d be the offset of the two two-dimensional Airy beams at the z position of the probe plane, and x0 be the parameter in the optical field of the two-dimensional Airy beam, representing any lateral scale.

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