Method for Detecting Local Orbital Angular Momentum of a Complex-Structure Light Beam under a Limited Aperture

Abstract

Method for detecting local orbital angular momentum of a complex-structured beam under a finite aperture. First, under the condition of a finite receiving aperture, the calculation method of the orbital angular momentum of the structured beam of the locally received energy is supplemented, filling the blank of the method for detecting the local orbital angular momentum of a complex radially structured beam under a finite receiving aperture. Second, according to the characteristics of the orbital angular momentum distribution of the complex radially structured optical field, the orbital angular momentum corresponding to each intensity ring of the complex radially structured optical field is refined. The present invention solves the calculation and detection of the local orbital angular momentum of a complex radially structured beam under a finite receiving aperture, and achieves the detection of the local orbital angular momentum of the complex radially structured optical field by extracting the radial position of the local optical field of the complex radially structure.

Description

Technical Field

[0001] The present invention belongs to the technical field of optics, and particularly relates to a method for detecting the local orbital angular momentum of a complex-structured light beam under a finite aperture. Background Art

[0002] The orbital angular momentum of light has received extensive attention in the past two decades. This quantum number has been applied in many fields, such as micro-mechanical operation, atomic trapping, and quantum and classical information transmission. For the calculation and measurement of orbital angular momentum, many literatures have given different schemes. However, the influence of the aperture has been ignored in this process. As is well known, under the condition of a finite receiving aperture, due to the divergence of light during propagation, we may not be able to receive all the light energy. In this case, the estimation and measurement of the orbital angular momentum of the received partial energy become a key issue. In addition, in recent years, many studies have been devoted to the research of the transverse quantum number - radial index of light. This quantum number increases the complexity of the light field in the radial structure and will divide the orbital angular momentum. Then, how to calculate and detect the orbital angular momentum corresponding to each intensity ring of the complex radial structure has also become a thorny problem. The existing calculation methods for orbital angular momentum focus on the calculation of the total orbital angular momentum of the overall light field, and its specific implementation form is the integral summation of the overall plane of the light field. For a structured light beam with a complex radial phase structure, the relationship between its orbital angular momentum and the radial structure cannot be known.

[0003] The disadvantages of the prior art mainly focus on the following two points: (1) Ignoring the aperture effect; the calculation result applicable to the prior art is the total orbital angular momentum, which is different from the local orbital angular momentum. Under the condition of a finite receiving aperture, it is very likely that only the local energy and information of the light field can be received. Therefore, the calculation and measurement of its local orbital angular momentum are lacking, and this problem has been ignored; (2) Not considering the orbital angular momentum distribution problem of a light field with a complex radial structure; for a light field with a complex radial structure, it has a screw dislocation in terms of phase and corresponds to a multi-ring structure in terms of intensity; the problem of the orbital angular momentum density distribution of such multi-intensity rings has not been considered in the prior art. Summary of the Invention

[0004] To overcome the deficiencies of the above-mentioned existing technologies, the object of the present invention is to propose a method for detecting the local orbital angular momentum of a complex-structured light beam under a finite aperture. First, under the condition of a finite receiving aperture, the calculation method of the orbital angular momentum of the structured light beam of the received local energy is supplemented, filling the gap in the method for detecting the local orbital angular momentum of a complex-structured light beam under a finite aperture. Secondly, according to the characteristics of the orbital angular momentum distribution of the complex radial structure light field, the orbital angular momentum corresponding to each intensity ring of the complex radial structure light field is refined. The present invention solves the calculation and detection of the local orbital angular momentum of a complex radial structure light beam under a finite receiving aperture, and achieves the detection of the local orbital angular momentum of the complex radial structure light field by extracting the radial position of the local light field of the complex radial structure.

[0005] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A method for detecting the local orbital angular momentum of a complex-structured light beam under a finite aperture, specifically including the following steps:

[0007] Step 1. Detection of the local orbital angular momentum of a complex radial structure light beam

[0008] 1.1. Give the light field expression of an arbitrarily structured light beam, including parameters: beam width, topological charge, wavelength of the light beam;

[0009] 1.2. Solve the light field expression given in Step 1.1 to obtain the intensity distribution of the structured light beam;

[0010] 1.3. Find the radial position of the light field intensity ring to be obtained from the result obtained in Step 1.2, and determine the upper and lower boundaries of the integral;

[0011] 1.4. By applying the radial position in Step 1.3, substitute it into the orbital angular momentum calculation formula, change the upper and lower bounds of the integral, and obtain the local orbital angular momentum of the corresponding structured light beam;

[0012] Step 2. Verify the local orbital angular momentum of the structured light beam obtained in Step 1.4 by corresponding the local orbital angular momentum with the intensity value obtained by summation

[0013] 2.1. Through a demonstration experiment for detecting the local orbital angular momentum of a spatial structured light field, detect the local orbital angular momentum of the intensity ring of the required structured light beam by superimposing the inverse phase, and obtain the corresponding intensity distribution map;

[0014] 2.2. Obtain the average intensity by accumulating the intensity values of each pixel point in the corresponding intensity distribution map obtained in Step 2.1. At this time, the average intensity corresponds to the local orbital angular momentum.

[0015] In Step 1, the structured beam described in Step 1.1 uses an LG beam with a non-zero radial index.

[0016] The specific method of Step 1.1 is to write the field distribution of the LG beam with a non-zero radial index as:

[0017]

[0018] where A 0 represents the radially correlated quantity of the structured beam, z represents the distance between the input plane and the receiving plane; r is the radial coordinate in the polar coordinate system; l and p respectively represent the topological charge and radial index of the LG beam; is the fundamental mode Gaussian beam width; w 0 is the beam width of the incident beam; is the Rayleigh distance; the wave number k = 2π / λ, where λ is the wavelength; is the associated Laguerre polynomial.

[0019] The specific method of Step 1.2 is to obtain the light field intensity distribution of the LG beam, which is represented by multiplying the light field of the LG beam by its complex conjugate;

[0020] For the determination of the orbital angular momentum of the paraxial light field, the axial projection Jz of the total orbital angular momentum vector and the total light field energy W are given by the relationships in Eqs. (2) and (3):

[0021]

[0022] where represents the complex conjugate of, Im represents the imaginary part operation, and the total standard orbital angular momentum of the beam is as follows in Eq. (4):

[0023]

[0024] The integral over the radial position r in Eq. (4) is taken to be infinite, representing integration over the entire plane, and the total orbital angular momentum of the LG beam is obtained, where A 0 represents the radially correlated quantity of the structured beam.

[0025] The specific method of Step 1.3 is as follows:

[0026] First, through the one-dimensional transverse intensity profile data of the complex radially structured beam, find the radial position r of each light field intensity ring;

[0027] Use this radial position r to determine the upper and lower boundaries of the integral;

[0028] Obtain the orbital angular momentum on each intensity ring by controlling the upper and lower boundaries of the integral in Eq. (4).

[0029] The specific method of step 1.4 of step 1 is: use the radial position r described in step 1.3 of step 1, set it as the upper and lower bounds of the integral, substitute it into equation (4), and obtain the local orbital angular momentum of the light field intensity ring corresponding to the radial position r.

[0030] The specific method of step 2.1 of step 2 is:

[0031] LG beams with different radial indices are generated by the spatial light modulator SLM1;

[0032] Then, an aperture is placed in front of the spatial light modulator SLM2 to filter the intensity of the light field of the outer ring and allow the light field of the inner ring to pass through SLM2;

[0033] Then, by superimposing the inverse phase, the local orbital angular momentum of the inner ring light field intensity is detected for the LG beams with different radial indices generated by SLM1, and the corresponding light field intensity distribution diagram is obtained.

[0034] The specific method of step 2.2 of step 2 is:

[0035] The average intensity is obtained by processing the data obtained in step 2.1 of step 2;

[0036] The intensity value data is summed and normalized, and compared with the theoretical calculation result obtained in step 1.3 of step 1, that is, formula (4); the average intensity is obtained by accumulating and summing the intensity value (one dimension) of each pixel point (two dimensions), and the average intensity corresponds to the local orbital angular momentum.

[0037] Compared with the prior art, the present invention has the following advantages:

[0038] 1) The existing methods for calculating the local orbital angular momentum of structured light fields are lacking, and there is no effective means for calculating and detecting the local orbital angular momentum of complex radial structured light fields. The present invention improves and supplements the calculation scheme of global orbital angular momentum, and calculates and detects the local orbital angular momentum by defining the upper and lower boundaries of the integral; the intensity distribution of the structured light field is obtained to define the received local energy, and the local orbital angular momentum of the received light is calculated by adjusting the upper and lower boundaries in the integral of the orbital angular momentum calculation formula by determining the radial position of the required intensity ring. This upper and lower boundaries can be adjusted according to actual problems and the radial radius of the received partial intensity, making the present invention extensible and flexible.

[0039] 2) Due to the universality and symmetry of complex radially structured beams, as long as the expression of the optical field is known, according to the above process, the local orbital angular momentum of all radially symmetric structured beams can be calculated. In addition, this calculation method refines the calculation method of the orbital angular momentum of complex radially structured beams. By improving the global orbital angular momentum calculation scheme and flexibly selecting the upper and lower integration boundaries, this method can more accurately calculate the orbital angular momentum density corresponding to each intensity ring of the structured beam, and can be widely applied in the field of optical technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 It is a flowchart of the present invention.

[0041] Figure 2 It is the spectrogram of LG beams with zero radial index and LG beams with non-zero radial index.

[0042] Figure 3 It is the intensity distribution profile of the calculated structured optical field, and the radial position data of the first ring is marked in the figure.

[0043] Figure 4 From Figure 3 The theoretical local orbital angular momentum value of the innermost ring calculated from the radial position data.

[0044] Figure 5 It is the optical path diagram for experimental verification, each modulation element, and the intensity diagram received by the CCD.

[0045] Figure 6 It is the intensity distribution of the optical field detected experimentally after blocking the outer ring and demodulating. The left column is the two-dimensional intensity distribution, and the right column is the one-dimensional data.

[0046] Figure 7 It is the comparison of the orbital angular momentum corresponding to the innermost ring obtained by theoretical and experimental calculations and detections. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0047] The following will further describe the present invention in detail with reference to the drawings and specific embodiments.

[0048] A method for detecting the local orbital angular momentum of complex structured beams under a finite aperture specifically includes the following steps: Refer to Figure 1

[0049] Calculating and measuring the orbital angular momentum of complex radially structured beams under a finite receiving aperture has important practical significance. In actual situations, it is very likely that only the energy of part of the optical field can be received. Then, how to calculate and measure the orbital angular momentum corresponding to the received part of the optical field becomes a key issue.

[0050] Step 1: Detection of the local orbital angular momentum of complex radially structured beams

[0051] 1.1. Write the field distribution of an LG beam with a non - zero radial index as:

[0052]

[0053] where A 0 represents the radially - related quantity of the structured beam, z represents the distance between the input plane and the receiving plane; r is the radial coordinate in the polar coordinate system; l and p represent the topological charge and the radial index of the LG beam respectively; is the fundamental - mode Gaussian beam width; w 0 is the beam width of the incident beam; is the Rayleigh distance; the wave number k = 2π / λ, where λ is the wavelength; is the associated Laguerre polynomial.

[0054] 1.2. Calculate the intensity distribution of the structured light field, which is expressed by multiplying the structured light field by its complex conjugate;

[0055] Obtain the intensity distribution of the LG beam, which is expressed by multiplying the light field of the LG beam by its complex conjugate;

[0056] For the determination of the orbital angular momentum of the paraxial light field, the axial projection Jz of the total orbital angular momentum vector and the total light field energy W are given by the relationships in Eqs. (2) and (3):

[0057]

[0058] where represents the complex conjugate of, Im represents the imaginary - part operation, and the total standard orbital angular momentum of the beam is as follows in Eq. (4):

[0059]

[0060] The integral over the radial position r in Eq. (4) is taken to be infinite, representing integration over the entire plane, and the total orbital angular momentum of the LG beam is obtained. A 0 represents the radially - related quantity of the structured beam. See Figure 2 .

[0061] 1.3. Find the radial position of the intensity ring to be calculated and determine the upper and lower boundaries of the integral; in the case of a finite aperture, only part of the light - field energy can be received, then the orbital angular momentum on each intensity ring can be calculated by controlling the upper and lower boundaries of the integral in Eq. (4).

[0062] A set of calculation examples are provided here. Taking \(l = 3\) and \(w_0 = 3\mathrm{mm}\) as examples, the transverse intensity distribution of the LG beam based on Equation (1) can be calculated. Since it has the characteristic of radial symmetry, only half of its side sectional view is shown. By searching for data points, the position \(r\) of its first intensity ring can be found, as Figure 3 . Substituting this radial position into Equation (4), the orbital angular momentum of the innermost ring can be calculated, and the theoretical calculation results are as Figure 4 shown.

[0063] 1.4. By applying the radial position in Step 1.3, substituting it into the orbital angular momentum calculation formula, and changing the upper and lower limits of the integral, the local orbital angular momentum of the corresponding intensity ring is obtained; A set of calculation examples are provided here. Taking \(l = 3\) and \(w_0 = 3\mathrm{mm}\) as examples, the transverse intensity distribution of the LG beam based on Equation (1) can be calculated. Since it has the characteristic of radial symmetry, only half of its side sectional view is shown. By searching for data points, the position \(r\) of its first intensity ring can be found, as Figure 3 . Substituting this radial position into Equation (4), the orbital angular momentum of the innermost ring can be calculated, and the theoretical calculation results are as Figure 4 shown.

[0064] Step 2: Verify the local orbital angular momentum of the structured beam obtained in Step 1.4 by corresponding the intensity values obtained by summation to the local orbital angular momentum

[0065] 2.1. Generate LG beams with different radial indices through the spatial light modulator SLM1;

[0066] Then, place a diaphragm in front of the spatial light modulator SLM2 to filter and block the light field intensity of the outer ring, allowing the light field of the inner ring to pass through SLM2;

[0067] Furthermore, by superimposing the inverse phase on the LG beams with different radial indices generated by SLM1, detect the local orbital angular momentum of the light field intensity of the inner ring and obtain the corresponding light field intensity distribution diagram.

[0068] 2.2. Obtain the average intensity by accumulating and summing the intensity values of each pixel point in the corresponding intensity distribution diagram obtained in Step 2.1: Sum the intensity value data, perform normalization processing, and compare it with the theoretical calculation result obtained in Step 1.3, namely Equation (4); The average intensity is obtained by accumulating and summing the intensity values (one-dimensional) of each pixel point (two-dimensional). At this time, this average intensity corresponds to the local orbital angular momentum.

[0069] For the experimental verification part, as Figure 4 is the designed experimental optical path diagram, Figure 5(b) shows the phase holograms displayed on each modulation component and the intensity distribution patterns received by the CCD. In the experiment, LG beams with different radial indices were generated by a Spatial Light Modulator (SLM) SLM1, and the local orbital angular momentum of the desired intensity ring was detected by superimposing the inverse phase on SLM2 to obtain the corresponding intensity distribution pattern; a diaphragm was placed in front of SLM2 to block the intensity of the outer ring, and the experimental results at this time are as shown in Figure 6 shown.

[0070] As shown in Figure 7 , by summing its one-dimensional data, normalization is performed at this time and compared with the theoretical calculation results. The average intensity is obtained by accumulating and summing each pixel point (two-dimensional) or intensity value (one-dimensional), and this average intensity corresponds to the local orbital angular momentum at this time.

[0071] The above description is only a specific example of the present invention and does not constitute any limitation to the present invention. Obviously, for professionals in the field, after understanding the content and principle of the present invention, various modifications and changes in form and details may be made without departing from the principle and structure of the present invention. However, these modifications and changes based on the idea of the present invention are within the scope of protection of the claims of the present invention.

Claims

1. A method for detecting the local orbital angular momentum of a complex radial structure beam under a limited receiving aperture, characterized in that it specifically includes the following steps: Step 1: Detection of the local orbital angular momentum of a complex radial structure beam 1.

1. Give the optical field expression of an arbitrary structured beam, including parameters: beam width, topological charge, wavelength of the beam; 1.

2. Solve the optical field expression given in Step 1.1 of Step 1 to obtain the intensity distribution of the structured beam; 1.

3. Find the radial position of the optical field intensity ring to be obtained from the result obtained in Step 1.2 of Step 1, and determine the upper and lower boundaries of the integral; 1.

4. By applying the radial position in Step 1.3 of Step 1, substituting it into the calculation formula of orbital angular momentum, changing the upper and lower bounds of the integral, and obtaining the local orbital angular momentum of the corresponding structured beam; Step 2. Verify the local orbital angular momentum of the structured beam obtained in Step 1.4 of Step 1 by corresponding the intensity value obtained by summation to the local orbital angular momentum; 2.

1. Through a demonstration experiment for detecting the local orbital angular momentum of a spatial structured optical field, detecting the local orbital angular momentum of the intensity ring of the required structured beam by superimposing the inverse phase, and obtaining the corresponding intensity distribution map; 2.

2. Obtain the average intensity by accumulating and summing the intensity values of each pixel point in the corresponding intensity distribution map obtained in Step 2.1 of Step 2. At this time, the average intensity corresponds to the local orbital angular momentum.

2. A method for detecting the local orbital angular momentum of a complex radially structured beam under a finite receiving aperture according to claim 1, wherein: the structured beam described in Step 1.1 of Step 1 adopts an LG beam with a non-zero radial index.

3. A method for detecting the local orbital angular momentum of a complex radially structured beam under a finite receiving aperture according to claim 1, characterized in that: The specific method of Step 1.1 of Step 1 is to write the field distribution of the LG beam with a non-zero radial index as: Among them, A 0 represents the radial correlation quantity of the structured beam, z represents the distance between the input plane and the receiving plane; r is the radial coordinate in the polar coordinate system; l and p respectively represent the topological charge and the radial index of the LG beam; is the fundamental mode Gaussian beam width; w 0 is the beam width of the incident beam; is the Rayleigh distance; the wave number k = 2π / λ, where λ is the wavelength; is the associated Laguerre polynomial.

4. A method for detecting the local orbital angular momentum of a complex radially structured beam under a finite receiving aperture according to claim 1, characterized in that: The specific method of Step 1.2 of Step 1 is to obtain the optical field intensity distribution of the LG beam, which is represented by multiplying the optical field of the LG beam by its complex conjugate; For the determination of the orbital angular momentum of a paraxial optical field, the axial projection Jz of the total orbital angular momentum vector and the total optical field energy W are given by the relationships in Equations (2) and (3); wherein denotes the complex conjugate of, Im denotes the imaginary part operation, and the total standard orbital angular momentum of the light beam is as shown in formula (4) below: The integration over the radial position r in Equation (4) is taken to be infinite, representing the integration over the entire plane and obtaining the total orbital angular momentum of the LG beam, A 0 represents the radially correlated quantity of the structured beam.

5. A method for detecting the local orbital angular momentum of a complex radially structured beam under a finite receiving aperture according to claim 4, characterized in that: The specific method of Step 1.3 of Step 1 is: First, through the one-dimensional transverse intensity profile data of the complex radially structured beam, find the radial position r of each optical field intensity ring; Determine the upper and lower boundaries of the integral with this radial position r; Obtain the orbital angular momentum on each intensity ring by controlling the upper and lower boundaries of the integral in Equation (4).

6. A method for detecting the local orbital angular momentum of a complex radially structured beam under a finite receiving aperture according to claim 4, characterized in that: The specific method of Step 1.4 of Step 1 is: use the radial position r described in Step 1.3 of Step 1, set it as the upper and lower bounds of the integral, substitute it into Equation (4), and obtain the local orbital angular momentum of the optical field intensity ring at the corresponding radial position r.

7. A method for detecting the local orbital angular momentum of a complex radially structured beam under a limited receiving aperture according to claim 1, characterized in that: The specific method of step 2.1 in step two is as follows: Generate LG beams with different radial indices through the spatial light modulator SLM1; Then, place a diaphragm in front of the spatial light modulator SLM2 to filter and block the light field intensity of the outer ring, allowing the light field of the inner ring to pass through SLM2; Furthermore, by superimposing the inverse phase on the LG beams with different radial indices generated by SLM1, detect the local orbital angular momentum of the light field intensity of the inner ring, and obtain the corresponding light field intensity distribution map.

8. A method for detecting the local orbital angular momentum of a complex radially structured beam under a limited receiving aperture according to claim 4, characterized in that: The specific method of step 2.2 in step two is as follows: Process the data obtained in step 2.1 of step two to obtain the average intensity; Sum the intensity value data, perform normalization processing, and compare it with the theoretical calculation result obtained in step 1.3 of step one, namely formula (4); obtain the average intensity by accumulating and summing the one-dimensional intensity values of each two-dimensional pixel point. At this time, this average intensity corresponds to the local orbital angular momentum.

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