Nonlinear optical orbital angular momentum spectrum analyzer and method
Through a nonlinear optical orbital angular momentum spectrum analyzer, single-stop orbital angular momentum complex spectral measurement and infrared spectral band detection are achieved through a nonlinear optical orbital angular momentum complex measurement and infrared spectral band detection, solving the problem of limitations in the existing technology and improving the accuracy and resolution of measurement.
Patent Information
- Application Number
- CN202510454819.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-04-11
- Publication Date
- 2025-07-04
AI Technical Summary
The prior art is difficult to achieve both broadening the spectral range and performing single-sequence orbital angular momentum complex spectroscopy measurements, especially outside the visible light band, such as the infrared spectral band detection technology is not yet mature.
Using a nonlinear optical orbital angular momentum spectrum analyzer, the reference light is vortex phase and rotation phase modulated by a combination of polarization beam splitter, nonlinear crystal, laser, reference light modulator, single-mode optical fiber and photodetector, and signal analysis is performed in the nonlinear crystal with the light to be measured, combined with the single-mode optical fiber filtering and data processing module.
Single-sequence orbital angular momentum complex spectroscopy measurement is realized, and the detection range is expanded to the infrared spectral band, which improves the accuracy and resolution of measurement and solves the problem of frequency shift degenerate.
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Figure CN120252955A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the field of optoelectronic information detection, and relates to a non-linear optical orbital angular momentum spectrum analyzer and method. Background Art
[0002] In 1992, Allen et al. discovered that light can carry orbital angular momentum (OAM). This kind of light has a helical phase term of , and the wavefront changes helically during propagation, so it is called a vortex beam. Each photon in it can carry an OAM of magnitude , which is often called the topological charge number. Common vortex beams include Laguerre-Gaussian light, Bessel-Gaussian light, etc.
[0003] Vortex beams have a wide range of applications, and can be used for measuring the rotational speed of rotating objects, mode multiplexing in the field of optical communication, particle manipulation such as optical tweezers and optical wrenches, laser processing, high-resolution imaging, and so on. The detection technology of vortex optical signals with a single topological charge number has been very mature, but for multiple There are still many problems in the detection technology of vortex spectral signals with value superposition. At present, many research teams have explored this issue. In 2017, the team led by Zhang Pei invented an OAM complex spectrum analysis method based on the rotational Doppler effect. A strong reference light and the OAM signal light to be measured are incident on a rotating object, causing the OAM signal to undergo frequency shift. The interference with the reference light generates a beat frequency signal, and the complex spectrum information of the OAM signal light is restored using the one-to-one correspondence between OAM and frequency shift. However, beat frequency signals will also be generated between different signals to be measured. The frequency shift of the signal-signal beat frequency is degenerate and indistinguishable from the above-mentioned signal-reference light beat frequency. Therefore, repeated measurements are required to eliminate the influence. The team led by Fu Shiyao used Gaussian light as a reference in 2020 and made coaxial interference with the light to be measured. The OAM of the light to be measured can be analyzed according to the interference pattern. Although this method is simple, it cannot eliminate the interference of the signal-signal beat frequency. In 2023, the team led by Yu Siyuan introduced the concept of the Kramers-Kronig receiver in coherent communication into OAM measurement, eliminated the above interference, and achieved strict single-shot OAM complex spectrum detection through imaging. In 2024, the team of Zhao Chengliang and Ren Yuan cooperated and adopted an unfiltered multimode receiving scheme. Using the product relationship between the rotational Doppler spectrum of the scattered light and the autocorrelation function of the OAM spectrum of the signal to be measured and the OAM spectrum of the rotating object, single-shot OAM spectrum measurement was achieved. Currently, most of the research is limited to the visible light band, and the actual demand drives researchers to broaden the spectral range of OAM spectrum signal detection. For example, in 2022, the team of Chen Lixiang and Ren Yuan cooperated to achieve frequency up-conversion detection of infrared rotational Doppler signals using a nonlinear crystal. In 2024, the team led by Zhang Pei used a cyclic structure of frequency doubling and difference frequency combination to achieve the conversion of any detection band and improved the detection resolution. The incident light of the above methods is the OAM superposition with the topological charge number , and by scanning different , the OAM spectrum distribution of the light to be measured can be restored. Currently, there is no solution in the field that can both broaden the spectral range and achieve single-shot OAM complex spectrum measurement. Summary of the Invention
[0004] The purpose of the present invention is to overcome the shortcomings of the above-mentioned existing technologies and provide a nonlinear optical orbital angular momentum spectrum analyzer and method, which can not only achieve single-shot orbital angular momentum complex spectrum measurement, but also expand the application range to the infrared spectral band.
[0005] To achieve the above purpose, the present invention adopts the following technical solutions: A nonlinear optical orbital angular momentum spectrum analyzer includes a polarization beam splitter, a nonlinear crystal, a laser, a reference light modulator, a single-mode optical fiber, a photodetector, and a data processing module; The laser is used to emit a reference light, which passes through a reference light modulator and reaches a polarization beam splitter. The reference light modulator is used to apply a vortex phase and a rotational phase modulation to the reference light. The polarization beam splitter is used to receive the signal light to be measured and the modulated reference light. The nonlinear crystal is arranged on the output optical path of the polarization beam splitter. The single-mode fiber is located on the output optical path of the nonlinear crystal. The single-mode fiber is connected to the input end of the detector. The output end of the photodetector is connected to the data processing module.
[0006] Preferably, after the signal light to be measured is focused and converged by a lens, it is transmitted to the polarization beam splitter.
[0007] Preferably, the reference light modulator adopts a light field controller such as a spatial light modulator, a digital micromirror device, a deformable mirror or a metasurface.
[0008] Preferably, the nonlinear crystal adopts barium metaborate, potassium dihydrogen phosphate, lithium niobate or periodically poled potassium titanyl phosphate.
[0009] Preferably, the single-mode fiber only couples Gaussian light with an orbital angular momentum of zero.
[0010] A nonlinear optical orbital angular momentum spectrum analysis method includes the following processes: S1, the signal light to be measured with an orbital angular momentum spectrum is incident on the polarization beam splitter; S2, the Gaussian light emitted by the laser is loaded with a phase by the reference light modulator and becomes a vortex light carrying an orbital angular momentum, and enters the polarization beam splitter to be combined with the signal light to be measured; S3, using the nonlinear crystal to sum-frequency the combined reference light and signal light to be measured to generate a beam of light with a new frequency; S4, after the light beam passes through the single-mode fiber for Gaussian filtering, the photodetector detects the intensity of the Gaussian light component and transmits the data to the data processing module; S5, the data processing module obtains the orbital angular momentum complex spectrum information of the signal light through data processing.
[0011] Preferably, the expression of the signal light to be measured is:
[0012] is the complex amplitude distribution to be measured, is the angular frequency of the signal light, is the topological charge number, is the time, is the azimuth angle, i is the imaginary number; The expression of the reference light:
[0013] is the known amplitude distribution, is the angular frequency, , is the fixed angular frequency of the reference light, is the rotational angular frequency; is the phase evolution velocity, is the topological charge number, is a function of, representing the relationship between the rotational angular frequency and the orbital angular momentum.
[0014] Preferably, the coupled wave equation of the sum frequency process is:
[0015] is the angular frequency, is the wave number, is the speed of light in vacuum, is the effective nonlinear susceptibility, i is the imaginary unit, is the propagation distance, , under the phase matching condition, , so:
[0016] is the complex amplitude distribution to be measured, is the angular frequency of the signal light, is the topological charge number, is the time, is the azimuth angle, is the known amplitude distribution, is the angular frequency, m is the topological charge number.
[0017] Preferably, the single-mode fiber filtering selects the component of, and the output electric field is:
[0018] is the complex amplitude distribution to be measured, is the angular frequency of the signal light, is the complex amplitude of the orbital angular momentum of the reference light, is the topological charge number, is the time, is the azimuth angle, is the angular frequency, i is the imaginary unit.
[0019] Preferably, the data processing module performs Fourier transform on the intensity of the Gaussian light component obtained by the photodetector to obtain the spectral distribution of different beat frequency signals. Based on the known amplitude values of each component of the reference light, the relative phase, and the spectral distribution of the beat frequency signals, the complex spectral information of the orbital angular momentum of the signal light to be measured is obtained.
[0020] Compared with the prior art, the present invention has the following beneficial effects: By modulating the reference light, the present invention adjusts the change speed of the time-varying phase of different orbital angular momenta of the reference light, changes the mathematical relationship between the frequency shift and the orbital angular momentum, and separates the originally indistinguishable degenerate cases. It can not only achieve strict single-shot orbital angular momentum complex spectrum measurement, but also utilize the nonlinear crystal on this basis to expand the applicable range to the infrared spectral band, complement other infrared light detection technologies, and achieve more accurate measurement. BRIEF DESCRIPTION OF THE DRAWINGS
[0021] Figure 1 It is a schematic diagram of the overall structure of the non-linear optical orbital angular momentum spectrum analyzer of the present invention; Figure 2 It is a flowchart of the non-linear optical orbital angular momentum spectrum analysis method of the present invention; Figure 3 It is a comparison diagram of the effects between the present invention and the traditional method. DETAILED DESCRIPTION OF THE EMBODIMENTS
[0022] The following details the embodiments of the present invention. The examples of the embodiments are shown in the drawings, where the same or similar reference numerals represent the same or similar elements or elements with the same or similar functions from beginning to end. The embodiments described below with reference to the drawings are exemplary only for explaining the present invention and should not be construed as limiting the present invention.
[0023] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the technical field to which the present invention belongs. The terms "installation", "connection", and "connection" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection, an electrical connection, or a connection that can communicate with each other; it can be directly connected, or indirectly connected through an intermediate medium, and can be the internal communication of two elements or the interaction relationship between two elements. The term "and / or" used herein includes any and all combinations of one or more of the related listed items. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific circumstances. The terms used in the description of the present invention in this specification are only for the purpose of describing specific embodiments and are not intended to limit the present invention.
[0024] The following disclosure provides many different embodiments or examples for implementing different structures of the present invention. To simplify the disclosure of the present invention, the components and settings of specific examples are described below. Of course, they are only examples and are not intended to limit the present invention. In addition, the present invention may repeat reference numerals and / or reference letters in different examples. This repetition is for the purpose of simplification and clarity, and does not itself indicate the relationship between the various embodiments and / or settings discussed. In addition, the present invention provides examples of various specific processes and materials, but those of ordinary skill in the art can be aware of the application of other processes and / or the use of other materials.
[0025] As Figure 1 shown, a non-linear optical orbital angular momentum spectrum analyzer according to the present invention includes a lens system, a polarization beam splitter, a non-linear crystal, a laser, a reference light modulator, a single-mode optical fiber, a detector, and a data processing module.
[0026] The receiving module is composed of a series of lenses with different focal lengths, and is used to receive the orbital angular momentum spectrum signal to be measured, focus and converge the signal light to be measured, and transmit it to the beam combining system.
[0027] The reference light module is composed of a laser and a reference light modulator. The laser generates Gaussian light with a specific frequency and hits the reference light modulator. The reference light modulator can be an optical field regulator such as a spatial light modulator, a digital micromirror device, a deformable mirror, or a metasurface. Its function is to perform specific modulation on the orbital angular momentum spectrum of the reference light. In addition to carrying the vortex phase modulation, the reference light modulator also needs to add a rotational phase modulation , so that the vortex light of different orders evolves at a specific speed, so that the beat frequency signals that are degenerate in frequency shift in the previous scheme can be distinguished. For example, if the relationship between the frequency shift and the orbital angular momentum is exponential, then the frequency shifts of the components with orbital angular momentum values of 1, 2, 3, and 4 are 2, 4, 8, and 16 respectively, and the final beat frequency shift will not be degenerate.
[0028] The beam combining module is composed of a polarization beam splitter, and can also be replaced by a combination of a common beam splitter and a series of polarization components such as a wave plate and a polarizer. The function of the polarization beam splitter is to transmit horizontally polarized light and reflect vertically polarized light. Here, it is used to receive the signal light to be measured and the modulated reference light, combine the signal light to be measured and the modulated reference light, and transmit it to the non-linear module.
[0029] The non-linear module is composed of a non-linear crystal. Common non-linear crystals include barium metaborate ( ), potassium dihydrogen phosphate ( ), lithium niobate ( ), and periodically poled potassium titanyl phosphate ( ), etc. The orbital angular momentum spectrum signal to be measured and the modulated reference light generate sum frequency at the nonlinear crystal, producing light of a new frequency.
[0030] The detection module consists of a single-mode fiber and a photodetector. The single-mode fiber is located on the output optical path of the nonlinear crystal. The single-mode fiber is connected to the input end of the detector. The single-mode fiber can only couple Gaussian light with zero orbital angular momentum, thus realizing the selection of a specific output mode and the frequency shift information contained therein. Then the photodetector detects the intensity of the Gaussian light component and transmits the data to the data processing module.
[0031] The data processing module performs Fourier analysis on the intensity data transmitted by the detection module to obtain the spectral distribution of different beat frequency signals. Combining with the known complex spectral distribution of the reference light orbital angular momentum, the complex spectral information of the signal light orbital angular momentum can be calculated.
[0032] As Figure 2 shown, the nonlinear optical orbital angular momentum spectrum analysis method of the present invention includes the following processes: Step 1, the lens system collects the signal light to be measured with an orbital angular momentum spectrum (carrying helical phase ), and converges into the polarization beam splitter.
[0033] Step 2, the Gaussian light emitted by the laser is loaded with a phase by the reference light modulator, making the reference light become vortex light carrying orbital angular momentum , and enters the polarization beam splitter to be combined with the signal light to be measured.
[0034] Step 3, use the nonlinear crystal to achieve sum frequency, generating a beam of light with a new frequency . This newly generated light also carries orbital angular momentum components and is obtained by the superposition of the signal light and the reference light OAM, that is .
[0035] Step 4, the single-mode fiber performs Gaussian filtering to select components. The output end of the single-mode fiber is connected to the photodetector. The photodetector measures the intensity of the Gaussian light component and converts the optical signal into an electrical signal, which is transmitted to the data processing module.
[0036] Step 5, perform Fourier transform on the intensity magnitude of the Gaussian light component obtained by the photodetector to obtain the spectral distribution of different beat frequency signals. After knowing the amplitude values, relative phases of each component of the reference light and the spectral distribution of the beat frequency signal, it is easy to obtain the OAM complex spectral information of the signal light to be measured.
[0037] In the above process, the expression of the signal light to be measured:
[0038] is the complex amplitude distribution to be measured, is the angular frequency of the signal light, is the topological charge number, is the time, is the azimuth angle, i is the imaginary number.
[0039] Expression of the reference light:
[0040] is the known amplitude distribution, m is the topological charge number (used to describe the value of the orbital angular momentum), is the angular frequency, , is the fixed angular frequency of the reference light, is the rotational angular frequency, relative to is very small. is the phase evolution velocity, is the topological charge number. is a function of, representing the relationship between the rotational angular frequency and OAM. In the traditional model, the modulation of the rotating object makes , in the present invention, arbitrary modulation can be achieved through the reference light modulator, for example , but this design is not unique.
[0041] Coupled wave equation of the sum frequency process:
[0042] is the angular frequency, is the wave number, is the speed of light in vacuum, is the effective nonlinear susceptibility, is the propagation distance, , under the phase matching condition, . Therefore,
[0043] The single-mode fiber filtering selects the component, and the output electric field is:
[0044] is the complex amplitude of the orbital angular momentum of the reference light.
[0045] The total light intensity is:
[0046] Wherein, , represents different OAM components in . The first term is the DC term, which can be used to normalize the spectral intensity. The second term is the cross-beat frequency term of different OAM components, containing all the information of the signal light to be measured.
[0047] When , the Fourier transform yields the spectral distribution of different beat frequency signals:
[0048]
[0049] is the frequency shift amount, and
[0050] is the intensity of the corresponding spectrum. The Gaussian light emitted by a 1550 nm laser is adjusted to horizontal polarization through a polarization device and then incident on the reference light modulator. At this time, the modulator is loaded with a specially designed phase diagram, which makes the reference light change from Gaussian light to vortex light carrying orbital angular momentum. , . For the convenience of explanation, is taken below. Then, the 1550 nm signal light to be measured received by the lens system is combined with the reference light in a polarization beam splitter and then enters a nonlinear crystal to generate sum frequency, producing a beam of 775 nm light. This converts the optical signal in the infrared band to the visible light band, reducing the detection cost. This newly generated light also carries orbital angular momentum components and is obtained by the OAM superposition of the signal light and the reference light, that is, . The single-mode fiber only allows the Gaussian light to pass through, realizing the filtering function, which is equivalent to selecting the component and obtaining . The photodetector is connected to the output end of the single-mode fiber to measure the total light intensity mixed with different beat frequency signals, . The detector converts the optical signal into an electrical signal and transmits it to the data processing module. This module performs Fourier analysis on the intensity data transmitted by the detection module to obtain the spectral distribution of different beat frequency signals, with a frequency shift of and the corresponding spectral intensity of . Combining with the known complex spectral distribution of the reference light orbital angular momentum , the OAM complex spectral distribution of the signal light to be measured can be easily obtained.
[0051] The detection of infrared - band signals uses sum - frequency detection technology, which satisfies energy and momentum conservation. The infrared signal light and the infrared reference light output a signal in the visible - light band through the sum - frequency of the nonlinear process.
[0052] Compared with the traditional method the proposed solution of the present invention can distinguish the previously indistinguishable degenerate cases and achieve more accurate measurement. As Figure 3 shown, the dark - blue broken line represents the result of directly measuring the signal light once by the traditional solution, and the light - blue broken line represents the measurement result of the proposed solution of the present invention. Assuming the OAM value of the signal light , it can be clearly seen that there is degeneracy in the spectral distribution of different beat - frequency signals in the traditional solution. For example have the same frequency shift amount and are degenerate. However, in this solution, the previously degenerate non - diagonal terms are distinguished, and only and other conjugate non - diagonal terms are degenerate. In this example, the orbital angular momentum order of the signal light is divided into 4 types. In principle, 7 equations need to be constructed according to the frequency shift to solve the complete relative amplitude and phase information and restore the OAM complex - spectrum distribution. Due to spectral degeneracy, the traditional solution has only 3 frequency - shift amounts, and only 4 equations can be constructed in combination with the normalization condition, which cannot solve this problem. While this solution has 6 frequency - shift amounts, and 7 equations can be constructed in combination with the normalization condition to solve the above - mentioned problem.
[0053] The serial numbers of the above - mentioned embodiments of this application are only for description and do not represent the advantages or disadvantages of the embodiments.
[0054] In the above - mentioned embodiments of this application, the descriptions of each embodiment have their own emphases. For the parts not detailed in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.
[0055] In several embodiments provided by this application, it should be understood that the disclosed technical content can be implemented in other ways. Among them, the device embodiments described above are only illustrative. For example, the division of the units can be a logical - function division. In actual implementation, there can be other division methods. For example, multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the displayed or discussed mutual coupling or direct coupling or communication connection can be through some interfaces. The indirect coupling or communication connection of units or modules can be in an electrical or other form.
[0056] The units described as separate components may or may not be physically separated. The components shown as units may or may not be physical units, that is, they can be located in one place or distributed to multiple units. Some or all of the units can be selected according to actual needs to achieve the purpose of the solution of this embodiment.
[0057] The above are only the preferred embodiments of the present application. It should be noted that for those of ordinary skill in the art of this technology, without departing from the principle of the present application, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present application.
[0058] It should be understood that the above description is for illustrative purposes and not for limitation. Many embodiments and many applications other than the examples provided will be apparent to those skilled in the art upon reading the above description. Therefore, the scope of this patent should not be determined by reference to the above description, but should be determined by reference to the full scope of the foregoing claims and the equivalents of those claims. For the sake of completeness, all articles and references, including patent applications and publications, are incorporated herein by reference. The omission of any aspect of the subject matter disclosed herein in the foregoing claims is not a waiver of that subject matter, nor should it be considered that the applicant has not considered that subject matter to be part of the disclosed inventive subject matter.
Claims
1. A non-linear optical orbital angular momentum spectrum analyzer, characterized in that, It includes a polarization beam splitter, a nonlinear crystal, a laser, a reference light modulator, a single-mode fiber, a photodetector, and a data processing module; The laser is used to emit a reference light that reaches the polarization beam splitter through the reference light modulator. The reference light modulator is used to apply a vortex phase and a rotational phase modulation to the reference light. The polarization beam splitter is used to receive the signal light to be measured and the modulated reference light. The nonlinear crystal is arranged on the output optical path of the polarization beam splitter. The single-mode fiber is located on the output optical path of the nonlinear crystal. The single-mode fiber is connected to the input end of the detector, and the output end of the photodetector is connected to the data processing module.
2. The non-linear optical orbital angular momentum spectrum analyzer according to claim 1, characterized in that The signal light to be measured is focused and collected by a lens and then transmitted to the polarization beam splitter.
3. The non-linear optical orbital angular momentum spectrum analyzer according to claim 1, characterized in that, The reference light modulator adopts a light field controller such as a spatial light modulator, a digital micromirror device, a deformable mirror, or a metasurface.
4. The non-linear optical orbital angular momentum spectrum analyzer according to claim 1, wherein The nonlinear crystal adopts barium metaborate, potassium dihydrogen phosphate, lithium niobate, or periodically poled potassium titanyl phosphate.
5. The non-linear optical orbital angular momentum spectrum analyzer according to claim 1, characterized in that The single-mode fiber only couples Gaussian light with an orbital angular momentum of zero.
6. A method for non-linear optical orbital angular momentum spectrum analysis based on the analyzer according to any one of claims 1-5, characterized in that, It includes the following processes: S1, The signal light to be measured with an orbital angular momentum spectrum is incident on the polarization beam splitter; S2, The Gaussian light emitted by the laser is loaded with a phase by the reference light modulator and becomes a vortex light carrying an orbital angular momentum, and enters the polarization beam splitter to be combined with the signal light to be measured; S3, The nonlinear crystal is used to sum-frequency the combined reference light and signal light to be measured to generate a beam of light with a new frequency; S4, After the light beam passes through the single-mode fiber for Gaussian filtering, the photodetector detects the intensity of the Gaussian light component and transmits the data to the data processing module; S5, The data processing module obtains the complex spectrum information of the orbital angular momentum of the signal light through data processing.
7. The non-linear optical orbital angular momentum spectrum analysis method according to claim 6, characterized in that The expression of the signal light to be measured is: is the complex amplitude distribution to be measured, is the angular frequency of the signal light, is the topological charge number, is the time, is the azimuth angle, i is the imaginary number; The expression of the reference light: is the known amplitude distribution, is the angular frequency, , is the fixed angular frequency of the reference light, is the rotational angular frequency; is the phase evolution velocity, is the topological charge number, is a function of, representing the relationship between the rotational angular frequency and the orbital angular momentum.
8. The non-linear optical orbital angular momentum spectrum analysis method according to claim 6, wherein The coupled-wave equation of the sum-frequency process is: is the angular frequency, is the wave number, is the speed of light in vacuum, is the effective nonlinear susceptibility, i is the imaginary unit, is the propagation distance, , under the phase-matching condition, , thus: is the complex amplitude distribution to be measured, is the angular frequency of the signal light, is the topological charge number, is the time, is the azimuth angle, is the known amplitude distribution, is the angular frequency, i is the imaginary number, m is the topological charge number.
9. The non-linear optical orbital angular momentum spectrum analysis method according to claim 6, characterized in that Selection of single-mode fiber filtering For the component of, the output electric field is: is the complex amplitude distribution to be measured, is the angular frequency of the signal light, is the complex amplitude of the orbital angular momentum of the reference light, is the topological charge number, is the time, is the azimuth angle, is the angular frequency, i is the imaginary number.
10. The nonlinear optical orbital angular momentum spectrum analysis method according to claim 6, characterized in that, The data processing module performs a Fourier transform on the intensity magnitude of the Gaussian light component obtained by the photodetector to obtain the spectral distribution of different beat-frequency signals. Based on the known amplitude values, relative phases of each component of the reference light, and the spectral distribution of the beat-frequency signals, the complex spectrum information of the orbital angular momentum of the signal light to be measured is obtained.