Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach
We studied theoretically and experimentally the propagation of structured Laguerre–Gaussian (sLG) beams through an optical system with general astigmatism based on symplectic ABCD transforms involving geometry of the second-order intensity moments symplectic matrices. The evolution of the coordinate...
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MDPI AG
2024-02-01
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| Online Access: | https://www.mdpi.com/2304-6732/11/3/191 |
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| author | Alexander Volyar Eugeny Abramochkin Mikhail Bretsko Yana Akimova |
| author_facet | Alexander Volyar Eugeny Abramochkin Mikhail Bretsko Yana Akimova |
| author_sort | Alexander Volyar |
| collection | DOAJ |
| description | We studied theoretically and experimentally the propagation of structured Laguerre–Gaussian (sLG) beams through an optical system with general astigmatism based on symplectic ABCD transforms involving geometry of the second-order intensity moments symplectic matrices. The evolution of the coordinate submatrix ellipses accompanying the transformation of intensity patterns at different orientations of the cylindrical lens was studied. It was found that the coordinate submatrix <b>W</b> and the twistedness submatrix <b>M</b> of the symplectic matrix <b>P</b> degenerate in the astigmatic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sLG</mi></semantics></math></inline-formula> beam with simple astigmatism, which sharply reduces the number of degrees of freedom, while general astigmatism removes the degeneracy. Nevertheless, degeneracy entails a simple relationship between the coordinate element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></semantics></math></inline-formula> and the twistedness elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></semantics></math></inline-formula> of the submatrix <b>M</b>, which greatly simplifies the measurement of the total orbital angular momentum (OAM), reducing the full cycle of measurements of the Hermite–Gaussian (HG) mode spectrum (amplitudes and phases) of the structured beam to the only measurement of the intensity moment. Moreover, we have shown that Fourier transform by a spherical lens enables us to suppress the astigmatic OAM component and restore the original free-astigmatic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sLG</mi></semantics></math></inline-formula> beam structure. However, with further propagation, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sLG</mi></semantics></math></inline-formula> beam restores its astigmatic structure while maintaining the maximum OAM. |
| first_indexed | 2024-04-24T17:54:46Z |
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| spelling | doaj.art-24bf7b42329c4fb9b9470c8665ae07bf2024-03-27T13:59:59ZengMDPI AGPhotonics2304-67322024-02-0111319110.3390/photonics11030191Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix ApproachAlexander Volyar0Eugeny Abramochkin1Mikhail Bretsko2Yana Akimova3Physics and Technology Institute, V.I. Vernadsky Crimean Federal University, Vernadsky Prospect 4, Simferopol 295007, RussiaLebedev Physical Institute, Samara 443011, RussiaPhysics and Technology Institute, V.I. Vernadsky Crimean Federal University, Vernadsky Prospect 4, Simferopol 295007, RussiaPhysics and Technology Institute, V.I. Vernadsky Crimean Federal University, Vernadsky Prospect 4, Simferopol 295007, RussiaWe studied theoretically and experimentally the propagation of structured Laguerre–Gaussian (sLG) beams through an optical system with general astigmatism based on symplectic ABCD transforms involving geometry of the second-order intensity moments symplectic matrices. The evolution of the coordinate submatrix ellipses accompanying the transformation of intensity patterns at different orientations of the cylindrical lens was studied. It was found that the coordinate submatrix <b>W</b> and the twistedness submatrix <b>M</b> of the symplectic matrix <b>P</b> degenerate in the astigmatic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sLG</mi></semantics></math></inline-formula> beam with simple astigmatism, which sharply reduces the number of degrees of freedom, while general astigmatism removes the degeneracy. Nevertheless, degeneracy entails a simple relationship between the coordinate element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></semantics></math></inline-formula> and the twistedness elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></semantics></math></inline-formula> of the submatrix <b>M</b>, which greatly simplifies the measurement of the total orbital angular momentum (OAM), reducing the full cycle of measurements of the Hermite–Gaussian (HG) mode spectrum (amplitudes and phases) of the structured beam to the only measurement of the intensity moment. Moreover, we have shown that Fourier transform by a spherical lens enables us to suppress the astigmatic OAM component and restore the original free-astigmatic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sLG</mi></semantics></math></inline-formula> beam structure. However, with further propagation, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sLG</mi></semantics></math></inline-formula> beam restores its astigmatic structure while maintaining the maximum OAM.https://www.mdpi.com/2304-6732/11/3/191structured lightHermite–Gaussian modesymplectic matrixorbital angular momentum |
| spellingShingle | Alexander Volyar Eugeny Abramochkin Mikhail Bretsko Yana Akimova Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach Photonics structured light Hermite–Gaussian mode symplectic matrix orbital angular momentum |
| title | Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach |
| title_full | Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach |
| title_fullStr | Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach |
| title_full_unstemmed | Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach |
| title_short | Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach |
| title_sort | engineering orbital angular momentum in structured beams in general astigmatic systems via symplectic matrix approach |
| topic | structured light Hermite–Gaussian mode symplectic matrix orbital angular momentum |
| url | https://www.mdpi.com/2304-6732/11/3/191 |
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