On the Twistability of Partially Coherent, Schell-Model Sources
In this paper, the problem of assessing the twistability of a given bona fide cross-spectral density is tackled for the class of Schell-model sources, whose shift-invariant degree of coherence is represented by a real and symmetric function, denoted as <inline-formula><math xmlns="http...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Photonics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2304-6732/12/1/42 |
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| Summary: | In this paper, the problem of assessing the twistability of a given bona fide cross-spectral density is tackled for the class of Schell-model sources, whose shift-invariant degree of coherence is represented by a real and symmetric function, denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>(</mo><mo>−</mo><mi mathvariant="bold-italic">r</mi><mo>)</mo><mo>=</mo><mi>μ</mi><mo>(</mo><mi mathvariant="bold-italic">r</mi><mo>)</mo></mrow></semantics></math></inline-formula>. By employing an abstract operatorial language, the problem of determining the highly degenerate spectrum of a twisted operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>W</mi><mo stretchy="false">^</mo></mover><mi>u</mi></msub></semantics></math></inline-formula> is addressed through a modal analysis based on the complete knowledge of the spectrum of the <i>sole</i> twist operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>T</mi><mo stretchy="false">^</mo></mover><mi>u</mi></msub></semantics></math></inline-formula>, as found by R. Simon and N. Mukunda. To this end, the evaluation of the complete tensor of the matrix elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⟨</mo><msup><mi>n</mi><mo>′</mo></msup><mo>,</mo><msup><mi>ℓ</mi><mo>′</mo></msup><mo>|</mo><msub><mover accent="true"><mi>W</mi><mo stretchy="false">^</mo></mover><mi>u</mi></msub><mo>|</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>⟩</mo></mrow></semantics></math></inline-formula> is carried out within the framework of the so-called <i>extended Wigner distribution function</i>, a concept recently introduced by M. VanValkenburgh. As a nontrivial application of the algorithm developed here, the analytical determination of the spectrum of saturated twisted astigmatic Gaussian Schell-model sources is also presented. |
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| ISSN: | 2304-6732 |