Physical Significance of the Determinant of a Mueller Matrix
The determinant of a Mueller matrix <b>M</b> plays an important role in both polarization algebra and the interpretation of polarimetric measurements. While certain physical quantities encoded in <b>M</b> admit a direct interpretation, the understanding of the physical and ge...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Photonics |
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| Online Access: | https://www.mdpi.com/2304-6732/9/4/246 |
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| author | José J. Gil Razvigor Ossikovski Ignacio San José |
| author_facet | José J. Gil Razvigor Ossikovski Ignacio San José |
| author_sort | José J. Gil |
| collection | DOAJ |
| description | The determinant of a Mueller matrix <b>M</b> plays an important role in both polarization algebra and the interpretation of polarimetric measurements. While certain physical quantities encoded in <b>M</b> admit a direct interpretation, the understanding of the physical and geometric significance of the determinant of <b>M</b> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula>) requires a specific analysis, performed in this work by using the normal form of <b>M</b>, as well as the indices of polarimetric purity (IPP) of the canonical depolarizer associated with <b>M</b>. We derive an expression for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula> in terms of the diattenuation, polarizance and a parameter proportional to the volume of the intrinsic ellipsoid of <b>M</b>. We likewise establish a relation existing between the determinant of <b>M</b> and the rank of the covariance matrix <b>H</b> associated with <b>M</b>, and determine the lower and upper bounds of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula> for the two types of Mueller matrices by taking advantage of their geometric representation in the IPP space. |
| first_indexed | 2024-03-09T04:18:02Z |
| format | Article |
| id | doaj.art-769c08b9e303438597eeb0542d0f0d92 |
| institution | Directory Open Access Journal |
| issn | 2304-6732 |
| language | English |
| last_indexed | 2024-03-09T04:18:02Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Photonics |
| spelling | doaj.art-769c08b9e303438597eeb0542d0f0d922023-12-03T13:51:34ZengMDPI AGPhotonics2304-67322022-04-019424610.3390/photonics9040246Physical Significance of the Determinant of a Mueller MatrixJosé J. Gil0Razvigor Ossikovski1Ignacio San José2Departmento de Física Aplicada, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, SpainLPICM, CNRS, Ecole Polytechnique, Université Paris-Saclay, 91128 Palaiseau, FranceInstituto Aragonés de Estadística, Gobierno de Aragón, Bernardino Ramazzini 5, 50015 Zaragoza, SpainThe determinant of a Mueller matrix <b>M</b> plays an important role in both polarization algebra and the interpretation of polarimetric measurements. While certain physical quantities encoded in <b>M</b> admit a direct interpretation, the understanding of the physical and geometric significance of the determinant of <b>M</b> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula>) requires a specific analysis, performed in this work by using the normal form of <b>M</b>, as well as the indices of polarimetric purity (IPP) of the canonical depolarizer associated with <b>M</b>. We derive an expression for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula> in terms of the diattenuation, polarizance and a parameter proportional to the volume of the intrinsic ellipsoid of <b>M</b>. We likewise establish a relation existing between the determinant of <b>M</b> and the rank of the covariance matrix <b>H</b> associated with <b>M</b>, and determine the lower and upper bounds of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula> for the two types of Mueller matrices by taking advantage of their geometric representation in the IPP space.https://www.mdpi.com/2304-6732/9/4/246Mueller matrixpolarization opticspolarimetrydepolarization |
| spellingShingle | José J. Gil Razvigor Ossikovski Ignacio San José Physical Significance of the Determinant of a Mueller Matrix Photonics Mueller matrix polarization optics polarimetry depolarization |
| title | Physical Significance of the Determinant of a Mueller Matrix |
| title_full | Physical Significance of the Determinant of a Mueller Matrix |
| title_fullStr | Physical Significance of the Determinant of a Mueller Matrix |
| title_full_unstemmed | Physical Significance of the Determinant of a Mueller Matrix |
| title_short | Physical Significance of the Determinant of a Mueller Matrix |
| title_sort | physical significance of the determinant of a mueller matrix |
| topic | Mueller matrix polarization optics polarimetry depolarization |
| url | https://www.mdpi.com/2304-6732/9/4/246 |
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