MMI Couplers and the Talbot Effect, Symmetries and Golden Ratio
The Talbot effect concerns the periodic self-imaging along an optical axis of a free-space optical field that is periodic in an initial transverse plane. It may be modeled by a shift-invariant linear system, fully characterized by the convolution of its impulse response. Self-imaging at integer and...
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2025-03-01
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| author | Gazi Mahamud Hasan Mehedi Hasan Karin Hinzer Trevor Hall |
| author_facet | Gazi Mahamud Hasan Mehedi Hasan Karin Hinzer Trevor Hall |
| author_sort | Gazi Mahamud Hasan |
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| description | The Talbot effect concerns the periodic self-imaging along an optical axis of a free-space optical field that is periodic in an initial transverse plane. It may be modeled by a shift-invariant linear system, fully characterized by the convolution of its impulse response. Self-imaging at integer and fractional Talbot distances of point sources on a regular grid in free space may then be represented by a transmission matrix that is circulant, symmetric, and persymmetric. The free-space Talbot effect may be mapped to the Talbot effect in a multimode waveguide by imposing the anti-symmetry of the mirror-like sidewalls created by the tight confinement of light within a high-index contrast multimode waveguide. The position of the anti-symmetry axis controls the distribution of discrete lattice points in a unit cell. For different distributions, interesting features such as conditional flexibility in the placement of access ports without altering amplitude and phase relationships, omitting ports without power penalty, closed form uneven splitting ratios, and offset access ports can be derived from the MMI coupler. As a specific example, a simple <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> MMI coupler is shown to provide a power-splitting ratio related to the golden ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi></mrow></semantics></math></inline-formula>. The structure is amenable to planar photonic integration on any high-index contrast platform. The predictions of the theory are confirmed by simulation and verified by experimental measurements on a golden ratio MMI coupler fabricated using an SOI process. |
| format | Article |
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| publishDate | 2025-03-01 |
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| spelling | doaj-art-dc0605979cd9482da293d99f1592a60b2025-08-20T02:10:24ZengMDPI AGPhotonics2304-67322025-03-0112322910.3390/photonics12030229MMI Couplers and the Talbot Effect, Symmetries and Golden RatioGazi Mahamud Hasan0Mehedi Hasan1Karin Hinzer2Trevor Hall3Photonic Technology Laboratory, University of Ottawa, Ottawa, ON K1N 6N5, CanadaPhotonic Technology Laboratory, University of Ottawa, Ottawa, ON K1N 6N5, CanadaSUNLAB, University of Ottawa, Ottawa, ON K1N 6N5, CanadaPhotonic Technology Laboratory, University of Ottawa, Ottawa, ON K1N 6N5, CanadaThe Talbot effect concerns the periodic self-imaging along an optical axis of a free-space optical field that is periodic in an initial transverse plane. It may be modeled by a shift-invariant linear system, fully characterized by the convolution of its impulse response. Self-imaging at integer and fractional Talbot distances of point sources on a regular grid in free space may then be represented by a transmission matrix that is circulant, symmetric, and persymmetric. The free-space Talbot effect may be mapped to the Talbot effect in a multimode waveguide by imposing the anti-symmetry of the mirror-like sidewalls created by the tight confinement of light within a high-index contrast multimode waveguide. The position of the anti-symmetry axis controls the distribution of discrete lattice points in a unit cell. For different distributions, interesting features such as conditional flexibility in the placement of access ports without altering amplitude and phase relationships, omitting ports without power penalty, closed form uneven splitting ratios, and offset access ports can be derived from the MMI coupler. As a specific example, a simple <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> MMI coupler is shown to provide a power-splitting ratio related to the golden ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi></mrow></semantics></math></inline-formula>. The structure is amenable to planar photonic integration on any high-index contrast platform. The predictions of the theory are confirmed by simulation and verified by experimental measurements on a golden ratio MMI coupler fabricated using an SOI process.https://www.mdpi.com/2304-6732/12/3/229talbot effectmultimode interferenceMMI couplergolden ratio |
| spellingShingle | Gazi Mahamud Hasan Mehedi Hasan Karin Hinzer Trevor Hall MMI Couplers and the Talbot Effect, Symmetries and Golden Ratio Photonics talbot effect multimode interference MMI coupler golden ratio |
| title | MMI Couplers and the Talbot Effect, Symmetries and Golden Ratio |
| title_full | MMI Couplers and the Talbot Effect, Symmetries and Golden Ratio |
| title_fullStr | MMI Couplers and the Talbot Effect, Symmetries and Golden Ratio |
| title_full_unstemmed | MMI Couplers and the Talbot Effect, Symmetries and Golden Ratio |
| title_short | MMI Couplers and the Talbot Effect, Symmetries and Golden Ratio |
| title_sort | mmi couplers and the talbot effect symmetries and golden ratio |
| topic | talbot effect multimode interference MMI coupler golden ratio |
| url | https://www.mdpi.com/2304-6732/12/3/229 |
| work_keys_str_mv | AT gazimahamudhasan mmicouplersandthetalboteffectsymmetriesandgoldenratio AT mehedihasan mmicouplersandthetalboteffectsymmetriesandgoldenratio AT karinhinzer mmicouplersandthetalboteffectsymmetriesandgoldenratio AT trevorhall mmicouplersandthetalboteffectsymmetriesandgoldenratio |