Wigner functions defined with Laplace transform kernels |
Optics Express, Vol. 19, Issue 22, pp. 21938-21944 (2011)
http://dx.doi.org/10.1364/OE.19.021938
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Abstract
We propose a new Wigner–type phase–space function using Laplace transform kernels—Laplace kernel Wigner function. Whereas momentum variables are real in the traditional Wigner function, the Laplace kernel Wigner function may have complex momentum variables. Due to the property of the Laplace transform, a broader range of signals can be represented in complex phase–space. We show that the Laplace kernel Wigner function exhibits similar properties in the marginals as the traditional Wigner function. As an example, we use the Laplace kernel Wigner function to analyze evanescent waves supported by surface plasmon polariton.
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1. Introduction
1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]
2. M. J. Bastiaans, “Wigner distribution function and its application to 1st-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]
3. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent-light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986). [CrossRef]
4. J. Paye and A. Migus, “Space–time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995). [CrossRef]
6. L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989). [CrossRef]
7. G. Boudreaux-Bartels, “Time-varying signal processing using Wigner distribution synthesis techniques,” in The Wigner Distribution–Theory and Applications in Signal Processing, W. Mecklenbruker and F. Hlawatsch, eds., (Elsevier Science, 1997), pp. 269–317. [PubMed]
2. Laplace kernel Wigner Function
3. Example: LWF of evanescent waves in SPP
4. Cross–terms in LWF
5. Conclusion
Acknowledgments
References and links
1. | E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef] |
2. | M. J. Bastiaans, “Wigner distribution function and its application to 1st-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef] |
3. | M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent-light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986). [CrossRef] |
4. | J. Paye and A. Migus, “Space–time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995). [CrossRef] |
5. | B. Boashash, Time Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier Science Ltd, 2003). |
6. | L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989). [CrossRef] |
7. | G. Boudreaux-Bartels, “Time-varying signal processing using Wigner distribution synthesis techniques,” in The Wigner Distribution–Theory and Applications in Signal Processing, W. Mecklenbruker and F. Hlawatsch, eds., (Elsevier Science, 1997), pp. 269–317. [PubMed] |
8. | M. J. Bastiaans, “Wigner distribution in optics,” in Phase-Space Optics: Fundamenals and Applications, M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, eds. (McGraw-Hill, 2009), pp. 1–44. |
9. | A. Oppenheim, A. Willsky, and S. Hamid, Signals and Systems, 2nd ed. (Prentice Hall, 1997). |
10. | G. Arfken and H. Weber, Mathematical Methods for Physicists (Academic Press, 1995). |
11. | S. Maier, Plasmonics: Fundamentals and Applications (Springer Verlag, 2007). |
OCIS Codes
(350.6980) Other areas of optics : Transforms
(050.5082) Diffraction and gratings : Phase space in wave options
(070.7425) Fourier optics and signal processing : Quasi-probability distribution functions
ToC Category:
Fourier Optics and Signal Processing
History
Original Manuscript: July 28, 2011
Revised Manuscript: September 17, 2011
Manuscript Accepted: September 30, 2011
Published: October 21, 2011
Citation
Se Baek Oh, Jonathan C. Petruccelli, Lei Tian, and George Barbastathis, "Wigner functions defined with Laplace transform kernels," Opt. Express 19, 21938-21944 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21938
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References
- E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev.40, 749–759 (1932). [CrossRef]
- M. J. Bastiaans, “Wigner distribution function and its application to 1st-order optics,” J. Opt. Soc. Am.69, 1710–1716 (1979). [CrossRef]
- M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent-light,” J. Opt. Soc. Am. A3, 1227–1238 (1986). [CrossRef]
- J. Paye and A. Migus, “Space–time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B12, 1480–1490 (1995). [CrossRef]
- B. Boashash, Time Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier Science Ltd, 2003).
- L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE77, 941–981 (1989). [CrossRef]
- G. Boudreaux-Bartels, “Time-varying signal processing using Wigner distribution synthesis techniques,” in The Wigner Distribution–Theory and Applications in Signal Processing, W. Mecklenbruker and F. Hlawatsch, eds., (Elsevier Science, 1997), pp. 269–317. [PubMed]
- M. J. Bastiaans, “Wigner distribution in optics,” in Phase-Space Optics: Fundamenals and Applications, M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, eds. (McGraw-Hill, 2009), pp. 1–44.
- A. Oppenheim, A. Willsky, and S. Hamid, Signals and Systems, 2nd ed. (Prentice Hall, 1997).
- G. Arfken and H. Weber, Mathematical Methods for Physicists (Academic Press, 1995).
- S. Maier, Plasmonics: Fundamentals and Applications (Springer Verlag, 2007).
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