## Optical phase space, Wigner representation, and invariant quality parameters

JOSA A, Vol. 17, Issue 12, pp. 2440-2463 (2000)

http://dx.doi.org/10.1364/JOSAA.17.002440

Acrobat PDF (355 KB)

### Abstract

Wigner’s quasi probability and related functional and operator methods of quantum mechanics have recently played an important role in optics. We present an account of some of these developments. The symmetry structures underlying the ray and wave approaches to paraxial optics are explored in some detail, and the manner in which the Wigner phase-space representation captures the merits of both approaches is brought out. A fairly self-contained analysis of the second or intensity moments of general astigmatic partially coherent beams and of their behavior under transmission through astigmatic first-order optical systems is presented. Geometric representations of the intensity moments that render the quality parameters or polynomial invariants manifest are discussed, and the role of the optical uncertainty principle in assigning unbeatable physical bounds for these invariants is stressed. Measurement of the ten intensity moments of an astigmatic partially coherent beam is considered.

© 2000 Optical Society of America

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.6600) Coherence and statistical optics : Statistical optics

(080.2720) Geometric optics : Mathematical methods (general)

(260.0260) Physical optics : Physical optics

(350.5500) Other areas of optics : Propagation

**Citation**

R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters," J. Opt. Soc. Am. A **17**, 2440-2463 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-12-2440

Sort: Year | Journal | Reset

### References

- E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
- N. L. Balazs and B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984); M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984); V. I. Tatarskii, “The Wigner representation of quantum optics,” Usp. Fiz. Nauk. 139, 587–619 (1983) [ Sov. Phys. Usp. SOPUAP 26, 311–327 (1983)].
- R. J. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
- A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
- R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), Chap. 2.
- S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1.
- A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary state of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
- E. P. Wigner, “Quantum mechanical distribution functions revisited,” in Perspectives in Quantum Theory, W. Yourgrau and A. van der Merwe, eds. (MIT Press, Cambridge, Mass., 1971), pp. 25–36.
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 5.
- E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
- E. G. C. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
- A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
- K. E. Cahill and R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969); “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
- G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970); “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: II. Quantum mechanics in phase space,” Phys. Rev. D 2, 2187–2205 (1970).
- A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
- G. S. Agarwal, J. T. Foley, and E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
- A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
- W. Schempp, “Analog radar signal design and digital signal processing—a Heisenberg nilpotent Lie group approach,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon and K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 1–27.
- M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
- M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, and A. Gonzalez-Urena, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 65–87.
- R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
- R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
- R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987); R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
- R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
- R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
- R. Simon, N. Mukunda, and B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
- R. Martinez-Herrero, P. M. Mejias, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993); A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed. Proc. SPIE 1224, 2–14 (1990).
- J. Serna, R. Martinez-Herrero, and P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
- M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).
- J. Serna, P. M. Mejias, and R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992); J. Yang and D. Fan, “Intensity-moments characterization of general pulsed paraxial beams with the Wigner distribution function,” J. Opt. Soc. Am. A 16, 2488–2493 (1999).
- F. J. Narcowich, “Geometry and uncertainty,” J. Math. Phys. 31, 354–364 (1990); Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimensional domain,” Opt. Lett. 18, 669–671 (1993).
- H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992); D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993); S. Lavi, R. Prochaska, and E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. APOPAI 27, 3696–3703 (1988).
- G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, and A. Gonzalez-Urena, eds. (Sociedad Espanola de Optica, Madrid, Spain, 1993), pp. 325–358.
- G. Nemes and A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
- S. Ramee and R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–94 (2000).
- B. Eppich, A. T. Friberg, C. Gao, and H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin and A. Giesen, eds., Proc. SPIE 2870, 260–267 (1996).
- J. Serna and G. Nemes, “Decoupling of coherent Gaussian beam with general astigmatism,” Opt. Lett. 18, 1774–1776 (1993).
- G. Nemes, “Synthesis of general astigmatic optical systems, the detwisting procedure, and the beam quality factors for general astigmatic laser beams,” in Laser Beam Characterization, H. Weber, N. Reng, J. Ludtke, and P. M. Mejias, eds. (Festkorper-Laser-Institut Berlin GmbH, Berlin, 1994), pp. 93–104.
- R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
- A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
- D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
- R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
- R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993); N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. (N.Y.) 228, 205–268 (1993) (an extensive set of references on geometric phase can be found here) ; “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. (N.Y.) APNYA6 228, 269–340 (1993).
- S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978), pp. 257–264, 401–407.
- H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
- R. Simon and N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
- G. S. Agarwal and R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
- R. Simon and N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
- R. Simon and N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998); “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16, 2465–2475 (1999).
- N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983); N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and application,” J. Opt. Soc. Am. A 2, 416–426 (1985).
- D. Gloge and D. Marcuse, “Formal quantum theory of light rays,” J. Opt. Soc. Am. 59, 1629–1631 (1969); D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 3.
- R. K. Luneburg, Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1964).
- V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984), Chap. 1.
- S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
- M. Kauderer, Symplectic Matrices (World Scientific, Singapore, 1994).
- E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
- D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993); H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
- A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993); S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
- G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
- See, for instance, O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288–294.
- R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
- M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 3.
- O. Castanos, E. Lopez-Moreno, and K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon and K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 159–182.
- T. Pradhan, “Maxwell’s equations from geometrical optics,” Phys. Lett. A 122, 397–398 (1987).
- M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971); “Canonical transformations and matrix elements,” 12, 1780–1783 (1971).
- K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chaps. 7 and 9.
- E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
- F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
- A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
- R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
- K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
- R. L. Hudson, “When is the Wigner quasi-probability nonnegative,” Rep. Math. Phys. 6, 249–252 (1974).
- A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 19.
- R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
- K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
- A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
- P. A. Belanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
- A. J. Dragt, F. Neri, and G. Rangarajan, “General moment invariants for linear Hamiltonian systems,” Phys. Rev. A 45, 2572–2585 (1992).
- V. V. Dodonov and V. I. Man’ko, Invariants and the Evolution of Nonstationary Quantum Systems, in Proceedings of the Lebedev Physics Institute, M. A. Markov, ed. (Nova Science, Commack, N.Y., 1989), Vol. 183; D. D. Holm, W. P. Lysenko, and J. C. Scovel, “Moment invariants for the Vlasov equation,” J. Math. Phys. 31, 1610–1615 (1990).
- J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
- N. Burgoyne and R. Cushman, “Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues,” Celest. Mech. 8, 435–443 (1974); A. J. Laub and K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 231–238 (1974).
- R. Simon, S. Chaturvedi, and V. Srinivasan, “Congruences and canonical forms for a positive matrix: application to Schweinler–Wigner extremum principle,” J. Math. Phys. 40, 3632–3640 (1999).
- A. T. Friberg, C. Gao, B. Eppich, and H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov and S. R. Rotman, eds., Proc. SPIE 3110, 317–328 (1997).
- R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
- F. Gori, M. Santarsiero, and V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
- B. Eppich, C. Gao, and H. Weber, “Determination of the ten second order moments,” Opt. Laser Technol. 30, 337–340 (1998).
- J. Serna, F. Encinas-Sanz, and G. Nemes, “Characterization of a doughnut beam using a cylindrical lens,” presented at the 5th Workshop on Laser Beam and Optics Characterization, Erice, Sicily, Italy, March 20–25, 2000.
- R. Simon and N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science VI: from the Rotation Group to Quantum Algebras, B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689.
- R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1988); R. Simon, N. Mukunda, and E. C. G. Sudarshan, “The theory of screws—a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
- J. Shamir and N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
- R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
- H. Weber, “Propagation of the higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992); D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
- H. Bacry and M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
- M. Nazarathy and J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
- R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian Maxwell beams,” J. Opt. Soc. Am. A 3, 356–360 (1986); “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.