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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 12 — Dec. 1, 2000
  • pp: 2288–2300

Phase-space distributions for high-frequency fields

M. A. Alonso and G. W. Forbes  »View Author Affiliations

JOSA A, Vol. 17, Issue 12, pp. 2288-2300 (2000)

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The Wigner distribution function and various windowed Fourier transforms are examples of phase-space distributions that are used, among other things, to formalize the link between ray and wave optics. It is well known that, in the limit of high frequencies, these distributions become localized for simple wave fields and therefore that the localization can be used to define the associated ray families. This localized form is characterized here for both the Wigner distribution function and a Gaussian windowed Fourier transform. Aside from the greater understanding of the distributions themselves, these results promise a clearer intuition of phase-space-based methods for optical modeling. In particular, regardless of the context, the geometric construction that is presented for estimating the Wigner distribution function gives a valuable appreciation of its highly structured and sometimes surprising form.

© 2000 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(070.2590) Fourier optics and signal processing : ABCD transforms
(350.6980) Other areas of optics : Transforms
(350.7420) Other areas of optics : Waves

Original Manuscript: February 3, 2000
Revised Manuscript: July 6, 2000
Manuscript Accepted: May 19, 2000
Published: December 1, 2000

M. A. Alonso and G. W. Forbes, "Phase-space distributions for high-frequency fields," J. Opt. Soc. Am. A 17, 2288-2300 (2000)

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.
  2. J. F. Price, “Uncertainty principles and sampling theorems,” in Fourier Techniques and Applications, J. F. Price, ed. (Plenum, New York, 1985), pp. 25–44.
  3. N. G. De Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.
  4. M. V. Berry, “Semi-classical mechanics in phase space: a study of Wigner’s function,” Philos. Trans. R. Soc. London 287, 237–271 (1977). [CrossRef]
  5. M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979). [CrossRef]
  6. M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219–231 (1989). [CrossRef]
  7. This function is a solution of the Hamilton–Jacobi (or eikonal) equation. See, for example, Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), pp. 3–9.
  8. E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]
  9. The WDF for an optical field was first proposed in A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968). [CrossRef]
  10. The analogous result to Eq. (2.5) for momentum representation also has this property: f˜(p)=1f˜*(p0)∫Wx,p+p02×exp[-ikx(p-p0)]dx.
  11. L. Cohen, Time-Frequency Analysis (Prentice-Hall PTR, Englewood Cliffs, N.J., 1995), pp. 93–112.
  12. This distribution is sometimes referred to as the coherent state representation. For an excellent overview and collection of key references, see J. R. Klauder, B. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).
  13. G. Torres-Vega, J. H. Frederick, “Quantum mechanics in phase space: new approaches to the correspondence principle,” J. Chem. Phys. 93, 8862–8874 (1990). [CrossRef]
  14. G. Torres-Vega, J. H. Frederick, “A quantum mechanical representation in phase space,” J. Chem. Phys. 98, 3103–3120 (1993). [CrossRef]
  15. J. E. Harriman, “A quantum state vector phase space representation,” J. Chem. Phys. 100, 3651–3661 (1994). [CrossRef]
  16. K. B. Möller, T. G. Jörgensen, G. Torres-Vega, “On coherent-state representations of quantum mechanics: wave mechanics in phase space,” J. Chem. Phys. 106, 7228–7240 (1997). [CrossRef]
  17. A phase-space representation that is closely related to the GWFT was proposed in V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform,” Commun. Pure Appl. Math. 14, 187–214 (1961). The connection between this transformation and the GWFT is discussed in Ref. 15. [CrossRef]
  18. In quantum optics and in the study of the quantum harmonic oscillator, there is a natural scaling factor between the position and the momentum axes. A Gaussian wave function that preserves its width is known as a coherent state. If, on the other hand, the width of the Gaussian changes in time, “breathing” in a periodic fashion, the wave function corresponds to a squeezed state. In the particular case when the WDF of a coherent state is used in the blurring of W(x, p), the resulting Husimi function is known as the Q function. See, for example, H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995). [CrossRef]
  19. See Ref. 11, Chap. 7, pp. 93–100.
  20. N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986) pp. 252–320.
  21. C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957). [CrossRef]
  22. This result follows from ∫ exp[i(xt+at3/3)]dt=2π|a|-1/3Ai[xa-1/3].   In the argument to the Airy function, the third power is always chosen to be the real root. For a discussion of Airy functions, see, e.g., M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Sec. 10.4.
  23. If we use only the real phase space, the geometric prescription gives us only the real-valued saddle points. However, through the expressions involving Airy functions, these are enough to account for those that have small imaginary parts; those with large imaginary parts turn out to be insignificant.
  24. Note that, as x1 approaches x2, the relation   p¯=ϕ′(x¯)+δ=12{ϕ′[x¯+(x2-x1)/2]+ϕ′[x¯-(x2-x1)/2]} leads to (x2-x1)2≈8δ/ϕ′′′(x¯). With this, the connection becomes straightforward since the argument of the cosine in Eq. (3.12) can be written as   ϕ[x¯+(x2-x1)/2]-ϕ[x¯-(x2-x1)/2]-(x2-x1)p¯≈-23σ[8δ3/ϕ′′′(x¯)]1/2.
  25. In fact, Eq. (3.12) can easily be written in terms of an Airy function, such that the matching with Eq. (3.9) is explicit. See, for example, M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog. (Oxford) 57, 43–64 (1969). However, the form used in Eq. (3.12) is more convenient for the purposes of this paper.
  26. See, for example, the reference in Note 7, pp. 22–26.
  27. Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993).
  28. In fact, Airy defined the function that bears his name within this context. See G. B. Airy, “On the intensity of light in a neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–403 (1838).
  29. The evaluation at (x, p)=(x¯+η, p¯+δ) of the exponent of the integrand in Eq. (3.1) gives ϕ(x+12x′)-ϕ(x-12x′)-px′≈Φ+[ϕ′(x2)-ϕ′(x1)]δ-(x2-x1)η-(δ-ϕ″η)s+124ϕ″′¯s3+O(s5),   where ϕ″=ϕ″(x1)=ϕ″(x2). The contribution to Wis then W(x¯+η, p¯+δ)≈8k2ϕ′′′¯1/3a(x1)a(x2)Ai-8k2ϕ′′′¯1/3sgn[ϕ′′′¯](δ-ϕ″η)×2 cos(k{Φ+[ϕ′(x2)-ϕ′(x1)]η-(x2-x1)δ}).   [Notice that these two expressions reduce to Eqs. (3.13) and (3.14) when η=0.] The argument of the cosine factor stays constant at all points that fall along a line parallel to the secant from x1 to x2.
  30. The Pearcey function is defined as Ip(x, y)=∫ exp[i(xt+yt2+t4)]dt.See, for example, Ref. 27, pp. 60–61.
  31. To our knowledge, the earliest reference to the FrFT is E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937). [CrossRef] [PubMed]
  32. A general overview on this subject is given by H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to thefractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.
  33. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]
  34. D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B 38, 209–219 (1996). [CrossRef]
  35. M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999). [CrossRef]
  36. G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A (to be published).
  37. M. A. Alonso, G. W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” submitted to J. Opt. Soc. Am. A (to be published).
  38. M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979). [CrossRef]
  39. H. J. Korsch, M. V. Berry, “Evolution of Wigner’s phase space density under a nonintegrable quantum map,” Physica D 3D, 627–636 (1981). [CrossRef]
  40. A. Voros, “Semiclassical approximations,” Ann. Inst. Henri Poincare 14, 31–90 (1976).
  41. When the phase of the wave field is unimportant, the projection property in Eq. (2.3a) can be used instead of the problematic inversion formula in Eq. (2.5). In this case, a useful property regarding the projection of Wigner functions containing Airy forms is given in Ref. 4 and is generalized for higher-order catastrophes inM. V. Berry, F. J. Wright, “Phase space projection identities for diffraction catastrophes,” J. Phys. A 13, 149–160 (1980). [CrossRef]
  42. Fields of this form, corresponding to energy eigenstates of quantum-mechanical bound systems, were considered by Berry in Ref. 4. There he recognized what is referred to here as the ghost curves. A generalization of Berry’s work for the case of two-dimensional bound states is given inA. M. Ozorio de Almeida, J. H. Hannay, “Geometry of two dimensional tori in phase space: projections, sections and the Wigner function,” Ann. Phys. (N.Y.) 138, 115–154 (1982). [CrossRef]
  43. One such method, based on the Bargmann representation defined in Ref. 17, is given in A. Voros, “Wentzel–Kramers–Brillouin method in the Bargmann representation,” Phys. Rev. A 40, 6814–6825 (1989). [CrossRef] [PubMed]

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