Abstract

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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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]

1999

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
[CrossRef]

1997

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]

1996

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B 38, 209–219 (1996).
[CrossRef]

1995

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]

1994

J. E. Harriman, “A quantum state vector phase space representation,” J. Chem. Phys. 100, 3651–3661 (1994).
[CrossRef]

1993

G. Torres-Vega, J. H. Frederick, “A quantum mechanical representation in phase space,” J. Chem. Phys. 98, 3103–3120 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

1990

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]

1989

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219–231 (1989).
[CrossRef]

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]

1982

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]

1981

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]

1980

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]

1979

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).
[CrossRef]

M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979).
[CrossRef]

1977

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]

1976

A. Voros, “Semiclassical approximations,” Ann. Inst. Henri Poincare 14, 31–90 (1976).

1969

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.

1968

1961

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]

1957

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

1937

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]

1932

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

1838

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).

Abramowitz, M.

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.

Airy, G. B.

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).

Alonso, M. A.

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
[CrossRef]

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).

G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A (to be published).

Balasz, N. L.

M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979).
[CrossRef]

Balazs, N. L.

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).
[CrossRef]

Bargmann, V.

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]

Berry, M. V.

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219–231 (1989).
[CrossRef]

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]

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]

M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979).
[CrossRef]

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).
[CrossRef]

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]

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.

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986) pp. 252–320.

Chester, C.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice-Hall PTR, Englewood Cliffs, N.J., 1995), pp. 93–112.

Condon, E. U.

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]

De Bruijn, N. G.

N. G. De Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

Forbes, G. W.

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
[CrossRef]

G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A (to be published).

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).

Frederick, J. H.

G. Torres-Vega, J. H. Frederick, “A quantum mechanical representation in phase space,” J. Chem. Phys. 98, 3103–3120 (1993).
[CrossRef]

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]

Friedman, B.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986) pp. 252–320.

Hannay, J. H.

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]

Harriman, J. E.

J. E. Harriman, “A quantum state vector phase space representation,” J. Chem. Phys. 100, 3651–3661 (1994).
[CrossRef]

Jörgensen, T. G.

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]

Klauder, J. R.

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).

Korsch, H. J.

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]

Kravtsov, Y. A.

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.

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993).

Kutay, M. A.

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.

Lee, H. W.

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]

Lohmann, A. W.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.

Mendlovic, D.

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.

Möller, K. B.

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]

Mustard, D.

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B 38, 209–219 (1996).
[CrossRef]

Orlov, Y. A.

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993).

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.

Ozaktas, H. M.

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.

Ozorio de Almeida, A. M.

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]

Price, J. F.

J. F. Price, “Uncertainty principles and sampling theorems,” in Fourier Techniques and Applications, J. F. Price, ed. (Plenum, New York, 1985), pp. 25–44.

Skagerstam, B.

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).

Stegun, I. A.

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.

Tabor, M.

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).
[CrossRef]

Torres-Vega, G.

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]

G. Torres-Vega, J. H. Frederick, “A quantum mechanical representation in phase space,” J. Chem. Phys. 98, 3103–3120 (1993).
[CrossRef]

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]

Ursell, F.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Voros, A.

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]

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).
[CrossRef]

A. Voros, “Semiclassical approximations,” Ann. Inst. Henri Poincare 14, 31–90 (1976).

Walther, A.

Wigner, E.

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.

Wright, F. J.

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]

Ann. Inst. Henri Poincare

A. Voros, “Semiclassical approximations,” Ann. Inst. Henri Poincare 14, 31–90 (1976).

Ann. Phys. (N.Y.)

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]

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).
[CrossRef]

Commun. Pure Appl. Math.

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]

J. Aust. Math. Soc. B

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B 38, 209–219 (1996).
[CrossRef]

J. Chem. Phys.

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]

G. Torres-Vega, J. H. Frederick, “A quantum mechanical representation in phase space,” J. Chem. Phys. 98, 3103–3120 (1993).
[CrossRef]

J. E. Harriman, “A quantum state vector phase space representation,” J. Chem. Phys. 100, 3651–3661 (1994).
[CrossRef]

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]

J. Math. Phys.

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

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]

M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979).
[CrossRef]

Philos. Trans. R. Soc. London

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]

Phys. Rep.

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]

Phys. Rev.

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A

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]

Physica D

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]

Proc. Cambridge Philos. Soc.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Proc. Natl. Acad. Sci. USA

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]

Proc. R. Soc. London Ser. A

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219–231 (1989).
[CrossRef]

Sci. Prog. (Oxford)

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.

Trans. Cambridge Philos. Soc.

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).

Other

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.

The Pearcey function is defined as Ip(x, y)=∫ exp[i(xt+yt2+t4)]dt.See, for example, Ref. 27, pp. 60–61.

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.

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.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.

J. F. Price, “Uncertainty principles and sampling theorems,” in Fourier Techniques and Applications, J. F. Price, ed. (Plenum, New York, 1985), pp. 25–44.

N. G. De Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A (to be published).

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).

See, for example, the reference in Note 7, pp. 22–26.

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993).

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.

L. Cohen, Time-Frequency Analysis (Prentice-Hall PTR, Englewood Cliffs, N.J., 1995), pp. 93–112.

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).

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.

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.

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.

See Ref. 11, Chap. 7, pp. 93–100.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986) pp. 252–320.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

The phase-space area under a segment of the PSC is given by the difference in ϕ for the corresponding points.

Fig. 2
Fig. 2

The Airy function Ai(t) has its global maximum at t-1.019.

Fig. 3
Fig. 3

The main ridge of the WDF is displaced toward the concave side of the PSC. Further into the concave side the WDF is oscillatory, while on the convex side it decays exponentially. In this gray-scale picture, the shade of gray within the uniform region at the bottom of the figure corresponds to zero; the lighter the shade, the more positive the value.

Fig. 4
Fig. 4

The point (x¯, p¯) is defined as the midpoint in phase space between [x1, ϕ(x1)] and [x2, ϕ(x2)].

Fig. 5
Fig. 5

A half-scale replica of the PSC is described by (x¯, p¯) when x1 is fixed and x2 varies.

Fig. 6
Fig. 6

Superposition of half-scale replicas of the PSC, for which the PSC (shown as a thick gray curve) is an envelope.

Fig. 7
Fig. 7

Phase-space-area element enclosed by two pairs of neighboring replicas.

Fig. 8
Fig. 8

The area of the region enclosed by the PSC and a secant joining [x1, ϕ(x1)] and [x2, ϕ(x2)] is given by Φ.

Fig. 9
Fig. 9

When the PSC (shown as a solid gray curve) has inflection points, the replicas (shown as thin black curves) form secondary envelopes, referred to as ghost curves. One such ghost curve is shown here as a dashed gray curve.

Fig. 10
Fig. 10

Superposition of replicas for the PSC given by the derivative of Eq. (3.15). Notice that two replicas touch each point of the ghost curve (shown as the dashed gray curve), and one such pair of replicas is highlighted here.

Fig. 11
Fig. 11

Regions over which the different forms must be used in the asymptotic construction of the WDF of the field with ϕ as defined in Eq. (3.15): I, Wf; II, Wc; III, Wc+Wf; IV, since each point in this region is the midpoint of two pairs of points along the PSC, the estimate is given by the sum of their corresponding contributions of the form given by Wf; V, Wg. Strictly speaking, asymptotic contributions in terms of Pearcey integrals should be used within the regions shown as gray ellipses at the inflection points.

Fig. 12
Fig. 12

Asymptotic estimate of the WDF of the field of constant amplitude and phase kϕ(x), for k=40 and ϕ defined in Eq. (3.15). The PSC is shown as a dashed curve. Notice the strong banding along the ghost curve, which leads to a low average value when W is locally averaged. The gray levels are explained in the caption for Fig. 3.

Fig. 13
Fig. 13

The second part of the phase of the left-hand side of Eq. (4.10) corresponds to k times the phase-space area shown in gray.

Fig. 14
Fig. 14

(a) Modulus and (b) real part of the asymptotic estimate of the GWFT of the field of constant amplitude and phase kϕ(x), for k=40, ϕ defined in Eq. (3.15), and l=0.5. The PSC is shown as a dashed curve.

Fig. 15
Fig. 15

(a) PSC and replicas for a function that satisfies Eq. (5.2). (b) Exact WDF of the bound solution to Eq. (5.2) for m=10. The gray levels are explained in the caption for Fig. 3.

Fig. 16
Fig. 16

(a) Modulus and (b) real part of the exact GWFT of the bound solution to Eq. (5.2) for m=10. The PSC is shown as a dashed curve. Comparison with Fig. 15 reemphasizes that the intuitive form of the GWFT allows a more direct interpretation. This is even more evident when the squared modulus is plotted in place of the modulus in each case.

Equations (62)

Equations on this page are rendered with MathJax. Learn more.

f(x)=a(x)exp[ikϕ(x)],
p=ϕx.
f˜(p)k2π f(x)exp(-ikpx)dx,
f(x)=k2π f˜(p)exp(ikpx)dp,
W(x, p) k2πfx+x2f*x-x2×exp(-ikpx)dx,
W(x, p)=k2πf˜p+p2f˜*p-p2exp(ikpx)dp.
|f(x)|2=W(x, p)dp,
|f˜(p)|2=W(x, p)dx.
W(x, p)exp(ikpx)dp=fx+x2f*x-x2.
f(x)=1f*(x0)Wx+x02, pexp[ikp(x-x0)]dp.
G(x, p) k2π l-1/4f(x)exp-k2l(x-x)2-ikpx-x2dx,
G(x, p)=k2π l1/4f˜(p)exp-kl2(p-p)2+ikxp-p2dp.
lx-ipG(x, p)=-k2(x-ilp)G(x, p).
f(x)=k2π l1/4 expk2l(x-x)2G(x, p)×expikpx-12xdp.
f(x)=k2π l1/4G(x, p)expikpx2dp.
f˜(p)=k2π l-1/4G(x, p)exp-ikpx2dx.
W1(x, p) * W2(x, p)=k2πf1(x)f2*(x-x)×exp(-ikpx)dx2.
H(x, p)W(x, p) * Wg(x, p)|G(x, p)|2.
W(x, p)=k2πax+12xax-12x×expikϕx+12x-ϕ(x-12x)-pxdx.
p=12[ϕ(x+12xs)+ϕ(x-12xs)],
ϕ(x+12x)-ϕ(x-12x)-px=[ϕ(x+12xs)-ϕ(x-12xs)-pxs]+12[ϕ(x+12xs)-ϕ(x-12xs)](s/2)2+16[ϕ(x+12xs)+ϕ(x-12xs)](s/2)3+O(s4).
δ=12ϕ(x)(xs/2)2+14!ϕ(v)(x)(xs/2)4+O[(xs/2)6]
xs(δ)=±22δϕ(x)[1+O(δ)].
ϕ(x+12x)-ϕ(x-12x)-px=-δx+124ϕ(x)x3+O(x5).
W[x, ϕ(x)+δ]k2πax+12xtax-12xt×expik124ϕ(x)x3-δ xdx.
δX2ϕ(x)=[k-2ϕ(x)]1/3=O(k-2/3).
Wc[x, ϕ(x)+δ]=8k2ϕ(x)1/3a2(x)Ai-8k2ϕ(x)1/3×sgn[ϕ(x)]δ[1+O(k-1/3)].
ϕ(x¯+12x)-ϕ(x¯-12x)-px¯=±{Φ+12[ϕ(x2)-ϕ(x1)](s/2)2+O(s3)},
Φ=ϕ(x2)-ϕ(x1)-(x2-x1)p¯.
Wf(x¯, p¯)=k2π1/24a(x1)a(x2)[|ϕ(x2)-ϕ(x1)|]1/2×cos(kΦ+σπ/4),
ϕ(x¯+12x)-ϕ(x¯-12x)-pxΦ-(x2-x1)δ-δ s+12ϕ¯s3+O(s5),
Wg(x¯, p¯+δ)=8k2ϕ¯1/3a(x1)a(x2)×Ai-8k2ϕ¯1/3 sgn(ϕ¯)δ×2 cos{k[Φ-(x2-x1)δ]}.
ϕ(x)=x2+115x3-94x4+25x5.
G(x, p)[1+O(k-1)]l-1/4a(xs)Ω2exp(-kΩ0),
Ω0 (xs-x)22l+ipxs-x2-ϕ(xs),
Ω2 1-ilϕ(xs)l,
xs-x+il[p-ϕ(xs)]=0.
+ilδ-ϕ-ϕ22+=0.
=-il1-ilϕδ+l3ϕ(1-ilϕ)3δ22+.
Ω0=-iϕ+ix2(ϕ+δ)+l2(1-ilϕ)δ2+O(δ3),
Ω2=1-ilϕl+O(δ).
l2(1+l2ϕ2)δ2+O(δ3)5k-1.
G[x, ϕ(x)+δ][1+O(k-1/2)]l1/4a(x)[1-ilϕ(x)]1/2×exp-klδ22[1+l2ϕ2(x)]×expikϕ(x)-xϕ(x)+δ2-l2ϕ(x)1+l2ϕ2(x)δ22,
β(x)arctan[lϕ(x)].
G(x, ϕ+δ)[1+O(k-1/2)]×l1/4acos β exp-klδ2 cos2 β2×exp-iβ2+ikϕ-xϕ+δ2-l sin β cos βδ22.
f(x)+k2h(x)f(x)=0,
f(x)+w-2[2m+1-(x/w)2]f(x)=0.
W(x, p)=12Δnδp-nπkΔ-ΔΔfx+x2×f*x-x2exp-inπΔxdx,
W(x, p)=12ΔnWax, p-nπkΔ-ΔΔfx+x2×f*x-x2exp-inπΔxdx,
fθ(u)f(x)Γθ(x, u)dx,
Γθ(x, u)k2πlF|sin θ| expi2θ mod π-π2×expik2lF sin θ[(x2+u2)cos θ-2xu].
f0(u)f(u),
fπ/2(u)1lFf˜(u/lF).
Wθu,vlFk2πfθu+u2fθ*u-u2×exp-ikuvlFduWu cos θ-v sin θ,1lF(v cos θ+u sin θ).
Gθu,vlk2π l-1/4fθ(u)exp-k2l(u-u)2-ikvlu-u2duGu cos θ-v sin θ,1l(v cos θ+u sin θ).
f˜(p)=1f˜*(p0)Wx,p+p02×exp[-ikx(p-p0)]dx.
 exp[i(xt+at3/3)]dt=2π|a|-1/3Ai[xa-1/3].
p¯=ϕ(x¯)+δ=12{ϕ[x¯+(x2-x1)/2]+ϕ[x¯-(x2-x1)/2]}
ϕ[x¯+(x2-x1)/2]-ϕ[x¯-(x2-x1)/2]-(x2-x1)p¯-23σ[8δ3/ϕ(x¯)]1/2.
ϕ(x+12x)-ϕ(x-12x)-pxΦ+[ϕ(x2)-ϕ(x1)]δ-(x2-x1)η-(δ-ϕη)s+124ϕ¯s3+O(s5),
W(x¯+η, p¯+δ)8k2ϕ¯1/3a(x1)a(x2)Ai-8k2ϕ¯1/3sgn[ϕ¯](δ-ϕη)×2 cos(k{Φ+[ϕ(x2)-ϕ(x1)]η-(x2-x1)δ}).
Ip(x, y)= exp[i(xt+yt2+t4)]dt.

Metrics