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.

© 2011 Optical Society of America

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References

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  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).

1995 (1)

1989 (1)

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[Crossref]

1986 (1)

1979 (1)

1932 (1)

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

Arfken, G.

G. Arfken and H. Weber, Mathematical Methods for Physicists (Academic Press, 1995).

Bastiaans, M. J.

Boashash, B.

B. Boashash, Time Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier Science Ltd, 2003).

Boudreaux-Bartels, G.

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]

Cohen, L.

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[Crossref]

Hamid, S.

A. Oppenheim, A. Willsky, and S. Hamid, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

Maier, S.

S. Maier, Plasmonics: Fundamentals and Applications (Springer Verlag, 2007).

Migus, A.

Oppenheim, A.

A. Oppenheim, A. Willsky, and S. Hamid, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

Paye, J.

Weber, H.

G. Arfken and H. Weber, Mathematical Methods for Physicists (Academic Press, 1995).

Wigner, E.

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

Willsky, A.

A. Oppenheim, A. Willsky, and S. Hamid, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Phys. Rev. (1)

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

Proc. IEEE (1)

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[Crossref]

Other (6)

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).

B. Boashash, Time Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier Science Ltd, 2003).

Supplementary Material (4)

» Media 1: MOV (1967 KB)     
» Media 2: MOV (1860 KB)     
» Media 3: MOV (1819 KB)     
» Media 4: MOV (1740 KB)     

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Figures (5)

Fig. 1
Fig. 1 The LWF is visualized in x–pR–pI space. Due to the ROC, the LWF may be limited in pR. The extents along x and pI are determined by the space–bandwidth product of signals. At pR = 0, the x–pI plane in the LWF corresponds to the x–p space in the traditional WF.

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Fig. 2
Fig. 2 Evanescent waves in SPP excited at the metal–air interface.

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Fig. 3
Fig. 3 (a) [ Media 1] The real part and (b) [ Media 2] the complex part of the LWF for the SPP given by Eq. (8) for k1/k0 = 0.336 and k2/k0 = 3.308.

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Fig. 4
Fig. 4 Traditional WDF of evanescent waves for k1/k0 = 0.336 and k2/k0 = 3.308. Absolute values are plotted.

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Fig. 5
Fig. 5 The LWF cross–terms of evanescent waves in SPP for k1/k0 = 0.336, k2/k0 = 3.308, k3/k0 = 0.225 and k4/k0 = 4.668. In (a) [ Media 3], the only one cross–term is shown whereas in (b) [ Media 4] the sum of the two cross–terms are shown. Real values are plotted.

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Tables (1)

Tables Icon

Table 1 Marginals of the WF and LWF. E is the energy of signals. C = ∫�� dpR, which is a scaling factor dependent on the width of the ROC, if |��| < ∞. ℒψ is the Laplace transform of ψ. (†): if pR ∈ ��.

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Equations (15)

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𝒲 ( q , p ) = ∫ − ∞ ∞ ψ ( q + q ′ 2 ) ψ * ( q − q ′ 2 ) e − i q ′ ⋅ p d q ′ ,
𝒲 ℒ ( x , s ) = ∫ − ∞ ∞ ψ ( x + x ′ 2 ) ψ * ( x − x ′ 2 ) e − s x ′ d x ′ ,
ψ ( x 1 ) ψ * ( x 2 ) = 1 2 π i ∫ γ − i ∞ γ + i ∞ 𝒲 ℒ ( x 1 + x 2 2 , s ) e s ( x 1 − x 2 ) d s ,
H y ( x , z ) = { A e i β x e − k 1 z if   z ≥ 0 A e i β x e k 2 z if   z < 0 ,
β = k 0 ɛ 1 ɛ 2 ɛ 1 + ɛ 2 ,
k 1 2 = β 2 − k 0 2 ɛ 1               and
k 2 2 = β 2 − k 0 2 ɛ 2 ,
𝒲 ℒ ( z , s ) = e − 2 k i | z | { e 2 s | z | k 1 + k 2 2 − s + e − 2 s | z | k 1 + k 2 2 + s + e 2 s | z | − e − 2 s | z | s } ,
k i = { k 1 if   z ≥ 0 k 2 if   z < 0 .
𝒲 ℒ [ f ( z ) + g ( z ) ] = 𝒲 ℒ [ f ( z ) ] + 𝒲 ℒ [ g ( z ) ]                                 + ∫ − ∞ ∞ f ( z + z ′ 2 ) g * ( z − z ′ 2 ) e − s z ′ d z ′ + ∫ − ∞ ∞ g ( z + z ′ 2 ) f * ( z − z ′ 2 ) e − s z ′ d z ′ .
f ( z ) = { e − k 1 z , if   z ≥ 0 e k 2 z , if   z < 0
g ( z ) = { e − k 3 z , if   z ≥ 0 e k 4 z , if   z < 0 ,
𝒲 ℒ f g ( z , s ) = ∫ − ∞ ∞ f ( z + z ′ 2 ) g * ( z − z ′ 2 ) e − s z ′ d z ′                                                                                                               = e − k m | z | e 2 ( k n − s ) | z | − e − 2 ( k n − s ) | z | k n − s                                                                                                                           + e ( − k 1 + k 4 ) z e − 2 ( k 1 + k 4 2 + s ) | z | k 1 + k 4 2 + s + e ( k 2 − k 3 ) z e − 2 ( k 2 + k 3 2 − s ) | z | k 2 + k 3 2 − s ,
k m = { k 1 + k 3 if   z ≥ 0 k 2 + k 4 if   z < 0
k n = { − k 1 + k 3 2 if   z ≥ 0 k 2 − k 4 2 if   z < 0 ,

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