Abstract

An analytic relation for recovering the mutual intensity by means of intensity information under the condition of the fractional Fourier transform is derived. The results may simplify the reconstruction of the mutual intensity in comparison with the Wigner tomographic method and can be regarded as an inverse transform formula that expresses output intensity in terms of input mutual intensity under partially coherent illumination.

© 1998 Optical Society of America

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References

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  1. K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992); G. Hazak, “Comment on ‘Wave field determination using three-dimensional intensity information’,” Phys. Rev. Lett. 69, 2874–2784 (1992).
    [CrossRef] [PubMed]
  2. F. Gori, M. Santarsiero, and G. Guattari, “Coherence and space distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993); T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  3. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  4. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  5. R. Gase, T. Gase, and K. Blüthner, “Complex wave-field reconstruction by means of the Page distribution function,” Opt. Lett. 20, 2045–2047 (1995).
    [CrossRef] [PubMed]
  6. G. Iaconis and I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [CrossRef] [PubMed]
  7. U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt. 42, 2183–2199 (1995).
    [CrossRef]
  8. T. Alieva and F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
    [CrossRef]
  9. J. Tu and S. Tamura, “Wave field determination using tomography of ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
    [CrossRef]
  10. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  11. H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  12. Haldum M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  13. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wang, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
    [CrossRef] [PubMed]
  14. K. H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta. 31, 213–223 (1984).
    [CrossRef]
  15. H. Kuhn, D.-G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobabilities,” J. Mod. Opt. 41, 1607–1613 (1994).
    [CrossRef]
  16. D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
    [CrossRef] [PubMed]
  17. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distribution for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
    [CrossRef] [PubMed]
  18. G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1805 (1995).
    [CrossRef] [PubMed]
  19. U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
    [CrossRef]
  20. A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1678–1681 (1996).
    [CrossRef] [PubMed]
  21. M. Fatih Erden, H. M. Ozaktas, and D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
    [CrossRef]

1997 (1)

J. Tu and S. Tamura, “Wave field determination using tomography of ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

1996 (4)

G. Iaconis and I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
[CrossRef] [PubMed]

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1678–1681 (1996).
[CrossRef] [PubMed]

M. Fatih Erden, H. M. Ozaktas, and D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

1995 (6)

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1805 (1995).
[CrossRef] [PubMed]

Haldum M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt. 42, 2183–2199 (1995).
[CrossRef]

T. Alieva and F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

R. Gase, T. Gase, and K. Blüthner, “Complex wave-field reconstruction by means of the Page distribution function,” Opt. Lett. 20, 2045–2047 (1995).
[CrossRef] [PubMed]

1994 (3)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

H. Kuhn, D.-G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobabilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

1993 (3)

D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wang, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

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

1989 (1)

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distribution for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

1984 (1)

K. H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta. 31, 213–223 (1984).
[CrossRef]

Agullo-Lopez, F.

T. Alieva and F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

Alieva, T.

T. Alieva and F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

Barshan, B.

Beck, M.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wang, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Blüthner, K.

Brenner, K. H.

K. H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta. 31, 213–223 (1984).
[CrossRef]

Clarke, L.

D’Ariano, G. M.

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1805 (1995).
[CrossRef] [PubMed]

Fatih Erden, M.

Gase, R.

Gase, T.

Iaconis, G.

Janicke, U.

U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt. 42, 2183–2199 (1995).
[CrossRef]

Kiss, T.

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

Kuhn, H.

H. Kuhn, D.-G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobabilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

Leonhardt, U.

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1805 (1995).
[CrossRef] [PubMed]

Lohmann, A. W.

Mayer, A.

McAlister, D. F.

Mendlovic, D.

Munroe, M.

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

Ojeda-Castaneda, J.

K. H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta. 31, 213–223 (1984).
[CrossRef]

Onural, L.

Ozaktas, H. M.

Ozaktas, Haldum M.

Paul, H.

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1805 (1995).
[CrossRef] [PubMed]

Raymer, M. G.

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wang, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Richter, Th.

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

Risken, H.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distribution for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Smithey, D. T.

D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Tamura, S.

J. Tu and S. Tamura, “Wave field determination using tomography of ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Tasche, M.

A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1678–1681 (1996).
[CrossRef] [PubMed]

Tu, J.

J. Tu and S. Tamura, “Wave field determination using tomography of ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Vogel, K.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distribution for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Vogel, W.

A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1678–1681 (1996).
[CrossRef] [PubMed]

H. Kuhn, D.-G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobabilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

Walmsley, I. A.

Wang, V.

Welsch, D.-G.

A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1678–1681 (1996).
[CrossRef] [PubMed]

H. Kuhn, D.-G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobabilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

Wilkens, M.

U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt. 42, 2183–2199 (1995).
[CrossRef]

Zucchetti, A.

A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1678–1681 (1996).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

U. Janicke and M. Wilkens, “Tomography of atom beams,” J. Mod. Opt. 42, 2183–2199 (1995).
[CrossRef]

H. Kuhn, D.-G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobabilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Acta. (1)

K. H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta. 31, 213–223 (1984).
[CrossRef]

Opt. Commun. (2)

T. Alieva and F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (3)

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distribution for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1805 (1995).
[CrossRef] [PubMed]

A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1678–1681 (1996).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. Tu and S. Tamura, “Wave field determination using tomography of ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Other (2)

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992); G. Hazak, “Comment on ‘Wave field determination using three-dimensional intensity information’,” Phys. Rev. Lett. 69, 2874–2784 (1992).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, and G. Guattari, “Coherence and space distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993); T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[CrossRef]

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Equations (40)

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Γ(r1, r2, τ)=Ψ(r1, t)Ψ*(r2, t+τ),
W(r1, r2, ω)=Γ(r1, r2, τ)exp(-iωτ)dτ.
J(r1, r2, z)=Φ(r1, z)Φ*(r2, z),
J(x, Δx, z)as
J(x, Δx, z)=J(x1, x2, z),
x=(x1+x2)/2,
Δx=x1-x2.
I(x, z)=J(x, Δx=0, z).
A(Δν, Δx, z)=J(x, Δx, z)exp(-i2πΔνx)dx,
W(x, ν, z)=J(x, Δx, z)exp(-i2πΔxν)dΔx,
A(Δν, Δx, z)=W(x, ν, z)×exp[i2π(Δxν-xΔν)]dxdν,
ξ=(Δν2+Δx2)1/2,
Δν=ξ cos θ,
Δx=ξ sin θ,
A(ξ cos θ, ξ sin θ, z)=W(x, ν, z)×exp[-i2πξ(x cos θ-ν sin θ)]dxdν.
M(z)=ABCD.
W(x, ν, z)=W(Ax+Bν, Cx+Dν, 0).
I˜(η, z)=I(x, z)exp(-i2πηx)dx.
I(x, z)=W(x, ν, z)dν,
xν=abcdxν,
η=η/a2+b2,
abcd=D-B-CA,
I˜(η/a2+b2, z)=W(x, ν, z=0)×exp[-i2πη(xa/a2+b2+νb/a2+b2)]dxdν.
I˜(η/a2+b2, z)=A(ηa/a2+b2, -ηb/a2+b2, 0),
J(x, Δx, 0)=A(Δν, Δx, 0)exp(i2πΔνx)dΔν.
Mϕ(z)=cos ϕsin ϕ-sin ϕcos ϕ,
W(x, ν, z)=A(Δν, Δx, z)×exp[-i2π(Δxν-xΔν)]dΔxdΔν.
W(x, ν, 0)=0π-+A(ξ cos θ, ξ sin θ, 0)×exp[-i2πξ(ν sin θ-x cos θ)]|ξ|dξdθ.
W(x, ν, 0)=0π-+-+Iϕ(x, z)×exp[-i2πξ(x+ν sin ϕ-x cos ϕ)]dx|ξ|dξdϕ,
W(x, ν, 0)=0π-+Iϕ(x, z)g(x+ν sin ϕ-x cos ϕ)dxdϕ,
g(x)=-+|ξ|exp(-i2πξx)dξ.
g(x)gm(x), =-ξm+ξm|ξ|exp(-i2πξx)dξ=2-1x2+cos(ξmx)x2+ξm sin(ξmx)x.
J(x1, x2, 0)=Wx1+x22,ν, z=0×exp[i2π(x1-x2)ν]dν.
J(x1, x2, 0)=0πdϕ-+dxIϕ(x, z)K(x1, x2, ϕ, x),
K(x1, x2, ϕ, x)=f(ξ sin ϕ+x2-x1)×exp-i2πξx-x1+x22cos ϕ|ξ|dξ,
f(x)= exp(-i2πνx)dν=δ(x).
K(x1, x2, ϕ, x)=|x1-x2|sin2 ϕexp-i2πx1-x2sin ϕ×x-12(x1+x2)cos ϕ.
K(x1, x2, ϕ, x)=-ξmξm sinc[νm(ξ sin ϕ+x2-x1)]×exp-i2πξx-x1+x22cos ϕ|ξ|dξ,
K(x1, x2, ϕ=nπ, x)=sinc[νm(x2-x1)]gmx-x1+x22.
J(x1, x1, 0)=-+dxIb=0(x, z)δ(x1-x/a),

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