Abstract

On the basis of the space–time Wigner distribution function (STWDF), we use the matrix formalism to study the propagation laws for the intensity moments of quasi-monochromatic and polychromatic pulsed paraxial beams. The advantages of this approach are reviewed. Also, a least-squares fitting method for interpreting the physical meaning of the effective curvature matrix is described by means of the STWDF. Then the concept is extended to the higher-order situation, and what we believe is a novel technique for characterizing the beam phase is presented.

© 1999 Optical Society of America

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  1. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  2. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  3. S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
    [CrossRef] [PubMed]
  4. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  5. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
  6. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).
  7. D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993).
    [CrossRef]
  8. D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
    [CrossRef]
  9. R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
    [CrossRef]
  10. R. Martı́nez-Herrero, P. M. Mejı́as, “On the fourth-order spatial characterization of laser beams: new invariant parameter through ABCD systems,” Opt. Commun. 140, 57–60 (1997).
    [CrossRef]
  11. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  12. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]
  13. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]
  14. M. A. Porras, J. Alda, E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31, 6389–6402 (1992).
    [CrossRef] [PubMed]
  15. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
    [CrossRef]
  16. Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18, 669–671 (1993).
    [CrossRef] [PubMed]
  17. P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
    [CrossRef] [PubMed]
  18. Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
    [CrossRef]
  19. E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
    [CrossRef]
  20. E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
    [CrossRef]
  21. K. Yoshimori, K. Itoh, “Interferometry and radiometry,” J. Opt. Soc. Am. A 14, 3379–3387 (1997);“On the generalized radiance function for a polychromatic field,” J. Opt. Soc. Am. A 15, 2786–2787 (1998).
    [CrossRef]
  22. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  23. M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
    [CrossRef]
  24. J. Paye, 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]
  25. A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
    [CrossRef]
  26. Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Spatial–temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 679–692 (1995).
    [CrossRef]
  27. K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
    [CrossRef]
  28. G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]

1997 (4)

R. Martı́nez-Herrero, P. M. Mejı́as, “On the fourth-order spatial characterization of laser beams: new invariant parameter through ABCD systems,” Opt. Commun. 140, 57–60 (1997).
[CrossRef]

Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

K. Yoshimori, K. Itoh, “Interferometry and radiometry,” J. Opt. Soc. Am. A 14, 3379–3387 (1997);“On the generalized radiance function for a polychromatic field,” J. Opt. Soc. Am. A 15, 2786–2787 (1998).
[CrossRef]

1995 (3)

1994 (1)

1993 (2)

1992 (3)

M. A. Porras, J. Alda, E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31, 6389–6402 (1992).
[CrossRef] [PubMed]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

1991 (4)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[CrossRef] [PubMed]

1990 (1)

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

1988 (2)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
[CrossRef] [PubMed]

1981 (1)

E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

1979 (2)

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

1976 (1)

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[CrossRef]

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

1968 (1)

Alda, J.

Bastiaans, M. J.

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

Bélanger, P. A.

Bernabeu, E.

Cao, Q.

Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Deng, X.

Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Dragoman, D.

Itoh, K.

Keren, E.

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

Lavi, S.

Lin, Q.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Spatial–temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 679–692 (1995).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18, 669–671 (1993).
[CrossRef] [PubMed]

Marti´nez-Herrero, R.

R. Martı́nez-Herrero, P. M. Mejı́as, “On the fourth-order spatial characterization of laser beams: new invariant parameter through ABCD systems,” Opt. Commun. 140, 57–60 (1997).
[CrossRef]

P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
[CrossRef] [PubMed]

R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Meji´as, P. M.

R. Martı́nez-Herrero, P. M. Mejı́as, “On the fourth-order spatial characterization of laser beams: new invariant parameter through ABCD systems,” Opt. Commun. 140, 57–60 (1997).
[CrossRef]

P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
[CrossRef] [PubMed]

R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Migus, A.

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Neira, J. L. H.

R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

Onciul, D.

Oughstun, K. E.

K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

Paye, J.

Porras, M. A.

Prochaska, R.

Sanchez, M.

R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

Serna, J.

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

Walther, A.

Wang, S.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Spatial–temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 679–692 (1995).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18, 669–671 (1993).
[CrossRef] [PubMed]

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Wolf, E.

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[CrossRef]

Xiao, H.

K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

Yoshimori, K.

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

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

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

Opt. Acta (1)

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

Opt. Commun. (3)

Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, “On the fourth-order spatial characterization of laser beams: new invariant parameter through ABCD systems,” Opt. Commun. 140, 57–60 (1997).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (3)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Spatial–temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 679–692 (1995).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

Optik (Stuttgart) (2)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

Phys. Rev. A (1)

E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

Phys. Rev. D (1)

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

Proc. IEEE (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Other (1)

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

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Equations (51)

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r˜oq˜o=Sr˜iq˜i,S=A˜B˜C˜D˜,
r˜i,o=xyt-t0i,o,q˜i,o=uxuyλ0(v0-v)i,o.
ST=JS-1J,
J=i0-II0
Wst(r˜, q˜)=1λ03 ψ(r˜+r˜/2)ψ*(r˜-r˜/2)×exp-j 2πλ0 q˜Tr˜dr˜,
Wist(r˜, q˜)=Wost(A˜r˜+B˜q˜, C˜r˜+D˜q˜),
M˜=m˜rrm˜rqm˜qrm˜qq=r˜r˜Tr˜q˜Tq˜r˜Tq˜q˜TWst(r˜, q˜)dr˜dq˜,
Wst(r˜, q˜)dr˜dq˜=1.
M˜o=SM˜iST.
(M˜oJ)n=S(M˜iJ)nS-1(n1).
det M˜,
tr(M˜J)n(n1).
tr(M˜J)2=2tr(m˜rrm˜qq-m˜rq2).
M˜Q4=4πλ02(m˜rrm˜qq-m˜rq2).
Wst(r, u; t, v)= 1λ2 ψ(r+r/2, t+t/2)×ψ*(r-r/2, t-t/2)×exp-j2πuTrλ-vtdrdt,
rouo=Ssriui,Ss=ABCD.
Ws(r, u; v)=Wst(r, u; t, v)dt,
Wis(r, u; v)=Wos(Ar+Bu, Cr+Du; v).
M=mrrmrumurmuu=rrTruTurTuuTWs(r, u; v)drdudv,
Ws(r, u; v)drdudv=1.
Mo=SsMiSsT,
(MoJ)n=Ss(MiJ)nSs-1(n1).
tr(MJ)2=2 tr(mrrmuu-mru2).
MQ4=(4π/λ¯)2(mrrmuu-mru2),
λ¯=λWs(r, u; v)drdudv.
Φ(r˜)=Φ0+12 r˜TR˜-1r˜,
θ˜b=Φ(r˜)=R˜-1r˜,
θ˜a=q˜Wst(r˜, q˜)dq˜/Wst(r˜, q˜)dq˜.
δ=(θ˜a-θ˜b)(θ˜a-θ˜b)TWst(r˜, q˜)dr˜dq˜.
δ=θ˜a2¯-2R˜-1m˜rq+R˜-1m˜rrR˜-1,
R˜-1=R˜eff-1m˜rq/m˜rr.
Reff-1=mru/mrr,
Φ(x)=Φ0+c2x2+c3x3++cjxj+
(j2).
θb=Φ(x)x=2c2x+3c3x2++jcjxj-1+
(j2).
δ=(θa-θb)2Ws(x, ux; v)dxduxdv=θa2¯-2j=2jcjuxxj-1¯+j=2j2cj2x2j-2¯+j=2ljjlcjclxj+l-2¯.
cj=uxxj-1¯-l=2,ljlclxj+l-2¯jx2j-2¯(j2).
xj¯z=juxxj-1¯,
cj=j-1 xj¯z-l=2,ljlclxj+l-2¯jx2j-2¯(j2).
2x2¯3x3¯nxn¯2x3¯3x4¯nxn+1¯2xn¯3xn+1¯nx2n-2¯×c2c3cn=2-1x2¯/z3-1x3¯/zn-1xn¯/z.
S=A˜B˜C˜D˜=A0BEG1HLC0DF0001,
2πρ2(v) x2+ρ2(v)2π 2πλ2u2Ws(x, u; v)dxdu
Ws(x, u; v)dxdu,
x2ρ02+ρ02u2Ws(x, u; v)dxdu
λ2π Ws(x, u; v)dxdu.
x2ρ02+ρ02u2Ws(x, u; v)dxdudv
12π λWs(x, u; v)dxdudv.
mxx/ρ02+ρ02muuλ¯/2π.
mxxmuu(λ¯/4π)2.
MQ4=(4π/λ¯)2(mxxmuu-mxu2).

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