Abstract

Several schemes and methods exist for frequency-resolved optical gating as a technique for the full characterization of ultrashort optical signals as complex electric fields. However, the uniqueness of the reconstructed fields has never been shown. Here we derive conditions that are sufficient for unique reconstruction of the complex pulses. Furthermore, we construct several examples of distinct pulse pairs with identical or essentially identical frequency-resolved optical-gating traces; even examples with identical spectral intensities of the pulse pairs were found.

© 2004 Optical Society of America

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References

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  1. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
    [CrossRef]
  2. R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
    [CrossRef]
  3. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, Dordrecht, The Netherlands, 2002).
  4. D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
    [CrossRef]
  5. V. Wong and I. A. Walmsley, “Ultrashort-pulse characterization from dynamic spectrograms by iterative phase retrieval,” J. Opt. Soc. Am. B 14, 944–949 (1997).
    [CrossRef]
  6. D. T. Reid, “Algorithm for complete and rapid retrieval of ultrashort pulse amplitude and phase from sonogram,” IEEE J. Quantum Electron. 35, 1584–1589 (1999).
    [CrossRef]
  7. M. Jütte, W. von der Osten, and H. Stolz, “Characterization of picosecond laser pulses by a bandwidth-limited time-resolved spectrum,” Opt. Commun. 157, 173–176 (1998).
    [CrossRef]
  8. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, Dordrecht, The Netherlands, 2002), p. 345.
  9. C. Dorrer and I. Kang, “Simultaneous temporal characterization of telecommunication optical pulses and modulators by use of spectrograms,” Opt. Lett. 27, 1315–1317 (2002).
    [CrossRef]
  10. B. Seifert and H. Stolz, “Effect of group velocity mismatch in nonlinear mixing schemes to recover ultrashort optical pulses,” Opt. Commun. 216, 413–418 (2003).
    [CrossRef]
  11. S. Linden, H. Giessen, and J. Kuhl, “XFROG: a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
    [CrossRef]
  12. A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, “On phase retrieval in electron microscopy from image and diffraction pattern,” Optik (Stuttgart) 45, 303–316 (1976).
  13. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  14. J. Aczél, Lectures on Functional Equations and their Applications (Academic, New York, 1966).
  15. K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463–2466 (1995).
    [CrossRef]
  16. D. T. Reid, P. Loza-Alvarez, C. T. A. Brown, T. Beddard, and W. Sibbett, “Amplitude and phase measurement of mid-infrared femtosecond pulses by using cross-correlation frequency-resolved optical gating,” Opt. Lett. 25, 1478–1480 (2000).
    [CrossRef]
  17. H. Stolz, Ch. Nacke, B. Seifert, M. Seemann, and F. Kieseling, “Phase sensitive femtosecond spectroscopy of semiconductors,” Adv. Solid State Phys. 39, 473–482 (1999).
    [CrossRef]
  18. L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
    [CrossRef]
  19. A. M. J. Huiser and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: II. On the uniqueness and stability,” Optik (Stuttgart) 46, 407–420 (1976).
  20. A. M. J. Huiser, P. van Toorn, and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).
  21. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

2003

B. Seifert and H. Stolz, “Effect of group velocity mismatch in nonlinear mixing schemes to recover ultrashort optical pulses,” Opt. Commun. 216, 413–418 (2003).
[CrossRef]

2002

2000

D. T. Reid, P. Loza-Alvarez, C. T. A. Brown, T. Beddard, and W. Sibbett, “Amplitude and phase measurement of mid-infrared femtosecond pulses by using cross-correlation frequency-resolved optical gating,” Opt. Lett. 25, 1478–1480 (2000).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

1999

H. Stolz, Ch. Nacke, B. Seifert, M. Seemann, and F. Kieseling, “Phase sensitive femtosecond spectroscopy of semiconductors,” Adv. Solid State Phys. 39, 473–482 (1999).
[CrossRef]

D. T. Reid, “Algorithm for complete and rapid retrieval of ultrashort pulse amplitude and phase from sonogram,” IEEE J. Quantum Electron. 35, 1584–1589 (1999).
[CrossRef]

1998

M. Jütte, W. von der Osten, and H. Stolz, “Characterization of picosecond laser pulses by a bandwidth-limited time-resolved spectrum,” Opt. Commun. 157, 173–176 (1998).
[CrossRef]

S. Linden, H. Giessen, and J. Kuhl, “XFROG: a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[CrossRef]

D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
[CrossRef]

1997

1995

1993

1977

A. M. J. Huiser, P. van Toorn, and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).

1976

A. M. J. Huiser and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: II. On the uniqueness and stability,” Optik (Stuttgart) 46, 407–420 (1976).

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, “On phase retrieval in electron microscopy from image and diffraction pattern,” Optik (Stuttgart) 45, 303–316 (1976).

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Beddard, T.

Brown, C. T. A.

DeLong, K. W.

Dorrer, C.

Drenth, A. J. J.

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, “On phase retrieval in electron microscopy from image and diffraction pattern,” Optik (Stuttgart) 45, 303–316 (1976).

Ferwerda, H. A.

A. M. J. Huiser, P. van Toorn, and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).

A. M. J. Huiser and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: II. On the uniqueness and stability,” Optik (Stuttgart) 46, 407–420 (1976).

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, “On phase retrieval in electron microscopy from image and diffraction pattern,” Optik (Stuttgart) 45, 303–316 (1976).

Gallmann, L.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Giessen, H.

S. Linden, H. Giessen, and J. Kuhl, “XFROG: a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

Huiser, A. M. J.

A. M. J. Huiser, P. van Toorn, and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).

A. M. J. Huiser and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: II. On the uniqueness and stability,” Optik (Stuttgart) 46, 407–420 (1976).

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, “On phase retrieval in electron microscopy from image and diffraction pattern,” Optik (Stuttgart) 45, 303–316 (1976).

Iaconis, C.

Jütte, M.

M. Jütte, W. von der Osten, and H. Stolz, “Characterization of picosecond laser pulses by a bandwidth-limited time-resolved spectrum,” Opt. Commun. 157, 173–176 (1998).
[CrossRef]

Kane, D. J.

D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
[CrossRef]

R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[CrossRef]

Kang, I.

Keller, U.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Kieseling, F.

H. Stolz, Ch. Nacke, B. Seifert, M. Seemann, and F. Kieseling, “Phase sensitive femtosecond spectroscopy of semiconductors,” Adv. Solid State Phys. 39, 473–482 (1999).
[CrossRef]

Kuhl, J.

S. Linden, H. Giessen, and J. Kuhl, “XFROG: a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

Linden, S.

S. Linden, H. Giessen, and J. Kuhl, “XFROG: a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

Loza-Alvarez, P.

Matuschek, N.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Nacke, Ch.

H. Stolz, Ch. Nacke, B. Seifert, M. Seemann, and F. Kieseling, “Phase sensitive femtosecond spectroscopy of semiconductors,” Adv. Solid State Phys. 39, 473–482 (1999).
[CrossRef]

Reid, D. T.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Seemann, M.

H. Stolz, Ch. Nacke, B. Seifert, M. Seemann, and F. Kieseling, “Phase sensitive femtosecond spectroscopy of semiconductors,” Adv. Solid State Phys. 39, 473–482 (1999).
[CrossRef]

Seifert, B.

B. Seifert and H. Stolz, “Effect of group velocity mismatch in nonlinear mixing schemes to recover ultrashort optical pulses,” Opt. Commun. 216, 413–418 (2003).
[CrossRef]

H. Stolz, Ch. Nacke, B. Seifert, M. Seemann, and F. Kieseling, “Phase sensitive femtosecond spectroscopy of semiconductors,” Adv. Solid State Phys. 39, 473–482 (1999).
[CrossRef]

Sibbett, W.

Steinmeyer, G.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Stolz, H.

B. Seifert and H. Stolz, “Effect of group velocity mismatch in nonlinear mixing schemes to recover ultrashort optical pulses,” Opt. Commun. 216, 413–418 (2003).
[CrossRef]

H. Stolz, Ch. Nacke, B. Seifert, M. Seemann, and F. Kieseling, “Phase sensitive femtosecond spectroscopy of semiconductors,” Adv. Solid State Phys. 39, 473–482 (1999).
[CrossRef]

M. Jütte, W. von der Osten, and H. Stolz, “Characterization of picosecond laser pulses by a bandwidth-limited time-resolved spectrum,” Opt. Commun. 157, 173–176 (1998).
[CrossRef]

Sutter, D. H.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Trebino, R.

van Toorn, P.

A. M. J. Huiser, P. van Toorn, and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).

von der Osten, W.

M. Jütte, W. von der Osten, and H. Stolz, “Characterization of picosecond laser pulses by a bandwidth-limited time-resolved spectrum,” Opt. Commun. 157, 173–176 (1998).
[CrossRef]

Walmsley, I. A.

White, W. E.

Wong, V.

Adv. Solid State Phys.

H. Stolz, Ch. Nacke, B. Seifert, M. Seemann, and F. Kieseling, “Phase sensitive femtosecond spectroscopy of semiconductors,” Adv. Solid State Phys. 39, 473–482 (1999).
[CrossRef]

Appl. Phys. B

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

IEEE J. Quantum Electron.

D. T. Reid, “Algorithm for complete and rapid retrieval of ultrashort pulse amplitude and phase from sonogram,” IEEE J. Quantum Electron. 35, 1584–1589 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

M. Jütte, W. von der Osten, and H. Stolz, “Characterization of picosecond laser pulses by a bandwidth-limited time-resolved spectrum,” Opt. Commun. 157, 173–176 (1998).
[CrossRef]

B. Seifert and H. Stolz, “Effect of group velocity mismatch in nonlinear mixing schemes to recover ultrashort optical pulses,” Opt. Commun. 216, 413–418 (2003).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, “On phase retrieval in electron microscopy from image and diffraction pattern,” Optik (Stuttgart) 45, 303–316 (1976).

A. M. J. Huiser and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: II. On the uniqueness and stability,” Optik (Stuttgart) 46, 407–420 (1976).

A. M. J. Huiser, P. van Toorn, and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern: III. The development of an algorithm,” Optik (Stuttgart) 47, 1–8 (1977).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Phys. Status Solidi B

S. Linden, H. Giessen, and J. Kuhl, “XFROG: a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

Other

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

J. Aczél, Lectures on Functional Equations and their Applications (Academic, New York, 1966).

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, Dordrecht, The Netherlands, 2002), p. 345.

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, Dordrecht, The Netherlands, 2002).

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Figures (9)

Fig. 1
Fig. 1

Example of a centrosymmetric FROG trace. The center positions are given by the function τ0(Ω). Centrosymmetry means inversion symmetry in τ with respect to the center τ0.

Fig. 2
Fig. 2

Transformation 1 is unique for noncentrosymmetric FROG traces and distinguishable spectral intensities of the pulses apart from the function F, which can be choosen arbitrarily. Transformation 2 is always unique.

Fig. 3
Fig. 3

(A) Two pulses with identical intensities and phases. A=-1/20, B=1/20. (B) Two pulses with different intensities and phases. a1=-1/10, b1=1/(102), a2=-1/30, b2=1/(302). The pulse pairs (A) and (B) yield analytically identical FROG traces.

Fig. 4
Fig. 4

(A) Pulse defined by a˜1=-8 and b˜1=-12. (B) Pulse defined by a˜2=-2 and b˜2=1. The pulse pair in this figure yields the same FROG trace as that generated by the pulse pair in Fig. 5.

Fig. 5
Fig. 5

(A) Pulse defined by a˜1=-8 and b˜1,new=5.6. (B) Pulse defined by a˜2=-2 and b˜2,new=5.4. The pulse pair in this figure yields the same FROG trace as that generated by the pulse pair in Fig. 4.

Fig. 6
Fig. 6

Original pulse pair. (A) |E1(t)| according to Eq. (54); (B) |E2(t)| is the modulus of the second harmonic of E1(t). For details see text.

Fig. 7
Fig. 7

Ambiguous pulse pair for the second FROG trace with a very small error G.

Fig. 8
Fig. 8

(A) FROG trace produced by the pulse (54) and its second harmonic. (B) Modulus of the difference between the FROG trace of the original pulse pair and the FROG trace of the ambiguous pulse pair. The maximum value in (B) is 13%. Note that differences exist only in a very small region of the FROG trace. The increase in signal from line to line corresponds to 5.0% of the maximum signal.

Fig. 9
Fig. 9

(A) |EF|=|EFROG(Ω, 0)| is the modulus of the FROG field at τ=0. Due to the low error G, the difference between |EF| and |EF,ambig| is below the width of the lines. (B) The argument of EFROG,ambig(Ω, 0)/EFROG(Ω, 0) changes in the Ω direction. Several π jumps appear.

Equations (58)

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zE˜3(Ω, Θ, z)
=-iχ(2) Ωc2k3 -E˜1(Ω)×E˜2(Ω-Ω)exp[-iΔk(Ω, Ω, Θ)z]dΩ,
E˜2(Ω-Ω)-E2(t)exp[i(Ω-Ω)t]dt
E˜1(Ω)-E1(t-τ)exp(iΩt)dt.
EFROG(Ω, τ)-dtE1(t-τ)E2(t)exp(iΩt)
-dωE˜1(ω)E˜2(Ω-ω)exp(iωτ)
-dt-dω exp(iΩt+iωτ)exp(-iωt)E˜1(ω)E2(t).
IFROG(Ω, τ)|EFROG(Ω, τ)|2.
IFROG(Ω, τ)-dωG(ω, Ω)exp{iP(ω, Ω)}exp(iωτ)2,
G(ω, Ω)=|E˜1(ω)||E˜2(Ω-ω)|,
P(ω, Ω)=ϕ1(ω)+ϕ2(Ω-ω),
|EF,Ω(τ)|=-dωGΩ(ω)exp{iPΩ(ω)}exp(iωτ),
P(ω, Ω)=ϕ1(ω)+ϕ2(Ω-ω),
h(ω, Ω)=ϕ1(ω)+ϕ2(Ω-ω)+F(Ω).
-PΩ(ω)-2ωτ0+C
-PΩ(2ω0-ω)+C
h(ω, Ω)=ϕ1(ω)+ϕ2(Ω-ω)+F(Ω)
h(ω, Ω)=ϕ˜1(ω)+ϕ˜2(Ω-ω)+F˜(Ω)
ϕ˜1(ω)=ϕ1(ω)+cω-a,
ϕ˜2(Ω)=ϕ2(Ω)+cΩ+a+b,
F˜(Ω)=F(Ω)-cΩ-b,
f(Ω-ω)=g(ω)+d(Ω),
f(Ω)=ϕ˜2(Ω)-ϕ2(Ω),
g(ω)=ϕ1(ω)-ϕ˜1(ω),
d(Ω)=F(Ω)-F˜(Ω).
f(Ω+ω)=f(Ω)+f(ω)-a-b.
Ψ(Ω+ω)=Ψ(Ω)+Ψ(ω).
E1(t)=exp[(a1+ib1)t2],
E2(t)=exp[(a2+ib2)t2].
EFROG(Ω, τ)=π×exp(a1+ib1)τ2-(2a1τ+2ib1τ-iΩ)24[a1+a2+i(b1+b2)]
[-a1-a2-i(b1+b2)]1/2.
IFROG(Ω, τ)exp{[4τ2(a12a2+a22a1)+a1×(Ω+2b2τ)2+a2(Ω-2b1τ)2]/[2((a1+a2)2+(b1+b2)2)]}.
(a1+a2)/2(a1+a2)2+(b1+b2)2,
2(a1b2-a2b1)(a1+a2)2+(b1+b2)2,
2[a1(a22+b22)+a2(a12+b12)](a1+a2)2+(b1+b2)2.
(a1+a2)/2(a1+a2)2+(b1+b2)2=A4(A2+B2),
2(a1b2-a2b1)(a1+a2)2+(b1+b2)2=0,
2[a1(a22+b22)+a2(a12+b12)](a1+a2)2+(b1+b2)2=A.
a2=-Aa1A-2a1,
b1=[(2a1-A)(A2+B2)-a12A]1/2A,
b2=-A[(2a1-A)(A2+B2)-a12A]1/2A-2a1.
E˜1(ω)=exp[(a˜1+ib˜1)ω2],
E˜2(ω)=exp[(a˜2+ib˜2)ω2],
IFROG(Ω, τ)exp({τ2(a˜1+a˜2)+4τΩ(a˜2b˜1-a˜1b˜2)+4Ω2[a˜1(a˜22+b˜22)+a˜2(a˜12+b˜12)]}/{2[(a˜1+a˜2)2+(b˜1+b˜2)2]}).
P1(ω, Ω)=b˜1ω2+b˜2(Ω-ω)2
=(b˜1+b˜2)ω2-b˜22Ωω+b˜2Ω2.
P2(ω, Ω)=-P1(ω, Ω)-2ωτ0(Ω)+Fc(Ω),
τ0(Ω)=2Ω(a˜1b˜2-a˜2b˜1)a˜1+a˜2=kΩ
k=2 a˜1b˜2-a˜2b˜1a˜1+a˜2.
Fc(Ω)=kΩ2
P2(ω, Ω)=-(b˜1+b˜2)ω2+(b˜2-k)2Ωω-(b˜2-k)Ω2
=b˜1,newω2+b˜2,new(Ω-ω)2.
b˜2,new=-(b˜2-k),
b˜1,new=-(b˜1+k),
a˜1,new=a˜1,
a˜2,new=a˜2.
G=1N2 i,j=1N[IFROG(Ωi, τj)-μIFROG,ambig(Ωi, τj)]21/2,
E1(t)=[cos2(at)+b]×{exp[(c+id)(t-t0,1)2]+exp[(c+id)(t-t0,2)2]}

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