Abstract

Pump-probe microscopy is an imaging technique that delivers molecular contrast of pigmented samples. Here, we introduce pump-probe nonlinear phase dispersion spectroscopy (PP-NLDS), a method that leverages pump-probe microscopy and spectral-domain interferometry to ascertain information from dispersive and resonant nonlinear effects. PP-NLDS extends the information content to four dimensions (phase, amplitude, wavelength, and pump-probe time-delay) that yield unique insight into a wider range of nonlinear interactions compared to conventional methods. This results in the ability to provide highly specific molecular contrast of pigmented and non-pigmented samples. A theoretical framework is described, and experimental results and simulations illustrate the potential of this method. Implications for biomedical imaging are discussed.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2012 (3)

2011 (4)

T. E. Matthews, I. R. Piletic, M. A. Selim, M. J. Simpson, and W. S. Warren, “Pump-probe imaging differentiates melanoma from melanocytic nevi,” Sci. Transl. Med.3(71), 71ra15 (2011).
[CrossRef] [PubMed]

T. E. Matthews, J. W. Wilson, S. Degan, M. J. Simpson, J. Y. Jin, J. Y. Zhang, and W. S. Warren, “In vivo and ex vivo epi-mode pump-probe imaging of melanin and microvasculature,” Biomed. Opt. Express2(6), 1576–1583 (2011).
[CrossRef] [PubMed]

F. E. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nat. Photonics5(12), 744–747 (2011).
[CrossRef] [PubMed]

F. E. Robles, L. L. Satterwhite, and A. Wax, “Nonlinear phase dispersion spectroscopy,” Opt. Lett.36(23), 4665–4667 (2011).
[CrossRef] [PubMed]

2010 (4)

2008 (3)

J. W. Wilson, P. Schlup, and R. A. Bartels, “Synthetic temporal aperture coherent molecular phase spectroscopy,” Chem. Phys. Lett.463(4-6), 300–304 (2008).
[CrossRef]

J. W. Wilson, P. Schlup, and R. Bartels, “Phase measurement of coherent Raman vibrational spectroscopy with chirped spectral holography,” Opt. Lett.33(18), 2116–2118 (2008).
[CrossRef] [PubMed]

D. Fu, T. E. Matthews, T. Ye, I. R. Piletic, and W. S. Warren, “Label-free in vivo optical imaging of microvasculature and oxygenation level,” J. Biomed. Opt.13(4), 040503 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

2005 (3)

2004 (1)

D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett.92(12), 123905 (2004).
[CrossRef] [PubMed]

2001 (1)

E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Femtosecond dynamics of intracellular water probed with nonlinear optical Kerr effect microspectroscopy,” Biophys. J.80(6), 3019–3024 (2001).
[CrossRef] [PubMed]

2000 (1)

1995 (1)

1993 (1)

Y. J. Chang and E. W. Castner, “Femtosecond dynamics of hydrogen-bonding solvents. Formamide and N-methylformamide in acetonitrile, DMF, and water,” J. Chem. Phys.99(1), 113 (1993).
[CrossRef]

1988 (1)

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron.24(2), 443–454 (1988).
[CrossRef]

Applegate, B. E.

Bartels, R.

Bartels, R. A.

J. W. Wilson, P. Schlup, and R. A. Bartels, “Synthetic temporal aperture coherent molecular phase spectroscopy,” Chem. Phys. Lett.463(4-6), 300–304 (2008).
[CrossRef]

Benalcazar, W. A.

P. D. Chowdary, Z. Jiang, E. J. Chaney, W. A. Benalcazar, D. L. Marks, M. Gruebele, and S. A. Boppart, “Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging,” Cancer Res.70(23), 9562–9569 (2010).
[CrossRef] [PubMed]

Boppart, S. A.

P. D. Chowdary, Z. Jiang, E. J. Chaney, W. A. Benalcazar, D. L. Marks, M. Gruebele, and S. A. Boppart, “Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging,” Cancer Res.70(23), 9562–9569 (2010).
[CrossRef] [PubMed]

D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett.92(12), 123905 (2004).
[CrossRef] [PubMed]

Buckup, T.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys.123(5), 054509 (2005).
[CrossRef] [PubMed]

Castner, E. W.

Y. J. Chang and E. W. Castner, “Femtosecond dynamics of hydrogen-bonding solvents. Formamide and N-methylformamide in acetonitrile, DMF, and water,” J. Chem. Phys.99(1), 113 (1993).
[CrossRef]

Chaney, E. J.

P. D. Chowdary, Z. Jiang, E. J. Chaney, W. A. Benalcazar, D. L. Marks, M. Gruebele, and S. A. Boppart, “Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging,” Cancer Res.70(23), 9562–9569 (2010).
[CrossRef] [PubMed]

Chang, Y. J.

Y. J. Chang and E. W. Castner, “Femtosecond dynamics of hydrogen-bonding solvents. Formamide and N-methylformamide in acetonitrile, DMF, and water,” J. Chem. Phys.99(1), 113 (1993).
[CrossRef]

Chen, B. J.

Choma, M. A.

Chowdary, P. D.

P. D. Chowdary, Z. Jiang, E. J. Chaney, W. A. Benalcazar, D. L. Marks, M. Gruebele, and S. A. Boppart, “Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging,” Cancer Res.70(23), 9562–9569 (2010).
[CrossRef] [PubMed]

Ciocca, M.

Correia, R. R. B.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys.123(5), 054509 (2005).
[CrossRef] [PubMed]

Creazzo, T. L.

Cunha, S. L. S.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys.123(5), 054509 (2005).
[CrossRef] [PubMed]

da Silveira, N. P.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys.123(5), 054509 (2005).
[CrossRef] [PubMed]

de Boeij, W. P.

E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Femtosecond dynamics of intracellular water probed with nonlinear optical Kerr effect microspectroscopy,” Biophys. J.80(6), 3019–3024 (2001).
[CrossRef] [PubMed]

deCruz, A.

Degan, S.

Drexler, W.

Ellerbee, A. K.

Fischer, M. C.

Fu, D.

D. Fu, T. E. Matthews, T. Ye, I. R. Piletic, and W. S. Warren, “Label-free in vivo optical imaging of microvasculature and oxygenation level,” J. Biomed. Opt.13(4), 040503 (2008).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, “High-resolution in vivo imaging of blood vessels without labeling,” Opt. Lett.32(18), 2641–2643 (2007).
[CrossRef] [PubMed]

Fujimoto, J. G.

Grant, G.

F. E. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nat. Photonics5(12), 744–747 (2011).
[CrossRef] [PubMed]

Gruebele, M.

P. D. Chowdary, Z. Jiang, E. J. Chaney, W. A. Benalcazar, D. L. Marks, M. Gruebele, and S. A. Boppart, “Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging,” Cancer Res.70(23), 9562–9569 (2010).
[CrossRef] [PubMed]

Heisler, I. A.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys.123(5), 054509 (2005).
[CrossRef] [PubMed]

Ippen, E. P.

Izatt, J. A.

Jacob, D.

Jiang, Z.

P. D. Chowdary, Z. Jiang, E. J. Chaney, W. A. Benalcazar, D. L. Marks, M. Gruebele, and S. A. Boppart, “Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging,” Cancer Res.70(23), 9562–9569 (2010).
[CrossRef] [PubMed]

Jin, J. Y.

Kärtner, F. X.

Kenney-Wallace, G. A.

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron.24(2), 443–454 (1988).
[CrossRef]

Kobayashi, T.

Li, B.

Li, X. D.

Lotshaw, W. T.

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron.24(2), 443–454 (1988).
[CrossRef]

Marks, D. L.

P. D. Chowdary, Z. Jiang, E. J. Chaney, W. A. Benalcazar, D. L. Marks, M. Gruebele, and S. A. Boppart, “Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging,” Cancer Res.70(23), 9562–9569 (2010).
[CrossRef] [PubMed]

D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett.92(12), 123905 (2004).
[CrossRef] [PubMed]

Matthews, T. E.

T. E. Matthews, I. R. Piletic, M. A. Selim, M. J. Simpson, and W. S. Warren, “Pump-probe imaging differentiates melanoma from melanocytic nevi,” Sci. Transl. Med.3(71), 71ra15 (2011).
[CrossRef] [PubMed]

T. E. Matthews, J. W. Wilson, S. Degan, M. J. Simpson, J. Y. Jin, J. Y. Zhang, and W. S. Warren, “In vivo and ex vivo epi-mode pump-probe imaging of melanin and microvasculature,” Biomed. Opt. Express2(6), 1576–1583 (2011).
[CrossRef] [PubMed]

D. Fu, T. E. Matthews, T. Ye, I. R. Piletic, and W. S. Warren, “Label-free in vivo optical imaging of microvasculature and oxygenation level,” J. Biomed. Opt.13(4), 040503 (2008).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, “High-resolution in vivo imaging of blood vessels without labeling,” Opt. Lett.32(18), 2641–2643 (2007).
[CrossRef] [PubMed]

McMorrow, D.

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron.24(2), 443–454 (1988).
[CrossRef]

Miller, A.

Morgner, U.

Perret, Z.

Piletic, I. R.

T. E. Matthews, I. R. Piletic, M. A. Selim, M. J. Simpson, and W. S. Warren, “Pump-probe imaging differentiates melanoma from melanocytic nevi,” Sci. Transl. Med.3(71), 71ra15 (2011).
[CrossRef] [PubMed]

D. Fu, T. E. Matthews, T. Ye, I. R. Piletic, and W. S. Warren, “Label-free in vivo optical imaging of microvasculature and oxygenation level,” J. Biomed. Opt.13(4), 040503 (2008).
[CrossRef] [PubMed]

Pitris, C.

Potma, E. O.

E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Femtosecond dynamics of intracellular water probed with nonlinear optical Kerr effect microspectroscopy,” Biophys. J.80(6), 3019–3024 (2001).
[CrossRef] [PubMed]

Robles, F. E.

Samineni, P.

Satterwhite, L. L.

Schlup, P.

J. W. Wilson, P. Schlup, and R. Bartels, “Phase measurement of coherent Raman vibrational spectroscopy with chirped spectral holography,” Opt. Lett.33(18), 2116–2118 (2008).
[CrossRef] [PubMed]

J. W. Wilson, P. Schlup, and R. A. Bartels, “Synthetic temporal aperture coherent molecular phase spectroscopy,” Chem. Phys. Lett.463(4-6), 300–304 (2008).
[CrossRef]

Selim, M. A.

T. E. Matthews, I. R. Piletic, M. A. Selim, M. J. Simpson, and W. S. Warren, “Pump-probe imaging differentiates melanoma from melanocytic nevi,” Sci. Transl. Med.3(71), 71ra15 (2011).
[CrossRef] [PubMed]

Shelton, R. L.

Simpson, M. J.

T. E. Matthews, I. R. Piletic, M. A. Selim, M. J. Simpson, and W. S. Warren, “Pump-probe imaging differentiates melanoma from melanocytic nevi,” Sci. Transl. Med.3(71), 71ra15 (2011).
[CrossRef] [PubMed]

T. E. Matthews, J. W. Wilson, S. Degan, M. J. Simpson, J. Y. Jin, J. Y. Zhang, and W. S. Warren, “In vivo and ex vivo epi-mode pump-probe imaging of melanin and microvasculature,” Biomed. Opt. Express2(6), 1576–1583 (2011).
[CrossRef] [PubMed]

Terasakiy, A.

Tokunaga, E.

Villafaña, T. E.

Wagner, W.

Warren, W. S.

P. Samineni, A. deCruz, T. E. Villafaña, W. S. Warren, and M. C. Fischer, “Pump-probe imaging of historical pigments used in paintings,” Opt. Lett.37(8), 1310–1312 (2012).
[CrossRef] [PubMed]

J. W. Wilson, P. Samineni, W. S. Warren, and M. C. Fischer, “Cross-phase modulation spectral shifting: nonlinear phase contrast in a pump-probe microscope,” Biomed. Opt. Express3(5), 854–862 (2012).
[CrossRef] [PubMed]

P. Samineni, B. Li, J. W. Wilson, W. S. Warren, and M. C. Fischer, “Cross-phase modulation imaging,” Opt. Lett.37(5), 800–802 (2012).
[CrossRef] [PubMed]

T. E. Matthews, J. W. Wilson, S. Degan, M. J. Simpson, J. Y. Jin, J. Y. Zhang, and W. S. Warren, “In vivo and ex vivo epi-mode pump-probe imaging of melanin and microvasculature,” Biomed. Opt. Express2(6), 1576–1583 (2011).
[CrossRef] [PubMed]

T. E. Matthews, I. R. Piletic, M. A. Selim, M. J. Simpson, and W. S. Warren, “Pump-probe imaging differentiates melanoma from melanocytic nevi,” Sci. Transl. Med.3(71), 71ra15 (2011).
[CrossRef] [PubMed]

P. Samineni, Z. Perret, W. S. Warren, and M. C. Fischer, “Measurements of nonlinear refractive index in scattering media,” Opt. Express18(12), 12727–12735 (2010).
[CrossRef] [PubMed]

D. Fu, T. E. Matthews, T. Ye, I. R. Piletic, and W. S. Warren, “Label-free in vivo optical imaging of microvasculature and oxygenation level,” J. Biomed. Opt.13(4), 040503 (2008).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, “High-resolution in vivo imaging of blood vessels without labeling,” Opt. Lett.32(18), 2641–2643 (2007).
[CrossRef] [PubMed]

M. C. Fischer, T. Ye, G. Yurtsever, A. Miller, M. Ciocca, W. Wagner, and W. S. Warren, “Two-photon absorption and self-phase modulation measurements with shaped femtosecond laser pulses,” Opt. Lett.30(12), 1551–1553 (2005).
[CrossRef] [PubMed]

Wax, A.

Wiersma, D. A.

E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Femtosecond dynamics of intracellular water probed with nonlinear optical Kerr effect microspectroscopy,” Biophys. J.80(6), 3019–3024 (2001).
[CrossRef] [PubMed]

Wilson, C.

F. E. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nat. Photonics5(12), 744–747 (2011).
[CrossRef] [PubMed]

Wilson, J. W.

Yang, C.

Ye, T.

Yurtserver, G.

Yurtsever, G.

Zhang, J. Y.

Biomed. Opt. Express (2)

Biophys. J. (1)

E. O. Potma, W. P. de Boeij, and D. A. Wiersma, “Femtosecond dynamics of intracellular water probed with nonlinear optical Kerr effect microspectroscopy,” Biophys. J.80(6), 3019–3024 (2001).
[CrossRef] [PubMed]

Cancer Res. (1)

P. D. Chowdary, Z. Jiang, E. J. Chaney, W. A. Benalcazar, D. L. Marks, M. Gruebele, and S. A. Boppart, “Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging,” Cancer Res.70(23), 9562–9569 (2010).
[CrossRef] [PubMed]

Chem. Phys. Lett. (1)

J. W. Wilson, P. Schlup, and R. A. Bartels, “Synthetic temporal aperture coherent molecular phase spectroscopy,” Chem. Phys. Lett.463(4-6), 300–304 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron.24(2), 443–454 (1988).
[CrossRef]

J. Biomed. Opt. (1)

D. Fu, T. E. Matthews, T. Ye, I. R. Piletic, and W. S. Warren, “Label-free in vivo optical imaging of microvasculature and oxygenation level,” J. Biomed. Opt.13(4), 040503 (2008).
[CrossRef] [PubMed]

J. Chem. Phys. (2)

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys.123(5), 054509 (2005).
[CrossRef] [PubMed]

Y. J. Chang and E. W. Castner, “Femtosecond dynamics of hydrogen-bonding solvents. Formamide and N-methylformamide in acetonitrile, DMF, and water,” J. Chem. Phys.99(1), 113 (1993).
[CrossRef]

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

Nat. Photonics (1)

F. E. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nat. Photonics5(12), 744–747 (2011).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (9)

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett.25(2), 111–113 (2000).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, “High-resolution in vivo imaging of blood vessels without labeling,” Opt. Lett.32(18), 2641–2643 (2007).
[CrossRef] [PubMed]

P. Samineni, B. Li, J. W. Wilson, W. S. Warren, and M. C. Fischer, “Cross-phase modulation imaging,” Opt. Lett.37(5), 800–802 (2012).
[CrossRef] [PubMed]

P. Samineni, A. deCruz, T. E. Villafaña, W. S. Warren, and M. C. Fischer, “Pump-probe imaging of historical pigments used in paintings,” Opt. Lett.37(8), 1310–1312 (2012).
[CrossRef] [PubMed]

M. C. Fischer, T. Ye, G. Yurtsever, A. Miller, M. Ciocca, W. Wagner, and W. S. Warren, “Two-photon absorption and self-phase modulation measurements with shaped femtosecond laser pulses,” Opt. Lett.30(12), 1551–1553 (2005).
[CrossRef] [PubMed]

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Other (1)

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Figures (6)

Fig. 1
Fig. 1

The PP-NLDS system combines a pump-probe setup with a Michelson interferometer. Dotted line in inset depicts the phase dynamics of the sample induced by the pump. SP: spectrometer, SMF: single mode fiber, T: time delay between reference and probe beams, τ: time delay between pump and probe beams.

Fig. 2
Fig. 2

Processing method for PP-NLDS. The raw signal measured by the spectrometer (a) is Fourier transformed and filtered in the time-domain to isolate the interferometric term (b). The signal is then transformed back to the frequency-domain to gain access to the complex form of the field (c). (d) Simulated OKE response for water and CS2 [23,24].

Fig. 3
Fig. 3

Experiment (a) and numerical simulations at τ = 0 fs (b) for CS2. The experiment and simulated signals are in excellent agreement. (c) Spectral phase Φ, which does not account for the random phase term, φr. Thin dashed lines represent the standard deviation of 10 measurements.

Fig. 4
Fig. 4

Experiment (a) and numerical simulations at τ = 0 fs (b) and τ = 75 fs (c) for water. A discrepancy is observed in the transmission signal of water at τ = 0. This can be rectified by considering small errors in the setup that may shift τ to different values (shift of +/− 100 fs are within experimental error, limited by the cross correlation of the pump and probe pulses ~200 fs). A simulated signal at τ = 75 fs provides excellent agreement (c). Thin dashed lines represent the standard deviation of 10 measurements.

Fig. 5
Fig. 5

Simulated nonlinear response of water, CS2 and an instantaneous electronic response [delta term in Eq. (10)] as a function of wavelength and time-delay between pump and probe, τ. Red dashed lines are located at τ = 0, corresponding to the delay of the experimental/simulated results shown in Figs. 3 and 4.

Fig. 6
Fig. 6

Experimentally measured nonlinear optical response of methanol and R6G. Phase (a) and transmission (b) changes as a function of wavelength, and transmission changes in the time-domain (c).

Equations (19)

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E ˜ pr (ω,τ)= E ˜ 0 (ω) e iωT e i( n ˜ NL (ω,τ)ω2 z r /c)
E ˜ pr (ω,τ)= E ˜ 0 (ω) e K(ω,τ) e i(ωTΦ(ω,τ))
I(ω,τ)= | E ˜ 0 (ω)+ E ˜ pr (ω,τ) | 2 = | E ˜ 0 (ω) | 2 + | E ˜ 0 (ω) | 2 e 2K(ω,τ) +2 | E ˜ 0 (ω) | 2 e K(ω,τ) cos(ωT+ φ r Φ(ω,τ))
I ˜ (ω,τ)= | E ˜ 0 (ω) | 2 e K(ω,τ) e i(ωT+ φ r Φ(ω,τ))
I ˜ off (ω)= | E ˜ 0 (ω) | 2 e i(ωT+ φ r )
K(ω,τ)=ln( | I ˜ (ω,τ) | | I ˜ off (ω) | )
ΔΓ Γ | E ˜ pr (ω,τ) | 2 | E ˜ 0 (ω) | 2 | E ˜ 0 (ω) | 2 = | I ˜ (ω,τ) | 2 | I ˜ off (ω) | 2 | I ˜ off (ω) | 2 = e 2K(ω,τ) 1
Φ'(ω,τ)Φ(ω,τ) Φ(τ) ¯ =( I ˜ (ω,τ) I ˜ (τ) ¯ )( I ˜ off (ω) I ˜ off ¯ )
E ˜ pr (t,τ)= E ˜ 0 (tT) e κ(t+τ)+iϕ(t+τ)
ϕ(t)= c 1 I pu (t)[ δ(t)+ m r m (t) ]
E ˜ 0 (t)= e a t 2 e i ω 0 t
E ˜ 0 (ω)= e a t 2 e i(ω ω 0 )t dt = 1 2a exp( Ω 2 /(4a) )
ϕ(t)= ϕ 0 +ϕ't+ϕ'' t 2 +...
ϕ(t)=ϕ'' ( t+τ ) 2
E ˜ pr (ω)= e a t 2 e iϕ'' ( t+τ ) 2 e i(ω ω 0 )t dt
= 1 2ai2ϕ'' exp( iaϕ'' τ 2 +ϕ''τΩ Ω 2 /4 aiϕ'' )
( Ω/2ϕ''τ ) 2 a iϕ'' ( Ω/2ϕ''τ ) 2 τ 2 ( ϕ' ' 2 a 2 ) a 2
E ˜ pr (Ω)= C ˜ 0 e ( Ω/2ϕ''τ ) 2 a e iϕ'' ( Ω/2ϕ''τ ) 2 a 2
E ˜ pr (Ω) C ˜ E ˜ 0 e ϕ''τΩ/a e iϕ'' ( Ω/2a ) 2

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