Abstract

We present a noninterferometric method for direct and complete characterization of the degree of first-order temporal coherence γ(τ) of low-coherence broadband sources both in amplitude and phase. The method is based on measuring the radio-frequency transfer function HRF(f) of amplitude-modulated partially coherent guided waves in a system with first-order dispersion. Experimental data for sources based on amplitude-modulated amplified spontaneous emission in the 1.55μm band having spectral widths up to 30nm and subsequently dispersed in a 12.77km standard single-mode fiber coil provide complete characterization of the temporal coherence with 6.5fs resolution and a dynamic range exceeding 20dB.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2006 (4)

J. H. Lee, C. H. Kim, Y.-G. Han, and S. B. Lee, “Broadband, high-power, erbium fiber ASE-based CW supercontinuum source for spectrum-sliced WDM PON applications,” Electron. Lett. 42, 549-550 (2006).
[CrossRef]

H. Gouraud, P. Di Bin, L. Billonnet, P. Faugeras, and B. Jarry, “Optical fiber dispersion properties management for optimized bandpass microwave photonics slicing filter applications,” Opt. Commun. 265, 506-512 (2006).
[CrossRef]

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially-coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
[CrossRef]

J. Mora, B. Ortega, A. Díez, J. L. Cruz, M. V. Andrés, José Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on a Mach-Zehnder interferometer,” J. Lightwave Technol. 24, 2500-2509 (2006).
[CrossRef]

2005 (2)

José Capmany, A. Martínez, B. Ortega, and D. Pastor, “Transfer function of analog fiber-optics systems driven by Fabry-Perot lasers,” J. Opt. Soc. Am. B 22, 2099-2106 (2005).
[CrossRef]

B. Vidal, J. L. Corral, and J. Martí, “All-optical WDM microwave filter with negative coefficients,” IEEE Photon. Technol. Lett. 17, 666-668 (2005).
[CrossRef]

2004 (1)

2003 (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical Coherence Tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

2001 (1)

2000 (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

1999 (2)

Y. Teramura, K. Suzuki, M. Suzuki, and F. Kannari, “Low-coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974-5980 (1999).
[CrossRef]

J. M. Schmitt, “Optical Coherence Tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205-1215 (1999).
[CrossRef]

1998 (1)

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 26, 757-771 (1998).

1996 (1)

1995 (1)

H. Schmuck, “Comparison of optical millimeter-wave system concept with regard to chromatic dispersion,” Electron. Lett. 31, 1848-1849 (1995).
[CrossRef]

1994 (2)

P. F. Wysocki, M. J. F. Digonnet, B. Y. Kim, and H. J. Shaw, “Characteristics of erbium-doped superfluorescent fiber sources for interferometric sensor applications,” J. Lightwave Technol. 12, 550-567 (1994).
[CrossRef]

D. D. Sampson and W. T. Holloway, “100 mW spectrally uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611-1612 (1994).
[CrossRef]

1993 (1)

1991 (1)

1990 (1)

P. F. Wysocki, M. J. F. Digonnet, and B. Y. Kim, “Spectral characteristics of high-power 1.5 mm broad-band superluminiscent fiber sources,” IEEE Photon. Technol. Lett. 2, 178-180 (1990).
[CrossRef]

1982 (1)

Andrés, M. V.

Billonnet, L.

H. Gouraud, P. Di Bin, L. Billonnet, P. Faugeras, and B. Jarry, “Optical fiber dispersion properties management for optimized bandpass microwave photonics slicing filter applications,” Opt. Commun. 265, 506-512 (2006).
[CrossRef]

Binjrajka, V.

Bone, D. J.

Capmany, José

J. Mora, B. Ortega, A. Díez, J. L. Cruz, M. V. Andrés, José Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on a Mach-Zehnder interferometer,” J. Lightwave Technol. 24, 2500-2509 (2006).
[CrossRef]

José Capmany, A. Martínez, B. Ortega, and D. Pastor, “Transfer function of analog fiber-optics systems driven by Fabry-Perot lasers,” J. Opt. Soc. Am. B 22, 2099-2106 (2005).
[CrossRef]

J. Mora, José Capmany, and L. R. Chen, “Tunable and reconfigurable single bandpass photonic microwave filter using a high-birefringence Sagnac loop and DWDM channel selector,” in Proc. of the 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (LEOS, 2007) (IEEE, 2007), pp. 192-193.
[CrossRef]

Capmany, Juan

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany are preparing a paper to be called “Hilbert and Blaschke phases in the coherence function of stationary broadband light.”

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany, “Complete determination of the first-order degree of coherence of amplified spontaneous emission sources with femtosecond resolution,” presented at the Conference on Lasers and Electro-Optics and the Quantum Electronics and Laser Science Conference (CLEO/QELS 08), San Jose, Calif., May 4-9, 2008, paper CMU5.

Chang, C.-C.

Chantada, L.

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially-coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
[CrossRef]

Chen, L. R.

J. Mora, José Capmany, and L. R. Chen, “Tunable and reconfigurable single bandpass photonic microwave filter using a high-birefringence Sagnac loop and DWDM channel selector,” in Proc. of the 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (LEOS, 2007) (IEEE, 2007), pp. 192-193.
[CrossRef]

Ciprut, P.

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 26, 757-771 (1998).

Corral, J. L.

B. Vidal, J. L. Corral, and J. Martí, “All-optical WDM microwave filter with negative coefficients,” IEEE Photon. Technol. Lett. 17, 666-668 (2005).
[CrossRef]

Cruz, J. L.

Derickson, D.

D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).

Di Bin, P.

H. Gouraud, P. Di Bin, L. Billonnet, P. Faugeras, and B. Jarry, “Optical fiber dispersion properties management for optimized bandpass microwave photonics slicing filter applications,” Opt. Commun. 265, 506-512 (2006).
[CrossRef]

Díez, A.

Digonnet, M. J. F.

P. F. Wysocki, M. J. F. Digonnet, B. Y. Kim, and H. J. Shaw, “Characteristics of erbium-doped superfluorescent fiber sources for interferometric sensor applications,” J. Lightwave Technol. 12, 550-567 (1994).
[CrossRef]

P. F. Wysocki, M. J. F. Digonnet, and B. Y. Kim, “Spectral characteristics of high-power 1.5 mm broad-band superluminiscent fiber sources,” IEEE Photon. Technol. Lett. 2, 178-180 (1990).
[CrossRef]

Dorrer, C.

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical Coherence Tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Emanuel, A. W. R.

Faugeras, P.

H. Gouraud, P. Di Bin, L. Billonnet, P. Faugeras, and B. Jarry, “Optical fiber dispersion properties management for optimized bandpass microwave photonics slicing filter applications,” Opt. Commun. 265, 506-512 (2006).
[CrossRef]

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical Coherence Tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Fernández-Pousa, C. R.

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially-coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
[CrossRef]

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany, “Complete determination of the first-order degree of coherence of amplified spontaneous emission sources with femtosecond resolution,” presented at the Conference on Lasers and Electro-Optics and the Quantum Electronics and Laser Science Conference (CLEO/QELS 08), San Jose, Calif., May 4-9, 2008, paper CMU5.

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany are preparing a paper to be called “Hilbert and Blaschke phases in the coherence function of stationary broadband light.”

Gisin, B.

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 26, 757-771 (1998).

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 26, 757-771 (1998).

Gómez-Reino, C.

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially-coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
[CrossRef]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Gouraud, H.

H. Gouraud, P. Di Bin, L. Billonnet, P. Faugeras, and B. Jarry, “Optical fiber dispersion properties management for optimized bandpass microwave photonics slicing filter applications,” Opt. Commun. 265, 506-512 (2006).
[CrossRef]

Han, Y.-G.

J. H. Lee, C. H. Kim, Y.-G. Han, and S. B. Lee, “Broadband, high-power, erbium fiber ASE-based CW supercontinuum source for spectrum-sliced WDM PON applications,” Electron. Lett. 42, 549-550 (2006).
[CrossRef]

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical Coherence Tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Holloway, W. T.

D. D. Sampson and W. T. Holloway, “100 mW spectrally uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611-1612 (1994).
[CrossRef]

Irshid, M. I.

Jackson, D. A.

Jarry, B.

H. Gouraud, P. Di Bin, L. Billonnet, P. Faugeras, and B. Jarry, “Optical fiber dispersion properties management for optimized bandpass microwave photonics slicing filter applications,” Opt. Commun. 265, 506-512 (2006).
[CrossRef]

Kannari, F.

Kim, B. Y.

P. F. Wysocki, M. J. F. Digonnet, B. Y. Kim, and H. J. Shaw, “Characteristics of erbium-doped superfluorescent fiber sources for interferometric sensor applications,” J. Lightwave Technol. 12, 550-567 (1994).
[CrossRef]

P. F. Wysocki, M. J. F. Digonnet, and B. Y. Kim, “Spectral characteristics of high-power 1.5 mm broad-band superluminiscent fiber sources,” IEEE Photon. Technol. Lett. 2, 178-180 (1990).
[CrossRef]

Kim, C. H.

J. H. Lee, C. H. Kim, Y.-G. Han, and S. B. Lee, “Broadband, high-power, erbium fiber ASE-based CW supercontinuum source for spectrum-sliced WDM PON applications,” Electron. Lett. 42, 549-550 (2006).
[CrossRef]

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical Coherence Tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Leaird, D. E.

Lee, J. H.

J. H. Lee, C. H. Kim, Y.-G. Han, and S. B. Lee, “Broadband, high-power, erbium fiber ASE-based CW supercontinuum source for spectrum-sliced WDM PON applications,” Electron. Lett. 42, 549-550 (2006).
[CrossRef]

Lee, S. B.

J. H. Lee, C. H. Kim, Y.-G. Han, and S. B. Lee, “Broadband, high-power, erbium fiber ASE-based CW supercontinuum source for spectrum-sliced WDM PON applications,” Electron. Lett. 42, 549-550 (2006).
[CrossRef]

Maestre, H.

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany, “Complete determination of the first-order degree of coherence of amplified spontaneous emission sources with femtosecond resolution,” presented at the Conference on Lasers and Electro-Optics and the Quantum Electronics and Laser Science Conference (CLEO/QELS 08), San Jose, Calif., May 4-9, 2008, paper CMU5.

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany are preparing a paper to be called “Hilbert and Blaschke phases in the coherence function of stationary broadband light.”

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Martí, J.

B. Vidal, J. L. Corral, and J. Martí, “All-optical WDM microwave filter with negative coefficients,” IEEE Photon. Technol. Lett. 17, 666-668 (2005).
[CrossRef]

Martínez, A.

Mora, J.

J. Mora, B. Ortega, A. Díez, J. L. Cruz, M. V. Andrés, José Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on a Mach-Zehnder interferometer,” J. Lightwave Technol. 24, 2500-2509 (2006).
[CrossRef]

J. Mora, José Capmany, and L. R. Chen, “Tunable and reconfigurable single bandpass photonic microwave filter using a high-birefringence Sagnac loop and DWDM channel selector,” in Proc. of the 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (LEOS, 2007) (IEEE, 2007), pp. 192-193.
[CrossRef]

Ning, Y. N.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice-Hall, 1989).

Ortega, B.

Passy, R.

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 26, 757-771 (1998).

Pastor, D.

Prieto, F.

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 26, 757-771 (1998).

Rao, Y. J.

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Saleh, B. E. A.

Sampson, D. D.

D. D. Sampson and W. T. Holloway, “100 mW spectrally uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611-1612 (1994).
[CrossRef]

Sato, M.

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice-Hall, 1989).

Schmitt, J. M.

J. M. Schmitt, “Optical Coherence Tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205-1215 (1999).
[CrossRef]

Schmuck, H.

H. Schmuck, “Comparison of optical millimeter-wave system concept with regard to chromatic dispersion,” Electron. Lett. 31, 1848-1849 (1995).
[CrossRef]

Shaw, H. J.

P. F. Wysocki, M. J. F. Digonnet, B. Y. Kim, and H. J. Shaw, “Characteristics of erbium-doped superfluorescent fiber sources for interferometric sensor applications,” J. Lightwave Technol. 12, 550-567 (1994).
[CrossRef]

Suzuki, K.

Suzuki, M.

Tanno, N.

Teramura, Y.

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Torregrosa, A. J.

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany are preparing a paper to be called “Hilbert and Blaschke phases in the coherence function of stationary broadband light.”

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany, “Complete determination of the first-order degree of coherence of amplified spontaneous emission sources with femtosecond resolution,” presented at the Conference on Lasers and Electro-Optics and the Quantum Electronics and Laser Science Conference (CLEO/QELS 08), San Jose, Calif., May 4-9, 2008, paper CMU5.

Vidal, B.

B. Vidal, J. L. Corral, and J. Martí, “All-optical WDM microwave filter with negative coefficients,” IEEE Photon. Technol. Lett. 17, 666-668 (2005).
[CrossRef]

Von der Weid, J. P.

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 26, 757-771 (1998).

Weiner, A. M.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Wysocki, P. F.

P. F. Wysocki, M. J. F. Digonnet, B. Y. Kim, and H. J. Shaw, “Characteristics of erbium-doped superfluorescent fiber sources for interferometric sensor applications,” J. Lightwave Technol. 12, 550-567 (1994).
[CrossRef]

P. F. Wysocki, M. J. F. Digonnet, and B. Y. Kim, “Spectral characteristics of high-power 1.5 mm broad-band superluminiscent fiber sources,” IEEE Photon. Technol. Lett. 2, 178-180 (1990).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford Univ. Press, 2007).

Yeh, P.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford Univ. Press, 2007).

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Zhang, Y.

Zimmer, C. W.

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 26, 757-771 (1998).

Appl. Opt. (2)

Electron. Lett. (3)

J. H. Lee, C. H. Kim, Y.-G. Han, and S. B. Lee, “Broadband, high-power, erbium fiber ASE-based CW supercontinuum source for spectrum-sliced WDM PON applications,” Electron. Lett. 42, 549-550 (2006).
[CrossRef]

D. D. Sampson and W. T. Holloway, “100 mW spectrally uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611-1612 (1994).
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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and Juan Capmany, “Complete determination of the first-order degree of coherence of amplified spontaneous emission sources with femtosecond resolution,” presented at the Conference on Lasers and Electro-Optics and the Quantum Electronics and Laser Science Conference (CLEO/QELS 08), San Jose, Calif., May 4-9, 2008, paper CMU5.

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Figures (9)

Fig. 1
Fig. 1

Scheme of the measurement technique.

Fig. 2
Fig. 2

Scheme of the experimental setup. Source: Erbium-doped amplified spontaneous emission source. dB↑: variable attenuator. Pol, fiber polarizer; MZM, Mach–Zehnder amplitude modulator; SSMF, 12.77 km of standard single-mode fiber; PD, photodetector; VNA, vector network analyzer.

Fig. 3
Fig. 3

Optical spectra (continuous curves) and Fourier transform of measured γ ( τ ) (dots) of (a) first and (b) second ASE sources.

Fig. 4
Fig. 4

(a) RF power response of the analog 12.77 km single-mode link operated with the monochromatic wavelength used for normalization (N), and first (1) and second (2) broadband ASE sources. (b) Relative phase response of first (1–N) and second (2–N) sources after subtracting the phase of the normalizing response H ¯ RF ( f ) . The thin continuous lines represent the linear asymptotic approximation.

Fig. 5
Fig. 5

Coherence function of the first ASE source: (a) Visibility in log scale and (b) phase of γ S ( τ ) . Thick continuous curve, RF measurement; dots, Fourier transform of the optical spectrum; thin continuous curve, simulation of Eq. (17) including second-order dispersion.

Fig. 6
Fig. 6

Coherence function of the second ASE source: (a) visibility in log scale and (b) phase of γ S ( τ ) . Thick continuous curve, RF measurement; dots, Fourier transform of the optical spectrum; thin continuous curve, simulation of Eq. (17) including second-order dispersion.

Fig. 7
Fig. 7

Visibility of (a) first and (b) second ASE sources.

Fig. 8
Fig. 8

Interferogram or real part Re ( γ ) of the first-order degree of coherence retrieved (top row) and differences Δ between Re ( γ ) and the Hilbert transform of Im ( γ ) (bottom row) corresponding to (a) first and (b) second ASE sources.

Fig. 9
Fig. 9

Dependence of the phase unwrapping with the padding factor F in the second source. Continuous curve, unwrapped phase from the VNA after subtracting the linear asymptotic phase; gray circles, FFT phase for the original 8 THz spectral span; gray inverted triangles, unwrapped FFT phase for the 8 THz spectral span; white triangles, unwrapped FFT phase for a zero-padded 16 THz spectral span; black squares, unwrapped FFT phase for a zero-padded 24 THz spectral span

Equations (40)

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H RF ( f ) γ ( 2 π ϕ f ) ,
E 3 ( ν ) = H ( ν ) d η M ( η ) E 1 ( ν η ) .
I ( f ) = R d ν E 3 * ( ν f 2 ) E 3 ( ν + f 2 ) ,
I ( f ) = R d ν H * ( ν f 2 ) H ( ν + f 2 ) d ρ 1 M * ( ρ 1 ) d ρ 2 M ( ρ 2 ) E 1 * ( ν f 2 ρ 1 ) E 1 ( ν + f 2 ρ 2 ) .
E 1 * ( ν ) E 1 ( ν ) = S ( ν ) δ ( ν ν ) .
I ( f ) = R d ν H * ( ν f 2 ) H ( ν + f 2 ) d f M * ( f ) S ( ν f f 2 ) M ( f + f ) .
H ( ν 0 + ν ) = exp [ j β ( ν 0 + ν ) L ] = exp j β 0 L j β 1 L ( 2 π ν ) j β 2 L ( 2 π ν ) 2 2 ,
I ( f ) = R P 0 γ 0 ( 2 π β 2 L f ) e j 2 π β 1 L f d t m * ( t + π β 2 L f ) m ( t π β 2 L f ) exp ( j 2 π f t ) ,
γ ( τ ) = γ 0 ( τ ) exp ( j 2 π ν 0 τ ) .
m ( t ) 1 2 [ 1 + μ 2 cos ( 2 π f 0 t ) ] ,
I ( f ) = 1 2 R P 0 [ δ ( f ) + μ 2 cos ( 2 π 2 β 2 L f 2 ) γ 0 ( 2 π β 2 L f ) e j 2 π β 1 L f ( δ ( f f 0 ) + δ ( f + f 0 ) ) ] .
i ( t ) = 1 2 R P 0 { 1 + μ cos ( 2 π 2 β 2 L f 0 2 ) γ 0 ( 2 π β 2 L f 0 ) cos [ 2 π f 0 ( t β 1 L ) + ϕ 0 ( 2 π β 2 L f 0 ) ] } .
H RF ( f ) = T RF e j 2 π β 1 L f γ 0 ( 2 π β 2 L f ) cos ( 2 π 2 β 2 L f 2 ) ,
S ( ν ) d f M * ( f ) M ( f + f ) S ( ν ) T ( f ) ,
I ( f ) R P 0 T ( f ) d ν H * ( ν f 2 ) s ( ν ) H ( υ + f 2 ) ,
T ( f ) = 1 2 δ ( f ) + μ 4 δ ( f f 0 ) + μ 4 δ ( f f 0 ) .
H RF ( f 0 ) = T RF d ν H * ( ν f 0 2 ) s ( ν ) H ( ν + f 0 2 ) .
H * ( ν f 0 2 ) H ( ν + f 0 2 ) exp [ 2 π j β 1 ( ν ) L f 0 ] .
H RF ( f ) = T RF d ν s ( ν ) exp [ 2 π j β 1 ( ν ) L f ] .
H RF ( f ) = T RF e j 2 π β 1 S L f γ S ( 2 π β 2 S L f ) ,
γ ( τ ) = γ S ( τ ) exp ( j 2 π ν S τ ) .
H RF ( f ) = T RF exp ( 2 π j β 1 S L f ) d ν s ( ν S + ν ) exp ( 4 π 2 j β 2 S L f ν ) exp ( 4 π 3 j β 3 S L f ν 2 ) .
π 3 β 3 L f Δ ν 2 π ,
Δ ν π 2 β 2 β 3 ,
H RF ( f ) = T RF d ν s ( ν ) exp [ 2 π j β 1 ( ν ) L f ] { P + ( ν ) exp [ π j f τ ( ν ) ] + P ( ν ) exp [ π j f τ ( ν ) ] } ,
P ± ( ν ) = ε ̂ ± ( ν ) + ε ̂ in 2 ,
f τ ( ν ) 1 .
τ ( ν S ) β 1 ( υ S ) L ,
τ ̇ ( ν S ) β 2 ( υ S ) L ,
H ¯ RF ( f ) = T RF e j 2 π β 1 L f cos ( 2 π 2 β 2 L f 2 ) .
H RF ( f ) H ¯ RF ( f ) = γ S ( 2 π β 2 S L f ) exp [ 2 π j ( β 1 S β 1 ) L f ] .
arg [ H RF ( f ) H ¯ RF ( f ) ] f 2 π ( β 1 S β 1 ) L f + arg ( γ S ) 2 π T d f + 2 π N ,
( β 1 S β 1 ) L 2 π β 2 S L ( ν S ν 0 ) + 1 2 ( 2 π ) 2 β 3 S L ( ν S ν 0 ) 2 = T d .
D ( λ ) = S 0 4 ( λ λ Z 4 λ 3 ) ,
γ ̃ ( τ ) = γ ( τ ) + δ γ ( τ ) = s ( ν ) exp ( 2 π j ν τ ) d ν + δ s ( ν ) exp ( 2 π j ν τ ) d ν .
e 1 ( t ) = m = N N E m exp [ 2 π j ( ν S + m Δ ν ) t ] ,
e 3 ( t ) = 1 2 m E m exp [ 2 π j ( ν S + m Δ ν ) ( t β 1 S L ) ] exp [ 2 π 2 j β 2 S L ( m Δ ν ) 2 ] { 1 + μ 4 exp ( 2 π 2 j β 2 S L f 0 2 ) [ exp [ 2 π j f 0 ( t β 1 S L 2 π β 2 S L m Δ ν ) ] + c c ] } ,
E m * E n = p m δ m n .
i ( t ) = 1 2 R m p m 1 + μ 2 exp [ 2 π 2 j β 2 S L ( m Δ ν ) 2 ] cos [ 2 π f 0 ( t β 1 S L 2 π β 2 S L m Δ ν ) ] 2 = 1 2 R m p m { 1 + μ cos ( 2 π 2 β 2 S L f 0 2 ) cos [ 2 π f 0 ( t β 1 S L 2 π β 2 S L m Δ ν ) ] } .
lim Δ ν 0 m p m exp ( 4 π 2 j f 0 β 2 S L m Δ ν ) = S ( ν S + ν ) exp ( 4 π 2 j f 0 β 2 S L ν ) d ν = P 0 γ S ( 2 π f 0 β 2 S L ) ,

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