Abstract

We show that the minimal phase of the temporal coherence function γ(τ) of stationary light having a partially-coherent symmetric spectral peak can be computed as a relative logarithmic Hilbert transform of its amplitude with respect to its asymptotic behavior. The procedure is applied to experimental data from amplified spontaneous emission broadband sources in the 1.55 µm band with subpicosecond coherence times, providing examples of degrees of coherence with both minimal and non-minimal phase. In the latter case, the Blaschke phase is retrieved and the position of the Blaschke zeros determined.

© 2008 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995), p. 384.
  2. J. Perina, Coherence of Light, 2nd ed. (Kluwer Ac. Pub., Dordrecht, 1985), p. 46.
  3. R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The Phase Problem," Proc. R. Soc. Lond. A 350, 191-212 (1976).
    [CrossRef]
  4. M. V. Klibanov, P. E. Sacks, and A. V. Tikhonravov, "The phase retrieval problem," Inverse Probl. 11, 1-28 (1995).
    [CrossRef]
  5. C. R. Fernández-Pousa, H. Maestre, A. J. Torregrosa, and J. Capmany, "Measurement of the first-order temporal coherence of broadband sources by use of the radio-frequency transfer function of fiber dispersive links," J. Opt. Soc. Am. B 25, 1242-1253 (2008).
    [CrossRef]
  6. R.A. Silverman, Introductory Complex Analysis (Dover, New York, 1985), p. 262.
  7. Here we follow the usual Fourier convention exp(−iω t) for the temporal complex oscillation from the physics literature [1, 2]. It is the opposite to that of signal theory, which is the natural for interpreting the experimental radio-frequency measurements [5].
  8. For a reference at νa, the contour should have been completed in the lhp, resulting in a change of sign in the Hilbert phase. The Blaschke zeros would have been located in the lhp and the curves γa(τ) , corresponding to those shown in Figs. 6 and 7, would then be clockwise.
  9. H. M. Nussenzveig, "Phase problem in coherence theory," J. Math. Phys. 8, 561-572 (1967).
    [CrossRef]
  10. A. V. Oppenheim and R. W. Schafer, Discrete-time signal processing (Prentice-Hall, Englewood Cliffs, NJ, 1989), p. 662.
  11. R. Barakat, "Moment estimator approach to the retrieval problem in coherence theory," J. Opt. Soc. Am. 70, 688-694 (1980).
    [CrossRef]

2008 (1)

1995 (1)

M. V. Klibanov, P. E. Sacks, and A. V. Tikhonravov, "The phase retrieval problem," Inverse Probl. 11, 1-28 (1995).
[CrossRef]

1980 (1)

1976 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The Phase Problem," Proc. R. Soc. Lond. A 350, 191-212 (1976).
[CrossRef]

1967 (1)

H. M. Nussenzveig, "Phase problem in coherence theory," J. Math. Phys. 8, 561-572 (1967).
[CrossRef]

Barakat, R.

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The Phase Problem," Proc. R. Soc. Lond. A 350, 191-212 (1976).
[CrossRef]

Capmany, J.

Fernández-Pousa, C. R.

Fiddy, M. A.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The Phase Problem," Proc. R. Soc. Lond. A 350, 191-212 (1976).
[CrossRef]

Greenaway, A. H.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The Phase Problem," Proc. R. Soc. Lond. A 350, 191-212 (1976).
[CrossRef]

Klibanov, M. V.

M. V. Klibanov, P. E. Sacks, and A. V. Tikhonravov, "The phase retrieval problem," Inverse Probl. 11, 1-28 (1995).
[CrossRef]

Maestre, H.

Nussenzveig, H. M.

H. M. Nussenzveig, "Phase problem in coherence theory," J. Math. Phys. 8, 561-572 (1967).
[CrossRef]

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The Phase Problem," Proc. R. Soc. Lond. A 350, 191-212 (1976).
[CrossRef]

Sacks, P. E.

M. V. Klibanov, P. E. Sacks, and A. V. Tikhonravov, "The phase retrieval problem," Inverse Probl. 11, 1-28 (1995).
[CrossRef]

Tikhonravov, A. V.

M. V. Klibanov, P. E. Sacks, and A. V. Tikhonravov, "The phase retrieval problem," Inverse Probl. 11, 1-28 (1995).
[CrossRef]

Torregrosa, A. J.

Inverse Probl. (1)

M. V. Klibanov, P. E. Sacks, and A. V. Tikhonravov, "The phase retrieval problem," Inverse Probl. 11, 1-28 (1995).
[CrossRef]

J. Math. Phys. (1)

H. M. Nussenzveig, "Phase problem in coherence theory," J. Math. Phys. 8, 561-572 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Proc. R. Soc. Lond. A (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, "The Phase Problem," Proc. R. Soc. Lond. A 350, 191-212 (1976).
[CrossRef]

Other (6)

A. V. Oppenheim and R. W. Schafer, Discrete-time signal processing (Prentice-Hall, Englewood Cliffs, NJ, 1989), p. 662.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995), p. 384.

J. Perina, Coherence of Light, 2nd ed. (Kluwer Ac. Pub., Dordrecht, 1985), p. 46.

R.A. Silverman, Introductory Complex Analysis (Dover, New York, 1985), p. 262.

Here we follow the usual Fourier convention exp(−iω t) for the temporal complex oscillation from the physics literature [1, 2]. It is the opposite to that of signal theory, which is the natural for interpreting the experimental radio-frequency measurements [5].

For a reference at νa, the contour should have been completed in the lhp, resulting in a change of sign in the Hilbert phase. The Blaschke zeros would have been located in the lhp and the curves γa(τ) , corresponding to those shown in Figs. 6 and 7, would then be clockwise.

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Figures (8)

Fig. 1.
Fig. 1.

Schemes of optical spectra with totally-coherent (a) and partially-coherent references (b).

Fig. 2.
Fig. 2.

Derivation of the relative logarithm transform for broadband spectra with symmetric partially-coherent references. (a) Deconvolution of the optical spectral density, implying the decomposition γ(τ)= γ ˜ (τ) γR (τ) of the complex degree of coherence. The corresponding amplitudes and phases are shown in (b) and (c), respectively. In (c), Δ(τ)=0 in an interval |τ|<τR , where τR is the reference’s coherence time.

Fig. 3.
Fig. 3.

Thick curves: spectra of the two ASE sources, (a) and (b) (shifted upwards). Thin curves: spectra of the ideally-isolated references. Dots: Fourier transforms of the measured degrees of coherence.

Fig. 4.
Fig. 4.

Experimental visibilities (thick curves), fitted asymptotic (thin curves) and Fourier transform of the spectral references (dots) corresponding to the ASE sources (a) and (b).

Fig. 5.
Fig. 5.

E: Experimental phases (thick curves), H: Hilbert phases (thin curves) corresponding to the ASE sources (a) and (b). In (b), D: difference between E and H (thin curve shifted a cycle upwards) and B: Blaschke phase (thin curve, also up-shifted).

Fig. 6.
Fig. 6.

Polar plot of the measured degree of coherence γb (τ≥0) for the first ASE source.

Fig. 7.
Fig. 7.

(a) Polar plot of the measured degree of coherence γb (τ≥0) for the second ASE source (black curve) and degree of coherence with minimal phase (gray curve). (b) Close-up of the measured degree of coherence for τ≥400 fs.

Fig. 8.
Fig. 8.

Black curve: difference between the experimental phase and the Hilbert phase (curves E and H in Fig. 5 (b)). Dots: result of the fit to the Blascke phase (curve B in Fig. 5 (b)).

Equations (8)

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γ ( τ ) = v a v b s ( v ) exp ( 2 π i v τ ) d τ .
γ b ( z ) = Δ v 0 s ( v + v b ) exp ( 2 π i v z ) d v = η [ 1 + γ L ( z ) C ] ,
ϕ H ( τ ) = H [ log γ b ( τ ) ] = 2 τ π 0 log γ b ( τ ' ) τ 2 τ ' 2 d τ ' .
γ b ( z ) k = 1 N ( z a k * z a k z + a k z + a k * ) = γ ( z ) exp [ i ϕ H ( z ) ] .
ϕ ( τ ) = arg γ ( τ ) = ϕ H ( τ ) + ϕ B ( τ ) 2 π v b τ ,
ϕ B ( τ ) = k = 1 N arg ( τ a k τ a k * τ + a k * τ + a k )
ϕ R ( τ ) = arg γ R ( τ ) = Δ ( τ ) 2 π v b τ ,
ϕ ( τ ) = H [ log ( γ ( τ ) γ R ( τ ) ) ] + ϕ B ( τ ) 2 π v b τ ,

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