Hilbert and Blaschke phases in the temporal ... - OSA Publishing

Hilbert and Blaschke phases in the temporal coherence function of stationary broadband light Carlos R. Fernández-Pousa,* Haroldo Maestre, Adrián J. Torregrosa and Juan Capmany Signal Theory and Communications, Department of Physics and Computer Science, Universidad Miguel Hernández, Av. Universidad s/n, E03202 Elche (Alicante) Spain * Corresponding author: [email protected]

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 OCIS codes: (030.1640) Coherence; (030.6600) Statistical optics; (100.5070) Phase retrieval; (060.5625) Radio frequency photonics.

References and links 1. 2. 3. 4. 5.

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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. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, “The Phase Problem,” Proc. R. Soc. Lond. A 350, 191-212 (1976). M. V. Klibanov, P. E. Sacks, and A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 128 (1995). 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). R.A. Silverman, Introductory Complex Analysis (Dover, New York, 1985), p. 262. Here we follow the usual Fourier convention exp(−i ω 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. H. M. Nussenzveig, “Phase problem in coherence theory,” J. Math. Phys. 8, 561-572 (1967). A. V. Oppenheim and R. W. Schafer, Discrete-time signal processing (Prentice-Hall, Englewood Cliffs, NJ, 1989), p. 662. R. Barakat, “Moment estimator approach to the retrieval problem in coherence theory,” J. Opt. Soc. Am. 70, 688-694 (1980).

1. Introduction The experimental measurement of the autocorrelation phase of stationary polychromatic light constitutes a longstanding problem in coherence optics [1, 2]. When an optical source is characterized by use of an interferometer, a difficulty arises because the interferogram is a bandpass amplitude and phase-modulated signal whose carrier frequency is the wave’s optical frequency. For near infrared and optical signals, phase details become obscured by a highfrequency oscillation. In principle, the autocorrelation phase can be measured from the

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Received 14 Jul 2008; revised 28 Sep 2008; accepted 16 Oct 2008; published 24 Oct 2008

27 October 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18397

maxima of the interferometric fringes, but in practice only the autocorrelation amplitude −which is simply the fringe visibility−, can be characterized. This fact limits the situations where a complete determination of the autocorrelation of a wave −including phase− can be performed. Only sources with an almost trivial phase structure, such as those with symmetric spectra, can be characterized. Making an analytic continuation of the autocorrelation Γ(τ ) into the complex z-plane adds a new perspective to this problem. It can be shown that under certain conditions, phase can be retrieved from amplitude by computing its logarithmic Hilbert transform, i.e., the socalled minimal, canonical or Hilbert phase [1-4]. In these cases, the amplitude characterizes the whole autocorrelation. In other cases, however, the autocorrelation phase may contain a number of Blaschke terms associated to the zeros of the autocorrelation lying inside the contour used to compute the logarithmic Hilbert transform [1-4], and therefore, the phase can only be determined if the influence of these Blaschke terms is known. Thus, the question still remains about under which conditions a source leads to a minimal phase, and only a few general criteria are presently known [4]. In other words, without additional conditions the autocorrelation amplitude may fail in completely determining the phase through the logarithmic Hilbert transform, a failure that is usually referred to as the one-dimensional phase problem in coherence theory. In a recent publication [5], we have demonstrated a direct method for determining the amplitude and phase of the autocorrelation Γ(τ ) in the case of broadband near-infrared sources. This is equivalent to determining the complex or first-order degree of coherence γ(τ ). In [5] the autocorrelation is measured as the radio-frequency transfer function of a linear firstorder dispersive link based on a single-mode fiber, driven by the low-coherence source, and where the global analytic properties of the experimentally measured phase follow from link causality. Our purpose, in this context, is to report on the retrieval of the Hilbert and Blaschke contributions to the autocorrelation phase of stationary light with spectra containing several peaks, including the determination of the position of the Blaschke zeros when present. Thus, the question addressed here is not how to retrieve phase from amplitude −the phase is actually known−, but how to separate and characterize the different contributions to the phase. To the best of the authors’ knowledge, such a direct characterization of the autocorrelation phase has not been previously reported. Two features in our phase characterization are worth being stressed. Firstly, since the fingerprint of a Blaschke term is a 2π phase shift that may occur at an infinitely small time scale, the measured phase should have sufficient temporal resolution. As shown below by means of some experimental examples, the temporal resolution of the traces analysed allow for determining whether a phase jump is Hilbert or Blaschke. Secondly, the Blaschke phase can only be accessed after subtracting the Hilbert phase, and a certain algorithm is necessary to implement the logarithmic Hilbert transform. In order to compute the Hilbert phase we make use of the asymptotic behaviour of γ(τ ), which is determined by the narrowest peak in the optical spectrum, which acts as a partially-coherent reference. Reference waves were originally introduced as a method to guarantee that the phase is minimal in a process that is equivalent to the well known off-axis holography [3]. Its use in our context can be explained as follows. In its simpler form, the Rouché theorem [6] states that the addition of a positive constant C to an analytic function γ(z) implies that the sum C + γ(z) has no zeros as long as C > |γ(z)|. Experimentally, this constant can be added by combining a coherent wave to one side of the spectrum under consideration. However, we will alternatively use here narrow (partially-coherent) peaks that readily appear in some broadband spectra. Since their properties mimic those of totally-coherent references, we first recall in Section 2 its analysis, to subsequently derive a generalization to partially-coherent references. In Section 3, we present the retrieval of the Hilbert and Blaschke phases in the first-order degree of coherence of two broadband amplified spontaneous-emission sources and, in Section 4, we end with our conclusions.

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Finally, we mention that besides its interest as an experimental verification of the predictions of the theory of analytic functions in coherence theory, the possible applications of the experimental methods and techniques for phase and amplitude characterization presented here are not limited to the determination of the autocorrelation of broadband sources. The only basic assumption underlying the experimental technique is the stationary character of the lightwave under consideration [5]. Therefore, the same procedures can be applied to the characterization of the autocorrelation of broadband stationary, scalar waves that might convey information on filters, reflectors or backscatterers from a specifically-designed setup. 2. Theory The complex degree of coherence γ(τ) = |γ(τ)| exp[iφ (τ)] of a stationary optical signal is the Fourier transform of its optical spectrum s(ν), normalized as γ(0) = 1 and φ (0) = 0. The spectrum s(ν) is band-limited and has compact support [νa, νb] (νa > 0) and optical width Δν =νb − νa . The complex degree of coherence can be mathematically expressed as [7]:

γ (τ ) = ∫

νb νa

s(ν ) exp (−2π iντ ) dτ .

(1)

Spectra with a monochromatic reference such as that outlined in Fig. 1(a) can be split as s(ν ) = ηδ (ν−νb) + (1−η) sL(ν ), where we assume that the reference carries a fractional power η and is located at the upper limit νb, and where sL(ν ) is the broadband or low-coherent part of the total spectrum s(ν ). The complex function used here for phase retrieval is the analytic continuation of the Fourier-shifted function γb(τ ) = γ(τ )exp(i2πνbτ ),

γ b ( z) = ∫

0 − Δν

s (ν + ν b ) exp (−2π iν z ) dν = η [1 + γ L ( z ) / C ] ,

(2)

where C =η /(1−η), and γL(τ ) is the shifted degree of coherence associated to sL(ν). γb(z ) is an entire function that tends to η in the real axis and along rays in the upper complex half-plane (uhp), and have exponential growth in the lower half-plane (lhp) [3]. Had the coherent reference been located at νa , the corresponding downshifted function γa(z ) would tend to η in the opposite half-plane.

Fig. 1.Schemes of optical spectra with totally-coherent (a) and partially-coherent references (b).

Due to the regular behaviour of γb(z ) in the uhp, the Hilbert phase can be expressed as a logarithm Hilbert transform of the amplitude |γb(z )| = |γ(z )| by computing the real part of the integral of Log γb(z )/(z−τ) in a contour composed of the real line, indented at z = τ, and a semicircle of infinite radius in the uhp [3,4]. The result is:

φ H (τ ) = H [log | γ b (τ ) | ] =

2τ

π

∞

∫ 0

log | γ b (τ ') |

τ 2 − τ '2

dτ ' 

(3)

where H stands for the Hilbert-transform operator and log is the real logarithm. Notice the sign change with respect to Ref. [1] due to the integration in the uhp. According to the Rouché theorem, if the reference has sufficient power so that C > 1 (η > ½), the argument of γb(τ )

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coincides with the Hilbert phase. However, if C ≤ 1 then γb(z ) may contain zeros, say M, in the uhp. Zeros in the imaginary axis are ruled out since s(ν) ≥ 0. Due to the symmetry γb*(z*) = γb(−z ) the zeros appear in pairs at ak and −ak* and can be removed with M = 2N Blaschke factors, leading to: N ⎛

z − ak* z + a k ⎞⎟ =| γ ( z ) | exp [ iφ H ( z )] . ⎜ z − a z + a* ⎟ k k ⎠ k =1 ⎝

γ b ( z) ∏ ⎜

(4)

Therefore, after reinserting the phase of the coherent reference at νb, the total phase is:

φ (τ ) = arg γ (τ ) = φ H (τ ) + φ B (τ ) − 2πν bτ ,

(5)

where

φB (τ ) =

∑ arg ττ −− aa N

k =1

⎛ ⎜ ⎜ ⎝

k * k

τ + ak* ⎞⎟ τ + ak ⎟⎠

(6)

is the Blaschke phase [8].

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.

As schematically shown in Fig. 1(b), practical references are only partially coherent and their spectra may extend beyond the reference peak νb, say up to νc >νb. As in the totallycoherent case, the reference is the sharpest spectral peak that accounts for the asymptotic behaviour of γ(τ ), so that the reference’s coherence time τR is larger than the total source’s coherence time τC. In an experiment [5], the existence of a reference with these properties can be assured through the observation of a smooth asymptotic decay of the amplitude |γ(τ )| and a linear asymptote in the total phase φ (τ ). The decomposition of the total phase φ (τ ) is not so straightforward since the contribution of Log γb(z )/(z−τ) in the infinite semicircle is nonzero. #98830 - $15.00 USD

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However, its value can be computed for a wide collection of asymptotes [4], and the final result coincides with (3)-(6). The analysis of the phase φ (τ ) can be simplified by introducing a relative logarithmic transform. To explain its meaning, we first isolate the spectrum of the partially-coherent reference sR(ν) from the total spectrum s(ν). We assume that sR(ν) is symmetric around νb, as is the case in the examples below, and define it as η sR(ν) = s(ν) for ν > νb. The total spectrum is decomposed as s(ν)=η sR(ν)+(1−η)sL(ν). In general, the reference spectrum sR(ν) may overlap with sL(ν), and therefore sL(ν) may become negative and the analytic continuation γL(z ) of the complex degree of coherence associated to sL(ν) could contain zeros in the imaginary z-axis. To exclude this situation, we restrict our analysis to spectra where sL(ν) > 0, i.e, to spectrally-isolated symmetric references or to symmetric references with negligible spectral overlap.

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

To proceed, we heuristically assume that sL(ν) can be expressed as a convolution sL(ν) = sR(ν) ⊗ s (ν ) for a relative spectrum s (ν ) , as shown in Fig. 2(a). Then, the complex degree of coherence is decomposed as γ(τ) = ~γ (τ ) γR(τ ) for a shifted degree of coherence ~γ (τ ) of the form (2). The autocorrelation phase is therefore divided into contributions due to the #98830 - $15.00 USD

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reference, arg(γR), and a relative phase of a spectrum with a totally-coherent reference, arg( ~γ ), as shown in Fig. 2(c). In practice, it is not necessary to assure the existence of the deconvolution; the relative logarithmic transform follows from the observation that, in practical situations of interest, arg(γR) is zero (up to a linear term −2πνbτ accounting for its central frequency) and is, in addition, a Hilbert phase. This situation is outlined in Fig. 2(c).

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).

Indeed, since sR(ν) is symmetric around νb, the analytic continuation γR(z) can be decomposed as γR(z) = γRb (z) exp(−i2πνbz) with γRb (τ) real on the real axis z =τ. Then, the reference’s phase only contains π-shifts associated to zeros in the real axis and is given by:

φ R (τ ) = arg γ R (τ ) = Δ(τ ) − 2πν bτ ,

(7)

where Δ(τ) = [1−sign γR(τ )] π/2 accounts for the real zeros. Now, according to (5), the reference’s phase can be decomposed as Δ(τ) =φRH (τ )+φRB (τ ). However, the experimental observation in a measurement range |τ | 400 fs. The amplitude at τ = 481 fs shows a clear dip, which means that the curve γb(τ ) in the γb-plane almost crosses zero. Due to the discrete nature of the data this almost zero-crossing implies a phase jump. Since this phase jump is Hilbert, the almost zero-crossing must occur for Re γb >0, as confirmed by the polar plot of γb(τ ) shown in Fig. 7. This plot also shows (gray curve) the complex degree of coherence that would have been retrieved if only the Hilbert phase were considered.

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)).

Subtraction of the Hilbert phase to the experimental phase leads to the D curve shown in Fig. 5(b), which is up-shifted a cycle for clarity. The numerical cloud at τ =481 fs is due to the evaluation of the Hilbert phase via the Fast Fourier Transform. Curve D indicates the presence of a pair of Blaschke zeros of order one in the proximity of the real axis τ. We have performed a numerical fit of the position of the Blaschke zeros to curve D by using the functional form of the Blaschke phase (eq. (6) with N =1).

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The resulting curve is shown in Fig. 5(b) labelled B and also up-shifted. If superimposed, curves D and B coincide within a maximum deviation of 6 deg (excluding the behaviour around τ = 481 fs, as shown in Fig. 8 for the range τ ≤ 800 fs). The retrieved position of the Blaschke zero pair is finally ± 200.8 + i 23.3 fs. 4. Conclusions In conclusion, we have analysed the properties of the first-order degree of coherence of sources with partially-coherent references. We have derived a simple numerical technique for the evaluation of the Hilbert phase after normalizing visibility through its asymptotic behaviour. We have subsequently applied this technique to broadband near-infrared sources having peaked spectra and subpicosecond coherence times, both with Hilbert and non-Hilbert phase. In the latter case, we have been able to isolate the Blaschke phase and determine the position of the Blaschke zeros. Finally, we point out that the procedures used here in the direct characterization of the autocorrelation of the lightwave generated by a broadband stationary source can be adapted to the temporal characterization of the autocorrelation of arbitrary broadband stationary light. Acknowledgments This paper has been supported by Ministerio de Educación y Ciencia, Spain (TEC-200507715), Bancaja-UMH, and Generalitat Valenciana, Spain (GV/2007/024).

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