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 high-frequency oscillation. In principle, the autocorrelation phase can be measured from the 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 so-called 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 first-order 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.

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 [ν_a, ν_b] (ν_a>0) and optical width Δν=ν_b-ν_a. The complex degree of coherence can be mathematically expressed as [7]:

Spectra with a monochromatic reference such as that outlined in Fig. 1(a) can be split as s(ν)=ηδ(ν-ν_b)+(1-η) s_L(ν), where we assume that the reference carries a fractional power η and is located at the upper limit ν_b, and where s_L(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 γ_b(τ)=γ(τ)exp(i2πν_bτ),

where C=η/(1-η), and γ_L(τ) is the shifted degree of coherence associated to s_L(ν). γ_b(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 ν_a, the corresponding downshifted function γ_a(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).

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Due to the regular behaviour of γ_b(z) in the uhp, the Hilbert phase can be expressed as a logarithm Hilbert transform of the amplitude |γ_b(z)|=|γ(z)| by computing the real part of the integral of Log γ_b(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:

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 γ_b(τ) coincides with the Hilbert phase. However, if C≤1 then γ_b(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 γ_b*(z*)=γ_b(-z) the zeros appear in pairs at a_k and -a_k* and can be removed with M=2N Blaschke factors, leading to:

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

where

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)=$\tilde{\gamma }$(τ) γ_R(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|<τ_R, where τ_R is the reference’s coherence time.

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As schematically shown in Fig. 1(b), practical references are only partially coherent and their spectra may extend beyond the reference peak ν_b, say up to ν_c>ν_b. As in the totally-coherent case, the reference is the sharpest spectral peak that accounts for the asymptotic behaviour of γ(τ), so that the reference’s coherence time τ_R is larger than the total source’s coherence time τ_C. 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 γ_b(z)/(z-τ) in the infinite semicircle is nonzero. 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 s_R(ν) from the total spectrum s(ν). We assume that s_R(ν) is symmetric around ν_b, as is the case in the examples below, and define it as ηs_R(ν)=s(ν) for ν>ν_b. The total spectrum is decomposed as s(ν)=ηs_R(ν)+(1-η)s_L(ν). In general, the reference spectrum s_R(ν) may overlap with s_L(ν), and therefore s_L(ν) may become negative and the analytic continuation γ_L(z) of the complex degree of coherence associated to s_L(ν) could contain zeros in the imaginary z-axis. To exclude this situation, we restrict our analysis to spectra where s_L(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.

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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).

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To proceed, we heuristically assume that s_L(ν) can be expressed as a convolution s_L(ν)=s_R(ν)⊗s̄(ν) for a relative spectrum s̄(ν), as shown in Fig. 2(a). Then, the complex degree of coherence is decomposed as γ(τ)=$\tilde{\gamma }$(τ)γ_R(τ) for a shifted degree of coherence $\tilde{\gamma }$(τ) of the form (2). The autocorrelation phase is therefore divided into contributions due to the reference, arg(γ_R), and a relative phase of a spectrum with a totally-coherent reference, arg($\tilde{\gamma }$), 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(γ_R) is zero (up to a linear term -2πν_bτ 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).

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Indeed, since s_R(ν) is symmetric around ν_b, the analytic continuation γ_R(z) can be decomposed as γ_R(z)=γ_Rb (z) exp(-i2πν_bz) with γ_Rb (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:

where Δ(τ)=[1-sign γ_R(τ)]π/2 accounts for the real zeros. Now, according to (5), the reference’s phase can be decomposed as Δ(τ)=ϕ_RH (τ)+ϕ_RB (τ). However, the experimental observation in a measurement range |τ|<τ _max of a zero-free asymptotic decay of the amplitude |γ(τ)| prevents the existence of jumps in Δ(τ) due to the reference’s real zeros. This is because the asymptotic behaviour of γ(τ) is determined by the reference, and therefore |γ(τ)|≅|γ_R(τ)| for sufficiently large τ. This means that Δ(τ)=0 for |τ|<τ _max. In addition, τ _max extends typically up to scales of the order of the reference’s coherence time, τ _max ~τ_R, and since the zeros of γ_R(z) are located [9] well beyond τ_R, we may assume ϕ_RB(τ)=0 for |τ|<τ _max. Then, in this interval the reference’s phase is zero and coincides with the Hilbert phase: Δ(τ)=ϕ_RH (τ)=H[log|γ_R(τ)|]=0, and subtracting this equality to (5) we obtain:

which is the desired generalization of (5) to symmetric partially-coherent references. It simply states that the Hilbert phase can be extracted as a relative logarithm transform. The logarithm in (8) tends to zero at the limits of the measurement range, |τ|→τ _max, and the transform can be evaluated by discrete-time techniques [10].

3. Results

We have applied (8) to the analysis of broadband C-band sources by identifying in the experimental visibility |γ(τ)| a smooth, zero-free asymptotic decay corresponding to |γ_R(τ)| and subsequently computing the Hilbert phase. The experimental data were obtained by using the method of Ref. [5]. In Fig. 3, we present the spectra of two amplified spontaneous emission (ASE) sources, labelled (a) and (b), based on ~10m of Er³⁺-doped fiber whose output is launched into a commercial booster amplifier operated at two different values of pump power [5]. Their coherence times are τ_C=601 and 271 fs, respectively. In Fig. 3, we also show the isolated references, centred at ν_b=195.3 THz and having coherence times τ_R=1.89 and 1.82 ps, respectively. In order to appreciate the global accuracy of the measured autocorrelation, the Fourier transform of the retrieved degree of coherence has also been plotted with circles in Fig. 3. The respective amplitude and phase of γ_b(τ) are shown in Figs. 4 and 5 (labelled with E), within a time span τ _max=3 ps. The resolution of the experimental traces is ~6.5 fs. In Fig. 4, using circles we show the asymptotic decay accounting for |γ_R(τ)|, which was obtained by Fourier-transforming the references in Fig. 3. For practical purposes, we have also computed a numerical decay from the data of the amplitude |γ(τ)| with τ>τ_R. The form of the fitted curve shown in Fig. 4 is |γ_R(τ)|=ηexp(-aτ ²-bτ ⁴-cτ ⁶), which is justified as the sixth-cumulant expansion of the amplitude [11]. The respective values of reference power, η=0.513 and 0.201, have been determined from the spectra and have not been subjected to any numerical fit. The agreement of both procedures for determining the amplitude decay is globally good.

 

Fig. 6. Polar plot of the measured degree of coherence γ_b (τ≥0) for the first ASE source.

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In Figs. 5(a) and (b) we have plotted the Hilbert phase with a thin curve labelled H. In source (a) the total phase is Hilbert (η>0.5), as indicated by the small differences between both curves (<3 deg). The global structure of the complex coherence function γ_b (τ≥0) can be appreciated in the polar plot of Fig. 6. This plot starts from γ_b(τ=0)=1 and progressively approaches to zero, asymptotically through the real axis, arg γ_b (τ→∞)=0. This just reflects the fact that the coherence function is asymptotically determined by a symmetric reference spectral peak. We also observe that this curve does not encircle the point γ_b=0, which again shows that the phase arg γ_b is Hilbert.

 

Fig. 7. (a) Polar plot of the measured degree of coherence γ_b (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.

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By contrast, the weak reference in Fig. 3(b) provides a recognizable asymptotic decay in Fig. 4(b), but the total phase in Fig. 5(b) is not Hilbert. Besides a global 2π difference, the Hilbert phase reproduces the experimental phase only for τ>400 fs. The amplitude at τ=481 fs shows a clear dip, which means that the curve γ_b(τ) in the γ_b-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 γ_b>0, as confirmed by the polar plot of γ_b(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)).

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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).

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.