1. Introduction

The characterization of coherence properties of radiation emitted by relativistic beams is of increasing importance due to the development of many radiation sources in the scientific community, both from synchrotron and Free Electron Laser (FEL) facilities [1, 2] and from laser-driven accelerators [3]. Spatial coherence is measured with several interferometric methods, as for example by diffraction from a slit or a fiber [4] or in a double-slit experiment [5]. Recently, a new and unconventional interferometric technique has been introduced to measure spatial coherence. This technique, named Heterodyne Near Field Scattering (HNFS), is based upon the self-referenced interference patterns generated by scattering from a colloidal suspension [6]. The superposition of the intense transmitted field to the fainter scattered radiation (heterodyne condition [7]) allows to characterize the incident wavefront. HNFS has been successfully applied to the measurement of spatial coherence of visible light emitted in Self Amplified Spontaneous Emission Free Electron Laser (SASE FEL) at National Laboratories of Frascati (LNF) [8] and to the characterization of the transverse source size of the high brilliance undulator beam line ID02 (ESRF, Grenoble) [9]. Such experiments were performed under the assumption of quasi-monochromaticity so that temporal coherence effects were negligible with respect to transverse coherence.

However the demand for temporal coherence measurements is increasing, both for temporal characterization of the radiation whenever pulsed emission is present and for retrieval of the power spectral density (the time correlation function is related to the emission spectrum of the source through the Wiener-Khinchine’s theorem [10]). We show in this work that it is possible to extend the HNFS technique to broad-spectrum radiation sources. This is the first time to our knowledge that this method is used to deduce temporal coherence properties and power spectral density of a light source. Moreover, the HNFS technique is wavelength independent as already experimentally demonstrated in [9] and [11]. This allows to realize optical modeling with visible table-top setups for much complex and expensive X-ray radiation sources. In particular, as it will be discussed in detail below, there is a strong similarity between our optical experiments with visible light and X-ray betatron radiation emitted by laser-driven accelerators [3]. We thus report a proof of principle of the extension of the technique to broadband light sources emitting in the visible range properly scaled to reproduce the main features of broad-spectrum betatron radiation emitted in the laser-plasma accelerator under development at SPARC LAB (Frascati) [12]. Our optical modeling will allow to properly design such experiments and to get a better insight into the main features of betatron radiation with limited temporal coherence. This has been demonstrated to be advantageous in characterizing the acceleration process of particles by means of the main properties of the power spectral density [13] and the transverse size of the radiation source by means of spatial coherence[14, 15].

More generally, we also show that the proposed method gives access to the spatio-temporal coherence function defined as

There exist of course several ways to measure the function μ(Δt) in static conditions to perform spectroscopy, as for example using Static Fourier Transform Spectrometers [16], but in general these techniques need specific optics and a precise alignment of the experimental set-up. On the contrary, the method discussed in this paper is free of any optics and does not require a severe alignment.

The paper is organized as follows: we describe the Heterodyne Near Field Scattering technique and the experimental setup in section 2 and 3, respectively; we accurately and quantitatively discuss the scalability to betatron radiation in section 4; section 5 is devoted to experimental results and finally we collect our conclusions in section 6.

2. HNFS technique

HFNS technique is sketched in Fig. 1. A broadband radiation illuminates a colloidal suspension of microparticles (0.5–1 μm in diameter) at low concentration (10⁻⁴ –10⁻⁵ volume fraction). Each particle scatters an almost spherical wave which interferes with the transmitted radiation (heterodyne configuration). The resulting interference pattern consists of a series of circular fringes (see Fig. 2(a) as an example) whose visibility appreciably decreases when the optical path difference between the two waves is comparable to the coherence length (Fig. 1(b)). This occurs at high scattering angles and allows to retrieve the temporal coherence function of the incoming radiation at once. We have assumed that the entire single-particle interferogram lies within a coherence area of the source in order to neglect spatial coherence effects. Notice however that for smaller coherence areas the visibility of the fringes carries information both on spatial and temporal coherence, neither being dominant over the other one. The sum of a number of interference patterns generated by many scatterers gives origin to a speckle field on top of the transmitted beam as shown in Fig. 2(b) (heterodyne conditions ensure that no interference occurs between spherical waves scattered by different particles). In such a condition information about fringe visibility cannot be easily obtained in the spatial domain, while coherence properties can be obtained in the spatial frequencies domain. In fact, as it has been shown in [9], the power spectrum of the speckle field S(q) can be written as

 

Fig. 1 a) Sketch of the experimental set-up. A collimated broadband source illuminates a colloidal suspension. Each particle scatters a spherical wave (blue) that interferes with the transmitted radiation (red) and the resulting interferogram is acquired with a CCD on a transverse detection plane. b) The optical path difference Δx₂ observed at high angles will be larger than at small angles (Δx₁). This behavior is the analogous of the variation of the optical path of the Michelson’s interferometer arms.

Download Full Size | PDF

 

Fig. 2 a) Calculated interferogram for the spherical wave scattered by a single particle. b) Speckle field of a colloidal suspension measured with a CCD.

Download Full Size | PDF

where I(q) is the form factor of particles, T(q) = sin²(zq²/2k) is the Talbot transfer function, C(q) is the squared modulus of the spatio-temporal coherence factor, H(q) is the transfer function of the detector used to acquire the interferogram and P(q) is the power spectrum of readout and shot noise. Here q ≈ θ k (θ is the scattering angle), z is the sample-detector distance and k = 2π/λ.

The square modulus of the coherence factor C(q) as a function of Δt = zq²/2k²c (c is the speed of light) is obtained from Eq. (2) because the functions I(q), T(q), H(q) and P(q) are experimentally known. Notice that the oscillating behaviour of the function T(q) is actually enveloped by the other functions.

3. Experimental setup

Experiments have been performed by using four different sources of white-light: three broadband thermal sources (halogen lamps) and a Light Emitting Diode (LED). The halogen lamps provide a continuous broadband spectrum of radiation while the LED provides a more structured spectrum characterized by two main peaks centered at 443 nm and 540 nm, respectively (Fig. 3). Halogen lamps used in the experiment have similar spectra with FWHM of 1.9 ·10¹⁴ Hz (halogen 1), 2.2 ·10¹⁴ Hz (halogen 2) and 2.0 · 10¹⁴ Hz (halogen 3). The spatial coherence of the incident radiation was increased by means of a circular pinhole 80 μm in diameter. The light has then been collimated by means of a convergent lens with a focal length z_f = 80mm and the sample positioned at a distance of 20 mm from the lens. Samples consist of pure water suspensions of calibrated polystyrene spherical particles. In order to reach the heterodyne condition, samples were diluted so that 95% of the intensity of incident radiation was transmitted. The scattered radiation has been collected with a PCO1600 Charge Coupled Device (CCD) with a resolution of 1600 X 1200 pixel (pixel size was 7.4 μm X 7.4 μm) at a distance ranging from 30 to 190 mm from the cell. For each distance 100 images were acquired with a rate of 1 Hz. Each image was normalized to its average value (after subtracting the CCD dark noise) to compensate intensity fluctuations of the white-light sources. Pairs of images with time delay of 50 s have been subtracted in order to remove the static stray-light contribution (double-frame analysis, see [17]). The time lag of 50 s between the two images ensures image decorrelation and increases the statistics. This data analysis is robust, being free of any adjustable parameters.

 

Fig. 3 Frequency spectra of the halogen 2 lamp (a) and the LED (c) measured with a grating spectrometer. b) and d) are the corresponding temporal coherence factors μ(Δt) obtained by means of Wiener-Khinchine’s theorem.

Download Full Size | PDF

4. Scaling to betatron radiation

As mentioned in the introduction, one of the advantages of the HNFS technique is that of being easily scalable to different wavelengths. This occurs because the method relies on the interference between the spherical wave scattered by a colloidal particle and the transmitted reference beam, so that coherence properties of the radiation are given by the visibility of the interference fringes. In the Fourier space, the power spectrum of the intensity distribution is characterized by oscillations, described by the Talbot transfer function $T(q)=si{n}^2\left(\frac{z{q}^2}{2k}\right)$, which are envloped by the coherence function [9]. The position of Talbot maxima as a function of the Fourier wavevector q (entirely determined by the detector size) is simply related to the other experimental parameters by $\frac{z\lambda q^2}{4\pi }=const$. The distance z can then be set according to the working wavelength (typically it is of the order of few mm or cm with visible light, of the order of some meters for X-ray wavelengths) in order to keep Talbot maxima at the same q.

Here we show that our proof of principle experiment performed with visible light actually reproduces the main features envisioned to be found at the laser wakefield accelerator at LNF. We provide quantitative arguments to perform a complete scaling of experiments with betatron radiation to be reproduced with visible light.

The coherence size of betatron radiation with wavelength λ = 2 nm emitted by a source of radius r_b = 5μm observed at a distance z = 1 m from the source is S_c = λz/2r_b = 200 μm. The size of the first Fresnel zone is $F_1=\sqrt{\lambda z}=30\mu \mathrm{m}$. Because F₁ < S_c temporal effects are dominant and about 3 interference fringes (λ/Δλ ≈ 3) will be observed for each scatterer in the spatial domain. This corresponds to 6 Talbot oscillations in the spatial frequencies domain [18]. In the present set-up the sample-detector distance, the source size and the radiation wavelength can be properly scaled to obtain the same power spectrum expected with betatron radiation. In fact, repeating the calculations above with λ = 500 nm, d_p = 80 μm and z = 80 mm the coherence area size becomes S_c = λz/d_p = 0.5 mm and again λ/Δλ ≈ 3 (here d_p is the diameter of a pinhole imposing the desired spatial coherence). In both cases Talbot oscillations are progressively depressed by the limited coherence length of radiation but at least six observable oscillations can be used to compute C(q) as in Eq. (2).

5. Results and discussion

First we show the accuracy of the method in reproducing the expected profiles of the function given by I(q), H(q), C(q), which envelope the Talbot transfer function T(q). Then we will discuss how to extract the coherence information from data.

The results of experiments are shown Fig. 4. We compare the radial profile of the power spectra (circles) to the expected envelopes of Talbot oscillation including both spatial and temporal coherence. The expected envelope is given by

 

Fig. 4 Power spectrum of the four white light sources used in experiments (circles). The envelope of the Talbot oscillations is well matched by the theoretical coherence factor C(Δt) (solid curves) given by the product of |μ(Δt)|² (dashed curves) and |C_s(Δt)|² (dotted curves).

Download Full Size | PDF

where |μ(q)|² (dashed lines in Fig. 4) is calculated from the measured spectra of the sources by means of the Wiener-Khinchine theorem and |C_s(q)|² is the Airy pattern with argument kd_p q/z_f (dot lines in Fig. 4) that is completely determined by the pinhole. Notice that no adjustable parameters are there. Results are in good agreement with theory since the calculated function C(q) (solid lines in Fig. 4) matches the envelope of the Talbot oscillations. Furthermore, as expected, the decay time of μ(Δt) increases as the spectral width of the source is reduced.

Note that μ(Δt) can be estimated from Eq. (3) if C_s is known a priori. This is not the case of betatron radiation or of X-ray radiation in general. However, |C_s(q)|² can be indipendently measured with the same experimental set-up by inserting a monochromator between the radiation source and the sample. Thus we have followed a similar experimental procedure for the measurement of the spatial coherence factor of the pinhole by using a dichroic filter centered at λ = 650nm with a bandwidth of 40nm.

The values of |μ(Δt)|² of the halogen lamps and LED, extracted from experimental data, are shown in Fig. 5 together with the emission spectra of sources retrieved via the Wiener- Khinchine’s theorem. Note that the shape of the spectrum can be retrieved exploiting the Wiener-Khinchine’s theorem, while the absolute value of frequencies cannot. However it can be estimated fitting data with the Talbot transfer function T(q). We roughly obtain that the peak frequency is ≈ 0.4 · 10¹⁵ Hz for halogen lamps and ≈ 0.52 · 10¹⁵ Hz for LED (first peak), in good agreement with the spectra of Fig. 3. Because of the depression of higher-order Talbot oscillations, the spectra retrieved with the present technique does not accurately provide the real spectra shape for high Δt. This is well observed when a LED source is used. The second peak of the spectrum has a FWHM of 3·10¹³ Hz which corresponds to Δt ≈ 30 fs. At such time delay, the power spectrum signal is comparable to the instrumental noise (see Fig. 4), so that the second peak is not visible in the retrieved spectrum. On the contrary the first peak is three times larger than the second peak and can be adequately retrieved by the present technique.

 

Fig. 5 Temporal coherence factors |μ(Δt)|² of the four sources (circles) extracted from the envelope of the Talbot oscillations (see Fig. 4). Dashed curves are the corresponding spatial coherence factors |C_s(Δt)|² and insets are the retrieved spectra.

Download Full Size | PDF

Relevant information about the bandwidth of the radiation are proven to be in good agreement with the real width of the spectra. Results are summarized in Table 1.

Table 1. Comparison of results from the spectrometer and HNFS experiments.

View Table

6. Conclusions

We have shown that the Heterodyne Near Field Scattering (HNFS) technique can be advantageously used to characterize the spectrum of broadband radiation. We have chosen an experimental set-up scaled to visible light that properly represents an analogical modeling tool for reproducing the experimental conditions of current and future X-ray sources. In particular, we have discussed in details the case envisioned to be observed with X-rays radiation emitted by the laser wakefield acceleration under development at LNF. The behaviour of Talbot oscillations has been accurately reproduced, in order to mimic the data and therefore the analysis procedures to extract coherence information. Since the envelope of Talbot oscillations appearing in the spatial frequencies domain is the product of the spatial and temporal coherence factors, the spectrum of the radiation can be retrieved because the former is imposed by the source size and shape. In case it is an unknown function to be obtained experimentally (as it will be likely done with X-rays), it can be measured by narrowing the radiation spectrum as we did in the visible. Results show that the spectral widths of four different sources are in good agreement with data provided by a grating spectrometer. The current limitation of such diagnostics to accurately recover the real spectrum at high Δt is the read-out and shot noise. To overcome such limitations one may increase image statistics. The proposed technique will be improved to obtain a noninvasive diagnostics of X-rays in single-shot, as it has been done in [8], which is useful when the source of radiation has a great spatial variability in time.