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Abstract
The challenge of astronomical intensity interferometry is to detect the small photon-bunching signals of distant sources with a broad optical bandwidth. We have built a Hanbury Brown-Twiss-like laboratory intensity interferometer with a focus on a relatively broad bandwidth ($1\,$nm FWHM optical filter) and high photon rates (up to $10\,$MHz) per channel compared to typical (non-astronomical) intensity interferometry applications. As a light source we use a green LED to simulate starlight. The LED has proven to be a compact high-power source of stochastic light with a special advantage of a small emission area, which favours spatial coherence. Using single-photon correlations, we detect a bunching signal in the second-order correlation function with a coherence time of $< 1\,$ps and an amplitude of $<4\cdot 10^{-4}$ and describe signal and background quantitatively for a $40\,$ hours measurement. In this paper we show our setup, present the correlation measurements and compare them to theoretical expectations.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, significant progress has been achieved in astronomical observations using radio and optical interferometry, such as creating an image of a supermassive black hole [1] or mapping an accretion flow that powers a quasar [2]. Optical and near infrared interferometry with sub-milliarcsecond resolution enables one to determine the geometry and the structure of winds from rapidly rotating stars [3,4], to study stellar winds from giant stars [5] and accretion in binary systems [6]. It will also enable one to directly determine the size of stars up to few hundreds or even thousands of light years away from Earth, which is important to reduce uncertainties in the determination of the size of transiting exoplanets [7].
In general, there are two methods for optical interferometry. Currently operating radio, infrared and optical interferometers employ amplitude and phase correlations (the first-order interference of light), e.g., ALMA [8], CHARA [9] or VLTI [10]. Intensity interferometry is an alternative technique originally developed by Hanbury Brown and Twiss [11,12], which exploits second-order coherence of light (for a recent review see, e.g., [13]). In this case, the square of the light-wave amplitude (the instantaneous intensity) is used to correlate light in one or more telescopes, i.e., instead of correlating the amplitude of the electromagnetic waves, one correlates the arrival times or fluxes of photons. The correlation of the signals for different pairs of telescopes is analyzed electronically either instantaneously or at a later time.
Intensity interferometry has been used to successfully measure sizes of several stars in the 1960s and early 1970s with the Narrabri Stellar Intensity Interferometer [14,15]. Further development of the method was impeded by a lack of large-size telescopes separated by several hundreds of meters and by the ability to record photons with timing resolution on the order of $\sim 1$ ns or better. These limitations are alleviated in Imaging Atmospheric Cherenkov Telescopes (IACTs), such as the High Energy Stereoscopic System (H.E.S.S [16]) or the Very Energetic Radiation Imaging Telescope Array System (VERITAS [17]), although the primary purpose of IACTs is to detect the showers created by very-high-energy gamma rays entering the Earth’s atmosphere. In essence, these are optical telescopes with very large collecting area and with cameras, which are designed to have exceptional timing resolution for arrival of individual photons (e.g. [18]), in order to record the flashes of the Cherenkov light produced by the showers lasting less than $\sim$ microseconds. Thus, already the currently operating IACTs and especially the future Cherenkov Telescope Array (CTA [19]) are ideal instruments for reviving the intensity interferometry program [20–24]. In this article, we develop and test in laboratory setting a system that can be used for intensity interferometry measurements on IACTs. The system contains optical elements, which guide the light to photomultipliers to a digital read-out system with a card that is able to record the data to disk. In view of applications for measuring the light from stars which have broad-band spectra and show only small photon correlation, instead of using sharp optical filters [25] or a narrow-line source such as a Hg lamp [25,26], we use an LED as a source of thermal light and determine its small intensity correlation signal with an optical-filtered linewidth of $1\,$nm.
2. Signal and noise in intensity interferometry
In intensity interferometry the observable is the second-order correlation function
The spatial correlation $g_r^{(2)} \left ( \vec {r_2}-\vec {r_1}\right )$ contains information on the morphology of the light source. In the case of astronomical sources, it is proportional to the square of the Fourier transform of the angular size and emission profile of the source [27,28]. Given a detector system with high sensitivity to temporal correlation, the spatial correlation can be sampled by an array of telescopes like CTA. High sensitivity to temporal correlation thus is the challenging parameter of the system.
The time integrated signal of the temporal correlation
for a Lorentz-shaped frequency power spectrum with a second-order-correlation of [27,29] is given as $S = \tau _{c}$, if the coherence time $\tau _{c}= 1/\Delta \nu$ with the optical frequency bandwidth $\Delta \nu$ (FWHM) is much shorter than the observation time $t_0$. For a Gaussian-shaped power spectrum it is reduced to $S \approx 0.664\,\tau _c$. As the coherence time $\tau _{c}$ for astrophysical objects is typically in the order of picoseconds or less, the correlation signal is smeared by the electronics time resolution $\tau _{e}$ and the measured time correlation signal’s amplitude A is reduced to the order ofThe signal to noise of measured temporal correlation is thus given by
The challenge in detecting photon bunching from light sources with broad wavelength spectra (as expected in most astrophysical objects) is that the signal amplitude $A = 1/\left ( \Delta \nu \tau _{\textrm {e}} \right )$ is of the order of $10^{-5}$ for 100$\,$nm spectrum width and $1\,$ns time resolution. Ideally the broad-band spectrum is filtered with a narrow optical filter and excellent time resolution is guaranteed by the system. Our goal is to use the large collection area ($C\approx 100\,\textrm {m}^2$) of modern Cherenkov telescopes such as H.E.S.S. or the future CTA which have an intrinsic anisochronicity of the mirror arrangement on the order of a few nanoseconds. With an optical filter of $1\,$nm width and $1\,$ns time resolution the expected enhancement of the coincidence rate due to photon bunching is $\frac {\tau _\textrm {c}}{\tau _\textrm {e}} \approx 10^{-3}$ setting the requirement on noise and stability of the electronics to detect the signal.In this paper we describe a setup for intensity interferometry from stars and characterize it in the laboratory. To simulate the thermal emission from a star, we used a light-emitting diode (LED) filtered at $\lambda _0 = 532\,$nm and $\Delta \lambda = 1\,$nm and a detector system with time resolution $\sigma _{\textrm {t}} \approx 650\,$ps which defines $\tau _{\textrm {e}}$ as $\tau _{\textrm {e}} = 4\sigma _{\textrm {t}}$. With this system we detected photon bunching from an LED.
3. Experimental setup
In Fig. 1 the experimental setup is sketched. As light source we use the LED LT W5SM-JYKY-25-0 (Osram) with a spectrum centered at $519\,$nm with a FWHM of $39\,$nm, $3.5\,$W power and a light emitting area of about 0.8 x 0.8$\,$mm. A pinhole of 0.3$\,$mm diameter is positioned directly in front of the LED. This pinhole defines the effective size of the light source.
A second pinhole of size $3\,$mm (diameter) is positioned 3.8$\,$m downstream. Behind this second pinhole the light passes through a Thorlabs FL532-1 interference filter with a measured Gaussian transmission profile of $\lambda _0 = \left ( 531.90 \pm 0.05 \right )$ nm and a FWHM of $\Delta \lambda = \left ( 1.06 \pm 0.05 \right )$ nm and a Thorlabs LPVISE050-A linear polariser, all parts mounted in a lens tube. The light is then split by a Thorlabs CCM5-BS016/M 50:50 beamsplitter and detected by two 5-mm Hamamatsu H10770-40 photomultipliers with high quantum efficiency (40% in the green). The photomultipliers produce single photon pulses of $2\,$ns FWHM. The electric power for the photomultipliers is provided by two Conrad CP 12240E lead accumulators and the adjustment of the gain voltage is done using a Maxim MAX5825A evaluation board.
The signals of the photomultipliers pass through low pass filters with a cut-off frequency of $350\,$MHz, PicoQuant PAM 102-P amplifiers and finally into a PicoQuant TimeHarp 260N TDC with a time binning of $250\,$ps which stores the photon arrival times. The maximum sustained count rate for the TimeHarp TDC is $20\,$MHz per channel.
An offline correlation analysis is carried out afterwards. We observed that cross talk between the electronics channels give rise to a correlation signal at zero time delay. In order to avoid disturbance due to this cross talk, we chose to shift the signal away from zero time delay by using different cable lengths from the PMTs to the electronics resulting in a $61\,$ns time shift between coincident photons.
We used a pulsed laser to determine the time resolution of the system and obtained a resolution of 456 ps per channel respectively $\sigma _t= \left (645 \pm 9 \right )\,$ps for the correlation.
We observed that the arrival times of the photons are not uniformly distributed within the TDC bins. This TDC non-linearity leads to a systematic periodicity in the $g^{(2)}$-function. Taking into account that the $g^{(2)}$-function is equal to 1 for large time differences a template of this periodicity was extracted to calibrate the $g^{(2)}$-function.
4. Results
As demonstration of the performance of the setup we first carried out a measurement with a mercury arc lamp with a very sharp emission line resulting in a strong correlation signal. This lamp is a low-pressure thermal light source emitting the atomic transition line of wavelength $546\,{\textrm {nm}}$ and bandwidth $\Delta \lambda < 0.1\,$nm resp. $\tau _c > 10\,$ps, as we checked with a spectral measurement close to the resolution of the spectrometer. We applied a Thorlabs FL543.5-10 filter ($\lambda _0 = (543.5 \pm 2)\,$nm, $\Delta \lambda = (10 \pm 2)\,$nm) in order to select the $546.07\,$nm line of mercury. Given the very narrow line and the resulting narrow optical bandwidth the signal is expected to be significantly larger than for the LED measurement. The correlation result for the mercury lamp is depicted in Fig. 2 for a 2 hour measurement at $2.9$ resp. $2.1\,$MHz rates in the two photomultipliers. The difference in photon rates is compatible with the difference of transmission of the two beamsplitter channels and the sensitivity of the PMTs. The photon bunching peak at $-61\,$ns (determined by the difference in cable length) is clearly visible. The integrated peak area is $S= (20.1\pm 0.6)\,$ps. The observed signal height $g^{(2)}(\Delta t=0)-1 = 0.012$ is consistent with the order of $\tau _{\textrm {c}}/ \tau _{\textrm {e}} \approx 0.010$.
Fig. 2. Temporal intensity correlation function $g^{(2)}$ measured for the mercury arc lamp and zooming to the signal region showing the photon bunching peak at $-61\,$ns
Following this demonstration of the general validity of the setup we performed a measurement with the LED. We measured 40 hours at photon rates of $R_{\textrm {LED}} = 10.0$ and $7.1\,$MHz. The resulting $g^{(2)}$-function is shown in blue in Fig. 3 (left).
Fig. 3. Left: Normalised time correlation between the two PMT-signals as function of delay time using the LED (blue) and the (spatially unfiltered) light bulb (grey). Right: Difference between the two $g^2$-functions shown on the left.
The observed noise of the correlation signal is determined from the correlation distribution $g^{(2)}$ as $N = G \sqrt {250\,\textrm {ps}\cdot 4 \sigma _{\textrm {t}}} = (46.7 \pm 0.3)\,$fs, with G being the RMS (Gauss-sigma) of the $g^{(2)}$-value per $250\,$ps time channel outside the signal region, and $\tau _{\textrm {e}} = 4 \sigma _{\textrm {t}}$ is taken as the $\pm 2 \sigma _{\textrm {t}}$-interval of the signal width. This noise is a factor of $3$ larger than the expected noise given by the left hand side of Eq. (6), namely
indicating that the system contains further noise contributions.In order to investigate the noise we performed a correlation measurement with a standard $100\,$W light bulb not covered by a pinhole which should not show any correlation peak due to the broad spatial emission region. We observed a systematic stable pattern of the $g^{(2)}$-distribution on the level of $10^{-4}$. We used this measured distribution as a background template.
Hence one ’observation’ now consists of two sub-measurements - one to determine the signal and one to estimate the background. The signal measurement uses the LED with the $0.3\,$mm pinhole directly adjacent, the background measurement the light bulb. The photon rates for the light bulb are $R_{\textrm {bulb}} = 10.3$ and $6.9\,$MHz, the observation time is 40 hours as well. The resulting $g^{(2)}$-function for the light bulb is also shown in Fig. 3 (left) in grey. The difference of the two measurements is shown in Fig. 3 (right). In this subtracted $g^{(2)}$-distribution the photon bunching peak is clearly visible at the expected time difference of $-61\,$ns matching the cable-induced delay between equal-time events of the two PMTs.
The noise as again given by $N_{\textrm {sub}} = G_{\textrm {sub}}\sqrt {250\,\textrm {ps}\cdot 4\sigma _{\textrm {t}}}$ with $G_{\textrm {sub}} \approx G\sqrt {2}$ due to subtraction of the two $g^{(2)}$ functions is $(24 \pm 1)\,$fs for the $40\,$h observation period. Figure 4 shows the evolution of the noise as a function of time (grey data points) along with the statistical expectation (solid grey line) given by Eq. (6), illustrating the excellent agreement between expectation and measurement for the background fluctuation.
Fig. 4. Cumulative signal (red) and background (grey) measurement for the difference between the two time correlations (LED - bulb). The signal is defined as the peak area of the $g^{(2)}$-function and therefore is in units of time. Also shown are the theoretical expectation for the peak area (red band) and for the background (solid grey line).
The fitted width of the Gaussian peak in the subtracted $g^{(2)}$-function is $\sigma _{\textrm {t}} = (678 \pm 47)\,$ps which is in agreement with the determined time resolution of $(645 \pm 9)\,$ps.
The fitted signal peak area is $S_{\textrm {lab}} = (0.481 \pm 0.044)\,$ps. As expected, this peak area is stable with time within statistical uncertainties as shown in Fig. 4 (red points). The expected peak area is given by $0.664 \tau _c = 0.664 \frac {\lambda _0^2}{c \Delta \lambda }$ where the factor 0.664 is due to the Gaussian shape - in contrast to Lorentz-shape - of the spectrum. Additionally, one needs to take into account the spatial coherence loss due to the size of the pinholes (see Fig. 1) which reduces the signal peak area by a factor of 0.887 as determined with Monte-Carlo simulations of the setup. Combining these factors, the expected signal peak area amounts to $S_{\textrm {theo}} =\,(0.524 \pm 0.025)\,$ps, the error stemming from the uncertainty of the geometrical parameters like pinhole sizes and distances and from the spectral width measurement. The expected signal peak area is in agreement with the measurement as shown in the red band in Fig. 4. More precisely, in our experiment the LED shows thermal light characteristics of $S_{\textrm {lab}}/S_{\textrm {theo}} = 92\% \pm 10\%$.
A Gaussian fit to the correlation peak in the subtracted $g^{(2)}$-function results in a statistical significance of $(20.3 \pm 1.4)\,\sigma$. This result is in good agreement with the expected signal-to-noise ratio $S/N = (22.1\pm 1.3)\,\sigma$ as given by Eq. (7).
5. Discussion and conclusion
We performed investigations of the temporal correlation $g^{(2)}$ of LED light. For our setup, we detected a systematic structure of the $g^{(2)}$-distribution which we calibrated with a light bulb. We achieved an excellent agreement between measurement and expectations for both the signal and background. Further, the LED in combination with the PMTs and electronics guarantees stable conditions over 40 hours. We were able to evaluate temporal correlations with a sensitivity down to $3 \cdot 10^{-5}$. From our measurement we conclude that the LED is a thermal light source with a stochastic light component of at least $92\% \pm 10\%$ very well suited for measurements of intensity interferometry signals. Combined with an optical bandpass filter of $1\,$nm width, the LED is bright enough to produce PMT count rates of the order of $10\,$MHz within a coherence cell of 0.89 average spatial correlation. In particular these properties correspond to a black body of the temperature $T \approx 3400\,\textrm {K}$. It can therefore compete with many broadband thermal light sources existing on the market, and has the big advantage of being cost-effective and easy to operate.
Funding
Erlangen Centre for Astroparticle Physics; Deutsche Forschungsgemeinschaft.
Acknowledgments
We would like to thank Joachim von Zanthier and Raimund Schneider for helpful discussions during the preparation of the manuscript and Felix Pfeifer and Katja Gumbert for participation in the project.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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