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We demonstrate the quantum Talbot effect using a beam of single photons produced by parametric down conversion. In contrast to the previous works, we use a programmable spatial light modulator to behave as a diffraction grating. Thus, the investigation of the Talbot diffraction patterns under the variation of grating structure can be easily performed. The influence of spectral bandwidth of the down-converted photons on the diffraction pattern is also investigated. A theoretical model based on the wave nature of photons is presented to explain the Talbot diffraction patterns under varying conditions. The measured diffraction patterns are in good agreement with the theoretical prediction. We are convinced that our study improves the understanding of the quantum Talbot effect.
© 2016 Optical Society of America
1. Introduction
The Talbot effect, which was introduced in 1836 by Henry Fox Talbot [1], is a near-field diffraction effect. The Talbot diffraction patterns appear when a coherent light illuminates a diffraction grating. The grating self-image is formed behind the grating at multiples of specific distance, so-called the Talbot length. The experiments of near-field effects have been implemented, including light [2], atom [3], molecule [4], surface plasmon [5], and non-linear optics [6,7].
The diffraction of entangled photons has been numerously studied in the context of the superposition of two Bessel beams [8], quantum lithography [9, 10], quantum ghost imaging [11], and with the experimental Talbot effect [12–14]. The first experiment of quantum Talbot effect [12] used both single photons and biphoton pairs which were generated by spontaneous parametric down conversion (SPDC). The experimental configuration is similar to previous quantum ghost imaging setup [11]. The signal and idler photons from SPDC were split into two separate optical paths. The grating was placed in the signal path before a detector. The Talbot self-imaging of the grating was observed by scanning either the signal or the idler detector. In contrast to those studies, the quantum decoherence experiment has also been performed with single photons [15].
A liquid crystal spatial light modulator (SLM) is a versatile device in a wide range of applications in optics, including the creation of holograms [16], polarization control [17], and the study of optical vortices [18]. In general, a SLM can control both amplitude and phase of the light.
In this paper we report on the experimental and theoretical investigation of the Talbot effect using pairs of single photons [19]. In contrast to the previous demonstrations [12], the SLM in our experimental setup is programmed to act as the diffraction grating. Our technique permits the real-time variation of the period and the opening fraction of the grating. We further investigate the effect of spectral bandwidth down-converted photons on the diffraction patterns. In order to explain the Talbot diffraction patterns under varying conditions, the theoretical model based on paraxial approximation is presented in the next section. We subsequently present our experimental setup, and compare the results to our theoretical findings.
2. Theory and method
In this section, we present the theoretical consideration of the Talbot effect for photon point source. Let a spherical wave from a point source encounters a grating at the distance z0 as shown in Fig. 1. Considering the propagation in xz plane, the field distribution with the leading order approximation , can be obtained as [20]
where k = 2π/λ is the magnitude of the wave vector with wavelength λ, and R0 is the distance from the source to a position on the grating x1 as shown in Fig. 1. The field is then incident onto the grating at z0. For the diffraction grating, which has the periodic modulation along the x1-axis with the period d, the wave is transformed to be where n = 0, ±1, ±2, ..., and kd = 2π/d. The factor An = sin(nπf)/nπ is the components of the Fourier decomposition of the periodicity for the grating with an opening fraction f [2]. Subsequently, we apply the Huygens-Fresnel integral to find the wave function behind the grating at distance z with the transverse axis x. The field distribution ψ(x, z) is given by According to the field ψ1,+(x1, z0) in Eq. (2), the integral can be done analytically and the result is, where the distance Z = zz0/(z + z0). Omitting the pattern independent factor , we obtain the intensity distribution ψ*ψ behind the grating in the form where LT = d2/λ is the Talbot length. The obtained intensity can be reduced to the case of a plane wave by taking z0 → ∞. The interference pattern, which is similar to the period d of the grating itself can be found when the longitudinal distance z equals to an integer number of LT.
Fig. 1 A spherical wave of signal photon beam with the divergence angle, θ ≈ δ/z0=0.5 mrad, diffracts through the grating which has the periodic modulation along the x1-axis to the single slit and a detector is placed behind this slit for photon detection.
In order to compare the theoretical model to the experimental data, the effects due to quantum behaviour of single photons have to be considered. Namely, the intensity pattern is corresponding to the probability distribution of the detecting photons along the x-axis. Therefore, the obtained intensity (Eq. (5)) can be considered as the number of incident photons. One can evaluate the count rate of photons propagated through a single slit by integrating the intensity over the slit interval, , where X denotes the position of slit with the slit width, Δ as shown in Fig. 1. With the intensity distribution in Eq. (5), the count rate in this scheme can be obtained as
A single photon detector is placed behind the slit. The interference pattern can then be revealed by scanning transversely the grating over the slit or vice versa [21].In addition, the effect of spectral bandwidth of the wavelength has to be involved. For a Gaussian distribution of the wavelength around λ0, the total count rate is given by
where β represents the radius of the Gaussian distribution.3. Experiment
We set the quantum Talbot experiment with the single photon source. Our single photon pairs are produced from spontaneous parametric down conversion with type I configuration. A beta barium borate (BBO) is pumped by a 50 mW diode laser (LQA405-50E, Newport) with the center wavelength of 405 nm, diameter of about 1.5 mm. The angle between the signal and idler photons of 6 degrees is selected for the photon pairs of the wavelength 810 nm, shown in Fig. 2. Each of photon pairs has the divergence angle (θ) of about 0.5 mrad. The signal photons encounter the SLM (LC2012, Holoeye Photonics AG with resolution of 1024×768 pixels, the pixel size of 36 μm), using as a diffraction grating, which can be varied both the period and the opening fraction. The azimuthal phase of the SLM varies from 0 to around 1π with 8 bit (256 gray levels) level at 810 nm. An unmodulated phase of photons can be blocked by a polarizer. The SLM application software is used to the grating mode. The grating mode of the periods, d=360 μm, and 180 μm and the opening fractions (the ratio between the grating window and the grating period) of f =0.1–0.5 is set for the experiments. The typical Talbot length is then calculated to be LT =160 mm for d=360 μm, and LT =40 mm for d=180 μm. The SLM is mounted on three translation stages. The first one (MTS50/M-Z8E, resolution 1.6 μm, Thorlabs) can move along the longitudinal direction in order to set the distance z. The second one (PT1/M, Thorlabs) can move the SLM in the y axis for adjusting the center of the grating. The third one (Z812B, Thorlabs) is used to explore the interference patterns automatically by scanning the SLM along the transverse x axis with the step of 12 μm. The coincidence count is recorded for each transverse (x) scan. Behind the SLM grating, a single slit with the width of about 115 μm is placed at the distance z, and subsequently a fiber-coupled avalanche photodiode module (SPCM-AQ4C, PerkinElmer) is used as the detection. The single slit can be removed by a manual stage (PT1/M, Thorlabs) when the back alignment is needed with a laser source, and can be roughly adjusted the longitudinal distance by the other stage (PT1/M, Thorlabs). The idler photon is employed as a trigger to insure the signal from noises. The time window of 25 ns is applied for coincidence counts. A long pass filter with FWHM of 50 nm is placed at front of both photon pairs. The broad filters are used for producing the broad wavelength photon source.
Fig. 2 Our single photon Talbot setup. (1) BBO; (2) 50 mW laser diode (λ=405 nm); (3) SLM; (4) first translation stage; (5) second stage; (6) third stage; (7) polarizer; (8) single slit; (9), (10) optical fibers for signal photon APD, and idler photon APD; (11), (12) signal photon filter, and idler photon filter; (13) single slit stage (PT1/M); (14) stage for signal photon detection (PT1/M); (15) optical table. The photon pairs of the wavelength 810 nm are produced from spontaneous parametric down conversion. The SLM is used in the grating mode: the details are described in the text.
4. Results and discussions
The effect of a point source dominates the phase modulation of the photon wave function and the interference patterns are therefore expanded and the effective Talbot length has been extended to 174 mm, which is longer than the typical Talbot length of LT =160 mm. Figure 3 presents the interference patterns at z=160 mm. The period of the grating, d=360 μm, the center wavelength, λ=810 nm, FWHM of 50 nm, and the single slit with the width of about 115 μm were used. In details, Figure 3(a) is the pattern with the opening fraction, f =0.1. The fringe width is about 47%, which is larger than a regular width of 10%. This also happens throughout the other opening fractions, f =0.2–0.5, shown in Figs. 3(b)–3(e). The influence of spectral bandwidth dominates the interference patterns and this effect makes the fringe width broader. A controlled visibility can be obviously observed with different opening fractions. We check the results with the simulations done with Eq. (7), and present by the solid lines in Fig. 3. They are in good agreement with each other. We also show the simulations with the same conditions as before but using a plane-parallel (θ=0 rad) and monochromatic beam. They are presented as the insets in Fig. 3. In this case, the width of the interference patterns becomes larger with the higher opening fractions. This cannot be seen with our recent experiments because the reason of broad beam and wavelength distribution. We continued our investigations with the extended Talbot length z=174 mm for d=360 μm, and f =0.1, displayed in Fig. 4(a), for example. The width is smaller than the pattern at z=160 mm, f =0.1 shown in Fig. 3(a). This confirms that the actual Talbot length is extended to 174 mm. Moreover, we changed the grating period to d=180 μm, with f =0.2, and recorded the pattern at z=3LT =123 mm, displayed in Fig. 4(b). This distance (z=123 mm) is already calculated with the extended Talbot length (41 mm) for d=180 μm. In our theoretical study, although we include various effects, i.e. spectral bandwidth, beam divergence, opening fraction, period and slit detection, but actual experimental results are only given for various values of the opening fraction; all the other parameters are fixed in Fig. 3, as well as for various grating periods in Fig. 4. All these effects influence on the phase modulation of the wave function as described in our theory and we can observe in our experimental fringe patterns.
Fig. 3 The experimental interference patterns at z=160 mm, d=360 μm, θ=0.5 mrad, λ=810 nm, and FWHM = 50nm (detail see text): f =0.1 (a), f =0.2 (b), f =0.3 (c), f =0.4 (d), f =0.5 (e). The solid lines represent the theory done with Eq. (7). The coincidence counts (CCs) of photon pairs were accumulated in 10 s. The transverse position (x) represents in the unit of grating period (d). The error bars on the y-axis represent the photon shot noise. The error bars on the x-axis are small. The insets show the simulation of the normalized intensity pattern with the condition of using a plane-parallel and monochromatic beam.
Fig. 4 The experimental interference patterns and the simulations (the solid lines) done with Eq. (7) at z=174 mm (d=360 μm, f =0.1) (a), z=3LT =123 mm (d=180 μm, f =0.2) (b).
5. Conclusion
The Talbot effect has been studied with a single photon pair source. A series of experiments was conducted for different opening fractions and periods of the grating with the help of SLM. The use of SLM in the single photon Talbot experiment has merits. The phase modulation can be controlled and observed. Our results show that the measured diffraction widths become wider as the filter bandwidths increase. In addition, the visibility of the interference patterns can be controlled with the opening fractions of the grating. The calculations are used to verify the experiments and fit to the data quite well. We hope that our study here is useful and provides a better understanding for quantum experiments, as well as further implementation of quantum spectroscopy or small signal spectroscopy such as molecule spectroscopy.
Funding
The office of the higher education commission, the Thailand research fund (TRF) (MRG5380264).
Acknowledgments
The authors would like to thank W. Temnuch for her help in a part of photon alignment.
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