1. Introduction

The beam splitter (BS) is one of the most commonly used optical components in both classical and quantum optical experiments [1]. It plays an important role in investigating the quantum nature of light [2] and multiphoton interference effects [3]. Interferometers composed of BSs and phase shifters can realize any discrete finite-dimensional unitary operator [4]. In particular, the BS-based Hong-Ou-Mandel (HOM) interferometer and measurement can induce nonlinearity among photons, which is the key component of the scheme proposed by Knill, Laflamme, and Milburn [5] in 2001, the pioneer scheme proving the possibility of linear optical quantum computation. Since then, BSs have been widely used in photonic quantum computation [6], quantum communication [7], quantum metrology [8, 9], quantum simulation [10], and many other photonic technologies [11,12].

Polarization-independent BSs are usually made of thin layers of dielectric films or metal deposited on glass [13], which have fixed split ratios. In many optical applications split-ratio-tunable BSs are required which enable the adjustment the “interaction strength” among the injected photons, for instance in quantum simulations [14, 15] and decoherence-free states preparation [16, 17]. With fast modulators the tunable BSs can also be used for photon shutter [18] and switch [19]. Except for some specific fabrication [14], such polarization-independent devices [18, 19] usually require phase-sensitive interferometers. Recently, Ma et al. [20] realized a split-ratio-tunable BS by varying the phase of a Mach-Zehnder (MZ) interferometer.

A combination of polarizing beam splitter (PBS) and half-wave plate (HWP) is a widely used method to obtain split-ratio-tunable beam splitter for light with fixed or known polarization [13]. The advantages of this method is the high-precision polarization rotation operation of the HWP and the high extinction ratio of the PBS. In this paper, we present an experimental realization of a 2 × 2 polarization-independent split-ratio-tunable BS by building a MZ-type interferometer and adopting such split-ratio tuning method. However, in our experiment the PBS is replaced by beam displacer (BD), because BD enables separating orthogonally polarized light into two parallel beams rather than two perpendicular beams output from PBS, and thus enables stable phase-sensitive interference [21, 22]. We showed the split-ratios for both input ports can be tuned from 0 to 1 by varying the angle of a HWP. High-fidelity polarization-preserving transmission from either input to both output ports was confirmed by complete quantum process tomography. To demonstrate interference ability of photons from two input ports, we observed two-photon HOM interference for four different input polarization states with the BS reflectivity set to eleven values, respectively, and all the measured interference visibilities are close to the theoretical values.

2. Description of a beam splitter

A 2 × 2 polarization-preserving BS with reflectivity r is illustrated in Fig. 1, where A, B represent the input path modes and C, D denote the output path modes. Here we use the following quantum mechanical model to describe the state evolution at the BS,

 

Fig. 1 Conceptual illustration of a beam splitter with reflectivity r.

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3. Primary scheme

The primary scheme is shown in Fig. 2. Here we write the state transformation at a HWP oriented at an angle of ϕ as

 

Fig. 2 Primary scheme for realizing the tunable beam splitter. The polarizing beam splitter (PBS) transmits horizontally polarized light and reflects vertically polarized light. The angle of each half-wave plate (HWP) is noted beside.

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From Fig. 2, we can see that the whole setup comprises two MZ interferometers. One is from PBS1 to PBS4, and the other is from PBS1 to PBS5. The two MZ interferometers should keep in the zero phase difference position. Since two output beams go out from a PBS perpendicularly, they need pass through different mirrors, and hence in experiment such MZ interferometers are very unstable and particularly sensitive to vibration. Consequently, realizing the primer scheme directly is challenging. A possible method to deal with the problem is using feed-forward phase shift operations, for instance, the apparatus in [20]. In our work we employ another method to cope with the problem, where we replace the PBS by the BD.

4. Experimental realization

The experimental setup for realizing a 2 × 2 polarization-independent split-ratio-tunable BS is shown in Fig. 3(a). Two input ports A and B are both connected with single-model fibers. The fiber-port in port A is mounted on a motorized translation stage with a minimum step size of 0.625 μm, which can adjust the optical length difference between two input ports. After exiting the single-mode fiber, the input light first passes through a polarization controller (PC) to compensate the polarization variation induced by the fiber, which comprises sequently a HWP, and a quarter-wave plate (QWP). The following state preparation apparatus composed of a HWP and a QWP allows preparation of any pure polarization state in each port. Then the light in both ports passes through HWP1 oriented at 45°. The reason for using HWP1 is that the following setup starting from BD1 to BD4 results in a flip operation between H and V polarization against the input polarization state. Before BD1, a phase shifter (PS1) consisting of two 45°-oriented QWP inserted by a HWP is placed at path B, which we will explain later.

 

Fig. 3 Experimental setup for (a) realizing the tunable beam splitter, (b) the spontaneous parametric down-conversion source, (c) tomographic measurement, (d) observing two-photon Hong-Ou-Mandel interference. PC: polarization controller. PS: phase shifter. HWP: half-wave plate. QWP: quarter-wave plate. BD: calcite beam displacer. BBO: beta-barium borate crystal. IF: 3-nm interference filter centered at 780 nm. Letters and numbers label the beams.

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The interferometer part of the setup mainly consists of four 28.2 mm-long calcite BDs. For normally incident light, ordinary light travels straight through the BD while the extraordinary light undergoes a 3 mm parallel displacement. Each BD has a clear aperture of 10 mm ×10 mm, which is large enough for supporting four beams as shown in Fig. 3(a). Explicitly, BD1 and BD4 both transmit straight vertically polarized light and displace horizontally polarized light to the right in the horizontal plane. While BD2 (BD3) transmits straight horizontally polarized light and displaces vertically polarized light to the downside (upside) in the vertically plane.

Then let’s see the polarization evolution in the interferometer. By adjusting the two input fiber-ports we make the two beams A and B travel horizontally and in a vertical plane displaced by a distance of 3 mm. After BD1, beams A and B are divided into A1, A2, and B1, B2, respectively. We put a 45°-oriented HWP in each of beams A1 and B2 while for compensating the length in the wave plate we let beams A2 and B1 pass through a 0°-oriented HWP. In this way the polarizations of the light from input ports A and B become to H and V polarizations, respectively. Then by adjusting the fiber-port in port B carefully, we can make beams A1 and B1 merge into one beam (A3) after BD2, and meanwhile, A2 and B2 join into one beam (B3). Afterwards, the two beams A3 and B3 pass through HWP2 the angle of which can be tuned and here is denoted by θ. BD3 then splits beams A3 and B3 into beams D2, C2, and D1, C1, respectively. After passing through a 45°- and 0°-oriented HWP, respectively, beams C1 and C2 are combined into one beam (C) after BD4. Likewise beams D2 and D1 merge into beam D after BD4.

The polarization-preserving transmission from two input ports to two output ports are determined by four MZ interferometers composed of BD1, BD4, and several HWPs. In the ideal case, by carefully tilting BD4 we can make sure the polarization-preserving transmission in the four travelling ways due to the inherent stable interference feature of the BD [21, 22]. However, in practice, the BD can not have an absolutely smooth surface, and thus, the incident angle at different position may not be the same. Hence, in experiment, we first input a diagonally polarized light in port A and observed the light intensity at port D with a Glan-Taylor polarizer transmitting anti-diagonally polarized light. When a minimal intensity was observed BD4 would be optimized. Then at port C we put a phase shifter PS2 that is a 0°-oriented HWP and horizontally tilted it until observing a minimal intensity after a Glan-Taylor polarizer with the same setting. To this step we have realized polarization-preserving transmission for ports A → C and A → D. With regard to input port B, as we introduced earlier, a phase shifter PS1 is placed before incidence on BD1. We can optimize the angle of the HWP in PS1 (φ in Fig. 3) to make sure polarization-preserving transmission for ports B → C and B → D, without adjusting the other elements in the setup. Under these arrangement, the polarization transformation of the whole BS setup arrives at Eq. (1), where the reflectivity r is given by r = cos²(2θ) and can be tuned via θ, namely, the angle of HWP2.

5. Experimental results

We first test the polarization-independent split-ratio tunableness of the BS. We input a fraction of laser light (7 mW) from a Ti:sapphire pulse laser with a pulse width of 150 fs and a repetition rate of 76 MHz to either of the input ports A and B. We measured the output light intensities at ports C and D as a function of θ, with the input light prepared to H and V polarization, respectively. The experimental results are shown in Fig. 4. The curves are all fitted to sinusoidal functions via the Origin software and all the visibilities are above 99.6%. The curve functions fitted for data denoted as “A → C”, “B → D” (“A→D”, “B→C”) are very close to the reflectivity function of r = cos²(2θ) (the transmissivity function of t = 1 r = sin²(2θ)). Hence we have demonstrated that the BS has almost the same split ratio for light with H and V polarization. However, combined the polarization-preserving feature demonstrated below, we can confirm the split-ratio tunableness is independent of polarization.

 

Fig. 4 Measured output intensities as a function of the angle of HWP2. The intensities are all normalized by the maximal intensities at the corresponding output port. Notations “i → j” with i = A, B, and j = C, D, mean measurement at port j when light is injected at port i. The input light is prepared in horizontal ((a) and (b)) and vertical ((c) and (d)) polarization, respectively. All the curves are fitted to sinusoidal functions.

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To characterize the polarization-preserving transmission property, we experimentally performed quantum process tomography [23, 24] on the four transmission ways. The operation of a quantum process ℰ acting on an arbitrary two-dimensional state ρ can be decomposed with a complete set of basis Ê_m for Pauli operators I, X, Y, Z, as $ℰ\left(\rho \right)={\displaystyle {\sum }_{m,n=0}^3{\chi }_{mn}}{\widehat{E}}_m\rho {\widehat{E}}_n^{†}$. The matrix χ completely and uniquely describes the process ℰ. We follow the method in [25] to reconstruct χ from experimental tomographic measurements.

The input light we used was produced from spontaneous parametric down-conversion (SPDC). The SPDC source setup is shown in Fig. 3(b). The Ti:sapphire laser light centered at 780 nm is first frequency-doubled by a beta-barium borate (BBO) crystal and then pumps an 1.5 mm long BBO crystal cut for type-II beamlike phase-matching [26]. The two parametric beams are coupled into two single-mode fibers which are connected to the input ports of the BS, respectively. With the state preparation apparatus we can obtain the tomographic input states, namely, $\left\{H,V,D\equiv \left(H+V\right)/\sqrt{2},R\equiv \left(H-iV\right)/\sqrt{2}\right\}$. The split-ratio was set to 1 : 1 by tuning θ to 22.5°. For each of the input state in either input port we performed the standard quantum state tomography [27] at each output port. The tomographic measurement setup is depicted in Fig. 3(c), where a sequence of a QWP, a HWP, and a Glan-Taylor polarizer transmitting vertically polarized light can realize projection measurements on the states {H, V, D, R}. The photon detection is realized by coupling light into a single-photon avalanche diode (SPAD) via a single-mode fiber with a 3-nm interference filter (IF) centered at 780 nm in front of them.

Figures 5(a)–5(d) show the real parts of the χ matrix for the transmission processes of A → C, A → D, B → C, B → D, respectively, with the corresponding imaginary parts given in Figs. 5(e)–5(h). The fidelities between the experimental χ matrices and the ideal unit operation calculated from $F\left(\chi ,{\chi }_{ideal}\right)={(Tr\sqrt{\sqrt{\chi }{\chi }_{ideal}\sqrt{\chi }})}^2$ [28] with χ_ideal = Î are (99.3 ± 0.3)%, (97.6±0.8)%, (98.6±0.5)%, (98.1±0.4)%, for A → C, A → D, B → C, B → D, respectively. The errors are estimated by performing 50 times reconstructions of the process matrix via simulated data Poissonianly distributing around the measured data. These results demonstrate that the 2 × 2 BS can preserve polarization of photons. The affection on the measured results mainly comes from the inaccurate wave-plate settings in state preparation and tomographic measurement apparatus.

 

Fig. 5 Real parts of χ matrix determined by quantum process tomography for polarization transmission in four ways (a) A → C, (b) A → D, (c) B → C, (d) B → D, with corresponding imaginary parts shown in (e), (f), (g), (h), respectively.

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Another essential performance of a 2 × 2 BS is the ability supporting interference for incidence light on the two input ports. To prove this feature we observed two-photon HOM interference with the BS reflectivity changed from 0 to 1 increased by 0.1. The input two-photon states are produced from the SPDC source and prepared to HH, VV, DD, and RR, respectively. The measurement setup is depicted in Fig. 3(d). After a 3-nm IF each output port is directly coupled into a SPAD with a single-mode fiber, and then a coincidence logic unit gives the twofold coincidence counts. We measured the coincidence counts in 2 seconds against the relative delay between the two photons adjusted by the motorized translation stage at port B. The experimental results are shown in Fig. 6, with error bars representing the square root of the counts. For the data measured in the case of r ≠ 0.0, 1.0, the curves are fitted with Gaussian functions determined by the 3-nm IFs. The corresponding interference visibilities are listed at the insets of each sub-figure. While for the case of r = 0.0, 1.0, since no obvious dip was observed, we fitted the data with linear functions and did not give the corresponding visibilities. All the visibilities measured are closed to the ideal values [29] which are given by V_th = 2rt/(r² + t²), and explicitly, V_th = 1, 0.923, 0.724, 0.471, 0.220, 0 for r = 0.5, 0.4(0.6), 0.3(0.7), 0.2(0.8), 0.1(0.9), 0(1), respectively. These results show that high-visibility photonic interference can be guaranteed by the split-ratio-tunable 2 × 2 BS, regardless of input polarization. In our experiment, the contamination on the visibilities mainly results from inaccuracy of HWP2 angle, spatial mismatch of the two photons, and discrimination of the polarization states of the two photons.

 

Fig. 6 Experimental results of twofold coincidence counts versus the relative delay for Hong-Ou-Mandel interferometer with input two-photon polarization states HH ((a) and (b)), VV ((c) and (d)), DD ((e) and (f)), and RR ((g) and (h)). The corresponding beam splitter reflectivities and measured visibilities are listed in the insets. The data of r ≠ 0, 1 are fitted with Gaussian curves, while the data of r = 0, 1 are fitted with linear curves.

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6. Conclusions and discussions

We would like to make some discussions on the advantages of the BD compared with the PBS in MZ interferometers. First, the BD-based MZ interferometers are insensitive to vibration on the optical elements. The BD separates orthogonally polarized light into two close parallel beams, and hence the two output beams can go through the same optical elements in the following circuits. In this way, the vibration is almost the same for the two beams and thus its affection on the phase difference is greatly reduced. In our experiment the phase difference can keep for several hours or even days. Second, the BD-based MZ interferometers can be aligned more easily than the PBS-based MZ interferometers. Because the beams go through the same optical elements, some of them do not need fine alignment. In our experiment, we only need to adjust BD4 very carefully. Finally, the BD-based MZ interferometers can easily get high-quality interference. For accurately fabricated BDs, the beam is displaced in a fixed distance, so mode overlap can be easily optimized from one BD to another.

In conclusion, we have presented an experimental realization of a 2 × 2 polarization-independent split-ratio-tunable BS via a stable interferometer. We showed the split-ratio can be finely tuned by the angle of a HWP from 0 : 1 to 1 : 0. The high-fidelity polarization-preserving transmission in the four supported ways was demonstrated by complete quantum process tomography. We observed nearly ideal two-photon HOM interference for a variety of split-ratios with different input polarizations. Hence the tunable BS can be used as a good element in both classical and quantum optical technologies.

Finally, the experimental setup can have some other generalized applications. For example, by replacing HWP2 in Fig. 3 with two respective HWPs in paths A3 and B3, we can realize a tunable polarization-dependent BS which can be used to realize the quantum controlled-not gates [21, 30–32] and to expand multi-photon entangled states [33, 34]. Moreover, we can also realize a photonic shutter or router by changing HWP2 to some fast modulators, such as electro-optic modulators.