Orbital angular momentum (OAM) is an important resource in high-dimensional quantum information processing, as its quantum number can be infinite. Dove prism (DP) is a most common tool to manipulate OAM light. However, the Dove prism changes the polarization of the photon states and decreases the sorting fidelity of the interferometer. In this work, we analyze the polarization-dependent effect of the DP on OAM light manipulation in the normal single-path Sagnac interferometers (SPSIs) with beam splitter (BS) and polarizing beam splitter (PBS). The results demonstrate that the BS SPSI is more sensitive to the input polarization and the specific parameters of the DP. We have also proposed and realized a modified BS SPSI, of which the sorting fidelity can be 100% in principle and is independent on the input polarization and the transmission matrix of the DP. The experiments demonstrate that the fidelity of the modified BS SPSI is about 5%~10% higher than that of the normal one. The modified BS SPSI is easy to implement (only two more half-wave plates are required) and is stable for free running at the scale of several hours. These merits make the structure suitable for applications in critical quantum information processing tasks, such as quantum cryptography.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The orbital angular momentum (OAM) of light  is an important resource in many fields, such as quantum simulation , quantum metrology [3–6], micromechanic [7, 8] and high-dimensional quantum information processing [9–20]. Recently, OAM photon states with quantum number larger than 104 have been created . Methods to separate tens of OAM modes simultaneously have been successfully demonstrated [22–25] and many other innovative creating and sorting methods have also been proposed [26–31]. However, optical interferometers with Dove prisms (DPs) are the dominant methods to accomplish the nondestructive sorting of OAM modes [17, 19, 32–34]. A DP converses the l-order OAM state |l〉 into exp (i2lα| −l〉, where α is the rotation angle of the DP. The l-order OAM mode can be separated from the l′-order by utilizing a Mach-Zehnder interferometer (MZI) with one DP in each path. The normal double-path MZI lacks in stability. This shortcoming can be overcome with double and single-path Sagnac interferometers [17, 35–37]. The single-path Sagnac interferometer (SPSI) is very robust and can be cascaded for several levels . Unfortunately, the DP does not preserve the polarization of the photon states [39, 40], which increases the crosstalk of the interferometer.
In this work, we analyze the polarization-dependent property of the DP on the beam splitter (BS) and polarizing beam splitter (PBS) SPSIs detailedly. The results indicate that both the output polarization and sorting fidelity of the BS SPSI vary with the rotation angle α and depends on the specific parameters of the inserted DP, which leads to an α-dependent polarization flipping at the output port of destructive interference when the input state is horizontal (|H〉) or vertical (|V〉) polarized. The sorting fidelity of the PBS SPSI is nearly independent on α. However, the PBS SPSI leads to several percents loss of intensity and requires the input polarization to satisfy the form . In order to eliminate the polarization-dependent property of the DP, we designed and realized a modified BS SPSI, which consists of a simple passive polarization-compensation structure and the normal BS SPSI. The experimental results show that the modified BS SPSI is polarization-independent and has high sorting-fidelity (the crosstalk of which is about 5%~10% lower than the normal one). The modified BS SPSI is stable for free running in the scale of several hours. These merits indicate that this scheme can be effectively used in quantum information, e.g., in quantum cryptography.
2. DP in Single-path Sagnac interferometersFig. 1) and Rs and are the forward and inverse transformations between the lab and the DP coordinate systems (Fig. 1). As has been studied in Refs.  and , the DP is polarization preserving when the input polarization is parallel or perpendicular to the input surface. Furthermore, according to the Fresnel equations, the transmission coefficients of the DP are different for the parallel and perpendicular polarized components and the total internal reflection of the DP will cause a relative phase shift Δφ between these two components . Δφ is determined by the incident angle θ1 and the refractive index of the DP. That is, JDP should satisfies Equation 2 requires JDP to be the following form
A general polarization can be expressed as the superposition of |//〉 and | ⊥〉. Hence, in general, JDP should satisfyFig. 1).
2.1. DP in normal BS SPSI
Considering the DP with a rotation angle α, two l-order OAM states incident from the opposite directions (directions a and b in Fig. 2(a)) will experience a relative phase shift 2l(−α) − 2lα = −4lα, as the parallel axes of directions a (nDP,a) and b (nDP, b) are different (see Fig. 2(b)). Correspondingly, the coordinate transformations Rs,a and Rs,b are different. According to Eq. 1, the polarization variations of these two states are different. Thus, the output state of a normal BS SPSI (Fig. 2(c)) is expressed as
where T is the transmissivity of the BS.
For an ideal DP that is polarization preserving, JDP(π − α) = JDP(α) = Is. The output state in Eq. 5 isEq. 4 and |ψ〉out is more complex. For simplicity, let exp(−i4lα) = 1 and T = 1/2, which is the condition for constructive interference in path d. For a general input state |ψ〉in = (γ|H〉 + β|V〉)s|d〉p|l〉o, the output state of the BS SPSI is Eq. 8. According to Eq. 8, the output polarizations of both ports are changed, and the output polarization with destructive interference is only determined by the input polarization: γ|H〉 + β|V〉 → β|H〉 + γ|V〉. The output polarization with constructive interference is complex and depends on the input polarization, the rotation angle and the specific parameters of the DP. Here we discuss two special cases: γ = 1 or β = 1. In these cases, the polarization at the output port with constructive interference is preserved, while the polarization at the output port with destructive interference is flipped. The output polarizations are independent on the DP and α.
For an ideal OAM sorter, the OAM mode |l〉 will output from only one port. However, crosstalk between different ports is unavoidable in a practical experiment. Here we define the sorting fidelity of the BS SPSI for OAM modes as
According to Eq. 8, once the fabrication parameters of the DP is determined, Pc = |p〈c|ψ〉out|2 and Pd = 1 − Pc only vary with the rotation angle α. The blue line in Fig. 3 gives the sorting fidelity of the normal BS SPSI with a rotation angle α. The DP-dependent parameters are experimental values of a practical DP (t// = 0.9877, t⊥ = 0.9475 and Δφ ≃ 0.159π). The minimum ℱ is about 93.9%.
According to Eqs. 8 and 10, the dip of the sorting fidelity also depends on the relative phase shift Δφ of the DP. The black dashed line in Fig. 3 gives the minimum sorting fidelity (α = π/4) with varying Δφ, where the input polarization is set as |H〉. The sorting fidelity is 1 and 0.5 when Δφ is nπ and (2n + 1)π/2, respectively. What should be noted is that Δφ is determined by the incident angle and the refractive index of the DP. Although Δφ ∊ [0, 2π] in Fig. 3, the relative phase shift of π/2 requires the refractive index of the DP being at least 2.41. Thus, for common cases, Δφ is less than π/2. For example, if the refractive index is 1.52 (BK7 glass), the maximum value of Δφ is 0.259π.
2.2. DP in PBS SPSI
The PBS SPSI (by replacing the BS with a PBS in Fig. 2(c)) is different from the BS SPSI. If the DP is moved away, the photon will output from port c only. Although the DP changes the polarization, the PBS acts as a filter so that, at port c, the polarization output from direction a (b) is pure |H〉 (|V〉). The PBS SPSI requires the polarization states in the form of . When the OAM mode-dependent phase shift satisfies e−i4lα · eiφ = ±1, the PBS SPSI works as an OAM sorter. According to Eq 4, for the PBS SPSI, the DP works as follows:
Then, the output state becomes
Hence, the state is partially output from port d and decreases the output intensity at port c. The state output from port c is OAM mode-dependent. For two OAM modes satisfying 4(l1 −l2)α = π, the overlap of the output polarizations at port c isEq 13, sorting fidelity of the PBS SPSI is 36]) after the Sagnac loop.
Different from that of the BS SPSI, the sorting fidelity of the PBS SPSI is 100% when α = (2n + 1)π/4 while becomes minimum when α = nπ/2. As t// − t⊥ is at the order of 10−2, the polarization overlap of the PBS Sagnac interferometer (Eq. 13) is negligible. As shown in Fig. 4, the sorting fidelity (the green dashed line) of a practical DP is larger than 0.9995 for arbitrary α. Although the polarization at port c is elliptically polarized, one HWP and two QWPs are enough to transform the light into linear polarization. Thus, the sorting fidelity of the PBS SPSI is unaffected by the DP and the only concern is the intensity loss at port d. The green dashed line in Fig. 4 gives the output probability of the photon at port c. The total output probability of ports c and d is renormalized as 1. The maximum intensity loss is about 3%. It should be noted that the sorting fidelity and intensity loss also depends on parameters of the DP. However, for a commercial DP, the loss is usually within several percents and is acceptable.
3. High sorting-fidelity BS SPSI for OAM states
As discussed in Section 2, the sorting fidelity of the PBS SPSI for OAM modes is almost unaffected by the DP. Although possessing high sorting fidelity for OAM modes, the PBS SPSI is only practicable for polarizations in the form of . The normal BS SPSI works on arbitrary polarization. However, the minimum sorting fidelity of the normal BS SPSI is about 93.9%, which means the BS SPSI alone leads to about 6% sorting error for OAM modes without considering other experimental deviations. This is unaccepted in some applications, such as in quantum key distribution (QKD) with BB84 protocol. The quantum bit error rate of a BB84-protocol QKD should not exceed 11% even using single photon source . For practical QKD system with weak coherent source, the upper bound of quantum bit error rate should be lower . In this section, we first measured the parameters of a practical DP and estimated the corresponding sorting fidelity of the normal BS SPSI in Fig. 2(c) to OAM modes. Second, we proposed the modified BS SPSI to acquire high sorting fidelity for OAM modes. Third, we implemented the modified and the normal BS SPSIs and analyzed the experimental data with experimental deviations.
3.1. JDP of a practical DP
A measurement device showing in Fig. 5(a) is implemented to determine JDP of a practical DP (the base angle is 45° and the length is 63 mm). We rotate the polarization of the light to fulfill the measurement. The method is equivalent to rotating the DP but easier to be implemented and more precise. By rotating a HWP before the DP, the output light intensity varies and the parameters t// and t⊥ can be determined. The measurement gives that t// = Iout,///Iin ≃ 987 and t⊥ = Iout,⊥/Iin ≃ 0.945, where Iin is the input intensity and Iout,//(⊥) is the output intensity for |//〉 (|⊥〉).
Figure 5(b) gives the experimental setup to measure the relative phase shift Δφ. The rotation angle α of the DP is π/4. For an input state |H〉, the output state becomesEqs. 8–9, the sorting fidelity of a normal BS SPSI with the practical DP above should not be higher than 93.9% when α = π/4.
3.2. The modified BS SPSI
According to Eq 5, the sorting fidelity decreases because of the asymmetry of coordinate transformation Rs for the opposite directions in the Sagnac loop. If we rotate the lab coordinate system to be coincident with the DP coordinate system, then there is no difference between polarizations in both directions. An equivalent way is to rotate the polarization γ|H〉+ β|V〉 into the form ±γ|//〉a(b) +β|⊥〉a(b) before the photon entering into the DP, where |//〉a(b) (| ⊥〉a(b)) is the polarization that is parallel (perpendicular) to nDP,a(b). That is, as shown in Fig. 6(a), if the transformations in the degree of freedom of polarization satisfyEq. 16 is not unique. A simple solution is
The unitary transformation of Eq. 17 can be implemented by a HWP. U1(2),a(α) represents a HWP with its fast axis rotating α/2 at its horizontal axis (HWPa in Fig. 6(b)). In the opposite direction, the rotation angle of the fast axis of the same HWP becomes π − α/2 (HWPb in Fig. 6(b)), which corresponds to U1(2),b. Thus, only two more HWPs are necessary to complete the modified BS SPSI (as shown in Fig. 6(c)) and the polarizations from both directions a and b are for a general input state γ|H〉 + β|V〉. Hence, the sorting fidelity of the BS SPSI is independent on the input polarization and can be 100% in principle, though the polarization output is not preserved. Since the parameters t//, t⊥ and Δφ are fixed. Hence, passive optical elements after the BS SPSI can compensate the polarization variation, if necessary.
3.3. Experiments and discussion
The proof-of-principle experimental setup of the modified BS SPSI for OAM photon states is demonstrated in Fig. 6(c). As the first-order coherence of a single photon can be simulated by coherence light, a laser diode is used as the light source. The corresponding wavelength is 780 nm. The OAM light is generated by the spatial light modulator (SLM) and then inputs into the modified BS SPSI. The rotation angle of the DP is π/4 so that the odd and even-order modes will output from ports c and d, respectively. As path d overlaps the input path, the corresponding intensity is detected by inserting a BS into the optical path. The intensities of both ports are monitored by power meters. The HWP and the quarter-wave plate (QWP) before the Sagnac loop are used to verify the polarization-independent property of the modified BS SPSI. As a proof-of-principle experiment, the sorting fidelities of eleven OAM modes (l = 0, 1,⋯,10) with six typical incident polarization states (|H〉, |V〉, |+〉 = |H〉 + |V〉, |−〉 = |H〉·− |V〉, |L〉 = |H〉 − i|V〉) are verified. Each sorting fidelity measuring in the figure was repeated for more than 150 times within 20 seconds to obtain the standard deviation (σ) and , where N is the times of repetition and 〈ℱi〉 is the average sorting fidelity. In order to make a comparison, we also performed the corresponding experiments of the normal BS SPSI by removing the HWPs within the Sagnac loop in Fig. 6(c).
Figure 7 gives the sorting fidelities of the normal (blue bars) and the modified (red bars) BS SPSIs for eleven OAM modes (l = 0, 1, 2, ⋯ , 10) and six polarizations. The black dashed lines in Fig. 7 is the upper bound for normal BS SPSI. All error bars in Fig. 7 are set as ±3σ. As discussed in Section 2.1, the sorting fidelity of the normal BS SPSI should not exceed 0.939 theoretically. Figures 7(b)–7(f) all satisfy the theoretical analysis. However, the corresponding fidelities (the blue bars) in Fig. 7(a) are almost all larger than 0.939. The reason is that the analysis in Section 2.1 only considers the polarization-dependent effect of the DP. In fact, some other non-ideal conditions also affect the sorting fidelity. First, the rotating precision of the DP is only 2 degree. Thus the rotation angle of the DP should be α = π/4 + δα, where the angle δα is the rotating error. Second, paths a and b of the Sagnac loop are hardly overlapping completely. Hence, there also exist a small relative phase shift δp between paths a and b.
Now, we analyze the practical fidelity of the normal BS SPSI with experimental deviations. As δα and δp are small, we neglect the polarization varying caused by the DP for |+〉 and |−〉 when α = π/4 + δα. According to Eq. 5, the output state for |+〉s |l〉o isEq. 10, the sorting fidelity is
Hence, ℱ+ increases with l if δα is positive. For l = 0, δα = 0, ℱ+ could also be larger than 0.939 if δp is positive. However, as the generation purity of |l〉 decreases and the proportion of |l ± 1〉 increases with l, ℱ+ increases firstly and turns to decrease when l > 7. Figure 7(a) also shows that there are differences between odd and even-order modes. The phenomenon will be discussed later. The odd (or even-order) modes fit the theoretical analysis. Analogously, the output state for |−〉s|l〉o is
Thus ℱ− decreases monotonously with l. The sorting fidelity for odd-order (or even-order) modes in Fig. 7(b) decreases monotonously with l. Other polarizations (|H〉, |V〉, |L〉 and |R〉) can be treated as the superpositions of |+〉 and |−〉. It is easy to get that the sorting fidelities for these four polarizations are the same
Equation 21 indicates that the fidelity decreases monotonously with l for these fours polarizations. The experimental results (blue bars in Figs. 7(c)–7(f)) fit well with the theoretical analysis though there are slightly differences between odd and even-order OAM modes.
The red bars in Fig. 7 show the fidelities of the modified BS SPSI. All sorting fidelities in the figure are larger than 0.939 and the highest sorting fidelity is about 0.982. There are no significant differences of the sorting fidelities for the modified BS SPSI in Figs. 7(a)–7(f), which demonstrates that the structure is independent on the input polarization. The sorting fidelity decreases as the OAM mode l increases. This is mainly due to the experimental deviation ϕerror = δp + 4lδα =⇒ ℱ = (1 + cosϕerror)/2 − δB, where δB is the background noise. For example, if δB = 0.01, δp = 3degree and δα = 1degree, then ℱ(l = 1) = 0.982, and ℱ(l = 2) ≃ 0.972. By rotating the HWP and QWP before the Sagnac loop, different input polarizations are prepared. However, rotating the wave plates will affect the light path afterward and the interference slightly. In order to evaluate the polarization-independent property of the modified BS SPSI, we have not adjusted the Sagnac loop for different polarizations and hence δp for different polarizations are slightly changed. When δp and δα have the same sign, the fidelity decreases with l (the red bars in Figs. 7(c)–7(f)). Otherwise, the fidelity will increase slightly before decreasing with l (the red bars in Figs. 7(a) and 7(b)). The background noise due to reflection of the SLM and other optical elements and is difficult to estimate since path d and the input path are overlapped and the Sagnac loop is single-path. This increases the deviations of the experimental results and decreases the sorting fidelity. Another reason that limits the sorting fidelity is the phase-only SLM. In order to generate high-purity OAM mode, both phase and amplitude of the light should be modulated. Only the phase is modulated in the experiment (Fig. 6(c)). The generation purity of mode l decreases as the mode number l increases. This leads to the decreasing of fidelity. Because of the complex factors above, the highest sorting fidelity is about 0.982. Figs. 7(b)-7(f) show that the sorting fidelity of the normal BS SPSI decreases rapidly with l and the maximum and minimum values of sorting fidelity of the normal BS SPSI are 0.934 and 0.845, respectively. Although the corresponding maximum and minimum values in Fig. 7(a) are 0.958 and 0.938, respectively, the fidelity depends on polarization and experimental deviations. The sorting fidelity of the modified BS SPSI is about 5%~10% higher than that of the normal one for OAM modes. Additionally, there is no significant difference between ℱ(l = 2n) and ℱ(l = 2n + 1) for the modified BS SPSI. The sorting fidelity can be improved further by using rotators with higher rotation precision. It can also be improved by optimizing the phase pattern added on the SLM . Figure 8 gives the real time fidelity of the modified BS SPSI for free running, where the input state is |+〉s|l = 1〉o. The average fidelity for one hour running is 0.979 ± 0.015 with three times of standard deviation, which indicates that the single-path Sagnac interferometer is stable for free running. Thus, the proposed structure is promising for practical applications of OAM photon states, such as QKD.
The sorting fidelity of the normal BS SPSI for odd-order modes is slightly different from that for even-order modes and is a little polarization-dependent. The blue bars in Fig. 7(a) (for |+〉) shows that the sorting fidelity for odd-order modes are slightly smaller than that for even modes. On the contrary, the sorting fidelities (the blue bars) for odd-order modes are larger than that for the adjacent even-order modes in Figs. 7(b)-7(f) (for other five polarizations). This is attributed to that the normal BS SPSI is not symmetric for paths a and b. For example, if the input polarization is |+〉 in the coordinate system of DP, it is |//〉 for path a and is | ⊥〉 for path b and the situation is reversed for |−〉, which may lead to δp being polarization-dependent. Additionally, polarization varying due to δα, which has been neglected in Eqs. 18–21, will introduce cross term to Eq. 18. The asymmetries above finally leads to the sorting fidelity being slightly dependent on the input polarization and the parity of the OAM mode. On the contrary, the modified BS SPSI is symmetric for light incident from both paths and is independent on polarization and the parity of the OAM mode.
The sorting fidelities (the blue bars) of the normal BS SPSI for l = 0 in Figs. 7(a) and 7(b) are 0.951 and 0.909, respectively. Then δp can be obtained immediately according to Eqs. 19–20. The corresponding values of δp are 3 degree and 6.5 degree, respectively. As many factors discussed in the previous paragraph cannot be estimated independently, the deviation between these two polarizations is more than 3 degree. Though, δp = 6.5degree only leads to about 0.3% crosstalk to an interferometer. As ℱ+(l = 2) = 0.955 and ℱ−(l = 2) = 0.901, we obtain that δα ≃ 1degree and 1.6 degree, respectively. The results are consistent with the experimental condition (the rotation precision of α is 2degree). It should be noted that δp and δα are sensitive to sorting fidelity. As many factors are difficult to be estimated independently, the values of δp and δα are qualitative rather than quantitative.
It should also be noted that Δφ can be compensated by adjusting δp in a double-path MZI, if the normals of the two DPs of the interferometer are parallel and perpendicular to the input polarization, respectively , which means that only the polarization changing affects the interference. That is why the visibility reduces less and becomes minimum when the relative angle between the two DPs is α1 − α2 = π/4 (the angle leading to maximum changing of polarization). Ref.  only considered the special case with the normal of one DP parallel to the input polarization (α1 = 0). The visibility will be lower if α1 = π/4 and α2 = π/4. Ref.  shows that the sorting fidelity can be improved by increasing the value of the base angle α0. However, it will also increase the length of the DP and lengthen the Sagnac loop, which will decrease the stability of the SPSI. The sorting fidelity of the modified BS SPSI here is independent on the specific Jones matrix JDP and therefore reduces the fabrication requirement of the DP.
In conclusion, we have given a detailed analysis of polarization-dependent properties of the BS and PBS SPSIs for OAM photon states, which is instructive to quantum information processing with OAM photon states. We have also demonstrated a high sorting-fidelity BS SPSI that is independent on the input polarization and the specific parameters of the DP. The modified BS SPSI improves the sorting fidelity for OAM modes by about 5%~10% and is stable for free running. The experiments demonstrates that the proposed structure is promising for practical high-dimensional quantum information processing.
National Natural Science Foundation of China (NSFC) (Grant Nos. 61675189, 61627820, 61622506, 61475148, 61575183); National Key Research And Development Program of China (Grant Nos. 2016YFA0302600, 2016YFA0301702); "Strategic Priority Research Program (B)" of the Chinese Academy of Sciences (Grant No. XDB01030100).
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