Single-path Sagnac interferometer with Dove prism for orbital-angular-momentum photon manipulation

Fang-Xiang Wang; Wei Chen; Ya-Ping Li; Guo-Wei Zhang; Zhen-Qiang Yin; Shuang Wang; Guang-Can Guo; Zheng-Fu Han

doi:10.1364/OE.25.024946

1. Introduction

The orbital angular momentum (OAM) of light [1] is an important resource in many fields, such as quantum simulation [2], 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 [21]. 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 [38]. 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 $\frac{1}{\sqrt{2}} (| H 〉 + exp (i φ) | V 〉)$ . 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 interferometers

A DP is shaped from a truncated right angle prism with a base angle α₀ (as shown in Fig. 1). The DP is not polarization preserving and can be expressed by Jones matrices [39, 40]

J_{D P} = R_{s}^{'} T_{o u t} R_{i n n e r} T_{i n} R_{s}

where T_in₍_out₎ is the refractive matrix of the input (output) surface, R_inner is the total internal reflective matrix (Fig. 1) and R_s and

R_{s}^{'}

are the forward and inverse transformations between the lab and the DP coordinate systems (Fig. 1). As has been studied in Refs. [39] and [40], 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 [41]. Δφ is determined by the incident angle θ₁ and the refractive index of the DP. That is, J_DP should satisfies

{\begin{matrix} J_{D P} | / / 〉 = \sqrt{t_{/ /}} | / / 〉 \\ J_{D P} | ⊥ 〉 = \sqrt{t_{⊥}} e^{i Δ φ} | ⊥ 〉, \end{matrix}

where |//〉 and | ⊥〉 are the polarizations that are parallel and perpendicular to the input surface, respectively. And t_//(⊥) is the transmission coefficient of the DP for the polarized light that parallel (perpendicular) to the input surface. Equation 2 requires J_DP to be the following form

J_{D P} = (\begin{matrix} \sqrt{t_{/ /}} & 0 \\ 0 & \sqrt{t_{⊥}} e^{i Δ φ} \end{matrix}) .

Fig. 1 The schematic diagram of a DP. There are two refractive and one total reflective interfaces for a light beam propagating through the DP. θ₁ is the incident angle at the input interface. n_DP is the normal of the base. There is a relative rotation angle α between the lab (with red color) and the DP (with black color) coordinate systems. The rotation of the DP is along Z axis.

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A general polarization can be expressed as the superposition of |//〉 and | ⊥〉. Hence, in general, J_DP should satisfy

J_{D P} (α) = R_{s} (- α) (\begin{matrix} \sqrt{t_{/ /}} & 0 \\ 0 & \sqrt{t_{⊥}} e^{i Δ φ} \end{matrix}) R_{s} (α), R_{S} (α) = (\begin{matrix} c o s α & s i n α \\ - s i n α & c o s α \end{matrix}) .

where α is the angle between the coordinate systems (Fig. 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 (n_DP,a) and b (n_{DP, b}) are different (see Fig. 2(b)). Correspondingly, the coordinate transformations R_s,a and R_s,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

\begin{array}{r} {| ψ 〉}_{o u t} = (U_{p}^{B S} \otimes I_{s} \otimes I_{o}) ({| b 〉}_{p} 〈 b | \otimes J_{D P} (π - α) \otimes P_{o} (- α)) \\ ({| a 〉}_{p} 〈 a | \otimes J_{D P} (α) \otimes P_{o} (α)) (U_{p}^{B S} \otimes I_{s} \otimes I_{o}) {| ψ 〉}_{i n}, \end{array}

where the subscripts p, s and o represent the degrees of freedom of the path, polarization and OAM, respectively. The input state is expressed as |ψ〉_in =|p〉_p|s〉_s|l〉_o_. I_o and I_s are the identity operators. P_o is the phase operator of the DP on the degree of freedom of OAM and satisfies P_o(α)|l〉_o = exp(i2lα)|l〉_o.

U_{p}^{B S}

is the unitary transformation of the BS

U_{p}^{B S} = (\begin{matrix} \sqrt{T} & - \sqrt{1 - T} \\ \sqrt{1 - T} & \sqrt{T} \end{matrix}),

Fig. 2 (a) The schematic diagram of a DP with two OAM states incident from opposite directions. (b) The coordinate systems of the DP for light incident from directions a (n_DP,a and $n_{D P, a}^{⊥}$ ) and b (n_DP,b and $- n_{D P, b}^{⊥}$ ). The minus sign before $n_{D P, b}^{⊥}$ is to satisfy the right-hand rule of the coordinate system. (c) The schematic setup for a normal BS SPSI. BS: beam splitter.

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where T is the transmissivity of the BS.

For an ideal DP that is polarization preserving, J_DP(π − α) = J_DP(α) = I_s. The output state in Eq. 5 is

{| ψ 〉}_{o u t}^{i d e a l} = \frac{e^{2 i l α}}{2} [(1 - e^{- 4 i l α}) {| c 〉}_{p} + (1 + e^{- 4 i l α}) {| d 〉}_{p}] {| s 〉}_{s} {| l 〉}_{o} .

{| ψ 〉}_{o u t}^{i d e a l}

only depends on the rotation angle α of the DP. For a practical DP, J_DP is determined by Eq. 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

{| ψ 〉}_{o u t} = \frac{e^{2 i l α}}{2 \sqrt{t_{/ /} + t_{⊥}}} {(\sqrt{t_{/ /}} - \sqrt{t_{⊥}} e^{i Δ φ}) s i n 2 α (β {| H 〉}_{s} + γ {| V 〉}_{s}) {| c 〉}_{p} + [2 γ (\sqrt{t_{/ /}} c o s^{2} α + \sqrt{t_{⊥}} e^{i Δ φ} s i n^{2} α) {| H 〉}_{s} + 2 β (\sqrt{t_{/ /}} s i n^{2} α + \sqrt{t_{⊥}} e^{i Δ φ} c o s^{2} α) {| V 〉}_{s}] {| d 〉}_{p}} {| l 〉}_{o},

where, |H〉 (|V〉) represents the polarization that parallel to the X (Y) axis. the complex amplitudes γ and β satisfy γ² + β² = 1. In the case of exp(−i4lα) = −1 (the condition for constructive interference in path c), the output state can be obtained by exchanging the output ports c and d of 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

ℱ = \frac{I_{m a x}}{I_{m a x} + I_{m i n}},

where I_max₍_min₎ is the intensity at the output port with constructive (destructive) interference. Then, the crosstalk between different ports is

1 - ℱ

. The intensity I_max₍_min₎ corresponds to the output probability of the state at the port with constructive (destructive) interference after normalization. That is,

ℱ = m a x {|_{p} 〈 c | ψ 〉 |^{2}, |_{p} 〈 d | ψ 〉 |^{2}},

According to Eq. 8, once the fabrication parameters of the DP is determined, P_c = |_p〈c|ψ〉_out|² and P_d = 1 − P_c 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%.

Fig. 3 The sorting fidelities of the BS SPSI with the rotation angle α (the blue line) and the relative phase Δφ(the black dashed line) added by the DP, respectively. The input polarization is |H〉. Δφ = 0.159π for the blue line. The rotation angle is α = π/4 for the black dashed line.

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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 $\frac{1}{\sqrt{2}} (| H 〉 + e^{i φ} | V 〉)$ . When the OAM mode-dependent phase shift satisfies e⁻ⁱ⁴^lα · e^iφ = ±1, the PBS SPSI works as an OAM sorter. According to Eq 4, for the PBS SPSI, the DP works as follows:

\begin{array}{l} | H 〉 \overset{D P, a}{\to} (\sqrt{t_{/ /}} c o s^{2} α + \sqrt{t_{⊥}} s i n^{2} α e^{i Δ φ}) | H 〉 + (\sqrt{t_{/ /}} - \sqrt{t_{⊥}} e^{i Δ φ}) s i n α c o s α | V 〉, \\ | V 〉 \overset{D P, b}{\to} (\sqrt{t_{/ /}} s i n^{2} α + \sqrt{t_{⊥}} c o s^{2} α e^{i Δ φ}) | V 〉 - (\sqrt{t_{/ /}} - \sqrt{t_{⊥}} e^{i Δ φ}) s i n α c o s α | H 〉 . \end{array}

Then, the output state becomes

{| ψ 〉}_{o u t} = \frac{e^{2 i l α}}{\sqrt{2 (t_{/ /} + t_{⊥})}} {[(\sqrt{t_{/ /}} c o s^{2} α + \sqrt{t_{⊥}} s i n^{2} α e^{i Δ φ}) | H 〉 \pm (\sqrt{t_{/ /}} s i n^{2} α + \sqrt{t_{⊥}} c o s^{2} α e^{i Δ φ}) | V 〉] {| c 〉}_{p} + (\sqrt{t_{/ /}} - \sqrt{t_{⊥}} e^{i Δ φ}) s i n α c o s α (| V 〉 \mp | H 〉) {| d 〉}_{p}} {| l 〉}_{o} .

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 is

\begin{array}{l} P_{o v e r l a p} = {| 〈 ψ {(l_{1})}_{o u t, c} | ψ {(l_{2})}_{o u t, c} 〉 |}^{2} \\ = \frac{1}{N^{2}} \cdot | [(\sqrt{t_{/ /}} c o s^{2} α + \sqrt{t_{⊥}} s i n^{2} α e^{- i Δ φ}) 〈 H | - (\sqrt{t_{/ /}} s i n^{2} α + \sqrt{t_{⊥}} c o s^{2} α e^{- i Δ φ}) 〈 V |] \\ [(\sqrt{t_{/ /}} c o s^{2} α + \sqrt{t_{⊥}} s i n^{2} α e^{- i Δ φ}) | H 〉 + (\sqrt{t_{/ /}} s i n^{2} α + \sqrt{t_{⊥}} c o s^{2} α e^{- i Δ φ}) | V 〉] |^{2} \\ = \frac{1}{N^{2}} {(t_{/ /} - t_{⊥})}^{2} c o s^{2} (2 α), \end{array}

where

N = (t_{/ /} + + t_{⊥}) (1 - \frac{1}{2} s i n^{2} (2 α)) + \sqrt{t_{/ /} t_{⊥}} s i n^{2} (2 α) c o s Δ φ

is the renormalized factor at port c. According to Eq 13, sorting fidelity of the PBS SPSI is

ℱ^{'} = 1 - P_{o v e r l a p},

if polarizing compensation is allowed (e.g., by wave plates [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.

Fig. 4 The sorting fidelity (the blue line) and the output probability P_c of the state at port c (the green dashed line) of the PBS SPSI with the rotation angle α. The input polarization is |H〉 + |V〉. The DP-dependent parameters are t_// = 0.9877, t_⊥ = 0.9475, Δφ = 0.159π.

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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 $\frac{1}{\sqrt{2}} (| H 〉 + exp (i φ) | V 〉)$ . 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 [42]. For practical QKD system with weak coherent source, the upper bound of quantum bit error rate should be lower [43]. 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. J_DP of a practical DP

A measurement device showing in Fig. 5(a) is implemented to determine J_DP 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_// = I_out_,///I_in ≃ 987 and t_⊥ = I_out_,⊥/I_in ≃ 0.945, where I_in is the input intensity and I_out,_//(⊥) is the output intensity for |//〉 (|⊥〉).

Fig. 5 The device for measuring (a) the transmissivity coefficients $\sqrt{t_{/ /}}$ , $\sqrt{t_{⊥}}$ the relative phase shift Δφ of the DP. PBS: polarizing beam splitter.

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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 becomes

{| ψ 〉}_{o u t} = \sqrt{\frac{1}{N}} J_{D P} (π / 4) | H 〉 = \frac{1}{2 \sqrt{N}} [(\sqrt{t_{/ /}} + \sqrt{t_{⊥}} e^{i Δ φ}) | H 〉 + (\sqrt{t_{/ /}} - \sqrt{t_{⊥}} e^{i Δ φ}) | V 〉],

where the normalization factor N = (t_// + t_⊥)/2. The output polarization of the light is then analyzed by a PBS to determine Δφ. The experiment gives

| 〈 H | {ψ 〉}_{o u t} |^{2} = \frac{1}{2 (t_{/ /} + t_{⊥})} (t_{/ /} + t_{⊥} + 2 \sqrt{t_{/ /} t_{⊥}} c o s Δ φ) = I_{o u t, H} / (I_{o u t, H} + I_{o u t, V}) ≃ 0.939 \Rightarrow Δ φ ≃ 0.159 π

by measuring the output intensities after the PBS, where I_out,H (I_out,V is the transmission (reflective) intensity. This completes the measurement of J_DP. According to Eqs. 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 R_s 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 n_DP,a₍_b_). That is, as shown in Fig. 6(a), if the transformations in the degree of freedom of polarization satisfy

\begin{array}{l} U_{1, a} (γ | H 〉 + β | V 〉) = \pm γ {| / / 〉}_{a} + β {| ⊥ 〉}_{a}, \\ U_{2, b} (γ | H 〉 + β | V 〉) = \pm γ {| / / 〉}_{b} + β {| ⊥ 〉}_{b}, \\ U_{2, a} J_{D P} (α) U_{1, a} = U_{1, b} J_{D P} (π - α) U_{2, b}, \end{array}

then the polarizations output from both directions are the same and the sorting fidelity is independent on polarization, where the subscript a (b) denotes the rotation transformation in direction a (b). It is worth noticing that the rotation transformation U₁₍₂₎ in direction a is usually not equivalent to that in direction b and the solution for Eq. 16 is not unique. A simple solution is

\begin{array}{l} U_{1, a} = U_{2, a}, U_{1, b} = U_{2, b}, \\ U_{2, b} = U_{2, b} (α), U_{1, a} = U_{1, a} (α) \\ U_{2, b} (α) = U_{1, a} (- α), U_{1, a} (α) = (\begin{matrix} c o s (α) & s i n (α) \\ s i n (α) & - c o s (α) \end{matrix}), \end{array}

Fig. 6 (a) The unitary transformations to rotate the polarization. (b) n_DP_,_a₍_b₎ and $n_{D P, a (b)}^{⊥}$ consist the coordinate system of the DP for direction a (b). $H W P_{i}^{a (b)}$ is the fast axis of the HWP in the BS SPSI for direction a (b). (c) The proof-of-principle experimental setup of the modified BS SPSI. SLM: spatial light modulator; QWP: quarter-wave plate.

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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 (HWP_a in Fig. 6(b)). In the opposite direction, the rotation angle of the fast axis of the same HWP becomes π − α/2 (HWP_b in Fig. 6(b)), which corresponds to U_1(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 $\frac{1}{\sqrt{t_{/ /} γ^{2} + t_{⊥} β^{2}}} (\sqrt{t_{/ /}} γ | H 〉 - \sqrt{t_{⊥}} β e^{i Δ φ} | V 〉)$ 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 $σ = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(ℱ_{i} - 〈 ℱ_{i} 〉)}^{2}}$ , where N is the times of repetition and 〈ℱ_i〉 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 + δ_α, where the angle δ_α 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 δ_p between paths a and b.

Fig. 7 The sorting fidelities of the normal (blue bars) and modified (red bars) BS SPSIs for eleven OAM modes and different polarizations.(a)-(f) correspond to the sorting fidelities with |+〉, |−〉, |H〉, |V〉, |L〉 and |R〉 polarizations, respectively. The black dashed line corresponds to 0.939, the upper bound of the normal BS SPSI. All error bars are set as ±3σ, where σ is the standard deviation.

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Now, we analyze the practical fidelity of the normal BS SPSI with experimental deviations. As δ_α and δ_p are small, we neglect the polarization varying caused by the DP for |+〉 and |−〉 when α = π/4 + δ_α. According to Eq. 5, the output state for |+〉_s |l〉_o is

{| ψ 〉}_{o u t} = \frac{e^{2 i l α}}{\sqrt{N}} [(\sqrt{t_{/ /}} - \sqrt{t_{⊥}} e^{i Δ φ} e^{- i δ_{p}} e^{- 4 i l α}) {| c 〉}_{p} + (\sqrt{t_{/ /}} - \sqrt{t_{⊥}} e^{i Δ φ} e^{- i δ_{p}} e^{- 4 i l α}) {| d 〉}_{p}] {| + 〉}_{s} {| l 〉}_{o} .

where

α = \frac{π}{4} + δ_{α}

and N = 2(t_// + t_⊥) is the normalization coefficient. Then, according to Eq. 10, the sorting fidelity is

ℱ_{+} = \frac{1}{2 (t_{/ /} + t_{⊥})} [t_{/ /} + t_{⊥} + 2 \sqrt{t_{/ /} t_{⊥}} c o s (Δ φ - δ_{p} - 4 l δ_{α})] .

Hence, ℱ₊ increases with l if δ_α is positive. For l = 0, δ_α = 0, ℱ₊ could also be larger than 0.939 if δ_p 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

ℱ_{-} = \frac{1}{2 (t_{/ /} + t_{⊥})} [t_{/ /} + t_{⊥} + 2 \sqrt{t_{/ /} t_{⊥}} c o s (- Δ φ - δ_{p} - 4 l δ_{α})] .

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

ℱ_{H, V, L, R} = \frac{1}{2 (t_{/ /} + t_{⊥})} [t_{/ /} + t_{⊥} + 2 \sqrt{t_{/ /} t_{⊥}} c o s (Δ φ) c o s (δ_{p} + 4 l δ_{α})] .

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 ϕ_error = δ_p + 4lδ_α =⇒ ℱ = (1 + cosϕ_error)/2 − δ_B, where δ_B is the background noise. For example, if δ_B = 0.01, δ_p = 3degree and δ_α = 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 δ_p for different polarizations are slightly changed. When δ_p and δ_α 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 [44]. 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.

Fig. 8 The real time sorting fidelity of the modified BS SPSI for free running. The input state is |+〉_s|l = 1〉_o.

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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 δ_p being polarization-dependent. Additionally, polarization varying due to δ_α, 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 δ_p can be obtained immediately according to Eqs. 19–20. The corresponding values of δ_p 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, δ_p = 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 δ_α ≃ 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 δ_p and δ_α are sensitive to sorting fidelity. As many factors are difficult to be estimated independently, the values of δ_p and δ_α are qualitative rather than quantitative.

It should also be noted that Δφ can be compensated by adjusting δ_p in a double-path MZI, if the normals of the two DPs of the interferometer are parallel and perpendicular to the input polarization, respectively [40], 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. [40] 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. [39] 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 J_DP and therefore reduces the fabrication requirement of the DP.

4. Conclusion

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.

Funding

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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Single-path Sagnac interferometer with Dove prism for orbital-angular-momentum photon manipulation

Abstract

1. Introduction